“SCHEME-THEORETIC IMAGES” OF MORPHISMS OF STACKS
MATTHEW EMERTON AND TOBY GEE

Abstract. We give criteria for certain morphisms from an algebraic stack toa (not necessarily algebraic) stack to admit an (appropriately defined) schemetheoretic image. We apply our criteria to show that certain natural moduli
stacks of local Galois representations are algebraic (or Ind-algebraic) stacks.

Contents
1. Introduction
2. Stacks in groupoids and Artin’s axioms
3. Scheme-theoretic images
4. Examples
5. Moduli of finite height ϕ-modules and Galois representations
References

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1. Introduction
The goal of this paper is to prove an existence theorem for “scheme-theoretic
images” of certain morphisms of stacks. We have put scheme-theoretic images inquotes here because, generally, the objects whose existence we prove will be certain
algebraic spaces or algebraic stacks, rather than schemes. Like the usual schemetheoretic images of morphisms of schemes, though, they will be closed substacks ofthe target, minimal with respect to the property that the given morphism factors
through them. This explains our terminology.
In the case of morphisms of algebraic stacks (satisfying appropriate mild finiteness conditions), the existence of a scheme-theoretic image in the preceding sense
follows directly from the basic results about scheme-theoretic images for morphismsof schemes. Our interest will be in more general contexts, namely, those in which
the source of the morphism is assumed to be an algebraic stack, but the target
is not; in particular, we will apply our results in one such situation to constructmoduli stacks of Galois representations.
1.1. Scheme-theoretic images. We put ourselves in the setting of stacks ingroupoids defined on the big étale site of a locally Noetherian scheme S all of
whose local rings OS,s at finite type points s ∈ S are G-rings. (See Section 1.5 forany unfamiliar terminology.) Recall that in this context, Artin’s representability
The first author was supported in part by the NSF grants DMS-1303450, DMS-1601871, andDMS-1902307. The second author was supported in part by a Leverhulme Prize, EPSRC grant
EP/L025485/1, Marie Curie Career Integration Grant 303605, and by ERC Starting Grant 306326.
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M. EMERTON AND T. GEE

theorem gives a characterisation of algebraic stacks which are locally of finite presentation over S among all such stacks: namely, algebraic stacks which are locallyof finite presentation over S are precisely those stacks in groupoids F on the big
étale site of S that satisfy:[1] F is limit preserving;[2] (a) F satisfies the Rim–Schlessinger condition (RS), and(b) F admits effective Noetherian versal rings at all finite type points;[3] the diagonal ∆ : F → F ×S F is representable by algebraic spaces;[4] openness of versality.
See Section 2 below for an explanation of these axioms, and Theorem 2.8.4 forArtin’s theorem. (In Subsection 2.4 we also introduce two further axioms, labelled [4a] and [4b], that are closely related to [4]. These are used in discussing theexamples of Section 4, but not in the proof of our main theorem.)
We will be interested in quasi-compact morphisms ξ : X → F of stacks on the
big étale site of S, where X is algebraic and locally of finite presentation over S,
and F is assumed to have a diagonal that is representable by algebraic spaces and islocally of finite presentation (i.e. F satisfies [3], and a significantly weakened formof [1]). In this context, we are able to define a substack Z of F which we call thescheme-theoretic image of ξ. (The reason for assuming that ξ is quasi-compact isthat this seems to be a minimal requirement for the formation of scheme-theoretic
images to be well-behaved even for morphisms of schemes, e.g. to be compatible
with fpqc, or even Zariski, localisation.) If F is in fact an algebraic stack, locallyof finite presentation over S, then Z will coincide with the usual scheme-theoretic
image of ξ. In general, the substack Z will itself satisfy Axioms [1] and [3].
Our main result is the following theorem (see Theorem 3.2.34).
1.1.1. Theorem. Suppose that ξ : X → F is a proper morphism, where X isan algebraic stack, locally of finite presentation over S, and F is a stack over S
satisfying [3], and whose diagonal is furthermore locally of finite presentation. Let
Z denote the scheme-theoretic image of ξ as discussed above, and suppose that Z
satisfies [2]. Suppose also that F admits (not necessarily Noetherian) versal ringsat all finite type points (in the sense of Definition 2.2.9 below).
Then Z is an algebraic stack, locally of finite presentation over S; the inclusion
Z ,→ F is a closed immersion; and the morphism ξ factors through a proper,
scheme-theoretically surjective morphism ξ : X → Z. Furthermore, if F 0 is a
substack of F for which the monomorphism F 0 ,→ F is representable by algebraic
spaces and of finite type (e.g. a closed substack) with the property that ξ factors
through F 0 , then F 0 contains Z.
We show that if F satisfies [2], then the assumption that Z satisfies [2] follows from the other assumptions in Theorem 1.1.1 (see Lemma 3.2.20 below). By
applying the theorem in the case when Z = F (in which case we say that ξ isscheme-theoretically dominant), and taking into account this remark, we obtainthe following corollary.
1.1.2. Corollary. If F is an étale stack in groupoids over S, satisfying [1], [2]and [3], for which there exists a scheme-theoretically dominant proper morphismξ : X → F whose domain is an algebraic stack locally of finite presentation over S,
then F is an algebraic stack.

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1.1.3. Some remarks on Theorem 1.1.1 and its proof. The proof of Theorem 1.1.1
occupies most of the first three sections of the paper. One way for the reader to
get an idea of the argument is to read Lemma 2.6.4, and then to turn directly tothe proof of Theorem 3.2.34, taking the various results referenced in the argument(including even the definition of the scheme-theoretic image Z) on faith.
It is often the case (and it is the case in the proof of Theorem 3.2.34) that themain problem to be overcome when using Artin’s axiomatic approach to provingthat a certain stack is algebraic is the verification of axiom [4] (openness of versality). Lemma 2.6.4 (which is a stacky version of [Art69a, Lem. 3.10]) shows that one
can automatically get a slightly weaker version of [4], where “smooth” is replaced
by “unramified”, if the axioms [1], [2], and [3] are satisfied. Morally, this suggeststhat anything satisfying axioms [1], [2], and [3] is already close to an Ind-algebraic
stack, because it admits unramified maps (which are at least morally quite close toimmersions) from algebraic stacks that are even formally smooth at any particularpoint. So to prove Theorem 1.1.1, one has to build on this idea, and then use
the extra hypothesis (namely, that there is a proper surjection from an algebraic
stack onto the stack Z) to prove axiom [4]. The argument is ultimately topological,
using properness to eliminate the possibility of having more and more components
building up Zariski locally around a point.
1.2. Moduli of finite height Galois representations. The results in this paper
were developed with a view to applications to the theory of Galois representations,and in particular to constructing moduli stacks of mod p and p-adic representations of the absolute Galois group Gal(K/K) of a finite extension K/Qp . These
applications will be developed more fully in the papers [EG22, CEGS19], and werefer the interested reader to those papers for a fuller discussion of our results andmotivations.
Let r̄ : Gal(K/K) → GLn (Fp ) be a continuous representation. The theoryof deformations of r̄ — that is, liftings of r̄ to continuous representations r :Gal(K/K) → GLd (A), where A is a complete local ring with residue field Fp — isextremely important in the Langlands program, and in particular is crucial for proving automorphy lifting theorems via the Taylor–Wiles method. Proving such theorems often comes down to studying the moduli spaces of those deformations which
satisfy various p-adic Hodge-theoretic conditions; see for example [Kis09b, Kis09a].
From the point of view of algebraic geometry, it seems unnatural to only study“formal” deformations of this kind, and Kisin observed about ten years ago that
results on the reduction modulo p of two-dimensional crystalline representations
suggested that there should be moduli spaces of p-adic representations (satisfying
certain p-adic Hodge theoretic conditions, for example finite flatness) in which theresidual representations r̄ should be allowed to vary; in particular, the special fibres of these moduli spaces would be moduli spaces of (for example) finite flat
representations of Gal(K/K). Unfortunately, there does not seem to be any simple way of directly constructing such moduli spaces, and until now their existence
has remained a mystery. (We refer the reader to the introduction to [EG22] for afurther discussion of the difficulties of directly constructing moduli spaces of mod p
representations of Gal(K/K).)
Mod p and p-adic Galois representations are studied via integral p-adic Hodge
theory; for example, the theories of (ϕ, Γ)-modules and Breuil–Kisin modules.
Typically, one begins by analysing p-adic representations of the absolute Galois

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M. EMERTON AND T. GEE

group Gal(K/K∞ ) of some highly ramified infinite extension K∞ /K. (In the theory of (ϕ, Γ)-modules this extension is the cyclotomic extension, but in the theoryof Breuil–Kisin modules, it is a non-Galois extension obtained by extracting ppower roots of a uniformiser.) Having classified these representations in terms ofsemilinear algebra (modules over some ring, equipped with a Frobenius), one thenseparately considers the problem of descending the classification to representationsof Gal(K/K).
More precisely, by the theory of [Fon90], continuous mod pa representations ofGal(K/K∞ ) are classified by étale ϕ-modules, which are modules over a Laurent
series ring, equipped with a Frobenius. Following the paper [PR09] of Pappas andRapoport, we consider a moduli stack R of étale ϕ-modules, which, for appropriate
choices of the Frobenius on the Laurent series ring can be thought of informallyas a moduli stack classifying Gal(K/K∞ )-representations. (To keep this paper ata reasonable length, we do not discuss the problem of descending our results torepresentations of Gal(K/K); this is addressed in the papers [CEGS19, EG22].)
Pappas and Rapoport prove various properties of the stack R (including that it isa stack, which they deduce from deep results of Drinfeld [Dri06] on the fpqc localityof the notion of a projective module over a Laurent series ring), including that its
diagonal is representable by algebraic spaces. In the present paper we prove thefollowing theorem about the geometry of R (see Theorem 5.4.20 below).
1.2.1. Theorem. The stack R is an Ind-algebraic stack. More precisely, we may
write R ∼R , where each RF is a finite type algebraic stack over Z/pa Z, and
= lim
−→F F
where each transition morphism in the inductive limit is a closed immersion.
The theorem is proved by applying Theorem 1.1.1 to certain morphisms CF → R(whose sources CF are algebraic stacks) so as to prove that their scheme-theoretic
images RF are algebraic stacks; we then show that R is naturally identified with
the inductive limit of the RF , and thus that it is an Ind-algebraic stack. (Theindex F is a certain element of a power series ring; replacing the indexing set witha cofinal subset given by powers of a fixed F , one can write R as an Ind-algebraic
stack with the inductive limit being taken over the natural numbers.)
The stacks CF were defined by Pappas and Rapoport in [PR09], where it isproved that they are algebraic, and that the morphisms CF → R are representableby algebraic spaces and proper. The definition of the stacks CF is motivated by
the papers [Kis09b, Kis06, Kis08], in which Kisin (following work of Breuil andFontaine) studied certain integral structures on étale ϕ-modules, in particular (whatare now called) Breuil–Kisin modules of height at most F , where F is a power ofan Eisenstein polynomial for a finite extension K/Qp . A Breuil–Kisin module is a
module with Frobenius over a power series ring, satisfying a condition (depending
on F ) on the cokernel of the Frobenius. Inverting the formal variable u in the power
series ring gives a functor from the category of Breuil–Kisin modules of height at
most h to the category of étale ϕ-modules. We say that the Galois representations
corresponding to the étale ϕ-modules in the essential image of this functor haveheight at most F .
With an eye to future applications, we work in a general context in this paper,and in particular we allow considerable flexibility in the choice of the polynomial F
and the Frobenius on the coefficient rings. In particular our étale ϕ-modules do not
obviously correspond to representations of some Gal(K/K∞ ) (but the case thatthey do is the main motivation for our constructions and theorems). A key point

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in the argument (since it is one of the hypotheses of Theorem 1.1.1) is that R admits versal rings at all finite type points, and that the scheme-theoretic images RF
satisfy [2]. In the setting of representations of Gal(K/K∞ ), versal rings for R are
given be the framed deformation rings associated to continuous mod p representations of Gal(K/K∞ ), as we show in [EG22, CEGS19]; then the basic input to theverification of [2] for RF is the theory of finite height framed Galois deformation
rings (which are proved to be Noetherian by Kim in [Kim11], reflecting the factthat the scheme-theoretic images RF turn out to be finite-type algebraic stacks).
In our more general setting we work instead with lifting rings for ϕ-modules.
1.3. Further remarks on the contents and organisation of the paper. In theremainder of Section 1, we describe our notation and conventions (Subsection 1.5),
and also record some simple lemmas in local algebra that will be needed later inthe paper (Subsection 1.6).
In Section 2, we explain Artin’s axioms (as listed above) in some detail, andpresent many related definitions and results. We do not expect the reader experienced in the theory of stacks to find much that is novel in this section, and indeed,
many of the results that we have included are simple variants of results that are
already in the literature. In light of this, it might be worthwhile to offer some justification for the length of this section. Primarily, we have been guided by the demandsof the arguments presented in Section 3; these demands have largely dictated the
choice of material presented in Section 2, and its organisation. Additionally, we
anticipate that the typical reader of this paper interested in the application of ourresults to the moduli of Galois representations will not already be completely familiar with the foundational results discussed in this section, and so we have made aneffort to include a more careful discussion of these results, as well as more references
to the literature, than might strictly be required for the typical reader interested
only in Theorem 1.1.1 and Corollary 1.1.2.
Moreover, the basic idea of our argument came from a careful reading of [Art69a,§3], especially the proof of Theorem 3.9 therein, which provided a model argumentfor deducing Axiom [4] (openness of versality) from a purely geometric assumption
on the object to be represented. We also found the several (counter)examples that
Artin presents in [Art69b, §5] to be illuminating. For these reasons, among others,
we have chosen to discuss Artin’s representability theorem in terms that are as
close as possible to Artin’s treatment in [Art69a, §3] and [Art69b, §5], making
the minimal changes necessary to adapt the statement of the axioms, and of thetheorem, to the stacky situation. Of course, such adaptations have been presented
by many authors, including Artin himself, but these works have tended to focus
on developments of the theory (such as the use of obstruction theories to verify
openness of versality) which are unnecessary for our purposes. Ultimately, we
found the treatment of Artin representability in the Stacks Project [Sta, Tag 07SZ]
to be closest in spirit to the approach we wanted to take, and it forms the basis
for our treatment of the theorem here. However, for the reason discussed above, of
wanting to follow as closely as possible Artin’s original treatment, we have phrased
the axioms in different terms to the way they are phrased in the Stack Project,terms which are closer to Artin’s original phrasing.
One technical reason for preferring Artin’s phrasing is the emphasis that it placeson the role of pro-representability (or equivalently, versality). As already noted,
the main intended application of Theorem 1.1.1 is to the construction of moduli

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M. EMERTON AND T. GEEof Galois representations, and phrasing the axioms in a way which emphasises
pro-representability makes it easy to incorporate the formal deformation theory of
Galois representations into our arguments (one of the main outputs of that formal
deformation theory being various pro-representability statements of the kind that
Theorem 1.1.1 requires as one of its inputs. In fact, in the interests of generality wework with ϕ-modules that do not evidently correspond to Galois representations, so
we do not directly invoke results from Galois deformation theory, but rather adapt
some arguments from that theory to our more general setting.)
On a few occasions it has seemed sensible to us to state and prove a result in its
natural level of generality, even if this level of generality is not strictly required forthe particular application we have in mind. We have also developed the analogueof Artin’s axioms [4a] and [4b] of [Art69a, §3] (referred to as [4] and [5] in [Art69b,§5]) in the stacky context; while not necessary for the proof of Theorem 1.1.1,
thinking in terms of these axioms helps to clarify some of the foundational resultsof Section 2 (e.g. the extent to which the unramifiedness condition of Lemma 2.6.4
can be promoted to the condition of being a monomorphism, as in Corollary 2.6.12).
Just to inventory the contents of Section 2 a little more precisely: in Subsections 2.1 through 2.4 we discuss each of Artin’s axioms in turn. In Subsection 2.5
we develop a partial analogue of [Art70, Prop. 3.11], which allows us to construct
stacks satisfying [1] by defining their values on algebras of finite presentation overthe base and then taking appropriate limits. In Subsections 2.6 and 2.7 we develop
various further technical consequences of Artin’s axioms. Of particular importanceis Lemma 2.6.4, which is a generalisation to the stacky context of one of the steps
appearing in the proof of [Art69a, Lem. 3.10]: it provides unramified algebraic
approximations to stacks satisfying Axioms [1], [2], and [3], and so is the key toestablishing openness of versality (i.e. Axiom [4]) in certain contexts in which it
is not assumed to hold a priori. In Subsection 2.8 we explain how our particular
formulation of Artin’s axioms does indeed imply his representability theorem foralgebraic stacks.
In Section 3, after a preliminary discussion in Subsection 3.1 of the theory ofscheme-theoretic images in the context of morphisms of algebraic stacks, in Subsection 3.2, we present our main definitions, and give the proof of Theorem 1.1.1.
In Subsection 3.3, we investigate the behaviour of our constructions with respect
to base change (both of the target stack F, and of the base scheme S); as well
as being of intrinsic interest, this will be important in our applications to Galois
representations in [CEGS19, EG22].
In Section 4, we give various examples of stacks and Ind-stacks, which illustrate
the results of Section 3, and the roles of the various hypotheses of Section 2 in the
proofs of our main results. We also prove some basic results about Ind-stacks which
are used in the proof of Theorem 1.2.1.
The paper concludes with Section 5, in which we define the moduli stacks ofétale ϕ-modules and prove (via an application of Theorem 1.1.1) that they are
Ind-algebraic stacks.
1.4. Acknowledgements. We would like to thank Alexander Beilinson, Laurent
Berger, Brian Conrad, Johan de Jong, Vladimir Drinfeld, Ashwin Iyengar, Wansu
Kim, Mark Kisin, Jacob Lurie, Martin Olsson, Mike Roth, Nick Rozenblyum andEvan Warner for helpful conversations and correspondence. We are particularly
grateful to Mark Kisin, whose has generously shared his ideas and suggestions about

SCHEME-THEORETIC IMAGES OF MORPHISMS OF STACKS

7

the moduli spaces of Galois representations over the years. We are also indebted to
Brian Conrad, Michael Harris, and Florian Herzig for their close readings of, andmany helpful comments on and corrections to, various versions of this manuscript.
The first author would like to thank the members of his working group at the Stacks
Project Workshop — Eric Ahlqvist, Daniel Bragg, Atticus Christensen, Ariyan
Javanpeykar, Julian Rosen — for their helpful comments. Finally, we thank theanonymous referee for their careful reading of the manuscript, and their many
corrections and suggestions for improvement.
1.5. Notation and conventions. We follow the conventions of [Sta] except where
explicitly noted, and we refer to this reference for background material. We notethat the references to [Sta] in the electronic version of this paper are clickable, andwill take the reader directly to the web page of the corresponding entry. We use
Roman letters X, Y, . . . for schemes and algebraic spaces, and calligraphic letters
X , Y, . . . for (possibly non-algebraic) stacks (and more generally, for categories
fibred in groupoids). We elide the difference between commutative diagrams and2-commutative diagrams, referring to both as simply commutative diagrams.
Since we follow [Sta], we do not assume (unless otherwise stated) that our algebraic spaces and algebraic stacks have quasi-compact or quasi-separated diagonals.
This is in contrast to references such as [Knu71, Art74, LMB00], and occasionallyrequires us to make some simple additional arguments; the reader interested only
in our applications to moduli stacks of Galois representations should feel free toimpose the additional hypotheses on the diagonal that are common in the stacks
literature, and will lose nothing by doing so.
1.5.1. Choice of site. One minor difference between our approach and that takenin [Sta] is that we prefer to only assume that the stacks that we work with arestacks in groupoids for the étale topology, rather than the fppf topology. Thisultimately makes no difference, as the definition of an algebraic stack can be madeusing either the étale or fppf topologies [Sta, Tag 076U]. In practice, this meansthat we will sometimes cite results from [Sta] that apply to stacks in groupoids forthe fppf topology, but apply them to stacks in groupoids for the étale topology. In
each such case, the proof goes over unchanged to this setting.
To ease terminology, from now on we will refer to a stack in groupoids for theétale topology (on some given base scheme S) simply as a stack (or a stack over S).
(On a few occasions in the manuscript, we will work with stacks in sites other thanthe étale site, in which case we will be careful to signal this explicitly.)
1.5.2. Finite type points. If S is a scheme, and s ∈ S is a point of S, we letκ(s) denote the residue field of s. A finite type point s ∈ S is a point such thatthe morphism Spec κ(s) → S is of finite type. By [Sta, Tag 01TA], a morphism
f : Spec k → S is of finite type if and only if there is an affine open U ⊆ S suchthat the image of f is a closed point u ∈ U , and k/κ(u) is a finite extension.
In a Jacobson scheme, the finite type points are precisely the closed points; moregenerally, the finite type points of any scheme S are dense in every closed subset
of S by [Sta, Tag 02J4]. If X → S is a finite type morphism, then a morphism
Spec k → X is of finite type if and only if the composite Spec k → S is of finite
type, and so in particular a point x ∈ X is of finite type if and only if the composite
Spec κ(x) → X → S is of finite type.

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1.5.3. Points of categories fibred in groupoids. If X is a category fibred in groupoids,
then a point of X is an equivalence class of morphisms from spectra of fields
Spec K → X , where we say that Spec K → X and Spec L → X are equivalent
if there is a field M and a commutative diagram/ Spec L
Spec MSpec K/ X.

(This is an equivalence relation by [Sta, Tag 04XF]; strictly speaking, this proves
the claim in the case that X is an algebraic stack, but the proof goes over identicallyto the general case that X is a category fibred in groupoids.) If X is furthermore an
algebraic stack, then the set of points of X is denoted |X |; by [Sta, Tag 04XL] there
is a natural topology on |X |, which has in particular the property that if X → Y isa morphism of algebraic stacks, then the induced map |X | → |Y| is continuous.
If X is a category fibred in groupoids, then a finite type point of X is a pointthat can be represented by a morphism Spec k → X which is locally of finite type.
If X is an algebraic stack, then by [Sta, Tag 06FX], a point x ∈ |X | is of finite typeif and only if there is a scheme U , a smooth morphism ϕ : U → X and a finite typepoint u ∈ U such that ϕ(u) = x. The set of finite type points of an algebraic stack
X is dense in any closed subset of |X | by [Sta, Tag 06G2].
If X is an algebraic space which is locally of finite type over a locally Noetherianbase scheme S, then any finite type point of X may be represented by a monomorphism Spec k → X which is locally of finite type; this representative is unique up tounique isomorphism, and any other morphism Spec K → X representing x factors
through this one. (See Lemma 2.2.14 below.)
1.5.4. pro-categories. We will make several uses of the formal pro-category pro -C
associated to a category C, in the sense of [Gro95]. Recall that an object of pro -C
is a projective system (ξi )i∈I of objects of C, and the morphisms between two
pro-objects ξ = (ξi )i∈I and ν = (ηj )j∈J are by definition
Mor(ξ, η) = lim lim Mor(ξi , ηj ).
←− −→j∈J i∈I

We will apply this definition in particular to categories of Artinian local rings
with fixed residue fields in Section 2.2, and to the category of affine schemes locallyof finite presentation over a fixed base scheme in Section 2.5, as well as to categories(co)fibred in groupoids over these categories.
1.5.5. G-rings. Recall ([Sta, Tag 07GH]) that a Noetherian ring R is a G-ring if
for every prime p of R, the (flat) map Rp → (Rp )∧ is regular. By [Sta, Tag 07PN],
this is equivalent to demanding that for every pair of primes q ⊆ p ⊂ R the algebra(R/q)∧
p ⊗R/q κ(q) is geometrically regular over κ(q) (where κ(q) denotes the residue
field of q; recall [Sta, Tag 0382] that if k is a field, a Noetherian k-algebra A isgeometrically regular if and only if A ⊗k k 0 is regular for every finitely generated
field extension k 0 /k). Excellent rings are G-rings by definition.
In our main results we will assume that our base scheme S is locally Noetherian,and that its local rings OS,s at all finite type points s ∈ S are G-rings. This isa replacement of Artin’s assumption that S be of finite type over a field or an
excellent DVR; this more general setting is permitted by improvements in ArtinSCHEME-THEORETIC IMAGES OF MORPHISMS OF STACKS

9

approximation, due essentially to Popescu ([Pop85, Pop86, Pop90]; see also [CdJ02]
and [Sta, Tag 07GB]). However, since this assumption will not always be in force,
we will indicate when it is assumed to hold.
1.5.6. Groupoids. We will make use of groupoids in algebraic spaces, and we will use
the notation for them which is introduced in [Sta, Tag 043V], which we now recall.
A groupoid in algebraic spaces over a base algebraic space B is a tuple (U, R, s, t, c)
where U and R are algebraic spaces over B, and s, t : R → U and c : R×s,U,t R → R
are morphisms of algebraic spaces over B whose T -points form a groupoid for any
scheme T → B. (The maps s, t, c give the source, target and composition laws forthe arrows of the groupoid.) Given such a groupoid in algebraic spaces, there are
unique morphisms e : U → R and i : R → R of algebraic spaces over B whichgive the identity and inverse maps of the groupoid, and we sometimes denote thegroupoid in algebraic spaces by the tuple (U, R, s, t, c, e, i).
1.5.7. Properties of morphisms. In most cases, we follow the terminology and conventions for properties of morphisms of stacks introduced in [Sta]. We recall someof the general framework of those conventions here.
An important concept, defined for morphisms of categories fibred in groupoids,
and so in particular for morphisms of stacks, is that of being representable byalgebraic spaces. Following [Sta, Tag 04SX], we say that a morphism X → Y
of categories fibred in groupoids is representable by algebraic spaces if for any
morphism T → Y with T a scheme, the fibre product X ×Y T is (representable by) an
algebraic space. (This condition then continues to hold whenever T is an algebraic
space [Sta, Tag 0300].) A morphism of algebraic stacks is representable by algebraic
spaces if and only if the associated diagonal morphism is a monomorphism [Sta,Tag 0AHJ].
If P is a property of morphisms of algebraic spaces which is preserved underarbitrary base-change, and which is fppf local on the target, then [Sta, Tag 03YK]
provides a general mechanism for defining the property P for morphisms of categories fibred in groupoids that are representable by algebraic spaces: namely, such amorphism f : X → Y is defined to have property P if and only if for any morphism
T → Y with T a scheme, the base-changed morphism X ×Y T → T (which is amorphism of algebraic spaces, by assumption) has property P (and it is equivalent
to impose the same condition with T being merely an algebraic space, because an
algebraic space by definition has an étale (and therefore fppf) cover by a scheme,
and P is fppf local on the target by assumption).
If P is a property of morphisms of algebraic spaces which is smooth local on thesource-and-target, then [Sta, Tag 06FN] extends the definition of P to arbitrary
morphisms of algebraic stacks (in particular, to morphisms that are not necessarily
representable by algebraic spaces): a morphism f : X → Y is defined to haveproperty P if it can be lifted to a morphism h : U → V having property P , where
U is a smooth cover of X and V is a smooth cover of Y. If P is furthermore
preserved under arbitrary base-change and fppf local on the target, so that thedefinition of [Sta, Tag 03YK] applies, then the two definitions coincide in the caseof morphisms that are representable by algebraic spaces [Sta, Tag 06FM].
Many additional properties of morphisms of algebraic stacks are defined in [Sta,Tag 04XM]. In Subsection 2.3 below, we further extend many of these definitions

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M. EMERTON AND T. GEE

to the case of morphisms of stacks whose source is assumed to be algebraic, butwhose target is assumed only to satisfy condition [3] of Artin’s axioms.
1.6. Some local algebra. In this subsection, we state and prove some results fromlocal algebra which we will need in what follows.
1.6.1. Lemma. If B → A and C → A are local morphisms from a pair of complete
Noetherian local rings to an Artinian local ring, C → A is surjective, and theresidue field of A is finite over the residue field of B, then the fibre product B ×A C
is a complete Noetherian local ring, and the natural morphism B ×A C → A is local.
Proof. Write R := B ×A C, and mR := mB ×mA mC . Since B → A and C → A
are local morphisms of local rings, we see that if (b, c) ∈ R, with both b and c lying
over the element a ∈ A, then if a ∈ mA , we have (b, c) ∈ mR , while if a 6∈ mA , then(b, c) 6∈ mR . In the latter case, we find that (b, c) is furthermore a unit in R. Thus
R is a local ring with maximal ideal mR , and the natural morphism R → A is local.
If we choose r ≥ 0 so that mrA = 0 (which is possible, since A is Artinian), theneach of mrB and mrC has vanishing image in A, and so we see that
(1.6.2)mrR ⊆ mrB × mrC ⊆ mR .

From this inclusion, and the fact that B × C is mB × mC -adically complete, itfollows that R is mR -adically complete. (Indeed, we see that R is open and closed
as a topological subgroup of B × C, and that the induced topology on R coincides
with the mR -adic topology.)
Finally, to see that R is Noetherian, we use the hypothesis that C → A is surjective, which implies that the residue fields kC and kA of C and A are isomorphic,
as are the residue fields kR and kB of R and B, which are subfields of kC = kA .
Then the inclusion mR = mB ×mA mC ,→ mB × mC induces an inclusion
mR /m2R ,→ mB /m2B × mC /m2C ,and since B and C are Noetherian, and the extension degree [kR : kB ] = [kC : kA ]is finite, the target of the inclusion is a finite-dimensional kR -vector space. It
follows that mR /m2R is also finite-dimensional, and therefore that R is Noetherian,as required.

1.6.3. Lemma. Let B → A be a local morphism from a complete Noetherian local
ring to an Artinian local ring, which induces a finite extension of residue fields.
Then this morphism admits a factorisation B → B 0 → A, where B → B 0 is afaithfully flat local morphism of complete local Noetherian rings, and B 0 → A issurjective (and so in particular induces an isomorphism on residue fields).
Proof. Let kB ⊆ kA be the embedding of residue fields induced by the given
morphism B → A. Let ΛkA denote a Cohen ring with residue field kA , andchoose (as we may) a surjection ΛkA [[x1 , . . . , xd ]]  A (for some appropriate valueof d). Let B denote the image of B in A, and let A0 denote the fibre product
A0 := B ×A ΛkA [[x1 , . . . , xd ]]; then Lemma 1.6.1 shows that A0 is a complete Noetherian local subring of ΛkA [[x1 , . . . , xd ]] with residue field kB . If ΛkB denotes a
Cohen ring with residue field kB , then we may find a local morphism ΛkB → A0
inducing the identity on residue fields. The composite(1.6.4)is flat.

ΛkB → A0 ⊆ ΛkA [[x1 , . . . , xd ]]SCHEME-THEORETIC IMAGES OF MORPHISMS OF STACKS

11

By [Gro64, Thm. 0.19.8.6(i)], the composite morphism ΛkB → A0 → B (thesecond arrow being the projection) may be lifted to a morphism ΛkB → B. Nowb Λk ΛkA [[x1 , . . . , xd ]] (the completed tensor product). By [Gro64,
define B 0 := B ⊗B
Lem. 0.19.7.1.2], B 0 is a complete local Noetherian ring, flat over B.
The given morphisms B → A and ΛkA [[x1 , . . . , xd ]] → A induce a surjection
B 0 → A, and B → B 0 → A is the required factorisation of our given morphism
B → A. (Note that flat local morphisms of local rings are automatically faithfully
flat.)

2. Stacks in groupoids and Artin’s axioms
Since Artin first introduced his axioms characterising algebraic spaces [Art69a],
many versions of these axioms have appeared in the works of various authors. In thispaper we have tried to follow Artin’s original treatment closely, and the labellingof our four axioms is chosen to match the labelling in [Art69a].
In this section we will discuss each of the four axioms, explain why they imply
representability (essentially, by relating them to the axioms given in [Sta, Tag
07SZ]) and also discuss some related foundational material.
As noted in the introduction, our basic setting will be that of stacks in groupoidson the big étale site of a scheme S. A general reference for the basic definitions
and properties of such stacks is [Sta]. As remarked in the introduction, from now
on we will refer to such a stack in groupoids simply as a stack. At times we willfurthermore assume that S is locally Noetherian, and in Subsection 2.8, where we
present Artin’s representability theorem, we will additionally assume that the local
rings OS,s are G-rings, for each finite type point s ∈ S.
2.1. Remarks on Axiom [1]. We begin by recalling the definition of limit preserving.
2.1.1. Definition. A category fibred in groupoids (e.g. an algebraic stack) F over
S is limit preserving if, whenever we have a projective limit T = lim Ti of affine←−
S-schemes, the induced functor(2.1.2)2 - lim F(Ti ) → F(T )−→is an equivalence of categories.
More concretely, as in [Sta, Tag 07XK] this means that each object of F(T ) is
isomorphic to the restriction to T of an object of F(Ti ) for some i; that for any
two objects x, y of F(Ti ), any morphism between the restrictions of x, y to T is
the restriction of a morphism between the restrictions of x, y to Ti0 for some i0 ≥ i;and that for any two objects x, y of F(Ti ), if two morphisms x ⇒ y coincide after
restricting to T , than they coincide after restricting to Ti0 for some i0 ≥ i. (Since weare considering categories fibred in groupoids, it suffices to check this last condition
when one of the morphisms is the identity.)
We have the following related definition [Sta, Tag 06CT].
2.1.3. Definition. A morphism F → G of categories fibred in groupoids (e.g. ofalgebraic stacks) over S is said to be limit preserving on objects if for any affine
S-scheme T , written as a projective limit of affine S-schemes Ti , and any morphism
T → F for which the composite morphism T → F → G factors through Ti for somei, there is a compatible factorisation of the morphism T → F through Ti0 , for somei0 ≥ i.

12

M. EMERTON AND T. GEE

Somewhat more precisely, given a commutative diagram
T

/F


Ti


/G

we may factor it in the following manner:
T

Ti0

/F


Ti


/G

We also make the following variation on the preceding definition.
2.1.4. Definition. We say that a morphism F → G of categories fibred in groupoids(e.g. of algebraic stacks) over S is étale locally limit preserving on objects if for any
affine S-scheme T , written as a projective limit of affine S-schemes Ti , and any
morphism T → F for which the composite morphism T → F → G factors throughTi for some i, then there is an affine étale surjection Ti00 → Ti0 , for some i0 ≥ i,and a morphism Ti00 → F, such that, if we write T 0 := Ti0 ×Ti T, then the resulting
diagram
T0

Ti00

Ti



/F

/G

commutes.
The following lemma relates Definitions 2.1.1 and 2.1.3.
2.1.5. Lemma. If F is a category fibred in groupoids over S, then the following are
equivalent:(1) F is limit preserving.
(2) Each of the morphisms F → S, ∆ : F → F ×S F, and ∆∆ : F →F ×F ×S F F is limit preserving on objects.
Proof. This is just a matter of working through the definitions. Indeed, the morphism F → S being limit preserving on objects is equivalent to the functor (2.1.2)
being essentially surjective (for all choices of T = lim Ti ), the diagonal morphism←−ibeing limit preserving on objects is equivalent to this functor being full, and the
double diagonal being limit preserving on objects is equivalent to this functor being
faithful.

SCHEME-THEORETIC IMAGES OF MORPHISMS OF STACKS

13

More precisely, by definition the morphism F → S is limit preserving on objects
if and only if for every T = lim Ti as above, any morphism T → F factors through←−isome Ti ; equivalently, if and only if every object of F(T ) is isomorphic to therestriction to T of an object of F(Ti ) for some i; equivalently, if and only if thefunctor (2.1.2) is essentially surjective. Similarly, the morphism ∆ : F → F ×S F islimit preserving on objects if and only if for any pair of objects of F(Ti ) (for somevalue of i), a morphism between their images in F(T ) arises as the restriction of amorphism between their images in F(Ti0 ), for some i0 ≥ i; or equivalently, if andonly if the functor (2.1.2) is full. The claim about the double diagonal is similar,
and is left to the interested reader.

We can strengthen the preceding lemma when the category fibred in groupoids
involved is actually a stack.
2.1.6. Lemma. If F is a stack over S, then the following are equivalent:(1) F is limit preserving.
(2) The morphism F → S is étale locally limit preserving on objects, while eachof ∆ : F → F ×S F and ∆∆ : F → F ×F ×S F F is limit preserving on
objects.
Proof. Taking into account Lemma 2.1.5, we see that the lemma will follow if
we show that the assumptions of (2) imply that F → S is limit preserving on
objects. Thus we put ourselves in the situation described in Definition 2.1.3 (taking
G = S), namely we give ourselves an affine S-scheme T , written as a projective limit
T = limi Ti of affine S-schemes, and we suppose given a morphism T → F; we must←−show that this morphism factors through Ti for some i. Applying the assumptionthat F → S is étale locally limit preserving on objects, we find that, for some i,we may find an étale cover Ti0 of Ti and a morphism Ti0 → F, for some value of i,through which the composite T 0 := Ti0 ×Ti T → T → F factors; our goal is then toshow that, for some i0 ≥ i, we may find a morphism Ti0 → F through which themorphism T → F itself factors.
For any i0 ≥ i, we let Ti00 := Ti0 ×Ti Ti0 . In order to find the desired morphism
Ti0 → F, it suffices to equip the composite Ti00 → Ti0 → F with descent data
to Ti0 , in a manner compatible with the canonical descent data of the composite
T 0 → T → F to T . That this is possible follows easily from the assumptions on the
diagonal and double diagonal of F (cf. the proof of Lemma 2.5.5 (2) below).

We will have use for the following finiteness results.
2.1.7. Lemma. If F → G → H are morphisms between categories fibred in groupoids
over S, and if both the composite morphism F → H and the diagonal morphism∆ : G → G ×H G are limit preserving on objects, then the morphism F → G is also
limit preserving on objects.
Proof. Let T = lim Ti be a projective limit of affine S-schemes, and suppose that←−
we are given a morphism T → F such that the composite T → F → G factors
through Ti for some i. We must show that there is a compatible factorisation of
T → F through Ti0 for some i0 ≥ i.
Since the composite F → H is limit preserving on objects, we may factor T → F
through some Tj , in such a way that the composites T → Tj → F → H and
T → Ti → G → H coincide. Replacing i, j by some common i00 ≥ i, j, we have two

14

M. EMERTON AND T. GEE

morphisms Ti00 → G (one coming from the given morphism Ti → G, and one fromthe composite Tj → F → G) which induce the same morphism to H, and which
agree when pulled-back to T . Since ∆ is limit preserving on objects, they agree
over some Ti0 , for some i0 ≥ i00 , as required.

2.1.8. Corollary. If F and G are categories fibred in groupoids over S, both of which
are limit preserving, then any morphism F → G is limit preserving on objects.
Proof. This follows directly from Lemmas 2.1.5 and 2.1.7 (taking H = S in thelatter).

The next lemma (which is essentially [LMB00, Prop. 4.15(i)]) explains why thecondition of being limit preserving is sometimes referred to as being locally of finite
presentation.
2.1.9. Lemma. If F is an algebraic stack over S, then the following are equivalent:(1) F is limit preserving.
(2) F → S is limit preserving on objects.
(3) F is locally of finite presentation over S.
Proof. Note that (1) =⇒ (2) by definition (since (2) is just the condition that thefunctor (2.1.2) be essentially surjective), while the equivalence of (2) and (3) is a
special case of Lemma 2.3.16 below. (The reader may easily check that the present
lemma is not used in the proof of that result, and so there is no circularity inappealing to it.)
It remains to show that (3) =⇒ (1). Assuming that (3) holds, we claim thatthe diagonal ∆ : F → F ×S F is locally of finite presentation. To see this, choose
a smooth surjection U → F whose source is a scheme (which exists because F isassumed algebraic). Since U → F and F → S are locally of finite presentation by
assumption, we see that U → S is locally of finite presentation by [Sta, Tag 06Q3].
Then the diagonal U → U ×S U is locally of finite presentation by [Sta, Tag 0464]
and [Sta, Tag 084P], and therefore the composite
U → U ×S U → F × S U → F × S F
is locally of finite presentation (note that the last two morphisms are base changesof the smooth morphism U → F). Factoring this morphism as∆

U → F → F ×S F,(where ∆ is representable by algebraic spaces, because F is algebraic), we see
from [Sta, Tag 06Q9] that ∆ is locally of finite presentation, as claimed. A similarargument shows that the double diagonal ∆∆ : F → F ×F ×S F F is locally of finite
presentation. Applying Lemma 2.3.16 below (or [Sta, Tag 06CX]), we find thateach of ∆, ∆∆ and the structure map F → S is limit preserving on objects. It now
follows from Lemma 2.1.5 that F is limit preserving, as required.

2.2. Remarks on Axiom [2]. Throughout this subsection, we assume that S is
locally Noetherian, and that F is a category fibred in groupoids over S, which is astack for the Zariski topology. We denote by Fb the restriction of F to the categoryof finite type Artinian local S-schemes.

SCHEME-THEORETIC IMAGES OF MORPHISMS OF STACKS

15

We begin by discussing Axiom [2](a), which is the Rim–Schlessinger condition(RS). Consider pushout diagrams
Y

/ Y0


Z


/ Z0

of S-schemes, with the horizontal arrows being closed immersions. We have an
induced functor(2.2.1)F(Z 0 ) → F(Y 0 ) ×F (Y ) F(Z).

2.2.2. Definition. We say that F satisfies condition (RS) if the functor (2.2.1)
is an equivalence of categories whenever Y, Y 0 , Z, Z 0 are finite type local ArtinianS-schemes.
2.2.3. Lemma. If F is an algebraic stack, then F satisfies (RS).
Proof. This is immediate from [Sta, Tag 07WN].



The same condition appears under a different name, and with slightly different
phrasing, in [HR19, Lem. 1.2]. We recall this, and some closely related notions that
we will occasionally use.
2.2.4. Definition. Following [HR19], we say that F is Artfin -homogeneous (resp.
Arttriv -homogeneous, resp. Artsep -homogeneous, resp. Artinsep -homogeneous) if
the functor (2.2.1) is an equivalence of categories whenever Y, Z are local ArtinianS-schemes of finite type over S (resp. with the induced extension of residue fields
being trivial, resp. separable, resp. purely inseparable), and Y → Y 0 is a nilpotent
closed immersion.
2.2.5. Lemma. F satisfies (RS) if and only if it is Artfin -homogeneous.
Proof. This is just a matter of comparing the two definitions. Precisely: a closed
immersion of local Artinian schemes is automatically nilpotent. Conversely, a finite
type nilpotent thickening of a local Artinian scheme is local Artinian, and the
pushout of local Artinian schemes of finite type over S is also local Artinian offinite type over S. (Recall from [Sta, Tag 07RT] that if above we write Y = Spec A,
Y 0 = Spec A0 , Z = Spec B, then the pushout Z 0 = Spec B 0 is just given by B 0 =B ×A A0 .)
The following lemma relates the various conditions of Definition 2.2.4.
2.2.6. Lemma. The condition of Artfin -homogeneity of Definition 2.2.4 implies
each of Artsep -homogeneity and Artinsep -homogeneity, and these conditions inturn imply Arttriv -homogeneity. If F is a stack for the étale site, then conversely,
Arttriv -homogeneity implies Artsep -homogeneity, while Artinsep -homogeneity implies Artfin -homogeneity. If F is furthermore a stack for the fppf site, then Arttriv homogeneity implies Artfin -homogeneity.
Proof. This follows immediately from [HR19, Lem. 1.6] and its proof (see also theproof of [HR19, Lem. 2.6]).
16

M. EMERTON AND T. GEE

We now discuss Axiom [2](b). To begin, we recall some definitions from [Sta,Tag 06G7].
Fix a Noetherian ring Λ, and a finite ring map Λ → k, whose target is a field.
Let the kernel of this map be mΛ (a maximal ideal of Λ). We let CΛ be the category
whose objects are pairs (A, φ) consisting of an Artinian local Λ-algebra A and aΛ-algebra isomorphism φ : A/mA → k, and whose morphisms are given by local Λalgebra homomorphisms compatible with φ. Note that any such A is finite over Λ,and that the morphism Λ → A factors through ΛmΛ , so that we have CΛ = C(ΛmΛ )in an evident sense.
There are some additional categories, closely related to CΛ , that we will also
consider. We let CbΛ denote the category of complete Noetherian local Λ-algebras A∼
equipped with a Λ-algebra isomorphism A/mA −→ k, while we let pro -CΛ denotethe category of formal pro-objects from CΛ in the sense of Section 1.5.4. If (Ai )i∈I
is an object of pro -CΛ , then we form the actual projective limit A := limi∈I Ai ,←−thought of as a topological ring (endowed with the projective limit topology, each Ai
being endowed with its discrete topology). In this manner we obtain an equivalenceof categories between pro -CΛ and the category of topological pro-(discrete Artinian)local Λ-algebras equipped with a Λ-algebra isomorphism between their residue fields
and k [Gro95, §A.5]. We will frequently identify an object of pro -CΛ with theassociated topological local Λ-algebra A. There is a fully faithful embedding of CbΛ
into pro -CΛ given by associating to any object A of the former category the proobject (A/miA )i≥1 . In terms of topological Λ-algebras, this amounts to regarding A
as a topological Λ-algebra by equipping it with its mA -adic topology.
2.2.7. Remark. We note that objects of pro -CΛ , when regarded as topological rings,
are examples of pseudo-compact rings, in the sense of [Gab62]. In particular, anymorphism of such rings has closed image, and induces a topological quotient map
from its source onto its image; consequently, a homomorphism A → B of suchrings is surjective if and only if it is induced by a compatible collection of surjective
morphisms Ai → Bi for projective systems (Ai )i∈I and (Bi )i∈I of objects in CΛ .
(See the discussion beginning on [Gab62, p. 390], especially the statement and proof
of Lem. 1 and of Thm. 3.)
We have the usual notion of a category cofibred in groupoids over CΛ , forwhich see [Sta, Tag 06GJ]. The particular choices of CΛ and categories cofibredin groupoids over CΛ that we are interested in arise as follows (see [Sta, Tag 07T2]
for more details). Let F be a category fibred in groupoids over S, let k be a field
and let Spec k → S be a morphism of finite type.
Let x be an object of F lying over Spec k, and let Spec Λ ⊆ S be an affine open
so that Spec k → S factors as Spec k → Spec Λ → S, where Λ → k is finite. (Forthe existence of such a Λ, and the independence of CΛ of the choice of Λ up to
canonical equivalence, see [Sta, Tag 07T2].) Write p : F → S for the tautologicalmorphism. We then let Fbx be the category whose:(1) objects are morphisms x → x0 in F such that p(x0 ) = Spec A, with A an
Artinian local ring, and the morphism Spec k → Spec A given by p(x) →p(x0 ) corresponds to a ring homomorphism A → k identifying k with theresidue field of A, andSCHEME-THEORETIC IMAGES OF MORPHISMS OF STACKS

17(2) morphisms (x → x0 ) → (x → x00 ) are commutative diagrams in F of theform
00
x0 ` o>xx
Note that the ring A in (1) is an object of CΛ . Under the assumption that F satisfies(RS), Fbx is a deformation category by [Sta, Tag 07WU]. By definition, this meansthat Fbx (Spec k) is equivalent to a category with a single object and morphism, andthat Fbx is cofibred in groupoids and satisfies a natural analogue of (RS) (moreprecisely, an analogue of Arttriv -homogeneity).
The category Fbx naturally extends to its completion, which by definition is
the pro-category pro -Fbx , which is a category cofibred in groupoids over pro -CΛ .
(Note that this is a more general definition than that of [Sta, Tag 06H3], which
restricts the definition to CbΛ .) There is a fully faithful embedding of Fbx into its
completion, which attaches to any object of Fbx lying over an Artinian Λ-algebra thecorresponding pro-object, and this embedding induces an equivalence between Fbx
and the restriction of its completion to CΛ . We therefore also denote the completionof Fbx by Fbx . If A is an object of pro -CΛ , then we will usually denote an object of
the completion of Fbx lying over A by a morphism Spf A → Fbx .
We also introduce the notation Fx to denote the following category cofibred ingroupoids over pro -CΛ : if A is an object of pro -CΛ , then Fx (A) denotes the groupoid
consisting of morphisms Spec A → F, together with an isomorphism between the
restriction of this morphism to the closed point of Spec A and the given morphismx : Spec k → F. If A is Artinian, then Fx (A) = Fbx (A). In general, there is anatural functor Fx (A) → Fbx (A) (the functor of formal completion); a morphism
Spf A → F lying in the essential image of this functor is said to be effective.
2.2.8. Remark. If A is an object of pro -CΛ , then we may consider the formal scheme
Spf A as defining a sheaf of sets on the étale site of S, via the definition Spf A :=limi Spec Ai (writing A as the projective limit of its discrete Artinian quotients Ai ,−→and taking the inductive limit in the category of étale sheaves; this is a special caseof the Ind-constructions considered in Subsection 4.2 below). Of course, we may
also regard the resulting sheaf Spf A as a stack (in setoids).
Giving a morphism Spf A → Fbx in the sense described above is then equivalent
to giving a morphism of stacks Spf A → F which induces the given morphismx : Spec k → F when composed with the natural morphism Spec k → Spf A.
This view-point is useful on occasion; for example, we say that Spf A → Fbx is aformal monomorphism if the corresponding morphism of stacks Spf A → F is amonomorphism. (Concretely, this amounts to the requirement that the induced
morphism Spec Ai → F is a monomorphism for each discrete Artinian quotient Aiof A.)
We now introduce the notion of a versal ring at the morphism x, which will be
used in the definition of Axiom [2](b). (See Remark 2.2.10 for a discussion of why
we speak of a versal ring at a morphism, rather than at a point.) As above, fix thefinite type morphism Spec k → S, an affine open subset Spec Λ → S through which
this morphism factors, and the lift of this morphism to a morphism x : Spec k → F.

18

M. EMERTON AND T. GEE

2.2.9. Definition. Let Ax be a topological local Λ-algebra corresponding to anelement of pro -CΛ . We say that a morphism Spf Ax → Fbx is versal if it is smooth,in the sense of [Sta, Tag 06HR], i.e. satisfies the infinitesimal lifting property withrespect to morphisms in CΛ . More precisely, given a commutative diagram
Spec A/ Spec B

 z
Spf Ax/ Fbx

in which the upper arrow is the closed immersion corresponding to a surjection
B → A in CΛ , and the left hand vertical arrow corresponds to a morphism in pro -CΛ(equivalently, it is continuous when Ax is given its projective limit topology, and A
is given the discrete topology) we can fill in the dotted arrow (with a morphismcoming from a morphism in pro -CΛ ) so that the diagram remains commutative.
We refer to Ax as a versal ring to F at the morphism x. We say that Ax isan effective versal ring to F at the morphism x if the morphism Spf Ax → Fbx iseffective.
We say that F admits versal rings at all finite type points if there is a versal
ring for every morphism x : Spec k → F whose source is a finite type OS -field.
We say that F admits effective versal rings at all finite type points if there is aneffective versal ring for every morphism x : Spec k → F whose source is a finite
type OS -field.
Then Axiom [2](b) is by definition the assertion that F admits Noetherian effective versal rings at all finite type points.
2.2.10. Remark. One complication in verifying Axiom [2](b) is that, at least apriori, it does not depend simply on the finite type point of F represented by agiven morphism x : Spec k → F (for a field k of finite type over OS ), but on
the particular morphism. More concretely, if we are given a finite type morphism
Spec l → Spec k, and if we let x0 denote the composite Spec l → Spec k → F, thenit is not obvious that validity of [2](b) for either of x or x0 implies the validityof [2](b) for the other. One of the roles of Axiom [2](a) in the theory is to bridge
the gap between different choices of field defining the same finite type point of F,
and indeed one can show that in the presence of [2](a), the property of Fbx admittinga Noetherian versal ring depends only on the finite type point of F underlying x;
see part (1) of Lemma 2.8.7 below.
The problem of showing that effectivity is independent of the choice of representative of the underlying finite type point of F seems slightly more subtle, andto obtain a definitive result we have to make additional assumptions on F, andpossibly on S. This is the subject of the parts (2), (3), and (4) of Lemma 2.8.7.
In the case when the morphism Spec l → Spec k is separable, it is possible to
make a softer argument to pass the existence of a versal ring from from x to x0 ,without making any Noetherianness assumptions. We do this in Lemma 2.2.13; wefirstly recall from [Sta, Tag 07W7,Tag 07WW] a useful formalism for consideringsuch a change of residue field.
2.2.11. Remark. Let F be a category fibred in groupoids over S, and fix a morphismx : Spec k → F, where k is a finite type OS -field. Suppose that we are given a finite

SCHEME-THEORETIC IMAGES OF MORPHISMS OF STACKS

19

type morphism Spec l → Spec k, so that l/k is a finite extension of fields, and let
x0 denote the composite Spec l → Spec k → F. Write CΛ,k for the category of local
Artinian Λ-algebras with residue field k, and similarly write CΛ,l for the categoryof local Artinian Λ-algebras with residue field l.
We let (Fbx )l/k denote the category cofibred in groupoids over CΛ,l defined by
setting (Fbx )l/k (B) := Fbx (B ×l k), for any object B of CΛ,l . If F satisfies [2](a),it follows from [Sta, Tag 07WX] that there is a natural equivalence of categories∼
cofibred in groupoids (Fbx )l/k −→ Fbx0 ; an examination of the proof shows that
if l/k is separable, the same conclusion holds if F is only assumed to be Artsep homogeneous.
2.2.12. Remark. Recall [Sta, Tag 06T4] that we say that a Noetherian versal morphism Spf Ax → Fbx is minimal if whenever we can factor this morphism througha morphism Spf A → Fbx , the underlying map A → Ax is surjective. This notion isclosely related to conditions on the tangent space of Fbx , in the following way.
By definition, the tangent space T Fbx of Fbx is the k-vector space Fbx (k[]). As
explained in [Sta, Tag 06I1], there is a natural action of DerΛ (k, k) on T Fbx , andthe versal morphism Spf Ax → Fbx gives rise to a DerΛ (k, k)-equivariant morphismd : DerΛ (Ax , k) → T Fbx .
By [Sta, Tag 06IR], a Noetherian versal morphism Spf Ax → Fbx is minimal
provided that the morphism d is bijective on DerΛ (k, k)-orbits. Conversely, if F
is Arttriv -homogeneous, then it follows from the proof of [Sta, Tag 06J7] that Fbx
satisfies the condition (S2) of [Sta, Tag 06HW], and it then follows from [Sta, Tag
06T8] that if Spf Ax → Fbx is minimal, then d is bijective on DerΛ (k, k)-orbits.
2.2.13. Lemma. Let F be a category fibred in groupoids over S which is Artsep homogeneous. Suppose given x : Spec k → F, with k a finite type OS -field, let l/k bexa finite separable extension, and let x0 denote the composite Spec l → Spec k −→ F.
Suppose also that Spf Ax → Fbx is a versal ring to F at the morphism x, so that inparticular Ax is a pro-Artinian local OS -algebra with residue field k. Let Ax0 /Ax
denote the finite étale extension of Ax corresponding (via the topological invarianceof the étale site) to the finite extension l of k, so that in particular Ax0 is a proArtinian local OS -algebra with residue field l.
Then the induced morphism Spf Ax0 → Fbx0 realises Ax0 as a versal ring to F at
the morphism x0 . If Ax is Noetherian, and the morphism Spf Ax → Fbx is minimal,
then so is the morphism Spf Ax0 → Fbx0 .
Proof. By Remark 2.2.11, since F is Artsep -homogeneous and l/k is separable,∼
we have a natural equivalence of groupoids (Fbx )l/k −→ Fbx0 . Suppose given a
commutative diagram/ Spec B
Spec A z
Spf Ax0/ Fbx0in which the upper arrow is the closed immersion corresponding to a surjection
B → A in CΛ,l ; we wish to show that we can fill in the dotted arrow in such a waythat resulting diagram remains commutative. Passing to the pushout with k over l,

20

M. EMERTON AND T. GEE

and noting that the morphism Ax → Ax0 factors through Ax0 ×l k, we obtain acommutative diagram
Spec A ×l k

/ Spec B ×l k


Spf Ax0 ×l k/ Spf Ax%

/ Fbx

where the dotted arrow exists by the versality of the morphism Spf Ax → Fbx . Now
consider the diagram/ Spec B
Spec ASpec B ×l k

Spf Ax0/ Spf Ax .

Since Spf Ax0 → Spf Ax is formally étale, we may fill in the dotted arrow so as
to make the resulting diagram commutative. This dotted arrow also makes theoriginal diagram commutative, so Ax0 is a versal ring to F at the morphism x0 .
Finally, suppose that the versal morphism Spf Ax → Fbx is minimal. By Remark 2.2.12, the natural morphism DerΛ (Ax , k) → T Fbx is a bijection on DerΛ (k, k)orbits, and we need to show that the natural morphism DerΛ (Ax0 , l) → T Fbx0 is abijection on DerΛ (l, l)-orbits.
By [Sta, Tag 06I0,Tag 07WB] there is a natural isomorphism of l-vector spaces∼b
T Fx ⊗k l −→ T Fbx0 . Since Ax0 /Ax is étale (and l/k is separable), restriction induces
isomorphisms DerΛ (Ax0 , k) ∼= DerΛ (Ax , k) and DerΛ (l, k) ∼= DerΛ (k, k), and thus we∼also have natural isomorphisms of l-vector spaces DerΛ (Ax , k) ⊗k l −→ DerΛ (Ax0 , l)∼and DerΛ (l, l) −→ DerΛ (k, k) ⊗k l. One easily checks that the base-change by l overk of the morphism d : DerΛ (Ax , k) → T Fbx coincides, with respect to these identifications, with the morphism d : DerΛ (Ax0 , l) → T Fbx0 , in a manner which identifies,
under the isomorphism DerΛ (k, k) ⊗k l ∼= DerΛ (l, l), the action of DerΛ (k, k) ⊗k lwith the action of DerΛ (l, l). The result follows.

We now engage in a slight digression; namely, we use the theory of versal ringsin order to define the notion of the complete local ring at a finite type point of analgebraic space locally of finite type over S.
In order to motivate this concept, we recall first that quasi-separated algebraic
spaces are decent, in the sense of [Sta, Tag 03I8], which is to say that any pointof such an algebraic space X is represented by a quasi-compact monomorphismx : Spec k → X. Given a point x of a quasi-separated algebraic space, thoughtof as such a quasi-compact monomorphism, Artin and Knutson ([Art69b, Defn.
2.5], [Knu71, Thm. 6.4]) define a Henselian local ring of X at x; completing thislocal ring gives our desired complete local ring in this case. Rather than imposing
a quasi-separatedness hypothesis at this point and appealing to these results, we
adopt a slightly different approach, which will allow us to define a complete local
ring at any finite type point x of a locally finite type algebraic space X over S.

SCHEME-THEORETIC IMAGES OF MORPHISMS OF STACKS

21(Note that throughout the discussion, we will maintain our assumption that S is
locally Noetherian.)
To begin with, we note that finite type points always admit representatives thatare monomorphisms (regardless of any separatedness hypothesis); indeed, we havethe following lemma.
2.2.14. Lemma. Any finite type point of an algebraic space X, locally of finite typeover the locally Noetherian scheme S, admits a representative Spec k → X which amonomorphism. This representative is unique up to unique isomorphism, the field
k is a finite type OS -field, and any other representative Spec K → X of the given
point factors through this monomorphic representative in a unique fashion.
Proof. We apply the criterion of part (1) of [Sta, Tag 03JU]. More precisely, wechoose an étale morphism U → X with U a scheme over S, which will again be
locally of finite type over S (since smooth morphisms are locally of finite type, and X
is assumed to be locally of finite type over S). We choose a pair of finite type points
u, u0 ∈ U lying over the given finite type point of X; as noted in the proof of thelemma just cited, we must verify that the underlying topological space of the schemeu ×X u0 is finite. But since the diagonal map X → X ×S X is a monomorphism (asX is an algebraic space), the fibre product u ×X u0 maps via a monomorphism intou ×S u0 . This latter scheme has a finite underlying topological space (since u and
u0 are each the Spec of some finite type OS -field). Since monomorphisms induce
embeddings on underlying topological spaces, we see that u ×X u0 also has a finite
underlying topological space. By the above cited lemma, this implies the existenceof the desired monomorphism Spec k → X representing the given point. Denote
this monomorphism by x.
If x0 : Spec K → X is any other representative of the same point, then we may
consider the base-changed morphism Spec k ×X Spec K → Spec K, which is again
a monomorphism. Since its source is non-empty (as x and x0 represent the same
point of X), and as its target is the Spec of a field, it must be an isomorphism;
equivalently, the morphism x0 must factor through x. Since x is a monomorphism,
this factorisation is unique. Since K can be chosen to be a finite type OS -field, we
see that k must in particular be a finite type OS -field. If x0 is also a monomorphism,
then we may reverse the roles of x and x0 , and so conclude that x is determineduniquely up to unique isomorphism. This completes the proof of the lemma.
The following proposition then constructs complete local rings at finite typepoints of locally finite type algebraic spaces over S.
2.2.15. Proposition. If X is an algebraic space, locally of finite type over the locally
Noetherian scheme S, and if x : Spec k → X is a monomorphism, for some field
k of finite type over OS , then there is an effective Noetherian versal ring Ax to X
at the morphism x with the property that the corresponding morphism Spf Ax → X
is a formal monomorphism. Furthermore, the ring Ax , equipped with its morphism
Spec Ax → X inducing the given morphism x, is unique up to unique isomorphism.
Finally, if A is any object of CΛ , then any morphism Spec A → X factors uniquely
through the morphism Spec Ax → X.
Proof. Let Ax be a minimal versal ring to X at the morphism x, in the senseof [Sta, Tag 06T4] (which exists, and is Noetherian, by virtue of [Sta, Tag 06T5]
and the fact that X, being an algebraic space, admits a Noetherian versal ring at

22

M. EMERTON AND T. GEE

the morphism x; see e.g. Theorem 2.8.4 below). We will show that the morphism
Spf Ax → X is a formal monomorphism. To this end, we choose an étale surjective
morphism U → X whose source is a scheme (such a morphism exists, since X isan algebraic space). It suffices (by [Sta, Tag 042Q], and the definition of a formal
monomorphism in Remark 2.2.8) to show that the base-changed morphism U ×X
Spf Ax → U is a formal monomorphism. We begin by describing this morphism
more explicitly.
Abusing notation slightly, we write x to denote the point Spec k, as well its
monomorphism into X. The pull-back of U over the monomorphism x → X is thenan étale morphism Ux → x with non-empty source. We may write Ux as a disjoint
union of points ui , each of which is of the form ui = Spec li , for some finite separable
extension li of k. Since Ux → U is a monomorphism (being the base-change of amonomorphism), each of the composites ui → Ux → U is also a monomorphism; inother words, for each i, the field li is also the residue field of the image of ui in U ;in light of this, we identify ui with its image in U .
For each i, let Ai denote the finite étale extension of Ax corresponding, via the
topological invariance of the étale site, to the finite extension li /k. The projection
U ×X Spf Ax → Spf Ax is formally étale (being the pull-back `of an étale morphism),and thus admits a natural identification with the morphism i∈I Spf Ai → Spf Ax .
Since
the points ui of U are distinct for distinct values of i, in order to show that
`
Spf
Ai ∼
= U ×X Spf Ax → U is a formal monomorphism, it suffices to showi∈Ithat each of the individual morphisms Spf Ai → U is a formal monomorphism; andwe now turn to doing this.
For each i, we let CΛ,li denote the category of local Artinian OS -algebras withresidue field li . We write xi to denote the composite Spec li = ui → U → X.
The infinitesimal lifting property for the étale morphism U → X shows that the
bu → Xbx is an equivalence of categories cofibred in groupoids
induced morphism Uiiover CΛ,li . Lemma 2.2.13 shows that Ai is a versal ring to X at xi , and, by thenoted equivalence, it is thus also a versal ring to U at ui . In fact, since Ax is aminimal versal ring at x, each of the rings Ai is a minimal versal ring at ui .
On the other hand, since ui is a point of the scheme U , the complete local ringbU,u is a minimal versal ring to U at ui ; thus we may identify Ai with O
bU,u , so thatOiibU,u → U .
the morphism Spf Ai → U is identified with the canonical morphism Spf OiThis latter morphism is a formal monomorphism, and thus so is the former.
The fact that the morphism Spf Ax → X is effective is a consequence of X
being an algebraic space; see [Sta, Tag 07X8]. The uniqueness of this morphism,
up to unique isomorphism, follows from Lemma 2.2.16 below; the fact that its
effectivisation is unique up to unique isomorphism again follows from [Sta, Tag
07X8].
Since Ax is versal to X at x, any morphism Spec A → X, for A an object of CΛ ,factors through the morphism Spf Ax → X. Of course, it then factors through theinduced morphism Spec B → X, for some Artinian quotient B of Ax , and hence
also through the morphism Spec Ax → X. The uniqueness of this factorisation
again follows from Lemma 2.2.16 below.

2.2.16. Lemma. Let F be a category fibred in groupoids, and suppose that Spf Ax →Fbx is a versal morphism at the morphism x : Spec k → F, where k is a finite typeOS -field. Suppose also that Spf Ax → Fbx is a formal monomorphism.

SCHEME-THEORETIC IMAGES OF MORPHISMS OF STACKS

23

Then if A is any object of CΛ , any morphism Spec A → Fbx factors uniquely
through the morphism Spf Ax → Fbx . Furthermore, the ring Ax , together with themorphism Spf Ax → Fbx , is uniquely determined up to unique isomorphism by theproperty of being a versal formal monomorphism.
Proof. Since the morphism Spf Ax → Fbx is versal by assumption, any morphism
Spec A → Fbx factors through this morphism; that this factorisation is unique isimmediate from the definition of a formal monomorphism. If Spf A0x → Fbx isanother versal formal monomorphism, then applying this property to the discrete
Artinian quotients of A0x , and then reversing the roles of Ax and A0x , we find therequired unique isomorphisms.

2.2.17. Definition. We refer to the ring Ax of Proposition 2.2.15 as the complete
local ring of X at the point x.
In Subsection 4.2 below, we generalise the notion of the complete local ring ata point to certain Ind-algebraic spaces; see Definition 4.2.13. We now state andprove a result which will be used in Section 3.3, and which uses this generalisation.
2.2.18. Lemma. If F admits versal rings at all finite type points (in the sense of
Definition 2.2.9), and if F 0 → F is representable by algebraic spaces and locally offinite presentation, then F 0 admits versal rings at all finite type points. In fact,
if x0 : Spec k → F 0 is a morphism from a finite type OS -field to F 0 , inducing themorphism x : Spec k → F, and if Spf A → F is a versal ring to F at x, thenX := F 0 ×F Spf A is defined as an Ind-locally finite type algebraic space over S (inthe sense of Definition 4.2.11 below), the morphism x0 induces a lift of x to X, andthe complete local ring of X at x is a versal ring to F 0 at x0 .
Proof. If we write A ∼A as a projective limit of finite type local Artinian= lim←−i∈I iOS -algebras, then we define X := limi∈I F 0 ×F Spec Ai ; thus X is an Ind-locally−→finite type algebraic space over S, which is clearly well-defined (as a sheaf of setoids
on the étale site of S) independently of the choice of description of A as a projective
limit. The only claim, then, that is not immediate from the definitions is that thecomposite morphism Spf OX,x → X → F 0 is versal. We leave this as an easy
exercise for the reader; it is essentially immediate from the versality of Spf A. 
We now introduce the notion of a presentation of a deformation category by aneffectively Noetherianly pro-representable smooth groupoid in functors, which is
closely related to the notion of admitting an effective Noetherian versal ring. Ourreason for introducing this notion is to prove Lemma 2.2.24 and Corollary 2.7.3;
under an appropriate hypothesis on the diagonal of F, these will enable us to deduce
Axiom [2](a) from [2](b).
We say that a set-valued functor on CΛ is pro-representable if it representableby an object of pro -CΛ . If A is the associated topological ring to the representing
pro-object, then we will frequently denote this functor by Spf A. We say that afunctor is Noetherianly pro-representable if it is pro-representable by an object Aof CbΛ .1
1In [Sta], what we call Noetherian pro-representability is called simply pro-representability;see [Sta, Tag 06GX]. However, we will need to consider more general pro-representable functors,
and so we need to draw a distinction between the general case and the case of pro-representability
by an object of CbΛ .

24

M. EMERTON AND T. GEE

We refer to [Sta, Tag 06K3] for the definition of a groupoid in functors over CΛ ,and then make the following related definitions.
2.2.19. Definition. (1) We say that a groupoid in functors over CΛ , say (U, R, s, t, c),is smooth if s, t : R → U are smooth2 ; equivalently, if the quotient morphism
U → [U/R] is smooth.
(2) We say that (U, R, s, t, c) is (Noetherianly) pro-representable if U and R areeach (Noetherianly) pro-representable.
∼
A presentation of Fbx is an equivalence [U/R] −→ Fbx of categories cofibred ingroupoids over CΛ , where (U, R, s, t, c) is a groupoid in functors over CΛ . Suppose given such a presentation by a groupoid in functors that is Noetherianly prorepresentable, in the sense of Definition 2.2.19, and let Ax ∈ Ob(CbΛ ) be an object
that pro-represents U . We then obtain an induced morphism(2.2.20)
Spf Ax = U → [U/R] → Fbx .

2.2.21. Definition. We say that the given presentation is effectively Noetherianly
pro-representable if the morphism (2.2.20) is effective, i.e. arises as the formal completion of a morphism Spec Ax → F.
The existence of an effectively Noetherianly pro-representable presentation by asmooth groupoid in functors is closely related to the property of having effective
versal rings, as we will now see.
∼
2.2.22. Lemma. If [U/R] −→ Fbx is a presentation of Fbx by a smooth groupoid infunctors, for which U is pro-representable by a topological local Λ-algebra A, thenthe morphism Spf A = U → Fbx is versal. Conversely, if A is the topological local
Λ-algebra corresponding to some element of pro -CΛ , and if Spf A → Fbx is versal,
then if we write U := Spf A and R = U ×Fbx U, and s, t for the two projections
R → U , then (U, R, s, t) is a smooth groupoid in functors, and the natural morphism[U/R] → Fbx is an equivalence, and thus equips Fbx with a presentation by a smoothgroupoid in functors.
∼
Proof. Essentially by definition, if [U/R] −→ Fbx is a presentation of Fbx by a smooth∼groupoid in functors, then the induced morphism U → [U/R] −→ Fbx is smooth (inthe sense of [Sta, Tag 06HR]), and so by definition is versal. The converse statement
follows from [Sta, Tag 06L1]).


2.2.23. Remark. Suppose that Fbx admits a presentation by a smooth Noetherianly
pro-representable groupoid in functors. Then, if U = Spf A → Fbx is a versal
morphism with A an object of CbΛ (so that U is Noetherianly pro-representable),
one finds that R := U ×Fbx U is also Noetherianly pro-representable (cf. the proof∼of [Sta, Tag 06L8]), and so the equivalence [U/R] −→ Fbx of Lemma 2.2.22 gives aparticular smooth Noetherianly pro-representable presentation of Fbx .
In Lemma 2.7.2 below we will show that if the diagonal of F satisfies an appropriate hypothesis, and if we are given a versal morphism U = Spf A → Fbx froma (not necessarily Noetherianly) pro-representable functor U , then (without any
2The term smooth is used here in the sense of [Sta, Tag 06HG]; i.e. we require the infinitesimal
lifting property with respect to morphisms in CΛ . Other authors might use the term versal here,
because the residue field is being held fixed.

SCHEME-THEORETIC IMAGES OF MORPHISMS OF STACKS

25

a priori hypothesis that Fbx admits a presentation by a smooth pro-representablegroupoid in functors) the fibre product R := U ×Fbx U is also pro-representable,∼
and thus (taking into account the isomorphism [U/R] −→ Fbx of Lemma 2.2.22) the
existence of a versal ring to F at the morphism x will imply that Fbx in fact admits
a presentation by a pro-representable smooth groupoid in functors.
We close our discussion of Axiom [2] by stating a lemma that relates the existenceof presentations by smooth pro-representable groupoids in functors to the conditionsof Definition 2.2.4.
2.2.24. Lemma. Suppose, for every morphism x : Spec k → F, with k a finite typeOS -field, that Fbx admits a presentation by a pro-representable smooth groupoid infunctors. Then F is Arttriv -homogeneous (in the sense of Definition 2.2.4).
Proof. By [Sta, Tag 06KT], we need only check that a pro-representable functor
is Arttriv -homogeneous (or in the language of [Sta], satisfies (RS)). In the case of
Noetherianly pro-representable functors, this is [Sta, Tag 06JB], and the proof in
the general case is identical.

2.3. Remarks on Axiom [3]. Recall that a category fibred in groupoids F satisfies Axiom [3] if and only if the diagonal ∆ : F → F ×S F is representable byalgebraic spaces; equivalently (by [Sta, Tag 045G]), if and only if X ×F Y is analgebraic space whenever X → F, Y → F are morphisms from algebraic spaces
X, Y . We begin with the following lemma.
2.3.1. Lemma. Let F be a category fibred in groupoids satisfying Axiom [3]. If
X and Y are categories fibred in groupoids satisfying [3], then for any morphismsof categories fibred in groupoids X , Y → F, the fibre product X ×F Y is again acategory fibred in groupoids satisfying [3]. If X and Y are furthermore (algebraic)stacks, then the fibre product is also an (algebraic) stack.
Proof. The claim for algebraic stacks is proved in [Sta, Tag 04TF]. (Note that, as
stated, that result actually deals with stacks in the fppf topology; here, as explainedin Section 1.5, we are applying the analogous result for the étale topology.) An
examination of the proof of that result also gives the claim for categories fibred ingroupoids satisfying [3]. The claim for stacks then follows from [Sta, Tag 02ZL]. 
Our next goal in this section is to extend the definition of certain propertiesof morphisms of algebraic stacks to morphisms of stacks X → F whose source isassumed algebraic, but whose target is merely assumed to satisfy [3]. To this end,
we first note the following lemma.
2.3.2. Lemma. Let X → Y → F be morphisms of stacks, with X and Y algebraic
stacks, and with F assumed to satisfy [3]. If P is a property of morphisms ofalgebraic stacks that is preserved under arbitrary base-change (by morphisms ofalgebraic stacks), then the morphism X → Y has the property P if and only if,for every morphism of stacks Z → F with Z being algebraic, the base-changed
morphism X ×F Z → Y ×F Z has property P .
Proof. The indicated base-change can be rewritten as the base-change of the morphism X → Y via the morphism (Y ×F Z) → Y. Since P is assumed to be preserved
under arbitrary base-changes, we see that if X → Y has property P , so does thebase-change X ×F Z → Y ×F Z.

26

M. EMERTON AND T. GEE

Conversely, suppose that all such base-changes have property P ; then, in particular, the morphism X ×F Y → Y ×F Y has property P . Thus so does thepull-back of this morphism via the diagonal ∆ : Y → Y ×F Y. This pull-back may
be described by the usual “graph” Cartesian diagram (letting f denote the given
morphism X → Y)X
Γf :=idX ×f

f

/Y
∆



f ×idY/ Y ×F Y
X ×F Y

from which we deduce that the original morphism f has property P .



2.3.3. Example. If X → F is a morphism of stacks, with X being algebraic and F
satisfying [3], then we may apply the preceding lemma to the diagonal morphism∆ : X → X ×F X . (Note that the source of this morphism is an algebraic stack by
assumption, and the target is an algebraic stack by Lemma 2.3.1.) If Z → F is any
morphism of stacks with Z being algebraic, then (since the formation of diagonals
is compatible with base-change), the base-change of the diagonal may be naturally
identified with diagonal of the base-change∆ : (X ×F Z) → (X ×F Z) ×Z (X ×F Z).
In particular, since the the properties of being representable by algebraic spaces,
and of being locally of finite type, are preserved under any base-change, and holdfor the diagonal of any morphism of algebraic stacks [Sta, Tag 04XS]), we see that
∆ : X → X ×F X is representable by algebraic spaces, and is locally of finite type.
(Proposition 2.3.17 below generalises the first of these statements to the case when
X is also assumed only to satisfy [3].)We now make the following definition.
2.3.4. Definition. Assume that F is a stack satisfying Axiom [3]. Given an algebraic stack X and a morphism X → F, and a property P of morphisms of algebraic
stacks that is preserved under arbitrary base-change, then we say that X → F has
property P , if and only if, for any algebraic stack Y equipped with a morphism
Y → F, the base-changed morphism X ×F Y → Y (which by Lemma 2.3.1 is amorphism of algebraic stacks) has property P .
2.3.5. Remark. In the spirit of [Sta, Tag 03YK], it might be better to restrict thisdefinition to properties that are furthermore fppf local on the target. Since, in any
case, we only apply the definition to such properties, we don’t worry about thissubtlety here.
2.3.6. Remark. We will apply the preceding definition to the following propertiesof morphisms of algebraic stacks:• Locally of finite presentation (this property is smooth local on the sourceand-target, so is defined by [Sta, Tag 06FN]).
• Locally of finite type (again, this is smooth local on the source-and-target,
so is defined by [Sta, Tag 06FN]).
• Quasi-compact, [Sta, Tag 050U].
• Finite type, which as usual is defined to be locally of finite type and quasicompact, [Sta, Tag 06FS].

SCHEME-THEORETIC IMAGES OF MORPHISMS OF STACKS27• Universally closed, [Sta, Tag 0513].
• Surjective, [Sta, Tag 04ZS].
• Separated, i.e. having proper diagonal [Sta, Tag 04YW].
• Proper, which as usual we define to be separated, finite type, and universallyclosed.
• Representable by algebraic spaces, which is equivalent to the condition thatthe diagonal morphism be a monomorphism [Sta, Tag 0AHJ].
• Monomorphism [Sta, Tag 04ZW], which is equivalent to the condition thatthe diagonal be an isomorphism, or that the morphism, thought of as afunctor between categories fibred in groupoids, is fully faithful [Sta, Tag
04ZZ]. Note that monomorphisms are necessarily representable by algebraic
spaces.
• Closed immersion, [Sta, Tag 04YL]. (Closed immersions also admit an alternative characterisation as being the proper monomorphisms. To see
this, note that by [Sta, Tag 045F] it is enough to prove the same statement for morphisms of algebraic spaces. Since closed immersions of algebraic spaces are representable by definition, and proper monomorphisms
are representable by [Sta, Tag 0418], we reduce to the case of morphismsof schemes, which is [Gro67, 18.12.6].)• Unramified, which is defined to be locally of finite type, with étale diagonal[Ryd11, Appendix B]. (Note that the diagonal morphism is representableby algebraic spaces [Sta, Tag 04XS], and so the notion of it being étaleis defined by [Sta, Tag 04XB]. Note also that an unramified morphism
is representable by algebraic spaces if and only if its diagonal is an open
immersion, since open immersions of algebraic spaces are precisely the étale
monomorphisms; see [Gro67, 17.9.1] for this statement in the context ofmorphisms of schemes, which implies the statement for algebraic spaces,
because open immersions of algebraic spaces are representable by definition,and étale monomorphisms of algebraic spaces are representable by [Sta, Tag0418].)
2.3.7. Remark. Some of the preceding properties, when interpreted via the mechanism of Definition 2.3.4, also admit a more direct interpretation. In particular, if
X → F is a morphism of stacks with X being algebraic and F satisfying [3], thenthe diagonal morphism ∆ : X → X ×F X is a morphism of algebraic stacks, which
is furthermore representable by algebraic spaces (as noted in Example 2.3.3), andso we know what it means for it to be proper, or étale (for example). The following lemma incorporates this, and some similar observations, to give more directinterpretations of some of the preceding properties.
2.3.8. Lemma. Let f : X → F be a morphism of stacks, with X being algebraic
and F satisfying [3].
(1) The morphism f is quasi-compact, in the sense of Definition 2.3.4, if andonly if, for every morphism Y → F with Y a quasi-compact algebraic stack,the algebraic stack X ×F Y is quasi-compact.
(2) The morphism f is universally closed, in the sense of Definition 2.3.4, if
and only if for every morphism Y → F with Y an algebraic stack, theinduced morphism |X ×F Y| → |Y| is closed.
(3) If P is a property which is preserved under arbitrary base-change, whichis smooth local on the source-and-target, and which is fppf local on the

28

M. EMERTON AND T. GEEtarget, then f satisfies P , in the sense of Definition 2.3.4, if and only if forsome (or, equivalently, any) smooth cover U → X of X by a scheme, thecomposite morphism U → F (which is representable by algebraic spaces,
since F satisfies [3]) satisfies condition P in the sense of [Sta, Tag 03YK].3(4) Let P 0 be a property of morphisms of algebraic stacks that are representableby algebraic spaces. Assume further that P 0 is preserved under arbitrarybase-change, and let P be the property of morphisms of algebraic stacks
defined by the requirement that the corresponding diagonal morphism should
satisfy P 0 . Then f satisfies P , in the sense of Definition 2.3.4, if and only
if the diagonal morphism ∆f : X → X ×F X satisfies P 0 .
(5) The morphism f is representable by algebraic spaces, in the sense of Definition 2.3.4 and Remark 2.3.6, if and only if it is representable by algebraic
spaces in the usual sense, i.e. if and only if for any morphism T → F
with T a scheme, the base-change X ×F T is an algebraic space; and these
conditions are equivalent in turn to the condition that the diagonal ∆f be
a monomorphism.
(6) If f : X → F is locally of finite type, in the sense of Definition 2.3.4, then∆f : X → X ×F X is locally of finite presentation.
Proof. Suppose that f is quasi-compact, in the sense of Definition 2.3.4. If Z → F isa morphism of stacks, with Z being algebraic and quasi-compact, then by definition
X ×F Z → Z is a quasi-compact morphism. Since Z is quasi-compact, it again
follows by definition that X ×F Z is a quasi-compact algebraic stack. Conversely,
suppose that X ×F Z is quasi-compact, for every morphism Z → F with Z being
a quasi-compact algebraic stack. Let Y → F be any morphism of stacks with Y
algebraic, and let Z → Y be a morphism of algebraic stacks with Z quasi-compact.
Then the base-change∼(X ×F Y) ×Y Z → Y ×Y Z −→ Z
may be naturally identified with the base-change X ×F Z → Z; in particular,(X ×F Y) ×Y Z is quasi-compact. The base-changed morphism X ×F Y → Y isthus quasi-compact by definition. Since Y → F was arbitrary, we conclude that f
is quasi-compact, in the sense of Definition 2.3.4; this proves (1).
The proofs of (2) and of the first claim of (5) proceed along identical lines tothe proof of (1). To prove (3), suppose first that f has property P , in the sense of
Definition 2.3.4. If T → F is any morphism from a scheme to F, and U → X is any
smooth surjection, then the base-changed morphism U ×F T → X ×F T (which isnaturally identified with the base-change of the morphism U → X by the morphismof algebraic stacks X ×F T → X ) is smooth, while the morphism X ×F T → Tsatisfies P by assumption. Thus the composite U ×F T → T satisfies P (as P isassumed to be smooth local on the source-and-target), and so the morphism U → F
satisfies P in the sense of [Sta, Tag 03YK]. Conversely, if this latter condition
holds, and if T → F is a morphism whose source is a scheme, then the morphism
U ×F T → T satisfies P . Since U ×F T → X ×F T is smooth, and P is assumed
to be smooth local on the source-and-target, we find that X ×F T → T satisfies P .
Now suppose that Y → F is any morphism of stacks whose source is algebraic, andlet T → Y be a smooth surjection. We have just seen that X ×F T → T satisfies P .
3The assumption that P be fppf local on the target is included purely in order for the definitionof [Sta, Tag 03YK] to apply to P .

SCHEME-THEORETIC IMAGES OF MORPHISMS OF STACKS

29

Since P is smooth local on the source-and-target, and since each of the morphisms
X ×F T → X ×F Y and T → Y are smooth and surjective (the latter by assumption
and the former because it is naturally identified with the base-change of the latterby the projection X ×F Y → Y), we find that X ×F Y → Y satisfies P , as required.
Thus, by definition, the morphism f satisfies P , in the sense of Definition 2.3.4.
To prove (4), note that the formation of diagonals is compatible with basechange, after which (4) follows from Lemma 2.3.2. The second claim of (5) follows
from (4), applied in the case when P 0 is the property of being a monomorphism(since [Sta, Tag 0AHJ] shows that the associated property P is then precisely theproperty of being representable by algebraic spaces).
Claim (6) also follows from Lemma 2.3.2, and the fact that the analogous claim
holds for morphisms of algebraic stacks. (For lack of a reference, we now give a
proof of this property; that is, we show that if f : X → Y is a morphism of algebraic
stacks which is locally of finite type, then ∆f is locally of finite presentation. Choose
a smooth surjection V → Y from a scheme V . By [Sta, Tag 06Q8], it is enough toprove that the base change of ∆f by V → Y is locally of finite presentation. Thisbase change is the diagonal of the base change X ×Y V → V , so we may replace X
by X ×Y V and Y by V , and therefore reduce to the case of a morphism X → V .
Now choose a smooth surjection from a scheme U → X . By [Sta, Tag 06Q9]
it suffices to show that the composite U → X → X ×V X is locally of finite
presentation. Factoring this as the composite
U → U ×V U → U ×V X → X ×V X
we see that it suffices to show that U → U ×V U is locally of finite presentation;that is, we have reduced to the case of schemes, which is [Sta, Tag 0818].)
2.3.9. Example. Lemma 2.3.8 shows that a morphism of stacks X → F, with X
being algebraic and F satisfying [3], is proper, in the sense of Definition 2.3.4, if
and only if the following properties hold: (i) for some (or, equivalently, any) smoothsurjection U → X , with U a scheme, the composite morphism U → F (which isa morphism of stacks representable by algebraic spaces) is locally of finite typein the sense of [Sta, Tag 03YK]; (ii) for any morphism of stacks Y → F with Y
algebraic, the base-changed morphism X ×F Y → Y induces a closed morphism|X ×F Y| → |Y|; (iii) in the context of (ii), if Y is furthermore quasi-compact,
then the base-changed algebraic stack X ×F Y is quasi-compact; (iv) the diagonal
morphism ∆ : X → X ×F X is proper.
2.3.10. Example. Lemma 2.3.8 shows that a morphism of stacks X → F, with X
being algebraic and F satisfying [3], is unramified, in the sense of Definition 2.3.4, if
and only if the following properties hold: (i) for some (or, equivalently, any) smoothsurjection U → X , with U a scheme, the composite morphism U → F (which isa morphism of stacks representable by algebraic spaces) is locally of finite type inthe sense of [Sta, Tag 03YK]; (ii) the diagonal morphism ∆ : X → X ×F X (which
is a priori locally of finite presentation, by (i) and Lemma 2.3.8 (6)) is étale. The
unramified morphism X → F is furthermore representable by algebraic spaces if
the diagonal is in fact an open immersion (taking into account (5) of the precedinglemma, and the fact that open immersions are precisely the étale monomorphisms).
2.3.11. Example. Lemma 2.3.8 (4) shows that a morphism of stacks f : X → F,with X being algebraic and F satisfying [3], is a monomorphism in the sense of

30

M. EMERTON AND T. GEE

Definition 2.3.4, if and only if the diagonal ∆f : X → X ×F X is an isomorphism.
In turn, this is equivalent to the condition that f , thought of as a functor between
categories fibred in groupoids, is fully faithful. Lemma 2.3.8 (5) shows that f isthen in particular necessarily representable by algebraic spaces in the usual sense.
2.3.12. Definition. We say that a substack F 0 of F is a closed substack if the
inclusion morphism F 0 ,→ F is representable by algebraic spaces, and is a closed
immersion in the sense of [Sta, Tag 03YK], i.e. has the property that for any morphism T → F with T an affine scheme, the base-changed morphism T ×F F 0 → T
is a closed immersion of schemes.
2.3.13. Lemma. Assume that S is locally Noetherian, and that F is a stack overS which satisfies [3], for which the morphism F → S is limit preserving on objects.
If Z ,→ F is a closed immersion, then it is limit preserving on objects.
Proof. By definition of what it means for Z → F to be a closed immersion, if T → F
is a morphism from an algebraic space, then the induced morphism Z ×F T → T isa closed immersion. Since F satisfies [3], the source is an algebraic stack; but if T
is a scheme, then in fact the source will be a scheme (being a closed substack of ascheme). In particular the morphism Z → F is representable by algebraic spaces,and so to check that it is limit preserving on objects, it suffices to check that it islocally of finite presentation [Sta, Tag 06CX]. For this, it suffices to check, in thepreceding context, that if T is an affine scheme then Z ×F T → T is locally of finite
presentation.
If we write T = limi Ti as the projective limit of finite type affine S-schemes Ti ,←−then, since F → S is limit preserving on objects and S is locally Noetherian, we
may factor the morphism T → F through one of the Ti , and hence reduce to thecase when T is finite type over the locally Noetherian scheme S. But in this case
T itself is Noetherian, and so the closed immersion Z ×F T → T is in fact of finite
presentation. This proves the lemma.

Our next goal is to state and prove a lemma which is a variant of [Sta, Tag 06CX](which states that for a morphism between categories fibred in groupoids that isrepresentable by algebraic spaces, being locally of finite presentation is equivalent
to being limit preserving on objects). Before doing this, we recall some standard
facts about descending finitely presented morphisms of affine schemes over S tomorphisms of finitely presented affine schemes over S.
2.3.14. Lemma. Suppose that T, T 0 are affine schemes over S, and that we are
given a morphism of finite presentation T 0 → T . Suppose that T may be written
as a limit lim Ti of affine schemes of finite presentation over S. Suppose also that←−
we are given a morphism T 0 → T 00 with T 00 a scheme locally of finite presentation
over S.
Then for some j we may find a factorisation of the given morphism T 0 → T 00
which fits into a commutative diagram
T0

/ T0


T


/ Tj

j

/ T 00SCHEME-THEORETIC IMAGES OF MORPHISMS OF STACKS

31

in which the square is Cartesian, Tj0 is an affine scheme of finite presentation
over S, and the morphism Tj0 → Tj is of finite presentation. Furthermore, if T 0 → T
is surjective, then Tj0 → Tj can be taken to be surjective; and if T 0 → T is assumed
to be étale (resp. fppf, resp. an open immersion), then Tj0 → Tj can also be taken
to be étale (resp. fppf, resp. an open immersion).
2.3.15. Remark. If S is quasi-separated, then the hypothesis that T may be written
as a limit of affine schemes of finite presentation is automatically satisfied; see
Theorem 2.5.1 below.
Proof of Lemma 2.3.14. By [Sta, Tag 01ZM], there is some j0 and a morphism offinite presentation Tj00 → Tj0 such that the pull-back of this morphism to T is thegiven morphism T 0 → T . For j ≥ j0 , set Tj0 := Tj ×Tj0 Tj00 . Since products commute
with limits, we have T 0 = lim Tj0 .
←−Since Tj0 is affine, and Tj00 → Tj0 is of finite presentation (hence quasi-compact
and quasi-separated), it follows that Tj00 is quasi-compact and quasi-separated. Itthen follows from [Sta, Tag 01ZN] that we may assume that Tj0 is affine for allj ≥ j0 . If T 0 → T is surjective (resp. étale, resp. fppf, resp. an open immersion),then by [Sta, Tag 07RR] (resp. [Sta, Tag 07RP], resp. [Sta, Tag 04AI,Tag 07RR],
resp. [Sta, Tag 07RP,Tag 07RQ] (recall that open immersions are precisely the étale
monomorphisms)) we may assume that Tj0 → Tj is also surjective (resp. étale, resp.
fppf, resp. an open immersion) for all j ≥ j0 .
Since T 00 is locally of finite presentation over S, it follows from [Sta, Tag 01ZC]
that T 0 → T 00 factors through Tj0 for some j ≥ j0 , as required.

2.3.16. Lemma. A morphism X → F, where X is an algebraic stack and F is astack satisfying [3], is locally of finite presentation, in the sense of Definition 2.3.4,
if and only if it is limit preserving on objects.
Proof. Let U → X be a smooth surjection from a scheme (which exists, since X is analgebraic stack). This morphism is representable by algebraic spaces (again, since
X is an algebraic stack), and is locally of finite presentation (since it is smooth), andhence it is limit preserving on objects by [Sta, Tag 06CX]. Thus if the morphism
X → F is limit preserving on objects, so is the composite U → F [Sta, Tag 06CW].
Since F satisfies [3], this morphism is representable by algebraic spaces, and so
we may apply [Sta, Tag 06CX] again to deduce that U → F is locally of finite
presentation. Hence the same is true of the morphism X → F, by part (3) ofLemma 2.3.8.
Conversely, suppose that X → F is locally of finite presentation, in the senseof Definition 2.3.4. Suppose given a morphism T → X , for an affine S-scheme T ,written as a projective limit T = limi Ti of affine S-schemes Ti , and suppose further←−
that the composite T → X → F factors through one of the Ti . We must show thatour given morphism T → X factors in a compatible manner through Ti0 → X forsome sufficiently large i0 ≥ i. Replacing X with X ×F Ti (which is an algebraic
stack, since F satisfies [3]) and F by Ti , we may in fact assume that F = Ti , which
we do from now on.
Let U → X be a smooth surjection, and write R := U ×X U , so that there is a∼natural isomorphism [U/R] −→ X . The assumption that X → Ti is locally of finite
presentation is by definition equivalent to supposing that U → Ti is locally of finite
presentation.

32

M. EMERTON AND T. GEE

Let p1 , p2 : R ⇒ U denote the two projections. Consider the pull-backs UT∼and RT . There is a natural identification RT −→ UT ×T UT , via which the basechanges of the natural projections pi become identified with the natural projections∼
UT ×T UT ⇒ UT . There is a natural isomorphism of stacks [UT /RT ] −→ T.
0We may find (for example by [Sta, Tag 055V]) an étale slice T of UT , i.e. amorphism T 0 → UT for which the composite T 0 → UT → T is étale and surjective.
We then define T 00 := T 0 ×T T 0 . We have the composite morphisms T 0 → UT → U
and T 00 → RT → R, with respect to which the two projections T 00 ⇒ T 0 arecompatible with the two projections pi : R → U. Thus there is an induced morphism[T 0 /T 00 ] → [U/R], which is naturally identified with our original morphism T → X .
Since étale morphisms are open, and since T is quasi-compact (being affine), we
may replace T 0 by a quasi-compact open subscheme for which the induced morphism
to T remains surjective. Finally, replacing T 0 by the disjoint union of the membersof a finite affine open cover, we may in fact assume that T 0 is also affine. The
morphism T 0 → T is then an affine morphism that is locally of finite presentation(being étale), and hence is actually of finite presentation. By Lemma 2.3.14 wemay write the affine étale morphism T 0 → T as the projective limit of affine étalemorphisms Ti00 → Ti0 (starting from a sufficiently large value of i0 ). We write
Ti000 := Ti00 ×Ti0 Ti00 .
Since U → Ti is locally of finitely presentation, as is R → Ti , by applying [Sta,Tag 01ZM] and [Sta, Tag 07SJ] we find that we may factor the morphisms T 0 → U
and T 00 → R through Ti00 and Ti000 , for some sufficiently large value of i0 , in such amanner that the projections Ti000 ⇒ Ti00 are compatible with the projections R ⇒ U.
Thus we obtain a morphism [Ti00 /Ti000 ] → [U/R], which we may identify with therequired morphism Ti0 → X .
We close this section with some further propositions, the first of which generalises
one of the observations of Example 2.3.3 to the case when both source and target
are assumed merely to satisfy [3].
2.3.17. Proposition. Let f : X → Y be a morphism between categories fibred ingroupoids satisfying [3]. Then ∆f : X → X ×Y X is representable by algebraic
spaces.
Proof. The proof is essentially identical to that of [Sta, Tag 04XS]. Let T → X ×Y X
be a morphism from a scheme T ; by definition, this is the data of a triple (x, x0 , α)where x, x0 are objects of X over T , and α : f (x) → f (x0 ) is a morphism in thefibre category of Y over T . Since X , Y satisfy [3], the sheaves Isom X (x, x0 ) and
Isom Y (f (x), f (y)) are algebraic spaces over T by [Sta, Tag 045G]. The morphism
α corresponds to a section of the morphism Isom Y (f (x), f (x0 )) → T .
If T 0 → T is a morphism of schemes, then we see that a T 0 -valued point of∼
X ×X ×Y X T is by definition an isomorphism x|T 0 −→ x0 |T 0 whose image under f is
α|T 0 . Putting this together, we see that there is a fibre product diagram of sheaves
over T/ Isom X (x, x0 )(2.3.18)
X ×X × X T
Y


T

α


/ Isom Y (f (x), f (x0 ))Thus X ×X ×Y X T is an algebraic space, as required.

SCHEME-THEORETIC IMAGES OF MORPHISMS OF STACKS

33

2.3.19. Proposition. If X is a category fibred in groupoids satisfying [3], and ∆ :X → X ×S X is limit preserving on objects, then ∆∆ : X → X ×X ×S X X isrepresentable by algebraic spaces and limit preserving on objects.
Proof. By Proposition 2.3.17, ∆∆ is representable by algebraic spaces. (It follows
from Lemma 2.3.1 that X ×S X satisfies [3], since X does, so that the proposition
does indeed apply.) By [Sta, Tag 06CX], we see that ∆ is locally of finite presentation, and that to show that ∆∆ is limit preserving on objects, it is equivalent
to show that it is locally of finite presentation. Consider the diagram (2.3.18) inthe case that f is ∆; we must show that for all choices of T , the left hand verticalarrow is locally of finite presentation.
It therefore suffices to show that the right hand vertical arrow is locally of finite
presentation; but this is by definition the diagonal map
Isom (x, x0 ) → Isom (x, x0 ) ×S Isom (x, x0 ),which is locally of finite presentation by [Sta, Tag 084P] (note that since ∆ is locallyof finite presentation, so is Isom (x, x0 )).

2.3.20. Proposition. Let X → Y be a morphism of categories fibred in groupoids
which is representable by algebraic spaces. Assume that Y satisfies [3]. Then:(1) X satisfies [3].
(2) If X → Y and ∆Y : Y → Y ×S Y are locally of finite presentation, then so
is ∆X : X → X ×S X .
(3) If X → Y is locally of finite presentation, and Y satisfies [1], then sodoes X .
Proof. Throughout the proof, we will freely make use of [Sta, Tag 06CX], theequivalence of being locally of finite presentation and being limit preserving on
objects for morphisms representable by algebraic spaces. We begin with (1); we
need to show that the morphism X → X ×S X is representable by algebraic spaces.
This may be factored as
X → X ×Y X → X ×S X ;since the second morphism is a base change of Y → Y ×S Y, it is enough to showthat X → X ×Y X is representable by algebraic spaces.
To this end, suppose that T → X ×Y X is a morphism whose source is an algebraic
space. This induces a morphism T → Y, and we may consider the base change of
our whole situation by this morphism. Writing XT for X ×Y T , we may reinterpretthe given morphism T → X ×Y X as a section of the morphism XT ×T XT → T,
and it is then straightforward to see that we in fact have
(2.3.21)

X ×X ×Y X T = XT ×XT ×T XT T.

Now, since X → Y is representable by algebraic spaces, XT is an algebraic space,so that XT ×XT ×T XT T is an algebraic space, as required.
We now consider (2). We may factor ∆X as the composite
X → X ×Y X → X ×S X .
Since the second arrow is a base change of ∆Y , which is locally of finite presentation
by assumption, it follows from [Sta, Tag 06CV,Tag 06CW] that it suffices to showthat X → X ×Y X is locally of finite presentation. By definition, we need to showthat all the base changes of this morphism by morphisms T → X ×Y X with source

34

M. EMERTON AND T. GEE

an algebraic space are locally of finite presentation; examining (2.3.21), we see that
we are reduced to the case that Y = T is an algebraic space, which is a special caseof Lemma 2.3.8 (6).
To prove (3), by Lemma 2.1.5 we need to show that each of X → S, ∆X : X →X ×S X , ∆∆ : X → X ×X ×S X X is limit preserving on objects; the same result
also shows that both Y → S and ∆Y are limit preserving on objects, so that ∆X
is limit preserving on objects by (2). By Proposition 2.3.19, it is then enough toshow that X → S is limit preserving on objects; but this is immediate from [Sta,Tag 06CW], applied to the composite X → Y → S.

2.3.22. Lemma. If F 0 → F is a monomorphism of categories fibred in groupoids,
and if ∆F is representable by algebraic spaces and locally of finite presentation,then ∆F 0 is also representable by algebraic spaces and locally of finite presentation.
Proof. Since F 0 → F is a monomorphism, F 0 → F 0 ×F F 0 is an equivalence, andtherefore ∆F 0 is the base change of ∆F via the natural morphism F 0 ×S F 0 →F ×S F. The result follows.

2.4. Remarks on Axiom [4]. We begin with some preliminary definitions andresults related to the concept of smoothness of morphisms.
2.4.1. Definition. As in Section 2.3, we will use the notions of unramified and étalefor morphisms between algebraic stacks that are not necessarily representable byalgebraic spaces. We say that a morphism of algebraic stacks is unramified if it islocally of finite type and has étale diagonal, and that it is étale if it is unramified,flat, and locally of finite presentation [Ryd11, Appendix B]. Note that, although thedefinition of unramified morphism includes the condition that the diagonal be étale,
there is no circularity, because diagonal morphisms are representable by algebraic
spaces, and in this case the notation of étale is defined following [Sta, Tag 04XB].
By [Ryd11, Prop. B2] a morphism of algebraic stacks is étale if and only if it issmooth and unramified.
2.4.2. Definition. Let f : X → Y be a morphism of stacks. Then we say thatf is formally unramified (resp. formally smooth, resp. formally étale) if for every
affine Y-scheme T , and every closed subscheme T0 ,→ T defined by a nilpotent ideal
sheaf, the functor HomY (T, X ) → HomY (T0 , X ) is fully faithful (resp. essentiallysurjective, resp. an equivalence of categories).
2.4.3. Proposition.
(1) A morphism of algebraic stacks is smooth if and only
if it is formally smooth and locally of finite presentation.
(2) A morphism of algebraic stacks is unramified if and only if it is formallyunramified and locally of finite type.
(3) A morphism of algebraic stacks is étale if and only if it is formally étaleand locally of finite presentation.
Proof. See [Ryd11, Cor. B.9].



2.4.4. Definition. Assume that S is locally Noetherian, let F be a category fibredin groupoids over S, let U be a scheme locally of finite type over S equipped witha morphism ϕ : U → F, and let u ∈ U be a point.

SCHEME-THEORETIC IMAGES OF MORPHISMS OF STACKS

35(1) We say that ϕ is versal at u if for any diagram
Z0

/U


Z


/ F,

ϕ

where Z0 and Z are Artinian local schemes with the latter being a nilpotent
thickening of the former, and where the closed point z ∈ Z0 maps to u,
inducing an isomorphism κ(u) ∼= κ(z), we may lift the morphism Z → F
to a morphism Z → U .
(2) We say that ϕ is formally smooth at u if for any diagram as in (1) whereZ0 and Z are Artinian local schemes with the latter being a nilpotent
thickening of the former, and where the closed point z ∈ Z0 maps to u,
inducing a finite extension κ(u) ,→ κ(z), we may lift the morphism Z → F
to a morphism Z → U .
(3) If ϕ is representable by algebraic spaces and locally of finite presentation,then we say that ϕ is smooth at u if for any finite type S-scheme X equippedwith a morphism X → F over S, there is an open neighbourhood U 0 of u
in U such the base-change morphism U 0 ×F X → X is smooth.
(4) If ϕ is is representable by algebraic spaces and locally of finite presentation,then we say that ϕ is smooth in a neighbourhood of u if there exists aneighbourhood U 0 of u such that the restriction ϕ|U 0 : U 0 → F (which isagain a morphism representable by algebraic spaces) is smooth.
2.4.5. Remark. If u ∈ U is a finite type point, then ϕ is versal at u in the sense of (1)bU,u → Fϕ(u)of the preceding definition if and only if the induced morphism Spf O
is versal in the sense discussed in Subsection 2.2 above. Note that we follow [Sta,Tag 07XF] in using the terminology versal at u, rather than formally versal at u,
as some other sources (e.g. [HR19]) do.
Our definition of formal smoothness at a point follows that of [HR19, Def. 2.1].
In [Art69a, Def. 3.1], Artin defines the notion of formally étale at a point (for amorphism from a scheme to a functor), but his definition is not quite the obvious
analogue of the definition of formal smoothness given here, in that he does not
impose any condition on the degree of the extension of the residue field at the closed
point of Z0 over the residue field at u. The condition on the residue field that we
impose (following [HR19]) makes it slightly easier to verify formal smoothness (seein particular Lemma 2.4.7 (2) below), and is harmless in practice. (Indeed, if F
is a stack satisfying [1] and [3], then part (3) of Lemma 2.4.7 below shows that at
finite type points the variant definition of formal smoothness, in which we impose
no condition on the extension of residue fields, is equivalent to the definition given
above.)
2.4.6. Remark. The notion of smoothness is defined for morphisms between algebraic stacks, or, more generally (via Definition 2.3.4), for morphisms from analgebraic stack to a stack satisfying [3]. Thus the notion of smooth at a point cannaturally be extended to morphisms whose source is an algebraic stack, and whosetarget satisfies [3].
The notion of versality at a point or formal smoothness at a point of an algebraic
stack is slightly more problematic to define, since a point of an algebraic stack is

36

M. EMERTON AND T. GEE

defined as an equivalence class of morphisms from the spectrum of a field, andso we can’t speak of the residue field at a point of a stack. In Definition 2.4.10,
we will give a definition of formal smoothness at a point for a morphism whose
source is an algebraic stack, under slightly restrictive conditions on the target ofthe morphism (which, however, will not be too restrictive for the applications of
this notion that we have in mind). The key to making the definition work is
Corollary 2.4.8, which shows (under suitable hypotheses) that formal smoothness
at a point (for morphisms from a scheme) can be detected smooth locally.
The following lemma, which is essentially drawn from [HR19, §2], relates thevarious notions of Definition 2.4.4. We remark that part (3) of the lemma provides an analogue, for formal smoothness at a point, of Artin’s [Art69a, Lem. 3.3],
which provides a characterisation of morphisms from a scheme to a functor thatare formally étale at a point.
2.4.7. Lemma. Suppose that we are in the context of Definition 2.4.4, and that u
is a finite type point.
(1) In general, we have that (4) =⇒ (3) =⇒ (2) =⇒ (1).
(2) If F satisfies (RS), then (1) ⇐⇒ (2); i.e. ϕ is versal at u if and only if it
is formally smooth at u.
(3) Consider the following conditions:(a) ϕ is formally smooth at u.
(b) For any finite type S-scheme X, and any morphism X → F over S,
the base-changed morphism U ×F X → X contains the fibre over u inits smooth locus.
(c) For any locally finite type algebraic stack X over S, and any morphism
X → F over S, the base-changed morphism U ×F X → X contains
the fibre over u in its smooth locus.
(d) For any algebraic stack X over S, and any morphism X → F over S,
the base-changed morphism U ×F X → X contains the fibre over u inits smooth locus.
If ϕ is representable by algebraic spaces and locally of finite type, then conditions (a), (b), and (c) are equivalent. If furthermore F is a stack satisfying [1] and [3], then all four of these conditions are equivalent.
(4) If F is an algebraic stack which is locally of finite presentation over S, thenconditions (1), (2), (3), and (4) of Definition 2.4.4 are equivalent.
Proof. That condition (2) of Definition 2.4.4 implies condition (1) is immediate, as
is the fact that condition (4) implies condition (3). To see that condition (3) implies
condition (2), note that if A is an Artinian local OS -algebra whose residue field isof finite type over OS , then Spec A is of finite type over S. Thus, if condition (3)holds and we are in the situation of condition (2), the base-changed morphism
U ×F Z → Z is smooth in a neighbourhood of the image of the induced map
Z0 → U ×F Z, and since smooth morphisms are formally smooth, we may find thedesired lift Z → U . Thus condition (3) of the definition implies condition (2). This
completes the proof of part (1) of the present lemma.
Part (2) is [HR19, Lem. 2.3]; we recall the (short) argument. As explained in theprevious paragraph, Z0 , Z are automatically of finite type over S. Let W0 be theimage of Z0 in Spec OU,u , and let W be the pushout of W0 and Z over Z0 . ThenSCHEME-THEORETIC IMAGES OF MORPHISMS OF STACKS

37

by (RS) we have a commutative diagram

Z0

/ W0

/U


Z


/W


/ F,

ϕ

and versality at u allows us to lift the morphism W → F to a morphism W → U .
The composite morphism Z → W → U gives the required lifting.
We turn to proving (3). We first note that, if F is a stack satisfying [1] and [3],and ϕ : U → F with U locally of finite type over S, then ϕ is representable byalgebraic spaces, by [3], and is locally of finite type, by Lemma 2.6.3 (1) below.
Also, if X → F is a morphism from an algebraic stack to S, then X admits asmooth surjection X → X whose source is a scheme (which is locally of finite typeover S if X is, since smooth morphisms are locally of finite presentation), and, sincesmoothness can be tested smooth-locally on the source, we see that (c) or (d) for
X is equivalent to the corresponding statement for X. Thus we need only consider
the case when X is a scheme X from now on.
If F is a stack (on the étale site, and hence also on the Zariski site), and if X → F
is a morphism whose source is a scheme, then condition (d) may be checked Zariski
locally on X, and thus the general case of (d) follows from the case when X isaffine. If F furthermore satisfies [1], then we may factor X → F through a finite
type S-scheme, and thus assume that X is locally of finite type over S.
Thus we see that if F is a stack satisfying [1] and [3], then (d) follows from (c),while clearly (d) implies (c). Again, it is clear that (c) implies (b), and an argument
essentially identical to the proof of (1) above shows that (b) implies (a). Thus itremains to show that (a) implies (c), for maps X → F where X is a scheme locallyof finite type over S, under the assumption that ϕ is representable by algebraic
spaces and locally of finite type.
We must verify that the smooth locus of U ×F X → X contains the fibre over u.
The projection U ×F X → X is locally of finite type, and hence U ×F X is locallyof finite type over S. As U is also locally of finite type over S, the projection
U ×F X → U is locally of finite type as well. Since U ×F X is an algebraic space,it admits an étale cover by a scheme V . Since smoothness may be checked smooth
locally on the source, it suffices to show that the fibre of V over u is contained in
the smooth locus of the composite V → U ×F X → X. Since V → U is locally offinite type, the fibre of V over u is a scheme locally of finite type over κ(u), and so
it suffices to show that every closed point of this fibre lies in the smooth locus of
V → X. By [Gro67, Prop. 17.14.2], it in fact suffices to show that this morphism
is formally smooth at each of these points.
Let v be such a point, and suppose given a commutative diagram

Z0

/V


Z


/X

38

M. EMERTON AND T. GEE

as in the definition of formal smoothness at v. We may fit this into the larger
diagram
/V
/U
Z0

Z


/F


/X

Our assumption that ϕ is formally smooth at u implies that we may lift the composite of the lower horizontal arrows to a morphism Z → U (it is here that we
use the assumption that v is a closed point of the fibre over u, so that the residue
field at v is a finite extension of the residue field at u), and hence to a morphism
Z → U ×F X. Since V is étale, and so in particular formally smooth, over U ×F X,
we may then further lift this morphism to a morphism Z → V , as required. This
completes the proof that (a) implies (c).
It remains to prove part (4) of the lemma. Lemma 2.2.3 and part (2) show thatconditions (1) and (2) of Definition 2.4.4 are equivalent when F is an algebraic stack.
Taking into account the statement of part (1), it suffices to show that condition (2)of Definition 2.4.4 implies condition (4). Thus we suppose that ϕ : U → F isformally smooth at the point u ∈ U . The equivalence between conditions (a) and(d) of part (3) of the present lemma (which holds, since F is an algebraic stack,locally of finite presentation over S, and thus satisfies [1], by Lemma 2.1.9, and [3],by definition) shows (taking X = F) that ϕ is smooth in a neighbourhood of u, as
required.

As a corollary of Lemma 2.4.7 (3), we next show (under mild assumptions on
the morphism ϕ) that formal smoothness at a point can be checked smooth locallyon U .
2.4.8. Corollary. Let ϕ : U → F be a morphism (over S) whose source U is alocally of finite type S-scheme, and whose target F is a category fibred in groupoid,and suppose that ϕ is representable by algebraic spaces, and is locally of finite type.4
If u ∈ U is a finite type point, then the following are equivalent:(1) ϕ is formally smooth at u.
(2) There is a smooth morphism of schemes V → U and a finite type pointv ∈ V mapping to u such that the composite V → U → F is formallysmooth at v.
(3) For any smooth morphism of schemes V → U , and any finite type pointv ∈ V mapping to u, the composite V → U → F is formally smooth at v.
Proof. If V → U is a smooth morphism, mapping the finite type point v ∈ V to u,
and X → F is a morphism from a finite type S-scheme X, then we consider thefollowing commutative diagram.
V ×F X/ U ×F X

/X


V


/U


/F

4As seen in the proof of Lemma 2.4.7 (3), these conditions on ϕ hold automatically if F is astack satisfying [1] and [3].

SCHEME-THEORETIC IMAGES OF MORPHISMS OF STACKS39

If (1) holds, then by the equivalence of (a) and (b) in part (3) of Lemma 2.4.7,
the projection U ×F X → X contains the fibre over u in its smooth locus. The
fibre over v (with the respect to the projection V ×F X → V ) is contained in thebase-change to V of this fibre over u, and since the morphism V ×F X → U ×F X
is smooth, the smooth locus of the projection V ×F X → X contains the fibre overv in its smooth locus. Employing Lemma 2.4.7 again, we see that (1) implies (3).
Considering the same diagram, and using the facts that smooth morphisms areopen, and that smoothness can be tested smooth-locally on the source, we see inthe same way that (2) implies (1).
Clearly (3) implies (2) (e.g. by taking V = U and v = u), so we are done.


2.4.9. Corollary. Suppose that F is a stack over S satisfying [1] and [3], that U
is an algebraic stack, locally of finite type over S, and that U → F is a morphismof stacks over S. If u ∈ |U| is a finite type point of U, then the following are
equivalent:(1) There exists a smooth morphism V → U whose source is a scheme, and afinite type point v ∈ V mapping to u, such that the composite V → U → F
is formally smooth at v.
(2) For any smooth morphism V → U whose source is a scheme, and any finite
type point v ∈ V mapping to u, the composite V → U → F is formallysmooth at v.
Proof. Since U is an algebraic stack, there does exist a smooth surjection V → U
whose source is a scheme, and the finite type points of V are dense in the fibre
over u. Thus if (2) holds, so does (1). Conversely, suppose that (1) holds, for somechoice of V and v, and suppose that V 0 → U is a smooth morphism from a scheme
to U, and that v 0 ∈ V 0 is a finite type point mapping to u. We must show thatV 0 → F is formally smooth at v 0 .
Consider the fibre product V ×U V 0 ; this is an algebraic space (since U is analgebraic stack, and so satisfies [3]), and so we may find an étale (and in particularsmooth) surjection W → V ×U V 0 whose source is a scheme. Since v and v 0 both
map to u, and since W → V ×F V 0 is surjective, we may find a finite type pointw ∈ W lying over both v and v 0 .
Since F is a stack satisfying [1] and [3], and since V and V 0 are each of finite typeover S, the morphisms V → F and V 0 → F are both representable by algebraic
spaces and locally of finite type. Thus, since W → V and W → V 0 are both smoothmorphisms, we conclude from Corollary 2.4.8, first that W → F is formally smoothat w, and then that V 0 → F is formally smooth at v 0 . This completes the proof
that (2) implies (1).


2.4.10. Definition. If U is an algebraic stack, locally of finite type over S, and F
is a stack over S satisfying [1] and [3], then we say that a morphism ϕ : U → F
of stacks over S is formally smooth at a finite type point u ∈ |U | if the equivalentconditions of Corollary 2.4.9 hold.

40

M. EMERTON AND T. GEE

2.4.11. Remark. Suppose we have a commutative diagram of morphisms of stacks
over S
ϕ0
/ U0
U 00
ψ0

ψ



ϕ
/F
U
in which U, U 0 , and U 00 are algebraic stacks, locally of finite type over S, and F
satisfies [1] and [3]. Suppose further that ψ 0 and ϕ0 are smooth. Let u00 ∈ |U 00 |be a finite type point, with images u ∈ |U| and u0 ∈ |U 0 |. Then it follows directlyfrom the definition (and Corollary 2.4.9) that ϕ is formally smooth at u if and only
if ψ is formally smooth at u0 . (Both conditions hold if and only if the composite
ϕ ◦ ψ 0 = ψ ◦ ϕ0 is formally smooth at u00 .)The following lemma extends those parts of Lemma 2.4.7 dealing with formalsmoothness at a point to the case of morphisms whose source is an algebraic stack.
2.4.12. Lemma. Let U be an algebraic stack, locally of finite type over S, and let
F be a stack over S satisfying [1] and [3]. Let ϕ : U → F be a morphism of stacks
over S. Let u ∈ |U| be a point of U.
(1) The morphism ϕ is formally smooth at u if and only if, for every morphism
X → F whose source is an algebraic stack, the base-changed morphism
U ×F X → X contains the fibre over u in its smooth locus.
(2) If F is also an algebraic stack, then ϕ is formally smooth at u if and only
if it is smooth in a neighbourhood of u.
Proof. Given our assumptions on F, conditions (3)(a) and (3)(d) of Lemma 2.4.7
are equivalent. The present lemma follows in a straightforward manner, taking
into account Definition 2.4.10, Lemma 2.4.7 (4) (which we can apply, because F
is locally of finite presentation by Lemma 2.1.9), and the fact that smoothness
for morphisms from an algebraic stack to F can be checked smooth-locally on thesource.

The preliminaries being dealt with, we now state Axiom [4], the condition of
openness of versality. We assume that S is locally Noetherian, and that F is a
category fibred in groupoids over S.
Axiom [4]. If U is a scheme locally of finite type over S, and ϕ : U → F is versal
at some finite type point u ∈ U , then there is an open neighbourhood U 0 ⊆ U of usuch that ϕ is versal at every finite type point of U 0 .
2.4.13. Alternatives to axiom [4]. Following Artin [Art69a] we introduce a pair of
axioms, labelled [4a] and [4b], which are closely related to Axiom [4]. These axioms
are not needed for our main results (and in particular for our applications to Galois
representations in Section 5), but as we have modelled our treatment of Artin’s
representability theorem on [Art69a], we have followed Artin in introducing anddiscussing these variants on Axiom [4]. We will also make use of these axioms when
discussing the various examples in Section 4.
In order to state [4a], we first introduce some notation. Suppose that A is a
DVR, with field of fractions K. If A0K denotes an Artinian local thickening of K,
i.e. an Artinian local ring equipped with a surjection A0K → K (which then induces
an isomorphism between the residue field of A0K and K), then we let A0 denote theSCHEME-THEORETIC IMAGES OF MORPHISMS OF STACKS

41

preimage of A under the surjection A0K → K; it is a subring of A0K , equipped witha surjection A0 → A, whose kernel is nilpotent.
We now state Axiom [4a]. In the statement, we suppose that S is locally Noetherian, and that F is a category fibred in groupoids over S.
Axiom [4a]. Let A be a DVR over OS , whose residue field is of finite type over OS ,and let K denote the field of fractions of A. Then for any morphisms Spec A → F,
Spec A0K → F, with A0K being an Artinian local thickening of K, for which thediagram/ Spec A
Spec K
Spec A0K

/F

commutes, there exists a morphism Spec A0 → F making the enlarged diagram/ Spec A
Spec K
Spec A0K

/ Spec A0"/F

commute.
2.4.14. Remark. Our formulation of Axiom [4a] is slightly different to Artin’s, who
only requires the condition of [4a] to hold for A that are essentially of finite typeover OS . To explain this, note that the key role of [4a] in the theory is to makeLemma 2.6.7 below true, and that, in the proof of that lemma, we apply [4a] after
taking a certain integral closure. If S is a Nagata scheme (see e.g. [Sta, Tag 033R])
then taking this integral closure keeps us in the world of essentially finite type OS algebras, and so in the case of a Nagata base (which holds in Artin’s setting, since
his base S is assumed to be excellent) the argument only requires Artin’s more
limited form of [4a].
We also observe that if A is essentially of finite type over OS , then A0 may be
written as the inductive limit A0 = limi A0i , where A0i runs over the essentially finite−→
type OS -subalgebras of A0 with the properties that A0i → A is surjective, and thatthe localisation of A0i at its generic point is equal to A0K . Thus, if F also satisfies [1],then if we can find a morphism Spec A0 → F making the diagram of Axiom [4a]commute, we may similarly find a morphism Spec A0i → F making the diagram
Spec K/ Spec ASpec A0K

/ Spec A0

i

"

/F

commute, for some (sufficiently large) value of i. In Artin’s context, this allows analternate phrasing of [4a], e.g. in the form of Axiom [4] of [Art69b].
Artin proves in [Art69a, Thm. 3.7] that every locally separated algebraic spacesatisfies (his formulation of) Axiom [4a]. In fact this is true of arbitrary algebraic
stacks (with the more expansive formulation of the axiom given here).
2.4.15. Lemma. Every algebraic stack satisfies Axiom [4a].

42

M. EMERTON AND T. GEE

Proof. Since Spec A0 may be thought of as the pushout of Spec A and Spec A0K overSpec K, this is a special case of [Sta, Tag 07WN].

We can now state Axiom [4b]. We assume that S is locally Noetherian, and that
F is a category fibred in groupoids over S.
Axiom [4b]. If ϕ : U → F is a morphism whose source is an S-scheme, locally offinite type, and ϕ is smooth at a finite type point u ∈ U , then ϕ is smooth in aneighbourhood of u.
2.4.16. Remark. If we assume that ϕ : U → F is representable by algebraic spaces
and locally of finite type, then Lemma 2.4.7 (3) shows that being formally smoothat the finite type point u is equivalent to requiring that for each morphism X → F
from a finite type S-scheme, there is an open set containing the fibre over u in
U ×F X at which the projection to X is smooth. The morphism is smooth at u if
this open set can in fact be taken to be the preimage of a neighbourhood of u ∈ U .
Finally, Axiom [4b] holds precisely when this neighbourhood of u can be chosen
independently of X.
2.4.17. Remark. In Corollary 2.6.11 below, following Artin [Art69a, Lem. 3.10],
we show that if F is a stack satisfying [1], [2](a), and [3], and whose diagonal
is furthermore quasi-compact, then Axioms [4a] and [4b] for F together imply
Axiom [4] for F. This in turn implies that in Artin’s representability theorem(Theorem 2.8.4) we may replace Axiom [4] by the combination of Axioms [4a]and [4b], provided that we add to [3] the condition that the diagonal be quasicompact (see Theorem 2.8.5).
2.4.18. Remark. As Artin notes [Art69a, p. 39], the two axioms [4a] and [4b] arequite different in nature. Axiom [4a] is finitary; for example, the fact that it holds
for algebraic stacks immediately implies that it also holds for Ind-algebraic stacks.
Axiom [4b] is not finitary in nature, and although it holds for algebraic stacks, itwill typically not hold for Ind-algebraic stacks. (See Subsection 4.2 below for afurther discussion of Ind-algebraic stacks and their comportment with regard to
Artin’s axioms.)
2.5. Imposing Axiom [1] via an adjoint construction. We want to be able
to apply our results to stacks that are not necessarily limit preserving, and we willneed a way to pass from such a stack to one which is limit preserving without losing
too much information.
Fix a quasi-separated base-scheme S. Let Aff /S denote the category of affine
S-schemes, and let Aff pf/S denote the full subcategory of finitely presented affine
S-schemes. The category Aff /S is closed under the formation of filtering projective
limits (see [Sta, Tag 01YX]). Let pro -Aff pf/S denote the formal pro-category of
Aff pf/S .
2.5.1. Theorem. The formation of projective limits induces an equivalence∼pro -Aff pf/S −→ Aff /S .
Proof. This is a consequence of the theory of absolute Noetherian approximation
developed in [TT90, Appendix C]; for convenience, in this proof we will refer to thetreatment of this material in [Sta, Tag 01YT], which develops the relative versionsof the statements of [TT90, Appendix C] that we need.

SCHEME-THEORETIC IMAGES OF MORPHISMS OF STACKS

43

Note firstly that if X is an affine S-scheme, then X is in particular quasi-compact
and quasi-separated, so that by [Sta, Tag 09MU], we may write X = lim Xi as the←−limit of a directed system of schemes Xi of finite presentation over S, such thatthe transition morphisms are affine over S. By [Sta, Tag 01Z6], we may assume
that the Xi are all affine. This proves that the purported equivalence is essentiallysurjective.
Let f : X → Y be a morphism of S-schemes with Y also affine. Then we can also
write Y = lim Yj with Yj a directed system of affine schemes of finite presentation←−
over S. Then we have
Morpro -Aff pf/S (lim Xi , lim Xj ) = lim lim MorAff /S (Xi , Yj )
←−
←−
←− −→
i

j

j

i

= lim MorAff /S (X, Yj )←−j= MorAff /S (X, Y ),where the first equality is the definition of a morphism in pro -Aff pf/S , the second
is [Sta, Tag 01ZB], and the third is by the universal property corresponding to thestatement that Y = lim Yj . Thus the purported equivalence is fully faithful, as←−
required.

Now let F → Aff pf/S be a category fibred in groupoids. Passing to the corresponding pro-categories, we obtain a category fibred in groupoids∼

pro -F → pro -(Aff pf/S ) −→ Aff /S .
On the other hand, given a category F 0 fibred in groupoids over Aff /S , we may
always restrict it to Aff pf/S , to obtain a category F 0 |Aff pf/S fibred in groupoids overAff pf/S .
Since Aff /S is closed under the formation of filtering projective limits, we seethat the same is true of any category fibred in groupoids F 0 over Aff /S , and so forany such F 0 , evaluating projective limits induces a functor(2.5.2)pro -(F 0 |Aff pf/S ) → F 0

over Aff /S .
2.5.3. Lemma. If F is a category fibred in groupoids over Aff pf/S , then the naturalembedding F → (pro -F)|Aff pf/S is an equivalence.
∼

Proof. This is formal: the fully faithful embedding Aff pf/S → pro -Aff pf/S −→Aff /S identifies the essential image of Aff pf/S with the subcategory of pro-systems
that are isomorphic to a pro-system which is eventually constant. Similarly, theessential image of F in (pro -F) consists of those pro-objects that are isomorphic
to a pro-system which is eventually constant, which by the previous remark are
precisely the pro-objects lying over elements of Aff pf/S .

2.5.4. Lemma. If F 0 is a category fibred in groupoids over Aff /S , then F 0 is limit
preserving if and only if the functor (2.5.2) is an equivalence.
Proof. This is almost formal. (The non-formal ingredient is supplied by [Sta, Tag
01ZC].)


44

M. EMERTON AND T. GEE

The previous two lemmas show that the formation of pro -F from F gives anequivalence between the 2-category of categories fibred in groupoids over Aff pf/S
and the full subcategory of the 2-category of categories fibred in groupoids over
Aff /S consisting of objects satisfying [1].
The following lemma establishes some additional properties of this construction.
2.5.5. Lemma.
(1) If F is a category fibred in groupoids over Aff pf/S , and F 0
is a category fibred in groupoids over Aff /S , then there is an equivalence of
categories∼

MorAff pf/S (F, F 0 |Aff pf/S ) −→ MorAff /S (pro -F, F 0 ).
(2) If F is a stack for the Zariski site (resp. the étale site, resp. the fppf site)on Aff pf/S , then pro -F is a stack for the Zariski site (resp. the étale site,
resp. the fppf site) on Aff /S .
(3) If F 0 is a category fibred in groupoids over Aff /S whose diagonal and double
diagonal are both limit preserving on objects, then the functor (2.5.2) is fully
faithful.
(4) If F 0 is a category fibred in groupoids over Aff /S whose diagonal is limit
preserving on objects and representable by algebraic spaces, then the functor (2.5.2) is fully faithful, and the diagonal of pro -(F 0 |Aff pf/S ) is also representable by algebraic spaces.
(5) If S is locally Noetherian, and if F 0 is a category fibred in groupoids overS which admits versal rings at all finite type points, then pro -(F 0 |Aff pf/S )also admits versal rings at all finite type points.
Proof. (1) Passing to pro-categories, and then composing with the functor
pro -(F 0 |Aff pf/S ) → F 0of (2.5.2), induces a functor
MorAff pf/S (F, F 0 |Aff pf/S ) → MorAff /S (pro -F, pro -(F 0 |Aff pf/S ))→ MorAff /S (pro -F, F 0 ),which is the required equivalence. A quasi-inverse is given by the functor
MorAff /S (pro -F, F 0 ) → MorAff pf/S ((pro -F)|Aff pf/S , F 0 |Aff pf/S )∼−→ MorAff pf/S (F, F 0 |Aff pf/S ),obtained by first restricting to Aff pf/S , and then taking into account the equivalenceof Lemma 2.5.3.
(2) This follows by a standard limiting argument, which we recall. We needto show that all descent data is effective, and that the presheaves Isom (x, y) aresheaves. We being with the argument for descent data.
We must show that if T is affine, and T 0 → T is a Zariski (resp. étale, resp. fppf )cover of T , equipped with a morphism T 0 → pro -F with descent data, then there
is a map T → pro -F inducing the given map T 0 → pro -F.

SCHEME-THEORETIC IMAGES OF MORPHISMS OF STACKS45

By definition, the map T 0 → pro -F factors as T 0 → T 00 → pro -F, with T 00 of
finite presentation over S. By Lemma 2.3.14, we can find a commutative diagram
T0

/ T0


T


/ Tj

j

/ T 00in which the square is Cartesian, and Tj0 is an affine scheme of finite presentation
over S. Furthermore (by the same lemma), if T 0 → T is a Zariski (resp. étale, resp.
fppf ) covering, then we may assume that the morphism Tj0 → Tj is a Zariski (resp.
étale, resp. fppf ) covering.
Since F is assumed to be a stack, it is enough to show that (after possibly
increasing j) the descent data for the morphism T 0 → pro -F arises as the basechange of descent data for the morphism Tj0 → F. This descent data is given by anisomorphism between the two maps T 0 ×T T 0 → pro -F given by the two projections,which satisfies the cocycle condition on the triple intersection.
Now, an isomorphism between the two maps T 0 ×T T 0 → pro -F is equivalent tofactoring the induced map T 0 ×T T 0 → pro -F ×S pro -F through ∆pro -F . So, we havea morphism Tj0 ×Tj Tj0 → pro -F ×S pro -F which factors through the diagonal after
pulling back to T , and we want to show that it factors through the diagonal after
pulling back to some Tj 0 . This will follow immediately provided that ∆pro -F is limit
preserving on objects. Similarly, to deal with the cocycle condition, it is enoughto show that the double diagonal ∆∆pro -F is limit preserving on objects. Since
pro -F is limit preserving by definition, the required limit preserving properties ofthe diagonal and double diagonal follow from Lemma 2.1.5.
We now turn to proving that if x, y : T ⇒ pro -F are two morphisms, thenIsom (x, y) is a sheaf on the Zariski (resp. étale, resp. fppf ) site of T . By definition,
we may find a morphism T → Tj , with Tj affine of finite presentation over S, andmorphisms xj , yj : Tj → F such that x and y are obtained as the pull-backs ofxj and yj . We may further find a projective system {Tj 0 } of affine S-schemes of∼finite presentation, having Tj as final object, and an isomorphism T −→ limj 0 Tj 0 ,←−inducing the given morphism T → Tj ; we then write xj 0 and yj 0 for the composites
xj ,yjTj 0 → Tj ⇒ F. We recall (see Lemma 2.3.14) that the Zariski (resp. étale,
resp. fppf ) site of T is then naturally identified with the projective limit of thecorresponding sites of the Tj 0 ; by [AGV72, Thm. VI.8.2.3], the same is true of thecorresponding topoi. In particular, the various Isom presheaves Isom (xj , yj ), whichby assumption are in fact sheaves on the Zariski (resp. étale, resp. fppf ) sites of
the Tj , form a projective system whose projective limit can be identified with a sheafon the Zariski (resp. étale, resp. fppf ) sites of the T . Unwinding the definitions,
one furthermore finds that this projective limit sheaf is naturally isomorphic to
Isom (x, y). Thus we find that Isom (x, y) is indeed a sheaf.
(3) To check that (2.5.2) is fully faithful, it suffices to check that, for any affine
S-scheme T , if we write T as a projective limit T = limi Ti of finitely presented←−
affine S-schemes, then the functor (2.1.2) is fully faithful. As was already noted in
the proof of Lemma 2.1.5, this follows from the assumption that the diagonal anddouble diagonal of F 0 are limit preserving on objects.

46

M. EMERTON AND T. GEE(4) Proposition 2.3.19 shows that our assumptions on F 0 imply that its double diagonal is also limit preserving on objects, and so it follows from part (3)that (2.5.2) is fully faithful. This in turn implies that the diagrampro -(F 0 |Aff pf/S )F0∆/ pro -(F 0 |Aff

pf/S

∆

) ×S pro -(F 0 |Aff pf/S )/ F 0 ×S F 0

is 2-Cartesian, and thus if the bottom arrow is representable by algebraic spaces,
the same is true of the top arrow.
(5) Since finite type Artinian OS -algebras are objects of Aff pf/S , we see thatthe functors F 0 and pro -F 0 |Aff pf/S induce equivalent groupoids when restricted tosuch algebras. Thus the finite type points of F 0 and pro -F 0 |Aff pf/S are in naturalbijection, in the strong sense that for each finite type OS -field k there is a naturalbijection between the morphisms x : Spec k → F 0 and the morphisms x : Spec k →pro -F 0 |Aff pf/S , and a versal ring to F 0 at such a morphism is also a versal ring to
pro -F 0 |Aff pf/S at the same morphism.
Finally, we note the following basic result.
2.5.6. Lemma. If F is a category fibred in groupoids over Aff pf/S , and F 0 is a fullsubcategory fibred in groupoids, then pro -F 0 is a full subcategory fibred in groupoidsof pro -F.
Proof. This is immediate from the definitions.



2.6. Stacks satisfying [1] and [3]. In this subsection we discuss some propertiesof stacks satisfying Artin’s axioms [1] and [3].
2.6.1. Lemma. If F satisfies [1] and [3], and X is an algebraic stack locally offinite presentation over S, then any morphism of S-stacks X → F is locally offinite presentation (in the sense of Definition 2.3.4).
Proof. Lemma 2.3.8 shows that we may verify this after composing the morphism
X → F with a smooth surjection U → X whose source is a scheme, and thus may
assume that X is in fact an S-scheme X, locally of finite presentation. It follows
from [Sta, Tag 06CX] that X → S is limit preserving on objects, so that X is limit
preserving (as it is a stack in setoids). It follows from Corollary 2.1.8 that X → F
is limit preserving on objects. Since this morphism is representable by algebraic
spaces (as F satisfies [3]), applying [Sta, Tag 06CX] again yields the lemma.

2.6.2. Lemma. If F satisfies [1] and [3], then the diagonal ∆ : F → F ×S F islocally of finite presentation.
Proof. Since F is limit preserving by assumption, Lemma 2.1.5 shows that ∆ : F →F ×S F is limit preserving on objects. Since ∆ is representable by algebraic spaces
by assumption, it is locally of finite presentation by [Sta, Tag 06CX].

We now establish some simple results related to finiteness conditions in the casewhen S is locally Noetherian. We first recall that a morphism X → S from analgebraic stack X to the locally Noetherian scheme S is locally of finite type if andonly if it is locally of finite presentation (since both conditions may be verified after

SCHEME-THEORETIC IMAGES OF MORPHISMS OF STACKS

47

composing this morphism with a smooth surjection from a scheme to X , which
reduces us to the case when X is itself a scheme, in which case the claim follows
immediately from the hypothesis that S is locally Noetherian).
2.6.3. Lemma. Suppose that F satisfies [1] and [3], that S is locally Noetherian, andthat X → F is a morphism whose source is an algebraic stack. Then the morphism
X → F is locally of finite type if and only if it is locally of finite presentation, andthese conditions are in turn equivalent to the composite X → F → S being locallyof finite type (or, equivalently, locally of finite presentation, as noted above).
Proof. If X → S is locally of finite type, and so locally of finite presentation, thenLemma 2.6.1 shows that X → F is locally of finite presentation. And certainly, if
X → F is locally of finite presentation then it is locally of finite type.
Next, we want to show that if X → F is locally of finite type, then it is locallyof finite presentation. To this end, let T be an affine S-scheme. Since F → S islimit preserving, and so in particular limit preserving on objects, we may factor
any S-morphism T → F as T → T 0 → F, where T 0 is of finite presentation
over S. (It follows from [Sta, Tag 09MV] that T may be written as the limit ofsuch T 0 .) The base-changed morphism XT → T is obtained by pulling back thebase-changed morphism XT 0 → T 0 . This latter morphism is locally of finite type,and its target is of finite presentation over the locally Noetherian scheme S (andhence locally Noetherian itself), and so it is in fact locally of finite presentation.
Thus the morphism XT → T is also locally of finite presentation, and since T andthe morphism T → F were arbitrary, we conclude that the morphism X → F islocally of finite presentation, as claimed.
It remains to show that if X → F is locally of finite presentation, then X → S is
locally of finite presentation. Morally, we would like to prove this by arguing thatsince F satisfies [1], the morphism F → S is locally of finite presentation, and soconclude that the composite X → S is locally of finite presentation. Unfortunately,while this is a valid argument if F is an algebraic stack, it does not apply in thegenerality we are considering here, where F is assumed simply to satisfy [1] and [3].
Indeed, in this level of generality, we haven’t defined what it means for F → S to
be locally of finite presentation.
Thus we are forced to make a slightly more roundabout argument. Since X → F
is locally of finite presentation, it is limit preserving on objects, by Lemma 2.3.16.
The morphism F → S is also limit preserving on objects (since F satisfies [1]),and hence the composite X → S is limit preserving on objects [Sta, Tag 06CW].
It then follows from Lemma 2.1.9 that X → S is locally of finite presentation (or,equivalently, locally of finite type).
Our next results are inspired by a lemma of Artin [Art69a, Lem. 3.10]. Our firstlemma isolates one of the steps in Artin’s argument, and generalises it to the stacky
context.
2.6.4. Lemma. Suppose that S is locally Noetherian, that T is a locally finite typeS-scheme, that F satisfies [1] and [3], that t ∈ T is a finite type point, and that
T → F is an S-morphism which is formally smooth at t. Then, replacing T by anopen neighbourhood of t if necessary, we may factor the morphism T → F as
T → F 0 → F,

48

M. EMERTON AND T. GEE

where F 0 is an algebraic stack, locally of finite presentation over S, the first arrowis a smooth surjection, and the second arrow is locally of finite presentation, unramified, representable by algebraic spaces, and formally smooth at the image t0 of
t in |F 0 |.
Proof. Write R := T ×F T . By the assumption that F satisfies [3], this is a locallyfinite type algebraic space over S; indeed, it is the base-change of the morphism
T ×S T → F ×S F via the diagonal ∆ : F → F ×S F, and Lemma 2.6.2 shows that
this latter morphism is locally of finite presentation. The projections R ⇒ T endow
R with the structure of a groupoid in algebraic spaces over T . Since the morphism
T → F is locally of finite presentation by Lemma 2.6.3 (1), each of these projections
is also locally of finite presentation (or equivalently, locally of finite type, since thebase S is locally Noetherian), and the formal smoothness of T → F at t implies thatboth projections are smooth in a neighbourhood of (t, t) by Lemma 2.4.7 (3). Thus,
applying Lemma 2.6.6 below, we see that, by shrinking T around t if necessary, we
may find an open subgroupoid V ⊆ R that is actually a smooth groupoid over T .
We then define F 0 := [T /V ], and let t0 denote the image of t in |F 0 |.
Certainly, since V ⊆ T ×F T, the map T → F factors through F 0 . Since T → F 0
is smooth and T is locally of finite type over S, so is F 0 . (The property of being
locally of finite type is smooth local on the source; see [Sta, Tag 06FR]). The
morphism F 0 → F is thus locally of finite presentation, by Lemma 2.6.3 (1). Since
T → F is formally smooth at t, and T → F 0 is smooth, the morphism F 0 → F isalso formally smooth at t0 by definition. (See Definition 2.4.10.)
It remains to show that the morphism F 0 → F is unramified and representableby algebraic spaces. We have already seen that it is locally of finite presentation,and so, in particular, locally of finite type, and (as explained in Example 2.3.10)
we must show furthermore that the diagonal morphism F 0 → F 0 ×F F 0 is an
open immersion. (Recall that since F satisfies [3] by assumption, the fibre productF 0 ×F F 0 is an algebraic stack.) Since the morphism R = T ×F T → F 0 ×F F 0
is smooth and surjective (as T → F 0 is), we can verify this after base-changing by
this latter morphism. Since R = T ×F T , and V = T ×F 0 T , one verifies that thebase-changed morphism
F 0 ×F 0 ×F F 0 R → R
is precisely the open immersion V → R. This establishes the claim.



2.6.5. Remark. In the case when F is simply a sheaf of sets (which is the context
of [Art69a]), the algebraic stack F 0 will simply be an algebraic space, and if we
replace it by an étale cover by a scheme X, we obtain an unramified morphism
X → F which is formally smooth at a point x above t. This is essentially theconclusion of the second paragraph of the proof of [Art69a, Lem. 3.10], and ourargument is an adaptation to the stacky context of the argument given there.
2.6.6. Lemma. Let (U, R, s, t, c, e, i) be a groupoid in algebraic spaces locally offinite type over a locally Noetherian scheme S, with U a scheme. If u ∈ U is a
finite type point such that s is smooth at the point e(u) (or equivalently, such thatt is smooth at the point e(u)), then there exists an open neighbourhood U 0 of u in
U such that, if (R0 , s0 , t0 , c0 ) denotes the restriction of the groupoid (R, s, t, c) to U 0 ,there is an open subgroupoid V ⊆ R0 such that s0|V and t0|V are smooth, i.e. suchthat V is a smooth groupoid over U 0 .

SCHEME-THEORETIC IMAGES OF MORPHISMS OF STACKS

49

Proof. Let V 0 denote the open subspace of R on which s is smooth. Write U 0 =e−1 (V 0 ); then U 0 is an open subscheme of U containing u. If we replace U by U 0
and R by R0 := R|U 0 = (s × t)−1 (U 0 × U 0 ), so that V 0 is replaced by V 0 ∩ R0 , thenwe may, and do, assume that e(U ) ⊆ V 0 .
Recall that we have the commutative diagram:U dt

R o prt

R ×s,U,t R

0

pr1

s


U o

/R

c


R

t


/U

s

s

of [Sta, Tag 043Z], in which each square (including the top square) is Cartesian.
Pulling back the right-hand square of this diagram via the morphism V 0 → R, weform the Cartesian diagram
R ×s,U,t V 0

c

pr1


V0

/R
s

s


/ U.

The bottom arrow is smooth by the definition of V 0 , and is furthermore surjective,
since we have put ourselves in a situation in which e(U ) ⊆ V 0 . Thus the locus ofsmoothness of the left-hand vertical arrow is precisely the preimage under c of thelocus of smoothness of the right-hand vertical arrow; i.e. the locus of smoothnessof pr1 : R ×s,U,t V 0 → V 0 is equal to c−1 (V 0 ). Now certainly pr1 : V 0 ×s,U,t V 0 → V 0
is smooth, since it is a base-change of the smooth morphism s : V 0 → U. Thus
V 0 ×s,U,t V 0 ⊆ c−1 (V 0 ), or equivalently, c(V 0 ×s,U,t V 0 ) ⊆ V 0 .
Now define V := V 0 ∩ i(V 0 ). Clearly c(V ×s,U,t V ) ⊆ V. Also e(U ) ⊆ V , andi(V ) = V. Taken together, these properties show that V is an open subgroupoidof R. Since V ⊆ V 0 , we see that s|V is smooth. Since t = si and i(V ) = V , we seethat t|V is smooth as well. Thus the lemma is proved.
In the remainder of the section, we return to Axioms [4a] and [4b]; none of thismaterial is needed for our main theorems. The following lemma, and its corollary, provide an analogue in the stacky context of [Art69a, Lem. 3.10] itself. The
argument is essentially the same as Artin’s.
2.6.7. Lemma. Suppose that S is locally Noetherian, that T is a locally finite typeS-scheme, and that F satisfies [1], [3], and [4a]. If X → F is a morphism offinite type from an algebraic stack to F (in the sense of Definition 2.3.4), and if
T → F is an S-morphism which is formally smooth at a finite type point t ∈ T ,then there exists a neighbourhood T 0 of t in T such that the base-changed morphism
T 0 ×F X → X is smooth.
Proof. Clearly we may replace T by an affine open neighbourhood of t, and thus
suppose that T is quasi-compact. We may apply Lemma 2.6.4 to the morphism
T → F, and so, replacing T by a neighbourhood of t in T if necessary, we factor
T → F as T → F 0 → F as in the statement of that lemma. We also choose a

50

M. EMERTON AND T. GEE

smooth surjection U → X whose source is a scheme. We then consider the diagram
T ×F U/ F 0 ×F U/UT ×F X/ F 0 ×F X


/X


T


/ F0


/F

Since F satisfies [3], the fibre product T ×F U is an algebraic space, while F 0 ×F U ,T ×F X , and F 0 ×F X are algebraic stacks. Since the morphism X → F is quasicompact, and since T (and hence also F 0 ) is quasi-compact, the fibre-products
T ×F X and F 0 ×F X are furthermore quasi-compact.
Since the upper right horizontal arrow is an unramified morphism, and so (bydefinition) has an étale diagonal, it is in particular a DM morphism (i.e. has anunramified diagonal). Since its target is a scheme, we thus find that F 0 ×F U is infact a DM stack. We may thus amplify the preceding diagram to a commutative
diagram/ V0VT ×F U/ F 0 ×F U/UT ×F X/ F 0 ×F X


/X


T


/ F0


/F

in which V and V 0 are quasi-compact schemes, the vertical arrows with V and V 0
as their sources are étale, and the composites V → T ×F X and V 0 → F 0 ×F X ,as well as the four horizontal arrows on the left half of the diagram, are smoothsurjections. (Note that the upper-most square in this diagram is not assumed to
be Cartesian.)
Let C 0 ⊆ V denote the complement of the smooth locus of the composite V → U.
It is a closed subset of V , and so its image C in T is a constructible subset of T . (Themorphism V → T is a locally finite type morphism between Noetherian schemes,
and hence is finite type. Thus it maps constructible sets to constructible sets.)
Lemma 2.6.3 (1) shows that U is locally of finite type over S, and so, byLemma 2.4.7 (3), the smooth locus of the morphism T ×F U → U contains thefibre over t, and thus the same is true of the smooth locus of the morphism V → U .
In other words, the point t does not lie in C. To prove the lemma we must showthat there is an open neighbourhood of t disjoint from C; that is, we must showthat t does not lie in the closure of C.
Suppose that t does lie in the closure of C. Then we claim that we may finda point t̃ ∈ C whose closure Y contains t, and such that A := OY,t is a onedimensional domain. To see this, note firstly that C is a finite union of irreducible
locally closed schemes, so t is in the closure of one of these, and we may replace

SCHEME-THEORETIC IMAGES OF MORPHISMS OF STACKS

51

C by this irreducible component (with its induced reduced structure), and thereby
assume that C is open in its closure. Replacing the closure of C by an affine open
subset containing t, and C by a distinguished open subset of this affine open, we
put ourselves in the situation of having a Noetherian domain A with a closed pointt ∈ Spec A, and an element f ∈ A such that Spec A[1/f ] is a proper non-empty
subset. Replacing A by its localisation at the maximal ideal corresponding to t, we
may assume that A is a local domain, and that f is in the maximal ideal of A.
Since A is local Noetherian, it is finite-dimensional, and we may choose a prime
P not containing f so that dim A/P is as small as possible. Replacing A by A/P ,we have a Noetherian local domain A containing a nonzero element f ∈ A, withthe properties that f is not a unit, and f is contained in every non-zero prime
ideal of A. It remains to check that A is one-dimensional. To see this, note thatthe height one primes of A are precisely the isolated associated primes of (f ), sothat A has only finitely many height one primes. Let Q be any prime ideal of A.
Take an element x ∈ Q; then by Krull’s principal ideal theorem, (x) is contained insome height one prime, so we see that Q is contained in the union of the height one
primes. Since this is a finite union, we see by prime avoidance that Q is containedin some height one prime, as required.
Since C 0 is closed in V , and so of finite type over T , we may find a point ṽ ∈ C 0
lying over t̃ such that residue field extension κ(t̃) ,→ κ(ṽ) is finite. We will obtaina contradiction by showing that the morphism V → U is in fact formally smoothat ṽ (and hence smooth at ṽ, by Lemma 2.4.7 (3)).
Consider a diagram
/Z
Z0

/U


V

as in the definition of formal smoothness at ve (Definition 2.4.4 (2)); so Z0 → Z isa closed immersion of Artinian local OS -algebras, and the residue field L at theclosed point of Z0 is a finite extension of κ(ṽ) (and thus also finite over K = κ(t̃).)We expand this diagram to the diagram
Spec L

Z0

V
#

(/
/ V0

Z


/U

The morphism V 0 → U factors as V 0 → F 0 ×F U → U, and hence, as the composite
of an unramified and an étale morphism, is itself unramified.
We will show that the morphism Z → U lifts to a morphism Z → V 0 , compatible
with the given morphism Spec L → V 0 . Assuming this, we note that this morphism,
when restricted to Z0 , must coincide with the given morphism Z0 → V 0 ; thisfollows from the fact that V 0 → U is unramified, and thus formally unramified.
Since V → V 0 is smooth, and hence formally smooth, we then see that we may
further lift the morphism Z → V 0 to a morphism Z → V compatible with the given

52

M. EMERTON AND T. GEE

morphism Z0 → V . This completes the proof that V → U is formally smooth at ṽ,and thus completes the proof of the lemma.
It remains to prove the existence of the lifting Z → V 0 . Since V 0 → F 0 ×F U
is étale, it suffices to obtain a lifting Z → F 0 ×F U . For this, it suffices in turn toobtain a morphism Z → F 0 lifting the composite Z → U → F, and, for this, itsuffices to obtain a morphism Z → T lifting Z → F.
Write Z = Spec BL0 , and let B denote the integral closure of A in L. Then by
the Krull–Akizuki theorem ([Mat89, Thm. 11.7] and its Corollary) B is a semi-local
Dedekind domain with field of fractions L, whose residue fields at its closed points
are finite extensions of the residue field of A at its closed point, i.e. of κ(t). Let B 0
be the inverse image of B ⊆ L in BL0 . Since F satisfies [4a], we may extend themorphism Spec BL0 → F to a morphism Spec B 0 → F, compatible with the given
morphism
Spec B → Spec A → T → F.
Lemma 2.6.8 below then shows that we may lift the morphism Spec B 0 → F to amorphism Spec B 0 → T . Passing to the local ring at the generic point, this gives
the required morphism Z → T.

2.6.8. Lemma. Suppose that S is locally Noetherian, that F is a stack over S
satisfying [1] and [3], that T is a scheme, locally of finite type over S, and that
T → F is a morphism over S which is formally smooth at a finite type point t ∈ T .
Suppose that B is a Noetherian local OS -algebra whose residue field is finite typeover OS , that B 0 is an OS -algebra which is a nilpotent thickening of B, and that
we have a commutative diagram of morphisms over S
Spec B/ Spec B 0


T


/F

such that the left-hand vertical arrow maps the closed point of Spec B to the given
point t ∈ T . Then there is a lifting of the right-hand vertical arrow to a morphism
Spec B 0 → T .
Proof. Consider the projection T ×F Spec B 0 → Spec B 0 . The commutative diagramin the statement of the lemma gives rise to a morphism(2.6.9)Spec B → T ×F Spec B 0

lifting the closed immersion Spec B ,→ Spec B 0 . Since the closed point of Spec B
maps to the point t of T , by assumption, it follows from Lemma 2.4.7 (3) thatthe image of the closed point of Spec B under this morphism lies in the smooth
locus of the projection, and thus that the morphism (2.6.9) itself factors throughthis smooth locus. Since smooth morphisms are, in particular, formally smooth, we
may thus lift this morphism to a section Spec B 0 → T ×F Spec B. Composing thissection with the projection onto T gives the morphism required by the statementof the lemma.

2.6.10. Corollary. Suppose that S is locally Noetherian, that T is a locally finite
type S-scheme, that F satisfies [1], [3], and [4a], and furthermore that the diagonalof F is quasi-compact. If T → F is an S-morphism which is formally smooth at a
finite type point t ∈ T , then this morphism is in fact smooth at t.

SCHEME-THEORETIC IMAGES OF MORPHISMS OF STACKS

53

Proof. Let X → F be a morphism whose source is a scheme of finite type over S,
and consider the usual “graph diagram”
X

F

Γ

/ X ×S F∆/ F ×S F/F

in which the square is Cartesian (by construction). Lemma 2.1.5, along with [Sta,Tag 06CX], shows that the diagonal ∆ : F → F ×S F is locally of finite presentation,and it is quasi-compact by assumption. Thus it is in particular of finite type, andso the same is true of the graph Γ : X → X ×S F. The projection X ×S F → F isalso of finite type, being the base-change of the finite type morphism of Noetherian
schemes X → S. Thus the morphism X → F is of finite type, and so Lemma 2.6.7
implies that we may find a neighbourhood U of t in T such that the base-changed
morphism U ×F X → X is smooth. By definition, then, the morphism T → F issmooth at t.

2.6.11. Corollary. Suppose that S is locally Noetherian, and that F is a stack overS satisfying [1], [2](a),[3], [4a], and [4b], whose diagonal is quasi-compact. Then Fsatisfies [4].
Proof. Suppose that ϕ : U → F is a morphism from a scheme locally of finite
type over S to F which is versal at a finite type point u ∈ U . Lemma 2.4.7 (2)then shows that ϕ is formally smooth at u. From Corollary 2.6.10 we conclude
that ϕ is in fact smooth at u, and Axiom [4b] then implies that ϕ is smooth in aneighbourhood of u. Finally, Lemma 2.4.7 (1) shows that ϕ is versal at each finite
type point in this neighbourhood, and thus F satisfies Axiom [4], as claimed. 
Our final result in this subsection strengthens the conclusion of Lemma 2.6.4, inthe presence of [4a].
2.6.12. Corollary. Suppose, in the context of Lemma 2.6.4, that F furthermore
satisfies [4a], and has quasi-compact diagonal. Then, in addition to the conclusionsof that lemma, we may impose the condition that the morphism F 0 → F be amonomorphism.
Proof. By Corollary 2.6.10, replacing T by a neighbourhood of t if necessary, we
may suppose that the projections R := T ×F T ⇒ T are in fact smooth. Thus, inthe proof of Lemma 2.6.4, we may take V = R. It is then easily verified that F 0 :=[T /V ] = [T /R] → F is in fact a monomorphism. Indeed, following the proof of
Lemma 2.6.4, we deduce from the fact that V = R that the diagonal F 0 → F 0 ×F F 0
is an isomorphism, which is equivalent to F 0 → F being a monomorphism.

2.6.13. Remark. Example 4.3.4 shows that we cannot remove the assumption that
F has quasi-compact diagonal from the preceding corollaries.
2.7. Relationships between [2] and [3]. Throughout this subsection we supposethat F is a category fibred in groupoids over the locally Noetherian base scheme S.
2.7.1. Lemma. Let x : Spec k → F be a finite type point of F (here k is a finite
type field over OS ).

54

M. EMERTON AND T. GEE(1) If F admits an effective Noetherian versal ring at x, then, for each object Aof CbΛ , the functor Fx (A) → Fbx (A) is essentially surjective. Conversely, if this functor is essentially surjective for each object A of CbΛ , and if F admits a Noetherianversal ring at x, then F admits an effective Noetherian versal ring at x.
(2) If F satisfies [3], then, for each object A of CbΛ , the functor Fx (A) → Fbx (A)is fully faithful.
Proof. We begin by proving (1). Suppose first that F admits an effective Noetherianversal ring at x, and let Ax ∈ CbΛ be an effective versal ring at the morphism x; thatis, we have an object η of Fx (Ax ) whose image in Fbx (Ax ) is versal. If ξ : Spf B → F
is any object of Fbx (B), for some object B of CbΛ , then by definition of versality
we may find a morphism f : Ax → B and an isomorphism ξ ∼= f ∗ η. Since η iseffective, i.e. lies in the image of the functor Fx (Ax ) → Fbx (Ax ), we see that ξ liesin the essential image of the functor Fx (B) → Fbx (B); i.e. this functor is essentiallysurjective. On the other hand, if F admits a Noetherian versal ring Ax at x, and if
furthermore the functor Fx (Ax ) → Fbx (Ax ) is essentially surjective, then F admits
an effective Noetherian versal ring at x by definition.
We turn to proving (2). Suppose that η and ξ are two objects of Fx (A) (forsome object A of CbΛ ). Their product is a morphism η × ξ : Spec A → F ×S F. The
morphisms η and ξ, when restricted to Spec k (the closed point of Spec A), both
induce the given morphism x : Spec k → F, and hence we have the outer square in
a commutative diagram/ Spec A
Spec k
x

 y
F

η×ξ

∆


/ F ×S F

The set of morphisms between η and ξ in the category Fx (A) may be identified withthe set of morphisms Spec A → F which continue to make the diagram commute.
If we let ηb and ξb denote the images of η and ξ in Fbx (A), then the set of morphisms
between ηb and ξb may similarly be identified with the set of morphisms Spf A → F
for which the diagram/ Spf ASpec k
x

 y
F

η
b×ξb

∆


/ F ×S F

commutes.
Since F satisfies [3], the fibre product F ×∆,F ×S F ,η×ξ Spec A is an algebraic spaceover Spec A. The claim of full faithfulness in (2) now follows from the following
general fact [Sta, Tag 0AQH]: if A is a complete local ring, if T → Spec A is amorphism from an algebraic space to Spec A, and if t : Spec k → T is a section ofthis morphism over the closed point, then the restriction map Mort (Spec A, T ) →Mort (Spf A, T ) from the set of sections over Spec A extending t to the set of sections
over Spf A extending t is a bijection.

We now prove the lemma promised in Remark 2.2.23.
2.7.2. Lemma. Suppose that F satisfies [3], and that the diagonal of F is furthermore locally of finite type. If, for some finite type point x of F, there exists

SCHEME-THEORETIC IMAGES OF MORPHISMS OF STACKS

55

a versal morphism U → Fbx with U being (Noetherianly) pro-representable, thenR := U ×Fbx U is also (Noetherianly) pro-representable, so that Fbx admits a presentation by a (Noetherianly) pro-representable smooth groupoid in functors.
Proof. Note firstly that (U, R) is a smooth groupoid in functors by Lemma 2.2.22,
so we only need to show that R is (Noetherianly) pro-representable. Suppose that
U is pro-represented by A = limi∈I Ai . If B ∈ CΛ , then giving a morphism←−
Spec B → U ×Fbx U is equivalent to giving a pair of morphisms Spec B ⇒ U
which induce the same morphism to F. By definition these morphisms factor
through Spec Ai for some i. Thus U ×Fbx U is the colimit over I of the fibre products Spec Ai ×Fbx Spec Ai (the fibre product denoting a fibre product of categories
cofibred in groupoids on CΛ ). Since a colimit of pro-representable functors is prorepresentable (by the projective limit of the representing rings), it suffices to showthat each of these fibre products is pro-representable.
If we let ξ : Spec Ai → F denote the composite Spec Ai → Spf A → Fbx ,→ F,
then Spec Ai ×F Spec Ai is an algebraic space locally of finite type over S (because
F satisfies [3], and its diagonal is locally of finite type) which represents the functor
Isom (ξ, ξ). The fibre product Spec Ai ×Fbx Spec Ai is the subfunctor of the restrictionof Isom (ξ, ξ) to CΛ consisting of isomorphisms which induce the identity from x toitself. The identity from x to itself is a k-valued point of Isom (ξ, ξ), and thissubfunctor may equally well be described as the formal completion of Isom (ξ, ξ) at
this point. Since Isom (ξ, ξ) is represented by a locally Noetherian algebraic space,this formal completion is represented by the formal spectrum of the complete local
ring of Isom (ξ, ξ) at x in the sense of Definition 2.2.17 and thus is pro-representable.
It remains to show that if A is Noetherian, then U ×Fbx U is Noetherianly prorepresentable. By [Gro95, Prop. 5.1], we need to check that if k is the residue
field at x, then the k-vector space of morphisms Spec k[]/2 → U ×Fbx U is finitedimensional.
Now, any such morphism factors through a pair of morphisms Spec A/m2A → U .
Since A is Noetherian, A/m2A is Artinian. Applying the argument of the previous
paragraph to Spec A/m2A ×Fbx Spec A/m2A , we find that it is pro-represented by a
Noetherian complete local ring (the completion of a locally Noetherian algebraic
space at a closed point); any such ring admits only a finite-dimensional space ofmaps to k[]/2 , so we are done.

2.7.3. Corollary. Suppose that F satisfies [3], that its diagonal is furthermore locally of finite type, and that F admits versal rings at all finite type points. Then F
is Arttriv -homogeneous.
Proof. This is immediate from Lemmas 2.2.24 and 2.7.2.

The next result shows that when F satisfies [2] and [3], we can strengthen condition [2](a) so as to allow Y 0 and Z to be complete Noetherian local rings, rather
than merely Artinian.
2.7.4. Lemma. Suppose that F satisfies [3] and [2](b), and that
Y

/ Y0


Z


/ Z0

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M. EMERTON AND T. GEE

is a pushout diagram of S-schemes, with the horizontal arrows being closed immersions, Y being a finite type Artinian OS -scheme, each of Z and Y 0 being thespectrum of a complete Noetherian local OS -algebra whose residue field is finite typeover OS , and the left-hand vertical arrow being closed (i.e. corresponding to a local
morphism of local OS -algebras). If either F satisfies [2](a), or if F is an étale stack
whose diagonal is locally of finite presentation and the extension of the residue fieldof Y over the residue field of Z is separable, then the induced functor
F(Z 0 ) → F(Y 0 ) ×F (Y ) F(Z)is an equivalence of categories.
Proof. Let x be the underlying closed point of Y . By Lemma 1.6.1, Z 0 is Noetherian(note that Y ,→ Y 0 is assumed to be a closed immersion, and the residue fieldsof Y, Y 0 , Z are all finite type OS -algebras, so that in particular the residue fieldof Y is a finite extension of that of Z), so by Lemma 2.7.1 and the assumptionthat F satisfies [3] and [2](b), we may replace F by Fbx , and by the definition of Fbx ,we can then reduce to the case that Y 0 and Z are Artinian. In the case that F
satisfies [2](a) we are then done by the very definition of that condition, and in thecase that F is an étale stack whose diagonal is locally of finite presentation and theextension of the residue field of Y over the residue field of Z is separable the result
follows from Corollary 2.7.3 and Lemma 2.2.6.

2.8. Artin’s representability theorem. In this subsection, we continue to assume that S is locally Noetherian. In the statement of the next lemma, as well as
for one direction of the main theorems, we will suppose further that, for each finite
type point s ∈ S, the local ring OS,s is a G-ring. As explained in Section 1.5.5, thisassumption is needed in order to apply Artin approximation.
We begin with a lemma, which is essentially a rephrasing of [Sta, Tag 07XH]. It is
the key application of Artin approximation which underlies Artin’s representability
theorem.
2.8.1. Lemma. Suppose that, for each finite type point s ∈ S, the local ring OS,sis a G-ring. If F is a category in groupoids satisfying [1] and [2](a), if k is a finite
type OS -field, and if x : Spec k → F is a morphism representing the finite typepoint t ∈ |F|, for which Fbx admits an effective Noetherian versal ring, then there
exists a morphism ϕ : U → F whose source is a scheme locally of finite type overS, and a finite type point u ∈ U such that ϕ(u) = t, and such that ϕ is formallysmooth at u.
Proof. It follows from [Sta, Tag 07XH] (i.e. Artin approximation), together with
our assumptions, that we may find a morphism ϕ : U → F with U a scheme locallyof finite type over S, containing a point u, such that φ(u) = t, and such that ϕ isversal at u. Lemma 2.4.7 (2), together with our assumption of [2](a), shows thatin fact ϕ is formally smooth at u.

2.8.2. Lemma. Suppose that F is a category fibred in groupoids which satisfies [3],and that there exists a morphism ϕ : T → F whose source is a scheme, locally offinite type over S, which is smooth in a neighbourhood of a finite type point t ∈ T .
Then, if ϕ0 : T 0 → F is any morphism from a scheme, locally of finite type over S,
to F, which is formally smooth at a finite type point t0 ∈ T 0 whose image in F
coincides with the image of t, there is a neighbourhood V of t0 in T 0 such that the
restriction of ϕ0 to V is smooth.

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57

Proof. Replacing T by the hypothesised neighbourhood of t, we may assume thatϕ is smooth. Note that since F satisfies [3], the morphism ϕ is representable byalgebraic spaces, and so this is to be understood in the sense of [Sta, Tag 03YK],
i.e. the base change of this morphism over any morphism from a scheme to F issmooth. In particular, the base-changed morphism of algebraic spaces T ×F T 0 → T 0
is smooth. Since the morphism T 0 → F is formally smooth at t0 , the projection
T ×F T 0 → T is smooth in a neighbourhood U of the point (t, t0 ), by Lemma 2.4.7 (3).
Now the the composite U → T → F is the composite of smooth morphisms, henceis smooth. Rewriting this morphism as U → T 0 → F, we see that this composite isalso smooth. If we let V denote the image of U in T 0 , then V is an open subset ofT 0 containing t0 , and U → V is a smooth surjection. Smoothness being a property
that is smooth local on the source, we see that V → F is a smooth morphism, as
required.

2.8.3. Remark. The preceding lemma shows that in order to verify [4] for a category
fibred in groupoids satisfying [2](a) and [3], it suffices to find, for each finite typepoint of x ∈ |F|, a morphism ϕ : T → F whose source is locally of finite typeover S, and contains a finite type point t mapping to x, such that ϕ restricts to asmooth morphism on some neighbourhood of t. (The role of [2](a) is to ensure, via
Lemma 2.4.7 (2), that versality at a point coincides with formal smoothness at apoint.)
2.8.4. Theorem. Suppose that S is locally Noetherian. Any algebraic stack, locallyof finite presentation over S, satisfies Axioms [1], [2], [3] and [4]. Conversely,
suppose further that for each finite type point s ∈ S, the local ring OS,s is a G-ring.
Then if F is an étale stack in groupoids over S satisfying [1], [2], [3] and [4], thenF is an algebraic stack, locally of finite presentation over F.
Proof. If F is an algebraic stack, locally of finite presentation over S, then it follows
from Lemma 2.1.9 that F satisfies [1], while Lemma 2.2.3, Lemma 2.7.1, [Sta, Tag
07WU], [Sta, Tag 07WV], [Sta, Tag 06IW], [Sta, Tag 07X8], and [Sta, Tag 07X1]
show that F satisfies [2]. (More precisely, F satisfies [2](a) by Lemma 2.2.3. By [Sta,Tag 07X8] and Lemma 2.7.1, in order to prove that F satisfies [2](b), it suffices toshow that F has Noetherian versal rings at each finite type point. [Sta, Tag 06IW]
gives a criterion for the existence of such versal rings, which is satisfied by [Sta,Tag 07WU], [Sta, Tag 07WV], and [Sta, Tag 07X1].) By definition F satisfies [3].
Again by definition, we may find a smooth surjection U → F with U a scheme. If
x ∈ |F | is a point of finite type, then we may find a finite type point u ∈ U lying
over x, and Remark 2.8.3 then shows that F satisfies [4].
For the converse, we follow the proof of [Sta, Tag 07Y4]. By definition, we needto show that X admits a smooth surjection from a scheme. Taking the union of themorphisms obtained from Lemma 2.8.1 as we run over all finite type points of X ,we obtain a smooth map U → X whose source is a scheme, whose image contains
all finite type points of X . It remains to show that this is surjective. As in the
proof of [Sta, Tag 07Y4], this may be checked by pulling back to affine schemes offinite presentation over S, where it is immediate (as smooth maps are open, andthe finite type points of a scheme are dense).

2.8.5. Theorem. Any algebraic stack, locally of finite presentation over S, satisfies[1], [2], [3], [4a], and [4b]. Conversely, suppose further that for each finite type points ∈ S, the local ring OS,s is a G-ring. Then if F is an étale stack in groupoids

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over S satisfying [1], [2], [3], [4a], and [4b], and if the diagonal of F is furthermore
quasi-compact, then F is an algebraic stack, locally of finite presentation over S.
Proof. For the first statement, in view of Theorem 2.8.4, we need to show that [4a]and [4b] are satisfied for algebraic stacks. Lemma 2.4.15 shows that [4a] is satisfied,
while Lemma 2.4.7 (4) shows that [4b] is satisfied.
To prove the second statement, note that Corollary 2.6.11 shows, under the given
hypotheses on F, that F furthermore satisfies Axiom [4]. Theorem 2.8.4 then showsthat F is an algebraic stack, locally of finite presentation over S.

Example 4.3.4 shows that the condition on the diagonal of F is necessary inorder to deduce that a stack satisfying [1], [2], [3], [4a], and [4b] is algebraic.
2.8.6. Remark. In the proof of Artin representability, we don’t require the fullstrength of Axiom [2](b); all we need is that Fbx admits an effective Noetherianversal ring for at least one morphism x representing any given finite type pointof F. However, the following result, which describes the extent to which thishypothesis is independent of the choice of representative of a finite type point,
implies in particular that, in the context of the preceding representability theorems,
this hypothesis holds for at least one representative of a given finite type point if
and only if it holds for every such representative.
2.8.7. Lemma. Suppose that F satisfies [2](a), and suppose given x : Spec k → F,with k a field which is a finite type OS -algebra.
(1) If Fbx admits a Noetherian versal ring, then for any other morphism x0 :Spec l → F, with l a field which is a finite type OS -algebra, representing
the same point of F, we have that Fbx0 again admits a Noetherian versal
ring.
(2) If Fbx admits an effective Noetherian versal ring, then for any finite extenx
sion l of k, if we let x0 denote the composite Spec l → Spec k −→ F, wehave that Fbx0 again admits an effective Noetherian versal ring.
(3) Suppose that F satisfies [3] (in addition to [2](a)), and that, for some finitex
extension l of k, if we denote by x0 the composite Spec l → Spec k −→ F,b
we have that Fx0 admits an effective Noetherian versal ring. Assume alsoeither that l/k is separable, or that F is an fppf stack. Then Fbx admits aneffective Noetherian versal ring.
(4) Suppose either that (a) F satisfies [3] (in addition to [2](a)), and that either
the residue field of the image of the composite Spec k → F → S is perfect, or
F is an fppf stack; or (b) that F satisfies [1] and [3] (in addition to [2](a)),
and that the local rings of S at finite type points are G-rings. Suppose alsothat Fbx admits an effective Noetherian versal ring. Then for any othermorphism x0 : Spec l → F, with l a field which is a finite type OS -algebra,
representing the same point of F, we have that Fbx0 again admits an effective
Noetherian versal ring.
Proof. Given l as in (1), we may find a common finite extension l0 of k and l. Afterappropriate relabelling, then, to prove (1) we may assume that l is a finite extensionof k and that x0 is the composite Spec l → Spec k → F, and we must show that Fbx
admits a Noetherian versal ring if and only if Fbx0 does.

SCHEME-THEORETIC IMAGES OF MORPHISMS OF STACKS59

We use the notation and results of Remark 2.2.11. Since F satisfies [2](a), there∼
is a natural equivalence of categories cofibred in groupoids (Fbx )l/k −→ Fbx0 . We will
deduce part (1) of the lemma from from [Sta, Tag 06IW] (which gives a criterion
for a category cofibred in groupoids over CΛ,k , or CΛ,l , to admit a Noetherian versal
morphism), as follows. Since F satisfies [2](a), it follows from [Sta, Tag 07WA,Tag
06J7] that both Fbx and (Fbx )l/k satisfy all of the hypotheses of [Sta, Tag 06IW],
except possibly the condition of having a finite-dimensional tangent space. By [Sta,Tag 07WB], this condition holds for Fbx if and only if it holds for (Fbx )l/k , so it
suffices to observe that the existence of a Noetherian versal ring implies the finitedimensionality of the tangent space, by [Sta, Tag 06IU].
∼
To prove part (2), we again use the equivalence (Fbx )l/k −→ Fbx0 . Taking
into account part (1), it suffices to show that if Ax0 denotes a versal ring at
the morphism x0 , then the versal morphism Spf Ax0 → Fbx0 arises from a morphism Spec Ax0 → F. The above-mentioned equivalence of categories shows that
this versal morphism arises as the composite of the natural morphism Spf Ax0 →Spf(Ax0 ×l k) and a morphism Spf(Ax0 ×l k) → Fbx . Lemma 2.7.1 (1), and oureffectivity assumption regarding Fbx , shows that this latter morphism is effective,
i.e. is induced by a morphism Spec(Ax0 ×l k) → F. The composite
Spec Ax0 → Spec(Ax0 ×l k) → Spec Ax → Fthen induces the original versal morphism Spf Ax0 → Fbx0 , and so (2) is proved.
In order to prove (3), we first fix a versal morphism Spf Ax → F at x. If l/k is
separable, then, by the topological invariance of the étale site, we may find a finiteétale local extension B of Ax which induces the extension l/k on residue fields.
b Λk Λl ;Otherwise, let Λk , Λl be Cohen rings for k, l respectively, and set B := Ax ⊗then B/Ax is a finite extension by the topological version of Nakayama’s lemma,
and it is a faithfully flat local extension of complete local Noetherian rings by[Gro64, Lem. 0.19.7.1.2]. In either case, the composite Spf B → Spf Ax → F may
be regarded as a morphism Spf B → Fbx0 , and so, by our effectivity assumption
regarding Fbx0 , together with Lemma 2.7.1 (1), it is effective, i.e. is induced by amorphism Spec B → F.
Consider the pull-backs of this morphism along the two projections Spec B ⊗Ax
B ⇒ Spec B. Since the morphism Spf B → F factors through Spf Ax , we see thatthe two pull-backs become isomorphic over Spf(B ⊗Ax B). Since F satisfies [3],Lemma 2.7.1 (2) shows that the two pull-backs are themselves isomorphic. We
may check the cocycle condition in a similar way, and hence obtain étale (or fppf,
in the case that l/k is inseparable) descent data on the morphism Spec B → F,
which (since F is an étale stack, and an fppf stack if l/k is inseparable) allows us
to descend it to a morphism Spec Ax → F, as required.
We turn to proving (4). Thus we suppose given a morphism x0 : Spec l →F representing the same point of F that the given morphism x : Spec k → F
represents. We are assuming that Fbx admits an effective Noetherian versal ring,
and we wish to prove the corresponding fact for Fbx0 . If we let l0 be a common
finite extension of k and l, and let x00 denote the composite Spec l0 → Spec k → Fbx ,then it follows from (2) that Fbx00 admits an effective Noetherian versal ring. So,
relabelling l0 as l and l as k, we are reduced to the following problem: we are givenx
the morphisms x : Spec k → F and x0 : Spec l → Spec k −→ F, for which Fbx0

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admits an effective Noetherian versal ring, and we would like to conclude that Fbx
also admits an effective Noetherian versal ring.
If either the residue field of the image of Spec k in S is perfect or F is an fppf
stack, and if F satisfies [3], then this follows from part (3). Otherwise, we assume
further that F satisfies [1], and that the local rings of S at finite type points are
G-rings. This allows us to apply Artin approximation, in the form of Lemma 2.8.1,
to the morphism x0 , so as to conclude that there exists a morphism ϕ : U → F
whose source is a scheme, and a finite type point u of U , lying over the image of x0(which is also the image of x), such that ϕ is formally smooth at u.
At this stage x0 and l have done their job in the argument, and so we dropthem from consideration; in fact, we will recycle them as notation, in a manner
which we now explain. Pulling back U over x, we obtain a k-scheme whose smooth
locus contains the fibre over U . This fibre is non-empty, by Lemma 2.4.7 (2), andthus contains a point defined over a finite separable extension l of k. If we let x0
denote the resulting composite Spec l → Spec k → F, then Fbx0 admits an effective
Noetherian versal ring (given by the complete local ring of this fibre at this point).
Part (3) now implies that the same is true for Fbx , as required.

2.8.8. Remark. A version of Lemma 2.8.7 for the condition of admitting a presentation by a smooth Noetherianly pro-representable groupoid in functors, rather thanadmitting a Noetherian versal ring, can be proved in an almost identical fashion.
2.8.9. Remark. Example 4.3.10 shows that parts (3) and (4) of Lemma 2.8.7 are
not true without the assumption of [3].
3. Scheme-theoretic images
Suppose that ξ : X → F is a proper morphism, where X is an algebraic stack,and F is a stack whose diagonal is representable by algebraic spaces and locallyof finite presentation. In Section 3.2 we define the scheme-theoretic image Z of ξ,which is initially a Zariski substack of F. Our main aim in this section is to prove
Theorem 1.1.1, giving a criterion for Z to be an algebraic stack, as well as to provea number of related properties of Z.
Interpreting Theorem 1.1.1 as taking the quotient by a proper equivalence relation.
Whether or not Z satisfies [2], we can show (under mild hypotheses on F) that themorphism ξ : X → F factors through a morphism ξ : X → Z (see Lemma 3.2.23
below), and this morphism is “scheme-theoretically surjective”. If we define R :=X ×F X , then R is an algebraic stack (because F satisfies [3]) which defines aproper equivalence relation on X . Thus, at least morally, we may regard Z as thequotient of X by the equivalence relation R, and Theorem 1.1.1 may be regarded
as providing a context in which the quotient X /R may be defined as an algebraic
stack.
Note that in general the quotient of an algebraic stack, or a scheme, or even a
variety, by a proper equivalence relation, may not admit a reasonable interpretation
as an object of algebraic geometry. (See e.g. Examples 4.1.1 and 4.1.2.) Our
result shows that when the desired quotient admits an interpretation as the schemetheoretic image of a morphism from X to some stack F, the quotient does indeed
have a chance to be of an algebro-geometric nature.
One well-known theorem which concerns taking a quotient by a proper equivalence relation is Artin’s result [Art70] on the existence of contractions. We close

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61

this discussion by briefly describing the relationship between [Art70] and the present
note.
The relationship with [Art70]. In [Art70], Artin proves the existence of dilatations and contractions of an algebraic space along a closed algebraic subspace, given
a formal model for the desired dilatation or contraction. (See also Remark 4.1.3 below.) In the case of contractions, his result can be placed in the framework of Corollary 1.1.2. Namely, taking X in that theorem to be the algebraic space on which one
wishes to perform a contraction, one can define a functor F which is supposed torepresent the result of the contraction, and a proper morphism ξ : X → F. One canthen show that F satisfies [1], [2], and [3], and hence conclude (via Corollary 1.1.2)
that F is representable by an algebraic space. This is in fact essentially how Artin
proceeds, although there are some slight differences between our approach and his:
Artin defines F concretely on the category of Noetherian OS -algebra, and then extends it to arbitrary OS -algebras by taking limits. Our approach would be to apply
Artin’s concrete definition only to finite type OS -algebras, and then to extend toarbitrary OS -algebras by taking limits. The difference between the two approaches
is that in Artin’s approach, the fact that condition [2] is satisfied by F is rather
automatic, whereas the proof that F satisfies [1] becomes the crux of the argument.
In our approach, while the arguments remain the same, their interpretation differs:
essentially by definition, F will satisfy [1], while the bulk of the argument can beseen as proving that it also satisfies [2]. However, this attempt to link Artin’s result to ours is a little misleading, since in Artin’s context, the verification that F
satisfies [4] is straightforward. For some other F (such as those that appear in thetheory of moduli of Galois representations discussed in Section 5), the verificationof [4] seems to be less straightforward, however, and indeed the only approach weknow is via the general arguments of the present paper.
3.1. Scheme-theoretic images (part one). We recall the following definition(see for example [Sta, Tag 01R8]; note that we do not include a quasi-separated
hypothesis, as it is not needed for the basic properties of scheme-theoretic images
that we use here).
3.1.1. Definition. Let f : Y → Z be a quasi-compact morphism of schemes. The
kernel of the natural morphism OZ → f∗ OY is a quasi-coherent ideal sheaf I on Z([Sta, Tag 01R8]) and we define the scheme-theoretic image of f to be the closed
subscheme V (I) of Z cut out by I.
We say that f is scheme-theoretically dominant if the induced morphism OZ →f∗ OY is injective; that is, if the scheme-theoretic image of f is Z.
We say that f is scheme-theoretically surjective if it is scheme-theoretically dominant, and surjective on underlying topological spaces.
3.1.2. Remark. (1) The morphism f factors through V (I), and V (I) is the minimal
closed subscheme of Z through which f factors. The induced morphism f 0 : Y →V (I) is scheme-theoretically dominant, and also has dense image [Sta, Tag 01R8].
Thus, for a closed morphism, the notions of scheme-theoretical dominance andof scheme-theoretical surjectivity are equivalent.
(2) The formation of scheme-theoretic images is compatible with arbitrary flatbase change [Sta, Tag 081I].

62

M. EMERTON AND T. GEE(3) It follows easily from (2) that the formation of scheme-theoretic images is fpqc
local on the target, so that the condition of being scheme-theoretically dominant,or surjective, may be checked fpqc locally on the target.
(4) If g : X → Y is quasi-compact then the scheme-theoretic image of f g is a
closed subscheme of the scheme-theoretic image of f . If g is furthermore schemetheoretically dominant (e.g. fpqc), then OZ → f∗ OY and OZ → (f g)∗ OX have thesame kernel, and hence the scheme-theoretic images of f and f g coincide.
In particular, if g : X → Y is quasi-compact and scheme-theoretically dominant(resp. surjective), then f is scheme-theoretically dominant (resp. surjective) if andonly if the composite f g is scheme-theoretically dominant (resp. surjective).
(5) An fpqc morphism is scheme-theoretically surjective.
Points (3), (4), and (5) of the preceding remark allow us to extend the notionof scheme-theoretically dominant (resp. scheme-theoretically surjective) morphisms
to the context of morphisms of algebraic stacks in the following way.
Recall (cf. Remark 2.3.6 and Lemma 2.3.8) that an algebraic stack Y is quasicompact if its underlying topological space |Y| is quasi-compact, or equivalently,
by [Sta, Tag 04YC], if there is a smooth surjection U → Y with U a quasi-compact
scheme. A morphism of algebraic stacks f : Y → Z is quasi-compact if for every
morphism V → Z with V a quasi-compact algebraic stack, the fibre product Y ×Z V
is also quasi-compact [Sta, Tag 050U].
3.1.3. Definition. Let f : Y → Z be a quasi-compact morphism of algebraic stacks.
Let V → Z be a smooth surjection from a scheme, and let V = ∪i Ti be a cover of
V by quasi-compact open subschemes. For each Ti , the fibre product Y ×Z Ti isquasi-compact, so admits a smooth surjection Ui → Y ×Z Ti from a quasi-compact
scheme.
The composite morphism Ui → Ti is quasi-compact, and we say that f : Y → Z
is scheme-theoretically dominant if for all i, the morphism Ui → Ti is schemetheoretically dominant. We say that f is scheme-theoretically surjective if themorphisms Ui → Ti are all scheme-theoretically surjective.
It follows from Remark 3.1.2 (3)–(5) that this notion is well-defined, independently of the choices of V and the Ti and Ui , and that it agrees with Definition 3.1.1
if Y and Z are schemes.
Similarly, we can extend the definition of scheme-theoretic images to quasicompact morphisms of algebraic stacks.
3.1.4. Definition. Let f : Y → Z be a quasi-compact morphism of algebraic stacks,and choose V , Ti and Ui as in Definition 3.1.3.
The composite morphism Ui → Ti is quasi-compact, hence admits a schemetheoretic image Ti0 . The smooth equivalence relation RTi := Ti ×Z Ti on Ti restricts
to a smooth equivalence relation RTi0 on Ti0 , and the quotient stack [Ti0 /RTi0 ] is a
closed substack of the quotient stack [Ti /RTi ], itself an open substack of Z. The
substacks [Ti0 /RTi0 ] glue together to form a closed substack of Z, which we define
to be the scheme-theoretic image of f .
Again, it follows from Remark 3.1.2 (2)–(5) that this notion is well-defined,
independently of the choices of V and the Ti and Ui , and that it agrees with
Definition 3.1.1 if Y and Z are schemes.
Given this definition, the following remarks are an easy consequence of Remark 3.1.2.

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63

3.1.5. Remark. (1) The morphism f factors through its scheme theoretic image X ,and X is the minimal closed substack of Z through which f factors. The induced
morphism f 0 : Y → X is scheme-theoretically dominant, and the induced map|Y| → |X | has dense image.
(2) The formation of scheme-theoretic images is compatible with arbitrary flatbase change.
(3) The formation of scheme-theoretic images is fpqc local on the target, so thatthe condition of being scheme-theoretically dominant, or surjective, may be checked
fpqc locally on the target.
(4) If g : X → Y is quasi-compact then the scheme-theoretic image of f g is a
closed substack of the scheme-theoretic image of f . If g is also scheme-theoreticallydominant, then the scheme-theoretic images of f and f g coincide.
In particular, if g : X → Y is quasi-compact and scheme-theoretically dominant(resp. surjective), then f is scheme-theoretically dominant (resp. surjective) if andonly if the composite f g is scheme-theoretically dominant (resp. surjective).
3.1.6. Remark. One can show that if f : Y → Z is a quasi-compact and quasiseparated morphism of algebraic stacks, then the kernel of OZ → f∗ OY is a quasicoherent ideal sheaf, and cuts out the scheme-theoretic image of f as defined above.
(In fact, presumably this is true even without the quasi-separatedness assumption,although, as in the scheme-theoretic context, such a statement would be slightly
more delicate to prove, since then f∗ OY need not be quasi-coherent.) However,since we do not need to consider sheaves on stacks elsewhere in this paper, except
very briefly in the proof of Lemma 3.2.1 below, we will not give the details here,
but rather simply establish the few basic facts that we need as we use them.
The one special case of this theory that we need is that if f : X → S is aquasi-compact algebraic stack over a Noetherian base scheme S, then f is schemetheoretically dominant if and only if OS → f∗ OX is injective. To see this, let
g : U → X be a smooth cover by a quasi-compact scheme; then by Definition 3.1.3,
f is scheme-theoretically dominant if and only if the composite f g : U → X → S
is scheme-theoretically dominant, or equivalently if and only if OS → (f g)∗ OU is
injective. It remains to show that f∗ OX → (f g)∗ OU is injective. Since f∗ is left
exact, it is enough to show that OX → g∗ OU is injective. By the definition of OX ,this can be checked after pulling back by a smooth cover of X by a scheme (forexample, U itself), so we reduce to the case of algebraic spaces, which is immediatefrom [Sta, Tag 082Z].
We have the following simple lemma concerning scheme-theoretic dominance.
3.1.7. Lemma. Suppose that f : Y → Z is a scheme-theoretically dominant quasicompact morphism of algebraic stacks, and that Z 0 ,→ Z is a closed immersion forwhich the base-changed morphism Y 0 → Y is an isomorphism. Then Z 0 ,→ Z is
itself an isomorphism.
∼

Proof. The assumption that Y 0 −→ Y can be rephrased as saying that the morphism Y → Z can be factored through the closed substack Z 0 of Z. Since Y → Z
is scheme-theoretically dominant, we see that necessarily Z 0 = Z. (This last statement is easily reduced to the case of schemes, where it is immediate.)
3.2. Scheme-theoretic images (part two). Our goal in this section is to generalise the construction of scheme-theoretic images to certain morphisms of stacks

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whose domain is algebraic, but whose target is of a possibly more general nature.
The general set-up, which will be in force throughout this section, will be as follows: we suppose given a morphism ξ : X → F of stacks over a locally Noetherianbase-scheme S, whose domain X is assumed to be algebraic, and whose target F
is assumed to have diagonal ∆F which is representable by algebraic spaces andlocally of finite presentation (so, in particular we are assuming throughout thissection that F satisfies [3]).
We will furthermore typically assume that either F satisfies [1], or else that X is
locally of finite presentation. We will sometimes reduce the latter situation to theformer by using the results of Section 2.5. (See Remark 3.2.13 below.) We will alsofrequently need to assume that ξ is proper. We work in maximal generality when
we can, only introducing these hypotheses at the points that they are needed (butsee Remark 3.2.10 (3) below).
We begin with a lemma whose intent is to capture the properties that will characterise when a morphism Spec A → F, with A a finite type Artinian local OS -algebra,
factors through the scheme-theoretic image of ξ.
3.2.1. Lemma. If Spec A is a finite type Artinian local S-scheme, and ϕ : Spec A →F is a morphism over S, then the following conditions are equivalent:(1) There exists a complete Noetherian local OS -algebra B, and a factorisationof ϕ into S-morphisms Spec A → Spec B → F, such that the base-changed
morphism XB → Spec B is scheme-theoretically dominant.
(2) There exists a complete Noetherian local OS -algebra B, and a factorisationof ϕ into S-morphisms Spec A → Spec B → F, such that the base-changed
morphism XB → Spec B is scheme-theoretically dominant, and such thatthe morphism Spec A → Spec B is a closed immersion.
If ξ is furthermore proper, then these conditions are equivalent to the following
further two conditions.
(3) There exists an Artinian local OS -algebra B, and a factorisation of ϕ into
S-morphisms Spec A → Spec B → F, such that the base-changed morphism
XB → Spec B is scheme-theoretically dominant.
(4) There exists an Artinian local OS -algebra B, and a factorisation of ϕ into
S-morphisms Spec A → Spec B → F, such that the base-changed morphism XB → Spec B is scheme-theoretically dominant, and such that themorphism Spec A → Spec B is a closed immersion.
3.2.2. Remark. We remark that if A is a Artinian local OS -algebra, which is furthermore of finite type (or, equivalently, whose residue field is of finite type over OS ),and if B is a local OS -algebra for which there exists a morphism of OS -algebras
B → A, then the residue field of B is necessarily also of finite type over OS . Thus,in the context of the preceding lemma, the residue field of the ring B appearing in
any of the conditions of the lemma will necessarily be of finite type over OS .
The proof of Lemma 3.2.1 will make use of the theorem on formal functions foralgebraic stacks in the form of the following theorem of Olsson. As in [Ols07], we
work with sheaves on the lisse-étale site.
3.2.3. Theorem. Let A be a Noetherian adic ring, and let I be an ideal of definitionof A. Let X be a proper algebraic stack over Spec A. Then the functor sendinga sheaf to its reductions is an equivalence of categories between the category of

SCHEME-THEORETIC IMAGES OF MORPHISMS OF STACKS

65

coherent sheaves G on X and the category of compatible systems of coherent sheaves
Gn on the reductions Xn := X ×Spec A Spec(A/I n+1 ).
Furthermore, if G is a coherent sheaf on X with reductions Gn , then the naturalmap
H 0 (X , G) → lim H 0 (Xn , Gn )←−nis an isomorphism of topological A-modules, where the left hand side has the I-adic
topology, and the right hand side has the inverse limit topology.
Proof. A proper morphism of stacks is separated, so has quasi-compact and quasiseparated diagonal, so that the more restrictive definition of an algebraic stackin [Ols07] is automatically satisfied. The claimed result is then a special caseof [Ols07, Thm. 11.1].
Proof of Lemma 3.2.1. Evidently (2) implies (1). Conversely, suppose (1) holds.
Bearing in mind Remark 3.2.2, we see that the residue field of A is finite over thatof B, so Lemma 1.6.3 shows that we may factor the morphism Spec A → Spec B
as Spec A → Spec B 0 → Spec B, where B 0 is again a complete Noetherian local
ring, such that the first morphism is a closed immersion, and such that the second
morphism is faithfully flat. Since scheme-theoretic dominance is preserved underflat base-change, we see that XB 0 → Spec B 0 is scheme-theoretically dominant, andso replacing B by B 0 , we find that (2) is satisfied.
It is evident that (4) implies (3) implies (1), with no assumption on ξ. Thus, tocomplete the proof of the lemma, it suffices to show that (2) implies (4), when ξ isproper. In fact, we will show that if Spec A → Spec B is a morphism as in (2), thenwe may find an Artinian quotient B 0 of B such that this morphism factors throughthe closed immersion Spec B 0 → Spec B, and such that the base-changed morphism
XB 0 → Spec B 0 is scheme-theoretically dominant. Certainly Spec A → Spec B
factors through the closed immersion Spec B 0 → Spec B for some Artinian quotientof B, since A itself is Artinian. Thus it suffices to show that B admits a cofinal
collection of Artinian quotients B 0 for which XB 0 → Spec B 0 is scheme-theoreticallydominant. This is the content of Lemma 3.2.4 below.

3.2.4. Lemma. Assume that ξ is proper, and suppose that Spec B → F is a morphism over S, where B is a complete Noetherian local OS -algebra. Then the basechanged morphism XB → Spec B is scheme-theoretically dominant if and only if B
admits a cofinal collection of Artinian quotients B 0 for which XB 0 → Spec B 0 is
scheme-theoretically dominant.
Proof. Suppose firstly that B admits a cofinal collection of Artinian quotients B 0
for which XB 0 → Spec B 0 is scheme-theoretically dominant. Then the schemetheoretic image of XB → Spec B is a closed subscheme of Spec B containing eachof the Spec B 0 , and since the quotients B 0 are cofinal, it must in fact be Spec B.
We now consider the converse. Since ξ is proper, so is the base-changed morphism
XB → Spec B. If XB → Spec B is scheme-theoretically dominant, then the natural
map B → H 0 (Spec B, ξ∗ OXB ) = H 0 (XB , OXB ) is injective by Remark 3.1.6. Set
Xn = XB ×Spec B Spec(B/mn+1
B ). Noting that the structure sheaf OXB is coherent,
Theorem 3.2.3 applies to show that we have an isomorphism of topological Bmodules∼
H 0 (X , OX ) −→ lim H 0 (Xn , OXn ).
←
−
n

66

M. EMERTON AND T. GEE

Let In be the kernel of the composite morphism
B → H 0 (X , OX ) → H 0 (Xn , OXn ),so that we have injections B/In → H 0 (Xn , OXn ). The natural map Xn →X ×Spec B Spec(B/In ) is then an isomorphism by construction, and so X ×Spec B
Spec(B/In ) → Spec B/In is scheme-theoretically dominant (again by Remark 3.1.6;
since B/In → H 0 (Xn , OXn ) is injective, the natural map Xn → Spec(B/In ) is
scheme-theoretically dominant). Thus the B/In are the sought-after cofinal collection of Artinian quotients of B.
b
We now define, in our present context, a subgroupoid Zb of F.
3.2.5. Definition. If Spec A is a finite type Artinian local S-scheme, then we letbZ(A)denote the full subgroupoid of F(A) consisting of morphisms Spec A → F
that satisfy the equivalent conditions (1) and (2) of Lemma 3.2.1. (If ξ is proper,bthen we note that the objects of Z(A)can equally well be characterised in termsof conditions (3) and (4) of that lemma.)
We now define the scheme-theoretic image Z of ξ, using the terminology andresults of Section 2.5 (note that since S is assumed to be locally Noetherian, it is inparticular quasi-separated, so the results of Section 2.5 apply to categories fibredin groupoids over S). As in Section 2.5, we let Aff /S denote the category of affine
S-schemes, and let Aff pf/S denote the full subcategory of finitely presented affine
S-schemes.
To begin with, we will consider the restriction F|Aff pf/S to a category fibred ingroupoids over Aff pf/S , and define a full subcategory of F|Aff pf/S which (by an abuseof notation which we will justify below) we denote Z|Aff pf/S .
3.2.6. Definition. Let Z|Aff pf/S be the full subcategory of F|Aff pf/S defined as follows: if A is a finite type OS -algebra, then we let Z|Aff pf/S (A) denote the full
subgroupoid of F(A) consisting of points η whose formal completion ηbt , at eachbfinite type point t of Spec A, factors through Z.
3.2.7. Lemma. Z|Aff pf/S is a Zariski substack of F|Aff pf/S .
Proof. To check that Z|Aff pf/S is a full subcategory fibred in groupoids of F|Aff pf/S ,we need to check that if T 0 → T is a morphism in Aff pf/S , and T → F satisfies
the condition to lie in Z(T ), then the same is true of the pulled-back morphism
T 0 → F; but this is immediate from the definition of Z. To see that it is actuallya substack of F|Aff pf/S , it is then enough (by [Sta, Tag 04TU]) to note that since
the condition of a morphism T → F factoring through Z is checked pointwise, it isin particular a Zariski-local condition.

We remind the reader that Theorem 2.5.1 gives an equivalence of categories∼
pro -Aff pf/S −→ Aff /S . We now give our definition of the scheme-theoretic imageof X → F; see Remark 3.2.10 (2) below for the relationship of this definition to theexisting one in the case that F is an algebraic stack.
3.2.8. Definition. We define Z to be the pro-category pro -Z|Aff pf/S , which (byLemmas 2.5.6 and 3.2.7) is a full subcategory in groupoids of pro -F|Aff pf/S . Via theequivalence of Theorem 2.5.1, we regard Z as a category fibred in groupoids over
Aff /S . We also note that Lemma 2.5.3 yields an equivalence between the restriction

SCHEME-THEORETIC IMAGES OF MORPHISMS OF STACKS

67

of Z to Aff pf/S and our given category Z|Aff pf/S , thus justifying the notation forthe latter.
Since ∆F is assumed to be representable by algebraic spaces and locally of finite
presentation, it follows from Lemma 2.5.5 (4) that pro -(F|Aff pf/S ) is equivalent
to a full subcategory of F, so that we may regard Z as a full subcategory of F(the latter being thought of as a category fibred in groupoids over the category ofaffine S-schemes). We refer to Z as the scheme-theoretic image of the morphismξ : X → F.
3.2.9. Lemma. Z is a Zariski substack of F over Aff /S , and satisfies [1].
Proof. This follows immediately from Lemmas 2.5.5 (2), 2.5.4, and 3.2.7.



3.2.10. Remark. (1) Since Z is a Zariski stack over Aff /S , it naturally extends
to a Zariski stack over S (and is again a Zariski substack of F). We will see inLemma 3.2.20 below that under some relatively mild additional assumptions, Z isin fact an étale substack of F.
(2) In the case when F is an algebraic stack which satisfies [1], then its substackZ coincides with the scheme-theoretic image of ξ (in the sense of Definition 3.1.4);
see Lemma 3.2.29 below.
(3) If neither X nor F satisfy [1], then Definition 3.2.6 is not a sensible one. E.g.
if F is an algebraic stack that does not satisfy [1], and ξ : F → F is the identity
morphism, then evidently the scheme-theoretic image of ξ (in the usual sense, i.e.
in the sense of Definition 3.1.4) is just F, and so doesn’t satisfy [1] (by assumption)— whereas we noted above that the substack Z of F given by Definition 3.2.6 does
satisfy [1], by stipulation.
Here and on several occasions below, we will have cause to consider how schemetheoretic images interact with monomorphisms of stacks F 0 ,→ F. In light of this,it will be helpful to recall that Lemma 2.3.22 implies that if F has a diagonal whichis representable by locally algebraic spaces and locally of finite presentation, thenthe same is true of any stack F 0 admitting a monomorphism F 0 ,→ F (i.e. any
substack F 0 of F).
3.2.11. Lemma. Suppose that X → F factors as X → F 0 ,→ F, where F 0 is a stack
which satisfies [3], and F 0 ,→ F is a monomorphism. Then the scheme-theoretic
image of X → F 0 is contained in the scheme-theoretic image of X → F.
Proof. A consideration of the definitions shows that we need only check that if
Spec B → F 0 is such that X ×F 0 Spec B → Spec B is scheme-theoretically dominant, then X ×F Spec B → Spec B is scheme-theoretically dominant; but this isimmediate, because X ×F 0 Spec B = X ×F Spec B.

3.2.12. Remark. Throughout the rest of this section we prove several variants andrefinements of Lemma 3.2.11. More specifically, Lemma 3.2.21 gives a criterion forthe scheme-theoretic images of X → F 0 and X → F to actually be equal, andLemma 3.2.22 gives a criterion for X → F to factor through Z. Lemmas 3.2.25
and 3.2.26 relate the existence of a factorisation X → F 0 ,→ F to the propertyof F 0 containing Z, and Proposition 3.2.31 shows that Z can be characterised by auniversal property if it is assumed to be algebraic.
3.2.13. Remark. As we explained at the beginning of this section, our main results
assume either that F satisfies [1], or that X satisfies [1] (equivalently, X is locally68

M. EMERTON AND T. GEEof finite presentation over S). In the proof of the main theorem, we will argue by
reducing the second case to the first case in the following way:
We first note that it follows from Lemmas 2.5.3, 2.5.4, and 2.5.5 (2) and (4) thatpro -(F|Aff pf/S ) is a substack of F which satisfies [1] and [3]. If we assume in addition
that F admits versal rings at all finite type points, then so does pro -(F|Aff pf/S ),by Lemma 2.5.5 (5).
If X satisfies [1], then it follows from Lemma 2.5.4 that there is an equivalence∼pro -X|Aff pf/S −→ X , and thus from Lemma 2.5.5 (1) that ξ : X → F can be factored
through a morphism X → pro -FAff pf/S . By construction the monomorphism Z ,→F also factors through pro -FAff pf/S , and an examination of the definitions showsthat Z is also the scheme-theoretic image (in the sense of Definition 3.2.8) of theinduced morphism X → pro -(F|Aff pf/S ). Taken together, the previous remarks willallow us to simply replace F by pro -(F|Aff pf/S ), and thus assume that we are in thefirst case.
3.2.14. Lemma. Assume that ξ : X → F is proper, and let x : Spec k → F be a
finite type point. Then x is a point of Z if and only if the fibre Xx is non-empty.
Proof. If Xx is non-empty, then Xx → Spec k is scheme-theoretically surjective, andx is a point of Z by definition. Conversely, if x is a point of Z, then by Lemma 3.2.1
we can factor x : Spec k → F through a closed immersion Spec k → Spec B with B
an Artin local ring and XB → Spec B scheme-theoretically dominant. This implies
that the fibre Xx is non-empty, as required.

3.2.15. Definition. If Spf Ax → Fbx is a versal ring at x, then (by Remark 3.1.2 (4))
if we let Ai run over the discrete Artinian quotients of Ax , the scheme-theoretic
images Spec Ri of the morphisms XAi → Spec Ai fit together to give a formal
subscheme Spf Rx := Spf limi Ri of Fbx , which we call the scheme-theoretic image←−of the base-changed morphism X ×F Spf Ax → Spf Ax .
It follows from Remark 2.2.7 that the natural morphism Ax → Rx is surjective,
and is a quotient map of topological rings. Thus Spf Rx is even a closed formal
subscheme of Spf Ax .
The following lemma shows that, when ξ is proper, versal rings for Z can beconstructed from versal rings for F by taking scheme-theoretic images.
3.2.16. Lemma. Assume that ξ : X → F is proper. Let x : Spec k → Z be a
finite type point, which we also consider as a finite type point of F. Suppose that
Spf Ax → Fbx is a versal ring at x, and let Spf Rx be the scheme-theoretic imageof XSpf Ax → Spf Ax . Then the morphism Spf Rx → Fbx factors through a versal
morphism Spf Rx → Zbx .
Proof. We claim that a morphism Spec A → Spf Ax , with A an object of CΛ , factors through Spf Rx if and only if the composite Spec A → Spf Ax → Fbx factors through Zbx . In the notation of Definition 3.2.15, if Spec A → Fbx factors
through Spf Rx , then it in fact factors through Spec Ri for some i, and hence
through Zbx by definition (as the morphism XRi → Spec Ri is scheme-theoreticallydominant).
Conversely, if the composite Spec A → Spf Ax → Fbx factors through Zbx , then by
Lemma 3.2.1, we have a factorisation Spec A → Spec B → F, where B is an object

SCHEME-THEORETIC IMAGES OF MORPHISMS OF STACKS

69

of CΛ , the morphism Spec A → Spec B is a closed immersion, and XB → Spec B isscheme-theoretically dominant. By the versality of Spf Ax → Fbx , we may lift themorphism Spec B → Fbx to a morphism Spec B → Spf Ax , which furthermore wemay factor as Spec B → Spec Ai → Spf Ax , for some value of i. Since XB → Spec B
is scheme-theoretically dominant, the morphism Spec B → Spec Ai then factors
through Spec Ri , and thus through Spf Rx , as claimed.
In particular, we see that the composite Spf Rx → Spf Ax → Fbx factors througha morphism Spf Rx → Zbx . It remains to check that this morphism is versal. Thisis formal. Suppose we are given a commutative diagram

Spec A0/ Spf Rx

/ Spf AxSpec A


/ Zbx


/ Fbx

where the left hand vertical arrow is a closed immersion, and A0 , A are objectsof CΛ . By the versality of Spf Ax → Fbx , we may lift the composite Spec A → Fbx
to a morphism Spec A → Spf Ax . Since the composite Spec A → Spf Ax → Fbx
factors through Zbx , the morphism Spec A → Spf Ax then factors through Spf Rx ,as required.


If Z were to behave like the scheme-theoretic image of a morphism of algebraic
stacks, i.e. like a closed substack, then we would expect to be able to test theproperty of a morphism to F factoring through Z by precomposing with a schemetheoretically dominant morphism. We cannot prove this general statement at thispoint of the development, but we begin our discussion of this general problem by
establishing some special cases.
3.2.17. Lemma. Let Y → Z → F be a composite of morphisms over S, with Y
and Z being local Artinian schemes of finite type over S, for which the morphism
Y → Z is scheme-theoretically surjective, and such that the composite Y → F
factors through Z. Suppose that:(a) either ξ is proper and F is Arttriv -homogeneous, or F satisfies [2](b); and(b) either F is Artfin -homogeneous, i.e. F satisfies (RS), or the residue field
extension in Y → Z is separable.
Then the morphism Z → F also factors through Z.
Proof. Set Y = Spec A and Z = Spec A0 ; the scheme-theoretically surjective morphism Y → Z then corresponds to an injective morphism A0 → A of OS -algebras.
By assumption we may find a surjection of OS -algebras B → A, whose source is acomplete Noetherian local OS -algebra, such that the given morphism Spec A → F
factors through a morphism Spec B → F for which the induced morphism XB →Spec B is scheme-theoretically surjective.
Let B 0 := A0 ×A B. We have a commutative diagram70

M. EMERTON AND T. GEESpec A/ Spec A0Spec B/ Spec B 0#-  
F

Suppose firstly that ξ is proper and F is Arttriv -homogeneous. By Lemma 3.2.1,
we may assume that B is Artinian. We claim that we may fill in this diagramwith a morphism Spec B 0 → F; this follows since we are assuming that either F is
Artfin -homogeneous, or that the residue field extension is separable, in which casethis follows from Lemma 2.2.6. Otherwise, if ξ is not proper, then by assumption
F satisfies [2](b). Lemma 2.7.4 and our hypotheses then show that we may once
again fill in this diagram with a morphism Spec B 0 → F.
It remains to show that XB 0 → Spec B 0 is scheme-theoretically dominant. But
this follows directly from the fact that each of XB → Spec B and Spec B → Spec B 0
are scheme-theoretically dominant (the latter because B 0 → B is injective), and aconsideration of the commutative diagram
XB/ XB 0Spec B/ Spec B 0(This is a particular case of Remark 3.1.5 (4).)



3.2.18. Lemma. Suppose that either ξ is proper and F is Arttriv -homogeneous,or that F satisfies [2](b). If U → Y is a smooth surjective morphism of algebraic
stacks over S, and if Y → F is a morphism of stacks over S such that the composite
U → Y → F factors through Z, then the morphism Y → F itself factors through Z.
Proof. Let T → Y be a morphism, with T a scheme over S. We must show that thecomposite T → Y → F factors through Z. By assumption the morphism U ×Y T →T → Y → F (which admits the alternate factorisation U ×Y T → U → Y → F)
factors through Z. Since U ×Y T is an algebraic stack, it admits a smooth surjection
from a scheme U .
The composite U → T is a smooth morphism, and so we may apply [Sta, Tag
055V] to find a morphism V → U for which the composite V → U → T is étaleand surjective. Replacing U by V , we are reduced to showing that if U → T is asurjective étale morphism of S-schemes, and the composite U → T → F factors
through Z, then the morphism T → F factors through Z. (This is essentially theétale sheaf property of Z; see Lemma 3.2.20 and its proof below.)
Since Z is a Zariski substack of F, we may check this statement Zariski locallyon T , and hence assume that T is affine. Since T is then quasi-compact, we mayfind a quasi-compact open subset of U which surjects onto T , and then replacing U
by the disjoint union of the members of a finite affine cover of this quasi-compact
open subset, we may further assume that U is affine. Since Z satisfies [1], we may
further reduce to the case that T and U are locally of finite presentation over S,
and hence of finite type over S (since S is locally Noetherian). (More precisely:

SCHEME-THEORETIC IMAGES OF MORPHISMS OF STACKS71

by Lemma 2.3.14, we may write T = limi Ti , U = limi Ui , with Ui → Ti an étale←−←−
surjection of affine schemes locally of finite presentation over S, such that U → T
is the pull-back of each Ui → Ti . Since Z satisfies [1], the morphism U → Z factors
through Ui for some i, and thus so does the composite U → Z → F. Arguing as
in the proof of Lemma 2.5.5 (2), the étale descent data for the morphism U → F
arises as the base change of étale descent data for the morphism Ui0 → F for somei0 ≥ i, and since F is an étale stack, it follows that T → F factors through Ti0 , asrequired.)In this case we are reduced to checking that if u : Spec l → U is a finite type point
bT,t → Fbt
lying over the finite type point t : Spec k → T , then the morphism Spf Obbb
factors through Zt provided that the morphism Spf OU,u → Fu factors through Zbu .
That this is true follows from Lemma 3.2.17 and the fact that the local étale (andbT,t → O
bU,u can be written as the inverse limit ofhence faithfully flat) morphism O
faithfully flat (and hence injective) morphisms of Artinian local rings with separable
residue field extensions (to which the cited lemma applies).
The following lemma gives a useful criterion for when a morphism T → F factors
through Z.
3.2.19. Lemma. Assume that either ξ is proper and F is Arttriv -homogeneous, orthat F satisfies [2](b). Let T be an algebraic stack over S, and let η : T → F bea morphism for which the base-changed morphism XT → T is scheme-theoreticallydominant. Assume further that either F satisfies [1], or that T is locally of finite
type over S. Then η factors through Z.
Proof. Since T is an algebraic stack, we may find a smooth surjection U → T with
U a scheme. Since the formation of scheme-theoretic images commutes with flatbase-change, we see that XU → U must be scheme-theoretically dominant. On theother hand, Lemma 3.2.18 shows that T → F factors through Z if and only if
U → T → F does. Thus we reduce to the case that T is a scheme T .
Since Z is a Zariski substack of F by Lemma 3.2.9, we can check the assertionof the lemma Zariski locally on T , and hence we may assume that T = Spec A isaffine. If T is locally of finite type over S, then we may assume that T is of finite
type over S. In the case that we are assuming instead that F satisfies [1], write
A = limi Ai as the inductive limit of its finitely generated OS -subalgebras. Then−→for some i there exists ηi : Spec Ai → F such that η is obtained as the compositeof ηi with Spec A → Spec Ai ; Remark 3.1.5 (4) then shows that XAi → Spec Ai
is scheme-theoretically dominant. Thus we may replace A by Ai , and hence again
assume that T is finite type over S.
Let t be a finite type point of T . Since XT → T is scheme-theoretically dominant,bT,t . By definition, then,the same is true of the (flat) base-change XObT ,t → Spec O
bT,t → T → F factors through Z.
b Since this holds forthe composite morphism Spf O
all finite type points t, the morphism T → F factors through Z by definition. 
Our next lemma spells out the basic properties satisfied by Z.
3.2.20. Lemma. Suppose either that ξ is proper and that F is Arttriv -homogeneous,or that F satisfies [2](b). Assume also either that F satisfies [1], or that X is locallyof finite presentation over S. Then the scheme-theoretic image Z forms a substackof F, and satisfies Axioms [1] and [3]. If F satisfies [2](a), then so does Z. If ξ isproper and F satisfies [1] and [2](b), then Z satisfies [2](b).

72

M. EMERTON AND T. GEE

Proof. As already noted, by its very definition, we see that Z satisfies [1]. It thenfollows by standard limit arguments that in order to show that Z is a stack on the
big étale site of S, it suffices to do so after restricting Z to the category of finite
type OS -algebras; more precisely, this follows easily from Lemma 2.5.5 (3) and [Sta,Tag 021E].
Thus, since F is a stack for the étale topology, and since Z is defined to be afull subcategory of F, in order to show that Z is a stack on the big étale site of S,
it suffices to verify that the property of a morphism η : T → F factoring throughZ (for an S-scheme T ) is étale local on T . This follows from Lemma 3.2.18.
Since Z (thought of as a category fibred in groupoids) is a full subcategory of F,
we see that the diagonal Z → Z ×S Z is the base-change under the morphism
Z ×S Z → F ×S F of the diagonal F → F ×S F. Thus, since F satisfies [3], thesame is true of Z.
Suppose now that F satisfies [2](a). To verify [2](a) for Z, suppose that we are
given a pushout diagram
/ Y0
Y

Z


/ Z0

of finite type Artinian S-schemes, with the horizontal arrows being closed immersions. By assumption, the induced functor
F(Z 0 ) → F(Y 0 ) ×F (Y ) F(Z)induces an equivalence of categories. Since Z is a substack of F, we see that theinduced functor
Z(Z 0 ) → Z(Y 0 ) ×Z(Y ) Z(Z)is fully faithful, and, furthermore, that in order to verify that it induces an equivalence of categories, it suffices to show that an element of F(Z 0 ) whose image underpull-back in F(Y 0 ) × F(Z) lies in Z(Y 0 ) × Z(Z) itself necessarily lies in Z(Z 0 ).
Suppose given such an element of F(Z 0 ). By hypothesis, we may find OS e each the spectrum of a complete Noetherian local OS -algebra
algebras Ye 0 and Z,e andwith finite type residue field, and closed immersions Y 0 ,→ Ye 0 and Z ,→ Z,00eemorphisms Y → F and Z → F inducing the given morphisms Y → F ande are schemeZ → F, such that the pulled-back morphisms XYe 0 → Ye 0 and XZe → Z
theoretically dominant. If ξ is proper, then we can and do assume that Ye 0 and Ze are`` eArtinian, and then Ye 0 Ze → F factors through Ze0 := Ye 0 Y Z
by our assumptionthat F satisfies [2](a). Otherwise, F satisfies [2](b) by assumption, and Lemma 2.7.4`e` eshows that the induced morphism Ye 0 Z→ F factors through Ze0 := Ye 0 Y Z.
`
e→Z
e0 is scheme-theoretically dominant, theSince the tautological morphism Ye 0 Z
e0 is scheme-theoretically dominant. Since the given
pulled-back morphism XZe0 → Z
`
0
0
0
e0
morphism Z := YY Z → F factors through the natural morphism Z → Z , by0definition the morphism Z → F does indeed factor through Z.
Finally, suppose that ξ is proper, and that F satisfies [1] and [2](b), and let
Spec Ax → F be an effective versal morphism at a finite type point x of Z (regarded as a finite type point of F). Let Spec Rx be the scheme-theoretic imageof XSpec Ax → Spec Ax ; then the morphism Spec Rx → F factors through Z, by

SCHEME-THEORETIC IMAGES OF MORPHISMS OF STACKS73

Lemma 3.2.19. Lemma 3.2.4, applied with B = Rx , then shows that the schemetheoretic image of XSpf Ax → Spf Ax is equal to Spf Rx , and it now follows from
Lemma 3.2.16 that the induced morphism Spf Rx → Z is versal at x.
The next lemma gives a refinement of Lemma 3.2.11.
ξ0

3.2.21. Lemma. Suppose that ξ admits a factorisation X −→ F 0 ,→ F, with F 0
also satisfying [3], and with the second arrow being a monomorphism. Suppose
furthermore that the monomorphism Z ,→ F factors through F 0 , and that either ξ
is proper and F is Arttriv -homogeneous, or that F satisfies [1] and [2](b). Thenthe scheme theoretic image of ξ 0 is equal to Z.
Proof. If we let Z 0 denote the scheme-theoretic image of ξ 0 , then Lemma 3.2.11
shows that Z 0 ,→ Z. We must prove the reverse inclusion; that is, we must showthat for any morphism T → Z, with T an affine scheme over S, the composite
T → Z ,→ F factors through Z 0 . By definition the morphism T → Z factors
as T → T 0 → Z, with T 0 of finite type over S, and so it is no loss of generality
to assume from the beginning that T is finite type over S. By assumption thecomposite T → Z ,→ F factors through F 0 , and now by definition of Z 0 , we seethat it suffices to check that this composite factors through Z 0 under the additional
hypothesis that T = Spec A for some finite type Artinian OS -algebra A.
Since the morphism T → F factors through Z, by definition there exists amorphism Spec A → Spec B → F as in Lemma 3.2.1 (1) such that XB → Spec B
is scheme-theoretically dominant. In the case that F does not satisfy [1], we areassuming that ξ is proper, so by Lemma 3.2.1 we can assume that B is Artinian,which by Remark 3.2.2 implies in particular that in this case, we can assume thatSpec B is of finite type over S. Lemma 3.2.19 then implies that Spec B → F factors
through Z, and so in particular through F 0 . It now follows from the definition that
T → F 0 factors through Z 0 , as required.

3.2.22. Lemma. Assume either that F satisfies [1] and [2](b), or that F is Arttriv homogeneous, ξ is proper, and either F satisfies [1], or X is locally of finite presentation over S. Then the morphism ξ : X → F factors through a morphismξ : X → Z, and the scheme-theoretic image of ξ is just Z.
Proof. Since the diagonal map X → X ×F X gives a section to the projection
X ×F X → X , it is immediate from Lemma 3.2.19 that ξ factors through Z. Thatthe scheme-theoretic image of ξ is Z is immediate from Lemma 3.2.21.

3.2.23. Lemma. Assume either that F satisfies [1], or that X is locally of finite
presentation over S. Suppose further that ξ is proper, and that F is Arttriv homogeneous. Then the morphism ξ is proper and surjective.
3.2.24. Remark. Note that Lemma 3.2.20 shows that Z is a stack satisfying [3],and so it makes sense (following Definition 2.3.4) to assert that ξ is proper andsurjective.
Proof of Lemma 3.2.23. Following Definition 2.3.4, we have to show that if Y isan algebraic stack, and if Y → Z is a morphism of stacks, then the base-changed
morphism of algebraic stacks Y ×Z X → Y is proper and surjective. Since bydefinition Z → F is a monomorphism, the fibre product over Z is isomorphic to74

M. EMERTON AND T. GEE

the fibre product Y ×F X , taken over F, and so the properness of ξ immediately
implies the properness of ξ.
To verify the surjectivity, we note that surjectivity can be checked after pullingback by a surjective morphism. Replacing Y by a cover of Y by a scheme, we may
assume that Y is in fact a scheme. Since we may also check surjectivity locally on
the target, we may in fact assume that Y is an affine scheme T . Since Z satisfies [1],we may factor the morphism T → Z through an affine S-scheme of finite type, andhence (since surjectivity is preserved under base-change) may further assume that
T is of finite type over S.
Now T ×Z X → T is a proper morphism of algebraic stacks, and so to check that
it is surjective, it suffices to check that its image contains each finite type pointof T . Thus it suffices to show that if t : Spec k → T is any finite type point, thenthe fibre of X over t is non-empty. Since ξ is proper, it follows from Lemma 3.2.14
that the fact that the composite t : Spec k → T → Z → F factors through Z
implies that the fibre Xt 6= ∅.

3.2.25. Lemma. If F 0 is a closed Zariski substack of F, and if the morphismξ : X → F factors through F 0 , then F 0 contains Z.
Conversely, assume either that F satisfies [1] and [2](b), or that ξ is proper, F
is Arttriv -homogeneous, and either X is locally of finite presentation over S, or
F satisfies [1]. Then if F 0 is a closed substack of F which contains Z, then themorphism ξ : X → F factors through F 0 .
Proof. Suppose that F 0 is a closed Zariski substack of F, and that ξ factors throughF 0 ; we will show that F 0 contains Z. Since Z is (by definition) a Zariski substackof F, it suffices to show that if T → Z is a morphism with T an affine S-scheme,
then the composite T → Z → F factors through F 0 . By definition, given such amorphism T → Z, we may find a finite type affine S-scheme T 0 and a factorisation
T → T 0 → Z; thus we may and do assume that T is of finite type over S.
In order to show that T → F factors through F 0 , it suffices to show that thebase-changed closed immersion T ×F F 0 ,→ T is an isomorphism. Since T is of finite
type over S, for this it suffices to show that if t ∈ T is a finite type point, then for
bT,t /mn → T factors through T ×F F 0 , or
any n ≥ 0, the induced morphism Spec Otbequivalently, that the composite Spec OT,t /mnt → T → F factors through F 0 . Thus
we are reduced to the case when T = Spec A, with A an Artinian OS -algebra offinite type.
By the definition of Z, we may find a complete Noetherian local OS -algebra
B, a morphism Spec A → Spec B over S, and a morphism Spec B → F inducing the given morphism Spec A → F, such that the base-changed morphism
XB → Spec B is scheme-theoretically dominant. Since ξ factors through F 0 by
assumption, the morphism XB → Spec B factors through the closed immersion
Spec B ×F F 0 ,→ Spec B. Since the former morphism is also scheme-theoreticallydominant, we see that this closed immersion is necessarily an isomorphism, andhence that the morphism Spec B → F factors through F 0 . Thus the morphism
Spec A → F also factors through F 0 , and we are done.
For the converse, we note that, under the additional hypotheses, the morphism
ξ factors through a morphism ξ : X → Z, by Lemma 3.2.22; so if F 0 contains Z,
we see that ξ factors through F 0 , as required.
SCHEME-THEORETIC IMAGES OF MORPHISMS OF STACKS

75

Under the assumption that ξ is proper, we may strengthen Lemma 3.2.25 asfollows.
3.2.26. Lemma. Assume either that F satisfies [1], or that X is locally of finite presentation over S. Assume also that ξ is proper and that F is Arttriv -homogeneous.
If F 0 is a substack of F, and if the monomorphism F 0 ,→ F is representable by algebraic spaces and of finite type, then the morphism ξ : X → F factors through F 0
if and only if F 0 contains Z.
Proof. The “if” direction was proved in Lemma 3.2.25. For the “only if” direction, set Z 0 := Z ×F F 0 . We begin by showing that the finite type monomorphism
Z 0 → Z is a closed immersion; it suffices to show that it is proper, and sincemonomorphisms are automatically separated, and since it is finite type by assumption, it in fact suffices to show that it is universally closed.
Firstly, note that since X → Z is proper and surjective by Lemma 3.2.23, thecomposite |X | → |Z 0 | → |Z| is surjective, while |Z 0 | → |Z| is injective. Thus |X | →|Z 0 | is surjective. Similarly, |X | → |Z| is closed, and a trivial topological argument
then shows that |Z 0 | → |Z| is closed. Since both properness and surjectivity arepreserved by arbitrary base changes, we conclude that Z 0 → Z is universally closed,
as claimed, and thus a closed immersion.
By Lemma 3.2.23, the morphism ξ : X → Z is proper and surjective; but it also
factors through the closed substack Z 0 of Z, so we must have Z 0 = Z. Thus F 0
contains Z, as required.

3.2.27. Remark. If F is an algebraic stack that is locally of finite presentation over S,
then in particular it satisfies Axioms [1], [2], and [3] (by Theorem 2.8.4), and so allof the previous results apply.
3.2.28. Remark. Note that at this point in the development of the theory, we do notknow that the monomorphism Z → F is representable by algebraic spaces, whichmeans for example that it is hard to use Lemma 3.2.26 to uniquely characterise Z
by a universal property. However, once we have shown that Z is an algebraic stack,
it is immediate from the assumption that F satisfies [3] that the monomorphism
Z → F is representable by algebraic spaces, and Proposition 3.2.33 below can thenbe applied to deduce that Z is in fact a closed substack of F.
3.2.29. Lemma. If F is an algebraic stack, locally of finite presentation over S, thenZ coincides with the scheme-theoretic image of ξ (in the sense of Definition 3.1.4).
Proof. Let Y ,→ F denote the scheme-theoretic image of ξ. Since Y is a closed
substack of F, and since the morphism ξ factors through Y, Lemma 3.2.25 showsthat Y contains Z. On the other hand, since Y → F is a monomorphism and ξ
factors through Y, we see that the projection XY := X ×F Y → Y is naturally
identified with the canonical morphism X → Y, and so in particular is schemetheoretically dominant. Lemma 3.2.19 then implies that Y is contained in Z. 
3.2.30. Remark. As noted in Remark 3.2.10 (3), the conclusion of this lemma needn’t
hold if F doesn’t satisfy [1].
The following result will be useful in Section 3.3.
3.2.31. Proposition. Suppose that there is a closed substack Y of F such that Y
is an algebraic stack, locally of finite presentation over S, and that the morphism

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M. EMERTON AND T. GEE

X → F factors as a composite X → Y → F. Then Z coincides with the schemetheoretic image of X in Y (in the sense of Definition 3.1.4).
Proof. Replacing Y by the scheme-theoretic image of X → Y, we may assume thatit coincides with this scheme-theoretic image; we must then show that Y equals Z.
By Lemma 3.2.25, we see that Y contains Z. To show the reverse inclusion, wenote first that Lemma 3.2.29 implies that Y is the scheme-theoretic image of themorphism X → Y; the desired inclusion now follows by Lemma 3.2.11.
The following lemma will be used to help prove openness of versality for Z.
3.2.32. Lemma. Assume that F satisfies [1], that ξ is proper, and that F is Arttriv homogeneous. If Y is an algebraic stack, locally of finite type over S, and if Y → Z
is a morphism over S which is formally smooth at a finite type point y ∈ |Y|, thenthere exists an open neighbourhood U of y in Y such that the induced morphism
XU → U is scheme-theoretically dominant (where XU denotes the base-change ofξ : X → Z via the composite U → Y → Z; or equivalently, the base-change ofξ : X → F via the composite U → Y → Z → F).
Proof. By definition (see Definition 2.4.10) we may find a smooth morphism V → Y
from a scheme V to Y, and a finite type point v ∈ V mapping to y, such that thebV,v , let XB → Spec B
composite V → Y → Z is formally smooth at v. Write B := O
denote the base-change of ξ : X → Z via the composite Spec B → V → Y → Z (or,equivalently, the base-change of ξ : X → F via the composite Spec B → V → Y →Z → F), and let Spec B/I denote the scheme-theoretic image of this base-change.
Let A be any Artinian quotient of B. We will show that the surjection B → A
necessarily contains I in its kernel. This will show that I = 0 (since A was arbitrary)and hence that XB → Spec B is scheme-theoretically dominant. From this one
sees that there is a neighbourhood U of v in V such that XU → U is schemetheoretically dominant (here XU has the evident meaning). Letting U denote theimage of U in Y (an open substack of Y), we see that XU → U is scheme-theoreticallydominant, as required. (Scheme-theoretic dominance can be checked fpqc locally,
and in particular smooth locally, on the target.)
The composite Spec A → Spec B → V → Y → Z → F factors through Z
by construction, and so by definition we may find a complete Noetherian local OS algebra C with finite type residue field, a closed immersion of OS -schemes Spec A →Spec C, and a morphism Spec C → F inducing the above morphism Spec A →F, such that the base-changed morphism XC → Spec C is scheme-theoreticallysurjective. The morphism Spec C → F factors through Z, by Lemma 3.2.19. Since
V → Z is formally smooth at v, we may lift the morphism Spec C → Z to amorphism Spec C → Spec B, extending the composite morphism Spec A → Spec B.
Since XC → Spec C is surjective, we conclude that the morphism Spec A → Spec B
factors through Spec B/I, as required.

3.2.33. Proposition. Assume either that F satisfies [1], or that X is locally of finite
presentation over S. Assume also that ξ is proper, that F is Arttriv -homogeneous,and that the monomorphism Z → F is representable by algebraic spaces. Then Z
is a closed substack of F.
Proof. We must show (under either set of finiteness assumptions on X and F) that
Z → F is a closed immersion. Since it is a monomorphism, it suffices to show

SCHEME-THEORETIC IMAGES OF MORPHISMS OF STACKS

77

that it is proper. As monomorphisms are separated, and as it is locally of finitely
presentation (this follows from Lemma 2.1.7, applied to the composite Z ,→ F → S,
together with Lemma 2.1.5 and [Sta, Tag 06CX]) it suffices to show that it isuniversally closed and quasi-compact. These are properties that (by definition) can
be checked after pulling back by a morphism W → F from a scheme.
Pulling back the morphisms X → Z → F via such a morphism, we obtainmorphisms XW → ZW → W with the composite being proper, and the first being
surjective (by Lemma 3.2.23). We must show that ZW → W is closed, and that,
if W is quasi-compact, the same is true of ZW . These are properties that canbe checked on the underlying topological spaces. So we consider the continuous
morphisms |XW | → |ZW | → |W |, with the first being surjective (by [Sta, Tag 04XI])
and the composite being closed. Furthermore, if |W | is quasi-compact, the sameis true of |XW |. It follows immediately that the second arrow is closed, and that
|ZW | is quasi-compact if |W | is. This completes the proof of the proposition.

We will now prove Theorem 1.1.1, as well as a variant where we assume that F
satisfies [1], but make no finiteness assumption on X . (In fact, as remarked above,
we will reduce Theorem 1.1.1 to this case.)
3.2.34. Theorem. Let S be a locally Noetherian scheme, all of whose local rings
OS,s at finite type points s ∈ S are G-rings. Suppose that ξ : X → F is a propermorphism, where X is an algebraic stack and F is a stack over S satisfying [3],and that F admits versal rings at all finite type points. Assume also either that
F satisfies [1], or that X is locally of finite presentation over S. Let Z denote thescheme-theoretic image of ξ as in Definition 3.2.6, and suppose that Z satisfies [2].
Then Z is an algebraic stack, locally of finite presentation over S; the inclusion
Z ,→ F is a closed immersion; and the morphism ξ factors through a proper,
scheme-theoretically surjective morphism ξ : X → Z. Furthermore, if F 0 is a
substack of F for which the monomorphism F 0 ,→ F is of finite type (e.g. a closed
substack) with the property that ξ factors through F 0 , then F 0 contains Z.
Proof. Note firstly that by Lemma 2.6.2, the diagonal of F is locally of finite presentation, so it follows from Corollary 2.7.3 that F is Arttriv -homogeneous. Taking into account Lemmas 3.2.22, 3.2.23 and 3.2.26, Remark 3.2.28, and Proposition 3.2.33, we see that we only need to prove that Z is an algebraic stack, locally offinite presentation over S. By Theorem 2.8.4, it is enough to show that it is an étale
stack over S, and satisfies Axioms [1], [2], [3] and [4]. It follows from Lemma 3.2.20
that Z is an étale stack and satisfies Axioms [1] and [3], and we are assuming that it
satisfies Axiom [2]. It remains to show that Z also satisfies Axiom [4], i.e. opennessof versality. We will assume from now on that we are in the case that F satisfies [1],and explain at the end how to reduce the other case to this situation.
Consider an S-morphism T → Z, where T is a locally finite type S-scheme,
and let t ∈ T be a finite type point at which this morphism is versal. In fact thismorphism is then formally smooth at t, by Lemma 2.4.7 (2). Lemma 2.6.4 allows
us, shrinking T if necessary, to factor the morphism T → Z as
T → Z 0 → Z,
where Z 0 is an algebraic stack, locally of finite presentation over S, the first arrowis a smooth surjection, and the second arrow is locally of finite presentation, unramified, and representable by algebraic spaces, and formally smooth at the imaget0 ∈ |Z 0 | of t.

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Consider the following diagram, both squares in which are defined to be Cartesian:(3.2.35)

T ×Z 0 X 0

/ X0

/X


T


/ Z0


/Z

The vertical arrows are proper, and hence closed, the first of the horizontal arrows are smooth surjections, and the second of the horizontal arrows are unramified, locally of finite presentation, and representable by algebraic spaces. By
Lemma 2.4.12 (1), there is an open substack U of X 0 , containing the fibre over t0 ,such that the induced morphism U → X is smooth.
Since X 0 → Z 0 is closed, the complement U 0 of the image in Z 0 of the complementin X 0 of U is an open substack of Z 0 , containing t0 , whose preimage in X 0 maps
smoothly to X . Thus, replacing Z 0 by U 0 , and T by its preimage, we may further
assume that X 0 → X is smooth, and hence (being also unramified) étale.
Let t1 ∈ T be another finite type point. We will show that the morphism T → Z
is formally smooth (and so also versal) at t1 . This will establish [4] for Z. Let t01
denote the image of t1 in Z 0 . Since T → Z 0 is smooth, it suffices to show thatZ 0 → Z is formally smooth at t01 .
Let t1 denote the image of t1 in Z. By Lemma 2.8.1, we may find an S-scheme T1 ,locally of finite type, a morphism T1 → Z, and a point t̃1 ∈ T1 mapping to t1 , suchthat the morphism T1 → Z is formally smooth at t̃1 . We apply Lemma 2.6.4
once more, to obtain a factorisation T1 → Z 00 → Z (possibly after shrinking T1
around t̃1 ), where Z 00 is an algebraic stack, locally of finite presentation over S,
the first arrow is a smooth surjection, and the second arrow is locally of finite
presentation, unramified, representable by algebraic spaces, and formally smoothat the image t00 ∈ |Z 00 | of t̃1 . If we let X 00 denote the base-change of ξ : X → Z
over Z 00 , then Lemma 3.2.32 shows that, shrinking Z 00 around t00 if necessary, we
may assume that the morphism X 00 → Z 00 is scheme-theoretically dominant, andthus scheme-theoretically surjective (being a base-change of the morphism ξ, which
is surjective by Lemma 3.2.23).
Now consider the Cartesian squareZ 0 ×Z Z 00

/ Z 00
Z0/Z

Applying Lemma 2.4.12, we find that the morphism Z 0 ×F Z 00 → Z 0 contains thefibre over t00 in its smooth locus. Since t01 and t00 map to the same point of |F|, thisfibre contains a point t000 lying over t01 and t00 , and so we may find an open substack
Z 000 ⊆ Z 0 ×F Z 00 , such that t000 ∈ |Z 000 |, and such that the morphism Z 000 → Z 0 issmooth, and hence (being also unramified) étale.
In summary, we have a diagram
Z 000/ Z 00
Z0/ZSCHEME-THEORETIC IMAGES OF MORPHISMS OF STACKS

79

in which the horizontal arrows are locally of finite presentation and unramified, theleft-hand vertical arrow is étale, the right-hand vertical arrow is formally smooth at
t00 , and for which there is a point t000 ∈ |Z 000 | lying over t01 ∈ |Z 0 | and t00 ∈ |Z 00 |. Our
goal is to show that the lower horizontal arrow is formally smooth at the point t01 .
For this, it suffices (by Remark 2.4.11) to show that the upper horizontal arrow issmooth (or equivalently, étale, since it is unramified).
Thus we turn to showing that Z 000 → Z 00 is étale. Form the Cartesian diagramof algebraic stacks/ X 00X 000
/ Z 00Z 000
Regarding the top arrow as being the composition of an open immersion with thebase-change of the top arrow of (3.2.35), we see that it is étale. Recall that theright-hand vertical arrow X 00 → Z 00 is scheme-theoretically surjective. Both verticalarrows are proper, thus universally closed, and the bottom arrow is locally of finite
presentation and unramified. Lemma 3.2.36 below thus implies that the bottom
arrow is étale, and so Axiom [4] for Z is proved.
We have therefore shown that if F satisfies [1], then Z is an algebraic stack,locally of finite presentation over S. Assume now that we are in the case that X
is locally of finite presentation over S. Recall that by Lemma 2.6.2, the diagonalof F is locally of finite presentation. By Remark 3.2.13, ξ factors as
ξ0

X → pro -(F|Aff pf/S ) ,→ F,and pro -(F|Aff pf/S ) is an étale stack which satisfies [1] and [3] and admits versal
rings at all finite type points; and the scheme-theoretic image of ξ 0 is Z. Regardingξ 0 as the pull-back of ξ along the embedding pro -(F|Aff pf/S ) ,→ F, we see that ξ 0 is
proper, and therefore satisfies the assumptions of the Theorem; so it follows fromthe case already proved that Z is an algebraic stack, locally of finite presentation
over S.

Although we have stated and proved the following lemma in what seems to be
its natural level of generality, the only application of it that we make is in the casewhen the morphism Z 0 → Z is in fact representable by algebraic spaces.
3.2.36. Lemma. Let Y → Z be a quasi-compact morphism of algebraic stacks
that is scheme-theoretically surjective and universally closed, and let Z 0 → Z be amorphism of algebraic stacks that is locally of finite presentation and unramified.
If the base-changed morphism Y 0 → Y is étale, then Z 0 → Z is also étale.
Proof. Since Z 0 → Z is unramified, its diagonal is étale and in particular unramified, so Z 0 → Z is a DM morphism in the sense of [Sta, Tag 04YW]. Choose ascheme mapping Z via a smooth surjection to Z, and pull everything back over Z;
in this way we reduce to the case Z = Z. As explained in [Sta, Tag 04YW], it
now follows from [Sta, Tag 06N3] that there is a scheme Z 0 and a surjective étalemorphism Z 0 → Z 0 .5
5As noted above, the only application of this lemma that we will make is in the case whenZ 0 → Z is representable by algebraic spaces. In this case, the pull-back of the algebraic stack Z 0
over the scheme Z is already an algebraic space, and so by definition admits an étale surjection

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Replacing Z 0 by Z 0 , we have reduced to the case that the morphism Z 0 → Z is
the morphism of schemes Z 0 → Z.
We must show that Z 0 → Z is étale at each point z 0 ∈ Z 0 . Fix such a point, withimage z ∈ Z. Then we may replace Z by its base change to the strict henselisationshSpec OZ,z, and thus assume that Z is a local strictly henselian scheme with closed
point z. Then the étale local structure of unramified morphisms [Sta, Tag 04HH]shows that after shrinking Z 0 around z 0 , we can arrange that Z 0 → Z is a closed
immersion, so that in particular Z 0 is also local with closed point z 0 = z. ThenZ 0 → Z is étale if and only if Z 0 = Z.
Since Z 0 → Z is a closed immersion, so is Y 0 → Y. Since Y 0 → Y is étale byassumption, it is an étale monomorphism, and therefore it is an open (as well as
closed) immersion. The complement in Y of the image of Y 0 → Y is therefore
closed, and has closed image in Z, as Y → Z is closed by assumption; but thisimage has empty special fibre, and so this complement is empty, and Y 0 = Y. It
now follows from Lemma 3.1.7 that Z 0 = Z, as required.

3.2.37. Remark. In the statement of Lemma 3.2.36, it would not suffice to assume
that Y → Z is merely scheme-theoretically surjective. For example, if Z = Z istaken to be the scheme given as the union of two lines Z1 and Z2 crossing at the
point y in`
the plane, if Y = Y is taken to be the scheme obtained as the disjoint
union Z1
Z2 \ {y}, if Y → Z is taken to be the obvious morphism (which isscheme-theoretically surjective), and if Z 0 = Z 0 is taken to be Z1 , then the closed
immersion Z 0 → Z is unramified, and the base-changed morphism Y 0 → Y is an
open immersion, and so in particular étale. On the other hand, the closed immersion
Z 0 → Z is certainly not étale.
3.3. Base change. In this section we study the behaviour of our scheme-theoretic
images under base change by a morphism which is representable by algebraic spaces.
We will only need to consider cases where we know (in applications, as a consequenceof Theorem 1.1.1) that the scheme-theoretic image of the morphism being basechanged is an algebraic stack, and we have therefore restricted to this case, andhave not investigated the compatibility with base change in more general situations.
We first of all consider the question of base change of the target stack F.
3.3.1. Proposition. Suppose that we have a commutative diagram of étale stacks
in groupoids over a locally Noetherian base scheme S
X0

/X


F0


/F

(which is not assumed to be Cartesian), in which X , X 0 are algebraic stacks, F 0 → F
is representable by algebraic spaces and locally of finite presentation, and ∆F isrepresentable by algebraic spaces and locally of finite presentation. Write Z and Z 0
for the scheme-theoretic images of X → F and X 0 → F 0 respectively, and assume
further that Z is an algebraic stack, that Z is a closed substack of F, and thatthe morphism X → F factors through a scheme-theoretically dominant morphism
X → Z.
from a scheme. Thus, in this case, we can avoid appealing to the theory of DM morphisms andtheir relationship to Deligne–Mumford stacks.

SCHEME-THEORETIC IMAGES OF MORPHISMS OF STACKS

81

Then Z 0 is an algebraic stack, locally of finite presentation over S. In fact,
the morphism X 0 → F 0 factors through the algebraic stack Z ×F F 0 (which is inturn a closed substack of F 0 ), and Z 0 is the scheme-theoretic image of the induced
morphism of algebraic stacks X 0 → Z ×F F 0 .
Proof. Since X → F factors through Z by assumption, the composite X 0 → X → F
factors through Z, and so X 0 → F 0 factors through the algebraic stack Z ×F F 0 ,which is a closed substack of F 0 . By Proposition 2.3.20, ∆F 0 is representable byalgebraic spaces and locally of finite presentation. The result then follows from
Proposition 3.2.31.

3.3.2. Corollary. Suppose that we have a commutative diagram of étale stacks ingroupoids over a locally Noetherian base scheme S
X0

/X


F0


/F

(which is not assumed to be Cartesian), in which X , X 0 are algebraic stacks, F 0 → F
is representable by algebraic spaces and locally of finite presentation, and ∆F isrepresentable by algebraic spaces and locally of finite presentation.
Suppose that X → F is proper, and that X is locally of finite presentation over S.
Write Z, Z 0 for the scheme-theoretic images of X → F, X 0 → F 0 respectively
Suppose that Z satisfies [2], and that F admits versal rings at all finite type points.
Then Z and Z 0 are both algebraic stacks, locally of finite presentation over S. In
fact, the morphism X 0 → F 0 factors through the algebraic stack Z ×F F 0 (whichis in turn a closed substack of F 0 ), and Z 0 is the scheme-theoretic image of theinduced morphism of algebraic stacks X 0 → Z ×F F 0 .
If x is a finite type point of Z 0 , let Rx be a versal ring for the corresponding finite
type point of F, and write Spf Sx for the complete local ring at x of Spf Rx ×F F 0 ,in the sense of Definition 4.2.13 below. Then a versal ring for Z 0 at x is given by0
the scheme-theoretic image of the induced morphism XSpf
Sx → Spf Sx .
Proof. Everything except for the claim about versal rings is immediate from Theorem 1.1.1 and Proposition 3.3.1. The description of the versal rings follows from
Lemmas 2.2.18 and 3.2.16.

We now consider base changes S 0 → S. It is of course unreasonable to expectthat the formation of scheme-theoretic images is compatible with such base changes
unless we make a flatness hypothesis; but under this condition, we are able to prove
the following general result.
3.3.3. Proposition. Suppose that X → F is a morphism of étale stacks over the
locally Noetherian base scheme S, where X is algebraic and ∆F is representableby algebraic spaces. Suppose that the scheme-theoretic image Z of X → F is analgebraic stack, that Z → F is a closed immersion, and that X → F factors througha scheme-theoretically dominant morphism X → Z.
Let S 0 → S be a flat morphism of locally Noetherian schemes, and write XS 0 ,FS 0 , and ZS 0 for the base changes of X , F, and Z to S 0 . Write Z 0 for the schemetheoretic image of XS 0 → FS 0 .

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Then Z 0 = ZS 0 ; so Z 0 is an algebraic stack, Z 0 → FS 0 is a closed immersion, and
XS 0 → FS 0 factors through a scheme-theoretically dominant morphism X → Z 0 .
Proof. Note firstly that ZS 0 is an algebraic stack, that ZS 0 → FS 0 is a closed
immersion, and that XS 0 → FS 0 factors through ZS 0 . Since S 0 → S is flat, we seethat XS 0 → ZS 0 is scheme-theoretically dominant by Remark 3.1.5 (2). Since theformation of the diagonal is compatible with base change, ∆FS0 is representable byalgebraic spaces and locally of finite presentation. The result follows immediatelyfrom Proposition 3.2.31.

4. Examples
4.1. Quotients of varieties by proper equivalence relations. We indicate
some simple examples of quotients of varieties (over C, so that we may form thequotients in the sense of topological spaces) by proper equivalence relations which
are not algebraic objects.
4.1.1. Example. Consider the equivalence relation on P2 which contracts a line to apoint. In the topological category, the quotient of P2 (C) by this equivalence relation
is S 4 (a 4-sphere), which is not a Kähler manifold (since it has vanishing H 2 ).
Thinking more algebraically, one can show that there is no proper morphism ofalgebraic spaces P2 → X for which P2 ×X P2 coincides with this equivalence relation;indeed, if there were such a morphism, then the theorem on formal functions would
show that the complete local ring of X at the point obtained as the image of thecontracted P1 is equal to the ring of global sections of the structure sheaf of theformal completion of P2 along P1 ; but since the conormal bundle of P1 in P2 equals
O(−1), this ring of global sections is just equal to C, and so cannot arise as thecomplete local ring at a point of a two-dimensional algebraic space. Thus there
isn’t any reasonable way to take the quotient of P2 by this equivalence relation in
the world of algebraic spaces over C.
4.1.2. Example. Consider the space X of endomorphisms of P1 of degree ≤ 1. Suchax + ban endomorphism is described by a linear fractional transformation x 7→
cx + da b
3for which the matrix
is non-zero, and so X is a quotient of P (thoughtc dof as the projectivisation of the vector space M2 (C) of 2 × 2 matrices). However,the locus of singular matrices (i.e. the subvariety cut out by the vanishing of thedeterminant), which gives rise to the constant endomorphisms, is two-dimensional(indeed a quadric surface in P3 ), while the space of constant endomorphisms is
obviously just equal to P1 . Thus X can be thought of as a quotient of P3 in which aquadric is contracted to a projective line (via one of the projections P1 × P1 → P1 ).
Similarly to the preceding example, the space X can’t be realised in the world ofalgebraic spaces. (The first author would like to thank V. Drinfeld for explaining
this example to him.)
4.1.3. Remark. It follows from [Art70] that in contexts like those considered in
the preceding examples, in which we wish to contract a closed subvariety of aproper variety in some manner, if we have a formal model for neighbourhood ofa contraction, then we can perform the contraction in the category of algebraic
spaces. The computation with complete local rings on the (hypothetical) quotient

SCHEME-THEORETIC IMAGES OF MORPHISMS OF STACKS

83

in Example 4.1.1 can be thought of as showing that, for this example, such a formal
model doesn’t exist.
A key intuition behind the main theorem of the present paper is that, as one of
the hypotheses of the theorem, we do assume that we have well-behaved complete
local rings at each finite type point of the quotient we are trying to analyse.
4.2. Ind-algebraic stacks. Ind-algebraic stacks give simple examples of stacks
which satisfy some but not all of Artin’s axioms, and which admit surjective morphisms from algebraic stacks. We refer the reader to [Eme] for the details of thetheory of Ind-algebraic stacks, contenting ourselves here with recalling some general facts, and giving various examples (several of which are taken from [Art69b])
which illustrate the roles of the various axioms in the results of Section 2. Example 4.3.12 also illustrates the necessity of the properness hypothesis in the statementof Theorem 1.1.1, while Example 4.3.7 illustrates the necessity of the hypothesisof scheme-theoretic dominance (rather than mere surjectivity) in the statement ofCorollary 1.1.2.
4.2.1. Definition. We say that a stack X over S is an Ind-algebraic stack if wemay find a directed system of algebraic stacks {Xi }i∈I over S and an isomorphism∼
limi∈I Xi −→ X , the inductive limit being computed in the 2-category of stacks.
−→
If X is a stack in setoids (i.e. is equivalent to a sheaf of sets), then we say that X
is an Ind-algebraic space, resp. an Ind-scheme, if the Xi can in fact be taken to bealgebraic spaces, resp. schemes.
4.2.2. Remark. More properly, the inductive limit is a 2-colimit, but as in [Eme],we find it more suggestive to use the usual notation for direct limits.
4.2.3. Remark. Our definitions of Ind-algebraic stacks, Ind-algebraic spaces, andInd-schemes are broader than usual. Indeed, at least in the case of Ind-schemes, it
is conventional to require the transition morphisms Xi → Xi0 to be closed immersions. We have adopted a laxer definition simply because it provides a convenient
framework in which to discuss many of the examples of Subsection 4.3 below.
4.2.4. Remark. Given a directed system {Xi }i∈I as in Definition 4.2.1, then for any
S-scheme T there is morphism of groupoids(4.2.5)
lim Xi (T ) → lim Xi (T ).
−→
−→
i∈I

i∈I

Although this is not an equivalence in general, it is an equivalence if T is quasicompact and quasi-separated (e.g. if T is an affine S-scheme). We record this fact
in the following lemma (which is presumably well-known); see e.g. [Sta, Tag 0738]
for the analogous statement in the context of sheaves.
Also, although in the above definition all stacks involved are (following our conventions) understood to be stacks for the étale site, in the statement of the lemma
we consider stacks for other topologies as well.
4.2.6. Lemma. Let {Xi }i∈I be an inductive system of Zariski, étale, fppf, or
fpqc stacks, and consider the inductive limit limi∈ Xi , computed as a stack for the−→
topology under consideration. If T is an quasi-compact S-scheme, then the morphism (4.2.5) is faithful. If, in addition, either T is quasi-separated, or the transition
maps in the inductive system are monomorphisms, then it is in fact an equivalenceof groupoids.

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Proof. We write X := limi∈I Xi (the inductive limit being taken as stacks). If−→
ξi , ηi are a pair of objects in Xi (T ), inducing objects ξi0 , ηi0 in Xi0 (T ) for eachi0 ≥ i, and objects ξ, η in X (T ), then the set of morphisms between ξi0 and ηi0 is
equal to the space of global sections of the sheaf Xi0 ×∆,Xi0 ×S Xi0 ,ξi0 ×ηi0 T , while
the set of morphisms between ξ and η is equal to the space of global sections of
X ×∆,X ×S X ,ξ×η T, which is also equal to the set of global sections of the inductive
limit of sheaves limi0 ≥i Xi0 ×∆,Xi0 ×S Xi0 ,ξi0 ×ηi0 T.
−→
If T is quasi-compact and quasi-separated, then the inductive limit of global
sections maps isomorphically to the global sections of the inductive limit [Sta, Tag0738], and so we find that the morphism (4.2.5) is fully faithful for such T . The
same reference shows that if T is merely assumed to be quasi-compact, then themorphism on global sections is injective, and hence that the morphism (4.2.5) isfaithful. Finally, if the transition morphisms in the inductive system of sheaves
under consideration are injective, then this reference again shows that the map
on global sections is an isomorphism provided merely that T is quasi-compact. If
the transition morphisms Xi0 → Xi00 are monomorphisms, then the correspondingtransition morphisms Xi0 ×∆,Xi0 ×S Xi0 ,ξi0 ×ηi0 T → Xi00 ×∆,Xi00 ×S Xi00 ,ξi00 ×ηi00 T are
indeed injective, and hence in this case the morphism (4.2.5) is again fully faithful.
We turn to considering the essential surjectivity of (4.2.5). If T is quasi-compact,
then for any object of X (T ), we may find a cover T 0 → T (in the appropriate
topology: Zariski, étale, fppf, or fpqc, as the case may be) such that T 0 is again
quasi-compact, and a morphism T 0 → Xi for some i such that the diagram
T0

/ Xi


T


/X

commutes. We obtain an induced morphism T 0 ×T T 0 → Xi ×X Xi . The target of
this morphism may be written as limi0 ≥i Xi ×Xi0 Xi . If T is also quasi-separated,−→so that T 0 ×T T 0 is again quasi-compact, then by what we have already observed,
this morphism arises from a morphism T 0 ×T T 0 → Xi ×Xi0 Xi for some sufficiently
large value of i0 . Consequently, we obtain a factorisation of the induced morphism
T 0 ×T T 0 → Xi0 ×S Xi0 through the diagonal. When we pass from Xi0 to X , we obtain
the descent data to T of the composite T 0 → T → X , and so the faithfulness result
already proved shows that, if we enlarge i0 sufficiently, this factorisation provides
descent data to T for the morphism T 0 → Xi0 . Since Xi0 is a stack, we obtain amorphism T → Xi0 inducing the original morphism T → X .
If the transition morphisms in the inductive system {Xi } are monomorphisms,
then the various fibre products Xi ×Xi0 Xi are all isomorphic to Xi (via the diagonal),
and hence the natural map Xi → limi0 ≥i Xi ×Xi0 Xi is an isomorphism. Thus in−→this case, the morphism T 0 ×T T 0 → Xi ×X Xi necessarily arises from a morphism
T 0 ×T T 0 → Xi , without any quasi-compactness assumption on T 0 ×T T 0 . Thus, inthis case, we obtain the essential surjectivity of (4.2.5) under the assumption that
T is merely quasi-compact.

4.2.7. Remark. The preceding lemma (in particular, the factthat the isomorphism (4.2.5) provides an explicit description of limi∈I Xi (T ) which makes no−→reference to the site over which the inductive limit is computed, when T is an affine

SCHEME-THEORETIC IMAGES OF MORPHISMS OF STACKS

85

S-scheme), shows that to form the stack limi∈I Xi , for an inductive systems of stacks−→
in the étale, fppf, or fpqc topologies, it in fact suffices to form the corresponding
inductive limit as a stack for the Zariski topology.
4.2.8. Remark. If {Xi }i∈I is an inductive system of stacks (for the étale topology),
and if Y is a category fibred in groupoids over S, then, analogously to (4.2.5), we
have a natural morphism of groupoids
lim Mor(Y, Xi ) → Mor(Y, lim Xi ).
−→
−→
i∈I

i∈I

In particular if Y is an algebraic stack, then the obvious extension of Lemma 4.2.6
holds: namely, if Y is quasi-compact then this morphism is faithful, and if furthermore either Y is quasi-separated, or the transition morphisms in the inductive
system are monomorphisms, then this morphism is an equivalence. (This can bechecked by choosing a smooth surjection T → Y from an S-scheme to Y, whose
domain can be taken to be affine if Y is quasi-compact, and using the fact thatmorphisms from Y to any stack on the étale site can be identified with morphisms
from T to the same stack with appropriate descent data; we leave the details to thereader.)4.2.9. Remark. We briefly discuss the comportment of Ind-algebraic stacks withregard to Artin’s axioms. Suppose that X := limi∈I Xi is an Ind-algebraic stack.
−→(1) If each Xi satisfies [1], then it follows from Remark 4.2.4 that X also satisfies [1].
(2) It follows from Remark 4.2.4 that X satisfies [2](a), since this is true ofeach Xi . The question of whether or not a particular Ind-stack satisfies [2](b) is
more involved, and we discuss it in some detail for Ind-algebraic spaces in 4.2.10
below.
(3) Typically, X need not satisfy [3]. If T is a quasi-compact and quasi-separated
S-scheme, then any morphism T → X ×S X factors through a morphism T →Xi ×S Xi for some i ∈ I, and thus X ×X ×S X T is the base-change with respect tothis morphism of
X ×X ×S X (Xi ×S Xi ) = Xi ×X Xi = lim Xi ×Xi0 Xi
−
→
0
i ≥i

(equipped with its natural morphism to Xi ×S Xi ); consequently the diagonal of
X is representable by Ind-algebraic spaces. If each of the transition morphisms
Xi → Xi0 is a monomorphism, then so is the morphism Xi → X , and we find that∼Xi −→ Xi ×X Xi . Thus in this case we find that X does satisfy [3]. One cangive examples in which the transition morphisms are not monomorphisms, but for
which nevertheless the diagonal of X is representable by algebraic spaces; see e.g.
Examples 4.3.4 and 4.3.12 below. However, certainly in general the diagonal of X is
not representable by algebraic spaces, even for examples which seem tame in many
respects; see e.g. Example 4.3.5.
(4) Typically an Ind-algebraic stack will not satisfy [4]. However, since algebraic
stacks satisfy [4a] (Lemma 2.4.15), we see that Ind-algebraic stacks will satisfy [4a].
Typically they will not satisfy [4b], though. In some cases, however, they may
satisfy both [4a] and [4b], but not [4]; Example 4.3.4 illustrates this possibility(and demonstrates that the quasi-separatedness hypotheses in Corollary 2.6.11 andTheorem 2.8.5 are necessary).

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4.2.10. Versal rings for Ind-algebraic spaces. If x is a finite type point of the Indalgebraic stack X := limi∈I Xi , then x arises from a finite type point xi of Xi for−→some i ∈ I, and if we write xi0 to denote the corresponding finite type point of Xi0 ,for each i0 ≥ i, then one can attempt to construct a versal deformation ring at x
by taking a limit over the versal rings of each xi0 . This limit process is complicatedin general by the non-canonical nature of the versal deformation ring of a finite
type point in a stack, and so we will restrict our precise discussion to the case of
Ind-algebraic spaces, for which we can find canonically defined minimal versal rings.
To be precise, suppose that S is locally Noetherian. Note that an algebraic
space X over S satisfies [1] if and only if it is locally of finite presentation over S(Lemma 2.1.9), which in turn holds if and only if it is locally of finite type over S[Sta, Tag 06G4].
4.2.11. Definition. We say that a sheaf of sets X on the étale site of S is an Indlocally finite type algebraic space over S if there is an isomorphism X ∼X,= lim−→i∈I iwhere {Xi }i∈I is a directed system of algebraic spaces, locally of finite type over S.
In this case, we will be able to show that each finite type point x of X admits arepresentative which is a monomorphism, unique up to unique isomorphism, and forsuch a monomorphism Spec k → X, we will construct a canonical minimal versal
ring to X at x. Despite this, it doesn’t follow that X satisfies [2](b). Loosely
speaking, there are two possible obstructions to [2](b) holding: Firstly, it may
be that the versal ring at a point x is not Noetherian. This happens when thedimension of the various Xi at the points xi increases without bound, so that the
Ind-algebraic space X is infinite dimensional; a typical example is given by infinitedimensional affine space (Example 4.3.3). Secondly, even if the versal ring x is
Noetherian, it may not be effective. This happens when the infinitesimal germsof Xi at the points xi collectively fill out a space of higher dimension than eachof the individual germs of Xi individually does; typical examples are given by theunion of infinitely many lines passing through a fixed point (Example 4.3.8), ora curve with a cusp of infinite order (Example 4.3.9). Roughly speaking, X will
satisfy [2](b) when the infinitesimal germs of the Xi at the points xi eventuallystabilise, or equivalently, when the projective limit defining the versal deformation
ring at x eventually stabilises. (This is not quite true; for example, in the inductive
system Xi , one could alternately add additional components, and then contract
them to a point. It will be true in all the examples we give below for which [2](b)is satisfied.)
We now present the details of the preceding claims.
4.2.12. Lemma. Any finite type point of an Ind-locally finite type algebraic space
X = limi∈I Xi over the locally Noetherian scheme S admits a representative x :−→Spec k → X which a monomorphism. This representative is unique up to unique
isomorphism, the field k is a finite type OS -field, and any other representative
Spec K → X of the given point factors through the morphism x in a unique fashion.
Furthermore, if i ∈ I is sufficiently large, then x : Spec k → X factors in a uniqueximanner as a composite Spec k −→Xi → X, and the morphism xi : Spec k → Xi isagain a monomorphism.
Proof. From the definitions, one sees that |X| = limi∈I |Xi |. Thus the given point−→of X arises from a point of Xi for some i ∈ I. Let xi : Spec k → Xi be theSCHEME-THEORETIC IMAGES OF MORPHISMS OF STACKS

87

monomorphic representative of this point whose existence is given by Lemma 2.2.14.
For each i0 ≥ i, let xi0 : Spec ki0 → Xi0 be the monomorphic representative of theimage of this point in Xi0 . Let k 0 denote the residue field of the image of this pointin S. Then we have natural containments k ⊇ ki0 ⊇ k 0 , for each i0 ≥ i. Thus, since
k is finite over k 0 , we see, replacing i by some sufficiently large i0 ≥ i if necessary,that we may assume that ki0 = k for all i0 ≥ i, and thus conclude that xi0 is simply
the composite xi : Spec k → Xi → Xi0 , for all i0 ≥ i. If we define x to be thecomposite x : Spec k → Xi → X, then, since xi0 is a monomorphism for each i0 ≥ i,one easily verifies that x is a monomorphism.
The remaining claims of the lemma are proved identically to the analogous claimsof Lemma 2.2.14.

If x is a finite type point of the Ind-locally finite type algebraic space X =limi∈I Xi over S, and if (by abuse of notation) we also write x : Spec k → X to−→denote the monomorphic representative of x provided by the preceding lemma, and(following the lemma) write xi : Spec k → Xi to denote the induced monomorphisms (for sufficiently large i), then we may consider the complete local rings
bX ,x (for sufficiently large i), in the sense of Definition 2.2.17. If i0 ≥ i (bothOi i
sufficiently large), then we obtain a canonical local morphism of complete local
bX 0 ,x 0 → O
bX ,x .
OS -algebras Oi iii
4.2.13. Definition. In the above context, we writebbX,x := lim O,
O←− Xi ,xi
i∈I

bX,x as the complete local ringconsidered as a pro-Artinian ring, and we refer to O
to X at x. By the following lemma it is canonically defined, independent of thedescription of X as an Ind-locally finite type algebraic space.
4.2.14. Lemma. If X is an Ind-locally finite type algebraic space over the locally
Noetherian scheme S, and if x : Spec k → X is the monomorphic representativeof a finite type point of X, which (by abuse of notation) we also denote by x, thenbX,x is a versal ring for X at x. Furthermore, the morphism Spf O
bX,x → X is aObformal monomorphism, and therefore the ring OX,x , equipped with this morphism,
is unique up to unique isomorphism.
Proof. Any commutative diagram
Spec A/ Spec B


bX,x
Spf O


bx
/X

in which A and B are finite type Artinian local OS -algebras, is induced by a commutative diagram/ Spec B
Spec Ab
Spf OXi ,xi


bi,x
/X

i

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M. EMERTON AND T. GEE

for some sufficiently large value of i. Since the lower horizontal arrow of this diagram is versal, we may lift the right-hand vertical arrow to a morphism Spec B →bX ,x , and hence obtain a lifting of the right-hand vertical arrow of the original
Spf O
i i
bX,x . This establishes the versality.
diagram to a morphism Spec B → Spf O
The property of being a formal monomorphism follows easily from the fact thatbX ,x → Xi are formal monomor(by Proposition 2.2.15) the morphisms Spf Oi ibbX,x → X isphisms. That the data of OX,x together with the morphism Spf O
unique up to unique isomorphism is then immediate from Lemma 2.2.16.

4.2.15. Ind-algebraic stacks as scheme-theoretic images. If X := limi∈I Xi is an Ind` −→algebraic stack, then there is an evident morphism of stacks i∈I Xi → X , whose
source is an algebraic stack (being the disjoint union of a collection of algebraic
stacks), and which can be verified to be representable by algebraic spaces precisely
when X satisfies [3]. Thus Ind-stacks give us examples of maps from algebraic stacks
to not-necessarily-algebraic stacks, to which we `can try to apply the machinery ofSection 3 (although the fact that the morphism i∈I Xi → X is not quasi-compact
in general is an impediment to such applications).
As one illustration of this, we note that Example 4.3.12 below shows that our
main results will not extend in any direct way to morphisms of finite type which
are not assumed to be proper.
4.3. Illustrative examples. Here we present various illustrative examples, andexplain how they relate to the general theory. Several of them are due originally toArtin [Art69b, §5].
4.3.1. Example. [Art69b, Ex. 5.11]: X is the sheaf of sets obtained by taking theunion of the two schemes Spec k[x, y][1/x] and Spec k[x, y]/(y) in the (x, y)-plane.
(More precisely, X is the sheaf obtained as the pushout of these schemes along
their common intersection.) The sheaf X satisfies [1], [2], [3], and [4b], but doesn’tsatisfy [4a]. (And hence is not an algebraic space, and so doesn’t satisfy [4].)We now give a series of examples of various kinds of Ind-algebraic stacks over a
field k. All the stacks we consider will be the inductive limit of stacks satisfying [1],and hence will satisfy [1], as well as [2](a) and [4a]. In fact, all the examples otherthan Example 4.3.2 will be Ind-locally finite algebraic spaces (in fact, even Indlocally finite type schemes) over k, and so will admit versal rings at finite typepoints by Lemma 4.2.14. (As discussed above, though, this doesn’t necessarily
imply that they satisfy [2](b).)

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89

4.3.2. Example. Let Xi be a directed system of algebraic stacks, locally of finite presentation over a locally Noetherian scheme S, for which the transition morphisms
are smooth, and consider the Ind-algebraic stack X := lim Xi . Since each Xi−→
satisfies [1], so does X , and being an Ind-algebraic stack, it also satisfies [2](a)and [4a]. Since the transition morphisms Xi → Xi0 (for i0 ≥ i) are smooth, andsince each of the algebraic stacks Xi satisfies [4], we find that X also satisfies [4](and hence also [4b]). If x : Spec k → X is a finite type point, then x factors as
xi
Xi → X for some i, and one easily verifies (again using the smoothness
Spec k −→of the transition morphisms) that if Spf R → Xi is a versal ring to xi , then thecomposite Spf R → Xi → X is a versal ring to X . Thus X satisfies [2](b). In conclusion, such an Ind-algebraic stack necessarily satisfies each of our axioms except
possibly [3].
One example of such an Ind-algebraic stack is obtained by taking {Gi } to be adirected system of smooth algebraic groups over a field k and setting Xi := [ ·/Gi ],so that X := lim [ ·/Gi ]. (Here “·” stands for “the point”, i.e. Spec k.) If we set−→
G := limi∈I Gi , then X may be regarded as the classifying stack [ ·/G ]. The fibre−→
product · ×X · is then naturally identified with G, and so if the Ind-algebraic group
G is not a scheme (as it typically will not be), then X does not satisfy [3].
4.3.3. Example. We take X to be “infinite dimensional affine space”. More formally, we write X := lim An , with the transition maps being the evident closed−→
immersions:An ∼= An × {0} ,→ An × A1 ∼= An+1 .
Since the transition maps are closed immersions, the Ind-scheme X satisfies [3] (andis quasi-separated). The complete local rings at points are non-Noetherian (they are
power series rings in countably many variables), and so X does not satisfy [2](b).
It does satisfy [4] (and so also [4b]) vacuously: because the complete local rings are
so large, one easily verifies that a morphism from a finite type k-scheme to X isnever versal at a finite type point.
4.3.4. Example. We take X to be the line with infinitely many nodes [Art69b, Ex.
5.8] (considered as an Ind-algebraic space in the evident way).

The Ind-scheme X satisfies [2](b), and, although the transition maps are not closed
immersions, it also satisfies [3]; however, it is not quasi-separated, since the diagonal
morphism X → X × X is not quasi-compact. Concretely, if A1 → X is the obvious
morphism, namely the one that identifies countably many pairs of points on A1
to nodes, then A1 ×X A1 is the union of the diagonal copy of A1 , and countably
many discrete points (encoding the countably many identifications that are madeto create the nodes of X).

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The Ind-scheme X doesn’t satisfy [4], but it does satisfy [4b], vacuously. (Onecan check that a morphism from a finite type k-scheme to X cannot be smoothat any point.) This example shows that the quasi-separatedness hypotheses arenecessary in Corollaries 2.6.11 and 2.6.12 and Theorem 2.8.5. 6
4.3.5. Example. We take X to be the line with infinitely many cusps.

The Ind-scheme X satisfies [2](b). As in the previous example, the transition
morphisms are not closed immersions, and in this case X does not satisfy [3];indeed, if A1 → X is the natural morphism contracting a countable set of points tothe cusps of X, then A1 ×X A1 is the Ind-scheme obtained by adding non-reduced
structure to the diagonal copy of A1 at each of the points that is contracted to acusp.
The Ind-scheme X does not satisfy [4], but just as in the previous example, itdoes satisfy [4b] vacuously.
4.3.6. Example. We take X to be the line with infinitely many lines crossing it[Art69b, Ex. 5.10].

This example satisfies [2](b) and [3] (and it is quasi-separated, since the transition
maps are closed immersions), but doesn’t satisfy [4b] (and hence doesn’t satisfy [4]).
4.3.7. Example. As a variation on the preceding example, we consider the Indscheme X given by adding infinitely many embedded points to a line.

As with the preceding example, this example satisfies [2](b) and [3], is quasiseparated, and doesn’t satisfy [4b] (and hence doesn’t satisfy [4]).
One point of interest related to this example is that the natural morphism A1 →X is a closed immersion (in particular it is both quasi-compact and proper), andis surjective; however it is not scheme-theoretically dominant. Since X is not an
algebraic space, this shows the importance of scheme-theoretic dominance (rather
than mere surjectivity) as a hypothesis in Corollary 1.1.2.
6Artin states that this example does not satisfy [4b], but seems to be in error on this point.

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91

4.3.8. Example. We take X to be the union of infinitely many lines through theorigin in the plane [Art69b, Ex. 5.9].

This example satisfies [3], but not [2](b): the complete local ring at the origin isequal to k[[x, y]], and the corresponding morphism Spf k[[x, y]] → X is not effective.
It satisfies [4] (and hence also [4b]) vacuously: the non-effectivity of the complete
local ring at the origin shows that one cannot find a morphism from a finite typek-scheme to X which is versal at a point lying over the origin in X.
nn

4.3.9. Example. Let Xn be the plane curve cut out by the equation y 2 = x2 +1 ,define the morphism Xn → Xn+1 via (x, y) 7→ (x2 , xy), and let X := lim Xn ; so X−→
is a line with a cusp of infinite order.

The Ind-scheme X does not satisfy [2](b): the complete local ring at the cusp isequal to k[[x, y]], and the morphism Spf k[[x, y]] → X is not effective. Also, X does
not satisfy [3]: if A1 = X0 → X is the natural morphism, then A1 ×X A1 is a formal
scheme, which is an infinite-order thickening up of the diagonal copy of A1 at theorigin.
4.3.10. Example. In [Art69b, Ex. 5.3], Artin gives the example of two lines meeting to infinite order as an Ind-scheme satisfying [2](b) and [4], but not [3]. Wegive a variant of Artin’s example here, which illustrates the necessity of [3] inLemma 2.8.7 (3).
We take k = R, and we define Xn := Spec R[x, y]/(y 2 + x2n ), with the transition
morphism Xn → Xn+1 given by (x, y) 7→ (x, xy), and set X := limn Xn .
−→

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complex conjugation

As with Artin’s example, the Ind-scheme X satisfies [4]. The complete local ringof X at the origin is equal to R[[x]], but the natural morphism Spf R[[x]] → X is
not effective; thus X does not satisfy [2](b). On the other hand, if we consider
the map Spec C → X induced by the origin, then we obtain a versal morphism
Spf C[[x]] → X which is effective, although the resulting morphism Spec C[[x]] → X
is not unique; we can map Spec C[[x]] along either of the branches through theorigin.

4.3.11. Example. We take X to be the indicated Ind-scheme (“the zipper”).

It has the same formal properties as the Ind-scheme of Example 4.3.6, namely itsatisfies [2](b) and [3], but not [4] or [4b].

SCHEME-THEORETIC IMAGES OF MORPHISMS OF STACKS

93

4.3.12. Example. We will give an example of an Ind-algebraic surface which contains
the zipper of the preceding example as a closed sub-Ind-scheme.

We begin by setting X 0 := A2 ; we also choose a closed point P0 ∈ X 0 . We let X 1
be the blow-up of X 0 at P0 ; it contains an exceptional divisor E1 , and we choose aclosed point P1 ∈ E1 . We proceed to construct surfaces X n inductively: each X n
is a smooth surface obtained by blowing up X n−1 at a point Pn−1 . The surface X ncontains an exceptional divisor En , as well as the strict transform of the exceptional
divisor En−1 on X n−1 . We choose a closed point Pn ∈ En which does not lie in thestrict transform of En−1 , and then define X n+1 to be the blow-up of X n at Pn .

94

M. EMERTON AND T. GEEThere are natural open immersions X n \ Pn ⊆ X n+1 \ Pn+1 ; the Ind-scheme X
obtained by taking the inductive limit of these open immersions is in fact a scheme,
and is the standard example of a locally finite type smooth irreducible surface which
is not of finite type. Inside X we have the union E := ∪n≥1 En \ {Pn }, which is an
infinite chain of P1 ’s. We now form an Ind-scheme X by identifying a countablecollection of points on E1 \ P1 with a point on each En \ {Pn } (for n ≥ 2); theimage of E in X is then a copy of the zipper of Example 4.3.11.
The Ind-surface X satisfies [2](b) and [3], and is quasi-separated. It does notsatisfy [4] or [4b].
We note that each of the composite morphisms X n \ {Pn } → X → X is of finite
type, so that our formalism of scheme-theoretic images applies to it. Each of thesemorphisms is in fact scheme-theoretically dominant, in the sense that its schemetheoretic image is all of X. (Morally, the Ind-surface X is irreducible.) Since X is
not an algebraic space, this example shows that Theorem 1.1.1 and Corollary 1.1.2
don’t extend in any direct way to the case of morphisms of finite type that are notproper.
5. Moduli of finite height ϕ-modules and Galois representations
In this section we will combine Theorem 1.1.1 with the results of [Kis09b, PR09]
to construct moduli stacks of finite height and finite flat representations of theabsolute Galois groups of p-adic fields.
We begin by proving some foundational results about ϕ-modules of finite heightand étale ϕ-modules. We then introduce various moduli stacks of finite height ϕmodules and étale ϕ-modules closely related to those considered by Pappas andRapoport in [PR09], to which we apply the machinery of the earlier parts of thepaper.
5.1. Projective modules over power series and Laurent series rings. We
begin with a discussion of some foundational results concerning finitely generated
modules over the power series ring A[[u]] and the Laurent series ring A((u)), where
A is an arbitrary (not necessarily Noetherian) commutative ring. In particular, werecall some deep results of Drinfeld [Dri06] on the fpqc-local nature of the projectivity of such modules. We are grateful to Drinfeld for sharing with us some of his
unpublished notes on the subject; several arguments in this section are essentiallydrawn from these notes.
5.1.1. Projective and locally free modules. The basic objects we are interested inare finitely generated projective modules over the rings A[[u]] and A((u)), where
A is some given ground ring. Of course, a finitely generated projective module
over A[[u]] (resp. over A((u))) is Zariski locally free on A[[u]] (resp. A((u))), andso we can speak of a finitely generated projective A[[u]]- or A((u))-module being ofrank d: this just means that it is Zariski locally free of rank d on Spec A[[u]] (resp.
Spec A((u))). However, our point of view is that we want to regard the “base” ofour modules as being Spec A, and so we will be interested in understanding the
behaviour of A[[u]]- or A((u))-modules locally on Spec A.
This prompts the following definition, which is used in [PR09].
5.1.2. Definition. If M is a finitely generated A[[u]]-module, then we say that that
M is fpqc locally free of rank d over A if there exists a faithfully flat A-algebra A0
such that M ⊗A[[u]] A0 [[u]] is free of rank d over A0 [[u]]. Similarly, if M is a finitely

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95

generated A((u))-module, then we say that that M is fpqc locally free of rank dover A if there exists a faithfully flat A-algebra A0 such that M ⊗A((u)) A0 ((u)) isfree of rank d over A0 [[u]].
We use analogous terminology for other topologies besides the fpqc topology.
E.g. if A0 can be chosen so that Spec A0 → Spec A is an fppf, étale, Nisnevich, or
Zariski cover (the last notion being understood in the sense that Spec A0 → Spec A
should be surjective, and locally on the source an open immersion), then we saythat M or M (as the case may be) is fppf, étale, Nisnevich, or Zariski locally freeof rank d over A.
5.1.3. Remark. One of the main objects of our discussion in this section is to understand, as best we can, the relationship between the various local freeness properties
just defined, and the property of being finitely generated and projective (over A[[u]]or A((u))).
As we note in Lemma 5.1.7 below, if A is Noetherian and A → B is an fppf
morphism, then the induced morphisms A[[u]] → B[[u]] and A((u)) → B((u)) arethemselves faithfully flat. Thus an A[[u]]- or A((u))- module which is fppf locally
free of finite rank in the sense of Definition 5.1.2 is in fact finitely generated andprojective (since being finitely generated and projective is a property of a modulethat can be checked fpqc locally).
In the case that A is not Noetherian, or that the morphism A → B is merely
faithfully flat, but not of finite presentation, we aren’t able to gain the same levelof control over either of the morphisms A[[u]] → B[[u]] or A((u)) → B((u)), andso the precise relationship between projectivity and the notions of local freenessintroduced in Definition 5.1.2 is not completely clear to us.
The most general statement that we were able to prove in the context of A[[u]]]modules is given in Proposition 5.1.9 below, in which we show that if an A[[u]]module is finitely generated and projective, then it is Zariski locally free (and thus
also fpqc locally free); and that if an A[[u]]-module is fpqc locally free, u-torsion free
and u-adically complete and separated, then it is finitely generated and projective.
In the context of A((u))-modules, the relationship between these notions is lessclear. In Lemma 5.1.23 we show that a finitely generated projective A((u))-module
is Nisnevich locally free as an A((un ))-module for all n sufficiently large, but in
Example 5.1.24 we give an example of a finitely generated projective A((u))-moduleof rank one which is not étale locally free as an A((u))-module (although it is Zariski
locally free as an A((un ))-module for all n ≥ 2).
5.1.4. Remark. The second main object of our discussion is to describe the extent towhich various notions of projectivity/local freeness for A[[u]]- and A((u))-modules
are genuinely local notions, and (closely related) the descent properties of thesenotions.
It is obvious that the various local freeness notions presented in Definition 5.1.2
are local in the relevant topology. Below, we will recall Drinfeld’s result that finitely
generated projective A((u))-modules satisfy descent in the fpqc topology.
Before turning to these main points of our discussion, we note the following
result.
5.1.5. Lemma. Let A be a commutative ring. Each of the natural maps Spec A[[u]] →Spec A and Spec A((u)) → Spec A induces an isomorphism on the Boolean algebrasof simultaneously open and closed subsets of its source and target.

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M. EMERTON AND T. GEE

Proof. We need to show that the injections A ,→ A[[u]] ,→ A((u)) induce bijections
on the sets of idempotents. Writing Idem(R) for the set of idempotents in the∼ring R, we note that the isomorphism A[[u]] −→ lim A[u]/(un ) induces a bijection←−∼
Idem(A[[u]]) −→ lim Idem(A[u]/un ). Since idempotents lift uniquely through nilpo←−
tent ideals, each of the transition morphisms in this latter projective system is abijection, and hence we find that the morphism Idem(A[[u]]) → Idem(A) (inducedby the map A[[u]] → A[u]/(u) = A) is a bijection. Its inverse is then given by themap Idem(A) → Idem(A[[u]]) induced by the inclusion A ,→ A[[u]]; in particular,
this map is also a bijection.
P∞
Now let e ∈ A((u)) be an idempotent, and write e = n=−∞ an un . We claimthat ai is nilpotent for each i < 0. Indeed, if A is reduced, then it is immediatefrom the equation e2 = e that ai = 0 when i < 0. The claim in general thenfollows by considering the image of e in Ared ((u)). Let I denote the ideal of A
generated by the ai for i < 0. By the claim we have just proved, together with thefact that ai = 0 for all but finitely many i < 0, we find that I is finitely generated
by nilpotent elements, and so is a nilpotent ideal of A. Thus the kernel of themorphism A((u)) → (A/I)((u)) is also nilpotent, and so we obtain a commutativesquare/ Idem A((u))
Idem(A)
Idem(A/I)/ Idem (A/I)((u))

in which the vertical arrows are bijections and the horizontal arrows are injections.
By the definition of I, and the discussion above, we see that the image of e in(A/I)((u)) in fact lies in (A/I)[[u]], and so, by what we have already proved in factlies in A/I. A consideration of the preceding commutative square then shows thate ∈ A, as required.

As a consequence of the preceding result, we have the following reassuring statement, which shows that in those contexts in which we have multiple ways to define
the locally free rank of an A[[u]] or A((u))-module, the definitions coincide.
5.1.6. Lemma. If M is a finitely generated projective A[[u]]- (resp. A((u))-)module,
and it is also fpqc locally free of rank d, then it is of rank d as a projective module
over A[[u]] (resp. A((u))).
Proof. The rank of a finitely generated projective module over a ring is locallyconstant, and so we may find a partition of Spec A[[u]] (resp. Spec A((u))) into
disjoint open subsets on each of which the rank of M is constant. Lemma 5.1.5
shows that this partition of Spec A[[u]] is induced by a corresponding partition ofSpec A. Working separately over each of these open and closed subsets of A, we may
assume that M is a projective A[[u]]- (resp. A((u))-)module of some fixed rank n.
By assumption there is a faithfully flat morphism A → B such that B[[u]] ⊗A[[u]]M (resp. B((u))⊗A((u)) M ) is free of rank d over B[[u]] (resp. B((u))). This moduleis also finitely generated projective of rank n, and thus we deduce that n = d, as
required.

We now develop those aspects of our discussion that don’t require Drinfeld’stheory of Tate modules.

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5.1.7. Lemma. If A is Noetherian and A → B is an fppf morphism, then each of
the induced morphisms A[[u]] → B[[u]] and A((u)) → B((u)) is faithfully flat.
Proof. Note that B[[u]] is flat over B ⊗A A[[u]] (being the u-adic completion of thelatter ring, which is finitely presented over the Noetherian ring A[[u]], and thus isitself Noetherian), which is in turn flat over A[[u]]. The maximal ideals of A[[u]]are all of the form (m, u), where m is a maximal ideal of A. Given such a maximal
ideal, since Spec B → Spec A is surjective, we may find a prime ideal p of B whichmaps to m, and then (p, u) is a prime ideal of B[[u]] which maps to the maximal
ideal (m, u) of A[[u]].
Thus A[[u]] → B[[u]] is a flat morphism for which the induced map Spec B[[u]] →Spec A[[u]] contains all maximal ideals in its image. Since flat morphisms satisfy
going-down, this morphism is in fact surjective, and thus A[[u]] → B[[u]] is faithfully
flat. The morphism A((u)) → B((u)) is obtained from this one via extending scalars
from A[[u]] to A((u)), and so is also faithfully flat.

As already discussed in Remark 5.1.3, for A[[u]]-modules, the conditions of being
finitely generated projective and of being fpqc locally free of finite rank are closely
related, the precise nature of this relationship being the subject of the following
results. The first part of the next proposition is closely related to [Kim09, Prop.
7.4.2].
5.1.8. Proposition. Let M be an A[[u]]-module. Then the following conditions areequivalent:(1) M is a finitely generated projective A[[u]]-module.
(2) M is u-torsion free and u-adically complete and separated, and M/uM isa finitely generated projective A-module.
Moreover if these conditions hold then there is an isomorphism of A[[u]]-modules∼(M/uM) ⊗A A[[u]] −→ M, which reduces to the identity modulo u. In particular if
furthermore M/uM is a free A-module, then M is a free A[[u]]-module.
Proof. If M is projective, then it is a direct summand of a finite free A[[u]]-module,
and is therefore u-torsion free and u-adically complete and separated; and certainly
M/uM is a projective A-module. For the reverse implication, note that by [GD71,
Prop. 0.7.2.10(ii)], we need only show that for each integer n ≥ 1, M/un M is a
projective A[u]/un -module.
To show this, note that firstly that since M is u-torsion free, for each m, n ≥ 1
we have a short exact sequence of A-modulesum

0 → M/un M → M/um+n M → M/um M → 0.
By the equivalence of conditions (1) and (4) of [Mat89, Thm. 22.3] (the local flatness
criterion), we see that M/un M is a flat A[u]/un -module for each n. It follows fromthe same short exact sequence and induction on n that M/un M is an A-moduleof finite presentation; so by [Sta, Tag 0561], it is an A[[u]]/un -module of finite
presentation, and is therefore projective, as required.
Finally, if these conditions hold, then since M/uM is projective, we can choose
an A-linear section to the natural surjection M → M/uM. We therefore have amorphism of projective A[[u]]-modules (M/uM) ⊗A A[[u]] → M, which reduces tothe identity modulo u. Write N := (M/uM)⊗A A[[u]]. Then the morphism N → M
is surjective by the topological version of Nakayama’s lemma, so we need only prove

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M. EMERTON AND T. GEE

that it is injective. For this, note that if K is the kernel of the morphism, thensince M is projective, the short exact sequence
0→K→N→M→0
splits, so that K is a finitely generated projective A[[u]]-module, and K/uK = 0.
Since K if finitely generated projective, it is in particular u-adically separated, so
we have K = 0, as required.

5.1.9. Proposition.
(1) If M is a finitely generated projective A[[u]]-module, then M is Zariski
locally on Spec A free of finite rank as an A[[u]]-module.
(2) If M is u-torsion free and is u-adically complete and separated, and M is
fpqc locally on Spec A free of finite rank as an A[[u]]-module, then M is a
finitely generated projective A[[u]]-module.
(3) If A is Noetherian, then M is a finitely generated projective A[[u]]-module
if and only if M is fppf locally on Spec A free of finite rank as an A[[u]]module.
Proof. We begin with (1). If M is a finitely generated projective SA -module,
then by Proposition 5.1.8, M/uM is a finitely generated projective A-module. It istherefore Zariski locally free. It follows that it is enough to show that if (M/uM)⊗A
B is a free B-module of finite rank, then M ⊗A[[u]] B[[u]] is a free B[[u]]-module offinite rank. But M ⊗A[[u]] B[[u]] is a finitely generated projective B[[u]]-module, so
it follows from Proposition 5.1.8 that M⊗A[[u]] B[[u]] is isomorphic to ((M/uM)⊗AB) ⊗B B[[u]], and is therefore free of finite rank.
For (2), if M is fpqc locally free, then M/uM is fpqc locally free of finite rank,
and is in particular fpqc locally finitely generated projective. By [Sta, Tag 058S],
the property of being finitely generated and projective is fpqc local, so we seethat M/uM is finitely generated projective. Thus M is finitely generated projective,by Proposition 5.1.8.
For (3), one implication is immediate from (1) (which furthermore allows us
to strengthen fppf locally free to Zariski locally free). The converse follows from
Lemma 5.1.7, again using [Sta, Tag 058S].

5.1.10. Tate modules. As discussed in Remark 5.1.3, for general faithfully flat morphisms A → B (i.e. outside the context of Lemma 5.1.7), we are not able to gainmuch direct control over the induced morphisms A[[u]] → B[[u]] or A((u)) →B((u)), and so we are not able to apply standard descent results in the context ofDefinition 5.1.2. However, in [Dri06], Drinfeld is able to establish descent results forthese morphisms, provided that one restricts attention to modules that are finitely
generated and projective. Drinfeld’s basic descent result is stated in the languageof Tate modules, and we begin by recalling the definition of this notion from [Dri06,§3], as well as some related definitions.
5.1.11. Definition. Let A be a commutative ring. (In fact, [Dri06] defines Tate
modules over not necessarily commutative rings, but we will only need the commutative case.) An elementary Tate A-module is a topological A-module which isisomorphic to P ⊕ Q∗ , where P, Q are discrete projective A-modules, and Q∗ :=HomA (Q, A) equipped with its natural projective limit topology (where we write Q∗
as the projective limit of (Q0 )∗ , where Q0 is a finite direct summand of Q, and giveSCHEME-THEORETIC IMAGES OF MORPHISMS OF STACKS99(Q0 )∗ the discrete topology). A Tate A-module is a direct summand of an elementary Tate A-module.
A morphism of Tate modules is a continuous morphism of the underlying Amodules.
5.1.12. Definition. A submodule L of a Tate module M is a lattice if it is open,
and if furthermore for every open submodule U ⊆ L, the quotient L/U is a finitely
generated A-module. We say that L is coprojective if M/L is a projective A-module(equivalently, a flat A-module; see [Dri06, Rem. 3.2.3(i)]). A Tate module contains
a coprojective lattice if and only if it is elementary [Dri06, Rem. 3.2.3(ii)].
The most important example of these definitions for our purposes is the following.
5.1.13. Example. We endow A[[u]] with its u-adic topology, and endow A((u)) withthe unique topology in which A[[u]] (equipped with its u-adic topology) is embedded1
as an open subgroup. Equivalently, we write A((u)) = limn n A[[u]], and endow−→ uA((u)) with its inductive limit topology, where each term in the inductive limit
is endowed with its u-adic topology. Or, again equivalently, we write A((u)) =1
A[[u]] ⊕ A[1/u], in which the first factor is endowed with its u-adic topology, anduthe second factor is discrete.
Both A[[u]] and A((u)) are then elementary Tate A-modules, and A[[u]] is a
coprojective lattice in A((u)). Furthermore, by [Dri06, Ex. 3.2.2], any finitely
generated projective A((u))-module has a natural topology, making it a Tate Amodule. (Indeed, we may write such a module M as a direct summand of A((u))nfor some n ≥ 1, where this latter module is endowed with its product topology.) In
fact, we have the the following theorem [Dri06, Thm. 3.10].
5.1.14. Theorem. There is a natural bijection between finitely generated projective
A((u))-modules, and pairs (M, T ) consisting of a Tate A-module M and a topologically nilpotent automorphism T : M → M , by giving M the A((u))-module structure
determined by um := T (m). (Here T is topologically nilpotent if and only if foreach pair of lattices L, L0 ⊆ M , we have T n L ⊆ L0 for all sufficiently large n.)
5.1.15. Descent. Drinfeld’s fundamental descent result is the following theorem [Dri06, Thm. 3.3], which shows that the notion of a Tate A-module is fpqc-local
on Spec A.
5.1.16. Theorem. If A0 is a faithfully flat A-algebra, then the functor M 7→b A M induces an equivalence between the category of Tate A-modules and theA0 ⊗category of Tate A0 -modules with descent data to A. Furthermore, the identification of Hom-spaces given by this equivalence respects the natural topologies.
The statement concerning topologies on Hom-spaces is left implicit in [Dri06],
but is easily checked.7
5.1.17. Remark. It is a theorem of Raynaud–Gruson [RG71, Ex. 3.1.4, Secondepartie] that the property of a module being projective can be checked fpqc locally.
7Any Tate module is a direct summand of one of the form P ⊕ Q∗ , where P and Q are bothfree A-modules. Since the formation of Homs is compatible with finite direct sums, this reduces
us to verifying the claim in the case of Hom(M, N ) where M and N are A-Tate modules which
are either free or dual to free, in which case it is straightforward.

100

M. EMERTON AND T. GEEThis implies that, for any faithfully flat morphism A → A0 , the functor M 7→
A0 ⊗A M induces an equivalence between the category of projective A-modules
and the category of projective A0 -modules equipped with descent data to A. Since
projective modules are particular examples of Tate modules (they are precisely thediscrete Tate modules), Drinfeld’s Theorem 5.1.16 incorporates this descent resultof Raynaud–Gruson as a special case.
The relationship between Tate modules and finitely generated projective A((u))modules given by Theorem 5.1.14 then implies that the notion of a finitely generated
projective A((u))-module is also fpqc local on Spec A; this is [Dri06, Thm. 3.11],
which we now recall. In addition, we prove some slight variants of this result that
we will need below.
5.1.18. Theorem. The following notions are local for the fpqc topology on Spec A.
(1) A finitely generated projective A((u))-module.
(2) A projective A((u))-module of rank d.
(3) A finitely generated projective A((u))-module which is fpqc locally free ofrank d.
(4) A finitely generated projective A[[u]]-module.
(5) A projective A[[u]]-module of rank d.
(6) A finitely generated projective A[[u]]-module which is fpqc locally free ofrank d.
5.1.19. Remark. More precisely, saying that the notion of a finitely generated projective A((u))-module is local for the fpqc topology on Spec A means the following(and the meanings of the other statements in Theorem 5.1.18 are entirely analogous):
If A0 is any faithfully flat A-algebra, set A00 := A0 ⊗A A0 . Then the category offinitely generated projective A((u))-modules is canonically equivalent to the category of finitely generated projective A0 ((u))-modules M 0 which are equipped with
an isomorphism∼

M 0 ⊗A0 ((u)),a7→1⊗a A00 ((u)) −→ M 0 ⊗A0 ((u)),a7→a⊗1 A00 ((u))
which satisfies the usual cocycle condition.
Proof of Theorem 5.1.18. Since the notion of being fpqc locally free of rank d is
fpqc local by definition, and the rank of a finitely generated projective module
can be computed fpqc locally by Lemma 5.1.6, it suffices to prove statements (1)and (4). The first of these is [Dri06, Thm. 3.11]. As noted above, it follows from
Theorem 5.1.16 together with Theorem 5.1.14. (The fact that the property ofan automorphism being topologically nilpotent satisfies descent follows from thecompatibility with topologies on Hom-spaces stated in Theorem 5.1.16.)
For (4), let A0 be a faithfully flat A-algebra, and let L0 be a projective A0 [[u]]module equipped with descent data. Then M 0 := L0 ⊗A0 [[u]] A0 ((u)) is a projective
A0 ((u))-module equipped with descent data, and L0 is a coprojective lattice in M 0
by Lemma 5.1.20 below. The short exact sequence of Tate A0 -modules
0 → L0 → M 0 → M 0 /L0 → 0
is split, and admits descent data to A. Thus, by Theorem 5.1.16, we may descend
this to a (split) short exact sequence of Tate A-modules. By (1) (or, perhaps better,by its proof), the endomorphism u of M 0 descends to an endomorphism of M , which

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101

equips M with the structure of a finitely generated and projective A((u))-module.
Since u preserves the submodule L0 of M 0 , it preserves the descended submodule
L of M .
Since M 0 /L0 is discrete (or, equivalently, a projective A0 -module), the sameis true of M/L, and thus we see that L is open in M , and coprojective (cf. Remark 5.1.17). In fact, L is a lattice in M . Indeed, since u is a topologically nilpotent
automorphism of M , the submodules un L (n ≥ 0) form a neighbourhood basis ofzero in L. Since
b A L)/un (A0 ⊗L)
b = L0 /un L0
A0 ⊗A (L/un L) = (A0 ⊗is finitely generated, and since the property of being a finitely generated A-module
is local for the fpqc topology on Spec A, we find that L/un L is finitely generated
over A. To complete the proof, we note that L is a projective A[[u]]-module byanother application of Lemma 5.1.20, as required.

We learned the following lemma from Drinfeld.
5.1.20. Lemma. Let M be a finitely generated projective A((u))-module, and let L
be an A[[u]]-submodule of M . Then the following are equivalent:(1) L is a finitely generated projective A[[u]]-module with A((u))L = M .
(2) L is a coprojective lattice in M .
Proof. If (1) holds, then L is certainly open in M , and each L/un L is a finitely
generated A-module, so L is a lattice in M . Since M/L ∼= L ⊗A[[u]] A((u))/A[[u]] ,and A((u))/A[[u]] is a free A-module, M/L is a projective A-module, so that L isa coprojective lattice, as required.
Conversely, suppose that L is a coprojective lattice. Since L is a lattice, wecertainly have A((u))L = M . The short exact sequence of A-modulesu

0 → L/uL → M/L → M/L → 0
splits (because M/L is projective), so that L/uL is a direct summand of the projective A-module M/L, and is thus itself projective. Since L is a lattice, L/uL isfinitely generated. It follows from Proposition 5.1.8 that L is a finitely generated
projective A[[u]]-module, as required.

5.1.21. Remark. As we proved in Proposition 5.1.9, a finitely generated and projective A[[u]]-module is fpqc (indeed, even Zariski) locally free of finite rank; thus the
only difference between the situations of parts (4), (5) and (6) of Theorem 5.1.18
is that, in parts (5) and (6), the locally free rank of the A[[u]]-module in question
is prescribed.
5.1.22. Lemma. Let M be a finitely generated A((u))-module, which is projective
as an A((un ))-module, for some n that is invertible in A. Then M is projective
over A((u)).
Proof. Identify A((u)) with A((un ))[X]/(X n − un ), so that we may regard M asa module over A((un ))[X]/(X n − un ) which is projective as an A((un ))-module.
Now consider the base-change M 0 := A((u)) ⊗A((un )) M ; this is a module overA((u))[X]/(X n − un ) which is projective as an A((u))-module. Since n andu are both invertible in A((u)), the quotient A((u)) := A((u))[X]/(X − u)of A((u))[X]/(X n − un ) is a direct summand of A((u))[X]/(X n − un ). Thus

102

M. EMERTON AND T. GEEM ∼= M 0 /(X − u)M 0 is a direct summand of M 0 , and hence is projective as anA((u))-module.

The following result relates the property of an A((u))-module being finitely generated and projective to the property of it being locally free, in the sense of Definition 5.1.2.
5.1.23. Lemma. Let M be a finitely generated projective A((u))-module. Then
there exists an n0 ≥ 1 such that for all n ≥ n0 , M is Nisnevich locally free as an
A((un ))-module.
Proof. By [Dri06, Thm. 3.4], we may make a Nisnevich localisation so that M isan elementary Tate A-module, i.e. contains a coprojective lattice L.
Since multiplication by u is topologically nilpotent, there is an integer n0 ≥ 0such that un L ⊆ L for all n ≥ n0 ; thus, for each such value of n, we see that L isnaturally an A[[un ]]-module, and that the natural morphism A((un ))⊗A[[un ]] L → M
is an isomorphism. By Lemma 5.1.20, L is a finitely generated projective A[[un ]]module. By Proposition 5.1.9 (1), after making a further Zariski localisation, we
may suppose that L is free of finite rank as an A[[un ]]-module, so that M is free offinite rank as an A((un ))-module, as required.

The following example shows that, in the context of the preceding lemma, we
can’t necessarily take n0 = 1.
5.1.24. Example. Let A = k[x, y]/(y 2 − x3 ), for some field k, and let I ⊆ A((u))
denote the ideal generated by (u2 − x, u3 − y). One can check that I is freely
generated over A((u2 )) by u2 − x and u3 − y, and if the characteristic of k isdifferent from 2, I is projective over A((u)) by Lemma 5.1.22.
Alternatively, and more conceptually, one can deduce this projectivity (with noassumption on the characteristic of k) by noting that (u2 , u3 ) is a smooth point (overthe complete non-archimedean field k((u))) of the rigid analytic curve y 2 = x3 lying
in the closed polydisk |x|, |y| ≤ 1 over k((u)), and that I is ideal sheaf of (u2 , u3 )in the Tate algebra A((u)) of the curve.
One can check that the A[[u2 ]]-submodule L of I generated by u2 − x and u3 − yis an A[[u2 , u3 ]]-submodule of I, and hence is closed under multiplication by un forany n ≥ 2. Since I is freely generated over A((u2 )) by u2 − x and u3 − y, we seethat I/L is a free A-module, so that L is a coprojective lattice in I. The proof of
Lemma 5.1.23 shows that L is then Zariski locally free over A[[un ]] for any n ≥ 2,
and thus that I is Zariski locally free over A((un )), for any n ≥ 2.
We claim that I is not an étale locally free A((u))-module. To see this, it suffices
to show that if R denotes the strict Henselisation of A at the maximal ideal (x, y),then I ⊗A((u)) R((u)) is not free over R((u)), i.e. that the ideal (u2 − x, u3 − y) is
not a principal ideal in R((u)). We prove this in the following lemma.
5.1.25. Lemma. If R denotes the strict Henselisation of k[x, y]/(y 2 − x3 ) at the
maximal ideal (x, y), then the ideal (u2 − x, u3 − y) of R((u)) is not principal.
Proof. Let f ∈ R[[u]] be non-zero, with non-zero constant term f0 ∈ R. (Any nonzero element of R((u)) may be be multiplied by some power of u so as to satisfy
this condition, and hence any principal ideal of R((u)) has a principal generator
satisfying this condition.) We claim, then, that f R((u)) ∩ R[[u]] = f R[[u]].

SCHEME-THEORETIC IMAGES OF MORPHISMS OF STACKS103

Indeed, if g ∈ f R((u)) ∩ R[[u]], then g = f h for some h ∈ R((u)). Choose N
minimally so that huN ∈ R[[u]], and suppose that N > 0. If we reduce the equation
guN = f huN modulo u, we find that
0 = f0 × the constant term of huN(an equation in R), and so (since R is a domain, as the cuspidal cubic y 2 = x3 isgeometrically unibranch at its singular point (0, 0)) we find that the constant termof huN is zero. This contradicts the minimality of N , and thus shows that N = 0,
so that in fact h ∈ R[[u]] and g ∈ f R[[u]], as claimed. Thus, if f R((u)) is any
principal ideal of R((u)), with f chosen as above, thenR[[u]]/(f R((u)) ∩ R[[u]] + uR[[u]]) = R[[u]]/(f, u)R[[u]] = R/f0 R.
Since R (which is a one-dimensional local ring) is not regular, the quotient R/f0 R
is necessarily of dimension > 1 over k.
On the other hand, we find thatR[[u]]/ (u2 − x, u3 − y) ∩ R[[u]] + uR[[u]]) = R/(x, y) = k.
Taking into account the result of the preceding paragraph, this shows that indeed (u2 − x, u3 − y) is not a principal ideal in R((u)).

5.2. Modules of finite height and étale ϕ-modules. In this section we discuss
finite height ϕ-modules and étale ϕ-modules. With an eye to future applications(for example, the case of Lubin–Tate (ϕ, Γ)-modules), we work in a more generalcontext than that usually considered.
5.2.1. Definitions. Fix a finite extension k/Fp , and write S := W (k)[[u]]. Let q besome power of p, and let ϕ be a ring endomorphism of S which is congruent to
the q-power Frobenius endomorphism modulo p.
5.2.2. Lemma. ϕ induces the usual q-power Frobenius on W (k).
Proof. It is enough to note that W (k) is generated as a Zp -algebra by a primitive (#k − 1)st root of unity, and that the (#k − 1)st roots of unity are distinct
modulo p.
5.2.3. Lemma. For each M, a ≥ 1, ϕ(uM +a−1 ) ∈ (uM q , pa ), and u(M +a−1)q ∈(ϕ(uM ), pa ). In particular, ϕ is continuous with respect to the (p, u)-adic topology.
Proof. Write ϕ(u) = uq + pY . Then ϕ(uM +a−1 ) = (uq + pY )M +a−1 , andu(M +a−1)q = (ϕ(u) − pY )M +a−1 , and the result follows from the binomial theorem.
We fix a finite extension E/Qp with ring of integers O and uniformiser $. If A is
an O/$a -algebra for some a ≥ 1, we write SA := (W (k) ⊗Zp A)[[u]]; we equip SA
with its u-adic topology. Let OE,A equal SA [1/u].
5.2.4. Lemma. ϕ admits a unique continuous A-linear extension to SA , which inturn admits a unique continuous A-linear extension to OE,A .
Proof. This follows immediately from Lemma 5.2.3.
5.2.5. Lemma. ϕ is faithfully flat and finite on SA and OE,A .



104

M. EMERTON AND T. GEE

Proof. It is enough to show this for SA , as it then follows for the localisation OE,A .
It then suffices to show that SA is a finite free ϕ(SA )-module of rank q, with abasis given by ui , 0 ≤ i ≤ q − 1.
If A is an Fp -algebra this is clear, and it follows from Nakayama’s lemma thatin general SA is generated as an ϕ(SA )-module by ui , 0 ≤ i ≤ q − 1. It remains
to check P
that the ui are linearly independent. To this end, suppose that we have aq−1
relation i=0 ui ϕ(ai ) = 0, where the ai are all contained in pn SA for some n ≥ 0.
We will show that in fact the ai are all contained in pn+1 SA ; since pn SA = 0 for n
sufficiently large, it follows that ai = 0 for all i, as required.
Write ϕ0 : SA → SA for the particular lift of the q-power Frobenius on SA/pA
defined via ϕ0 (u) = uq . Then for any x ∈ pn SA , we have ϕ(x) − ϕ0 (x) ∈ pn+1 SA ,Pq−1
so our assumptions imply that i=0 ui ϕ0 (ai ) ∈ pn+1 SA . Writing the ai out as
power series in u, and equating coefficients, we see immediately that ai ∈ pn+1 SA ,as required.

We fix a polynomial F ∈ (W (k) ⊗Zp O)[u] that is congruent to a positive powerof u modulo $. The following elementary lemma will be useful below.
5.2.6. Lemma. For all integers a, h ≥ 1 there is an integer n(a, h) ≥ 1 depending
only on a, h and F such that if A is an O/$a -algebra, then un(a,h) is divisible
by F h in SA , and F n(a,h) is divisible by uh .
Proof. Write F = un − $X for some n ≥ 1 and X ∈ SA . By the binomial
theorem, F a+h−1 is divisible by unh in SA , and thus by uh ; similarly, writing
un = F + $X, un(a+h−1) is divisible by F h . Putting these together, we see that
we can take n(a, h) := n(a + h − 1).
We will also use the following result.
5.2.7. Lemma. If A is a O/$a -algebra, then for each M ≥ 1, u(M +a−1)q is divisibleby ϕ(uM ), and ϕ(u(M +a−1) ) is divisible by uM q .
Proof. This is immediate from Lemma 5.2.3.



If M is an SA -module (resp. M is an OE,A -module) then we write ϕ∗ M
for M ⊗SA ,ϕ SA (resp. ϕ∗ M for M ⊗OE,A ,ϕ OE,A ). Since ϕ is faithfully flat,
the functors M 7→ ϕ∗ M, M 7→ ϕ∗ M are exact.
5.2.8. Corollary. If A is a O/$a -algebra, M is an SA -module, and uM M 6= 0,
then u(M −a+1)q ϕ∗ M 6= 0.
Proof. If u(M −a+1)q ϕ∗ M = 0, then by Lemma 5.2.7 we have ϕ∗ (uM M) =ϕ(uM )ϕ∗ M = 0. Since ϕ is faithfully flat, this implies that uM M = 0, a contradiction.

The following lemma is a straightforward generalisation (with a very similar
proof) of [PR09, Prop. 2.2] to our setting. Let R be an O/$a -algebra, and let u ∈R be a nonzerodivisor, such that R is u-adically complete and separated. (Forexample, we could take R = SA for some O/$a -algebra A.) For n ≥ 0, write
Un = 1 + un Md (R),
Vn = {A ∈ GLd (R[1/u])|A, A−1 ∈ u−n Md (R)}.
5.2.9. Lemma. Suppose that n > (2m + (a − 1)q)/(q − 1).

SCHEME-THEORETIC IMAGES OF MORPHISMS OF STACKS

105(1) For each g ∈ Un , A ∈ Vm , there is a unique h ∈ Un such that g −1 Aϕ(g) =h−1 A.
(2) For each h ∈ Un , A ∈ Vm there is a unique g ∈ Un such that g −1 Aϕ(g) =h−1 A.
Proof. We follow the proof of [PR09, Prop. 2.2]. For the first part, note that we
can solve for h−1 , namely h−1 = g −1 Aϕ(g)A−1 , so that the uniqueness of h isclear, and what we must show is that h ∈ Un , or equivalently, that h−1 ∈ Un .
We can write g −1 = I + un X with X ∈ Md (R), and by Lemma 5.2.7 we canwrite ϕ(g) = I + u(n−a+1)q Y with Y ∈ Md (R). Then g −1 Aϕ(g)A−1 = (I +un X)(I + u(n−a+1)q AY A−1 ). Since A ∈ Vm we have AY A−1 ∈ u−2m Md (R), sothat g −1 Aϕ(g)A−1 ∈ Un , as required.
For the second part we begin by showing uniqueness of g, for which it is enoughto show that if g −1 Aϕ(g) = A then g = 1. Write g = I + X, so that the preceding
relation between A and g may be rewritten as X = Aϕ(X)A−1 . It is enoughto check that we have X ∈ us Md (R) for all s ≥ n. We prove this by induction
on s, the case s = n being by hypothesis. If X ∈ us Md (R), then as above wehave Aϕ(X)A−1 ∈ u(s−a+1)q−2m Md (R), and since s ≥ n we have (s−a+1)q−2m >s, as required.
Finally we must show existence of g. For this, let A0 = h−1 A, and set
A0 = A, h0 = h. We inductively define sequences (hi ), (Ai ) by setting Ai =0 −1
h−1
Ai . These equalities imply that hi = A0 ϕ(hi−1 )(A0 )−1 ,i−1 Ai−1 ϕ(hi−1 ), hi = (A )0so since A ∈ Vm and h0 ∈ Un , an easy induction as above shows that hi ∈ Un+ifor all i. If we now set gi = h0 h1 · · · hi , then gi tends to some limit g ∈ Un . Since
for all i we have gi−1 Aϕ(gi−1 ) = h−1 A, in the limit we have g −1 Aϕ(g) = h−1 A, as
required.

5.2.10. Definition. Let h be a non-negative integer, and let A be an O/$a -algebra.
A ϕ-module of height F with A-coefficients is a pair (M, ϕM ) consisting of a finitely
generated u-torsion free SA -module M, and a ϕ-semilinear map ϕM : M → M,with the further properties that if we writeΦM := ϕM ⊗ 1 : ϕ∗ M → M,then ΦM is injective, and the cokernel of ΦM is killed by F . A ϕ-module of finite
height with A-coefficients, or a finite height ϕ-module with A-coefficients, is a ϕmodule with A-coefficients which is of height F for some F .
A morphism of finite height ϕ-modules is a morphism of the underlying SA modules which commutes with the morphisms ΦM .
We say that a finite height ϕ-module is projective of rank d if it is a finitely
generated projective SA -module of constant rank d.
5.2.11. Remark. We will primarily be interested in finite height ϕ-modules (resp.
étale ϕ-modules) that are furthermore projective over SA (resp. over SA [1/u]), towhich the base-change and descent results of Drinfeld [Dri06] recalled in Subsection 5.1.15 are applicable. However, we sometimes need to make constructions thattake us outside the category of projective modules, and in particular we need toconsider finite height ϕ-modules which are not projective, but become projective
after inverting u.
5.2.12. Definition. Let A be a O/$a -algebra. An étale ϕ-module with Acoefficients is a pair (M, ϕM ) consisting of a finitely generated OE,A -module M , and106

M. EMERTON AND T. GEE

a ϕ-semilinear map ϕM : M → M which induces an isomorphism of OE,A -modules
ΦM := ϕM ⊗ 1 : ϕ∗ M → M .
A morphism of étale ϕ-modules is a morphism of the underlying OE,A -modules
which commutes with the morphisms ΦM .
We say that M is projective (resp. free) of rank d if it is a finitely generated
projective (resp. free) OE,A -module of constant rank d. If τ is any topology on
the category of O/$a -modules lying between the Zariski topology and the fpqc
topology, then we say that M is τ -locally free of rank d if it is projective of rank d,and if τ -locally on Spec A, it is free of rank d.
5.2.13. Remark. If (M, ϕ) is a ϕ-module of height F with A-coefficients, then by
Lemma 5.2.6, (M[1/u], ϕ) is an étale ϕ-module with A-coefficients.
We will sometimes prove results about projective étale ϕ-modules by reducing
to the free case, using the following lemma.
5.2.14. Lemma. If M is a projective étale ϕ-module with A-coefficients, then M isa direct summand of a free étale ϕ-module with A-coefficients.
Proof. Since M is projective, we may find another finitely generated projective∼
OE,A -module P such M ⊕ P −→ F, for some finite rank free module F . Then∼ϕ∗ M ⊕ ϕ∗ P −→ ϕ∗ F,and since M is an étale ϕ-module, we have ϕ∗ M ∼= M, while since F is free, wehave ϕ∗ F ∼= F . Thus
M ⊕P ∼= M ⊕ ϕ∗ P,
and so taking the direct sum with another copy of P, we find thatF ⊕P ∼= F ⊕ ϕ∗ P ∼= ϕ∗ (F ⊕ P ).
In other words, the finitely generated projective module Q := F ⊕ P admits the
structure of an étale ϕ-module, and
Q⊕M ∼
=F ⊕F
is free of finite rank.



5.2.15. Lemma. Let A be a Noetherian O/$a -algebra, and let M be an étale ϕmodule with A-coefficients. Then for some F , there is a ϕ-module M of height F
with A-coefficients such that M[1/u] = M . If M is furthermore a free OE,A -module,
then we may choose M to be a free SA -module.
5.2.16. Remark. Note that in the case when M is projective but not necessarily
free, we do not claim that the ϕ-module M in Lemma 5.2.15 can be chosen to be
projective.
Proof of Lemma 5.2.15. By definition M is finitely generated as an OE,A -module,
so we may choose a generating set, and let M be the SA -span of this generating
set; if M is free then we may and do also choose M to be free. It follows easily from
Lemma 5.2.7 that if we scale M by a large enough power of u, we may assume that M
is ϕ-stable. Since M is u-torsion free, so is M, and since ΦM is an isomorphism,
ΦM is injective and the cokernel of ΦM is killed by some power of u. Thus M is a
finite height ϕ-module, as required.
SCHEME-THEORETIC IMAGES OF MORPHISMS OF STACKS

107

5.2.17. Remark. If q = p, then for certain choices of ϕ and F , the theories of finite
height ϕ-modules and étale ϕ-modules with Artinian coefficients admit interpretations in terms of Galois representations; more precisely, in terms of representationsof the absolute Galois groups of certain perfectoid fields. We refer to [EG22, §2] for amore thorough discussion of this; in Section 5.4.23 below we explain the connection
in a particular case, that of Breuil–Kisin modules.
5.3. Lifting rings. We now prove the existence of universal lifting rings for étale ϕmodules, as well as variants for finite height ϕ-modules. These will be used below
to show that our moduli stacks admit versal rings, and to verify the effectivity
hypothesis in our application of Theorem 1.1.1.
We remark that if we were in one of the settings mentioned in Remark 5.2.17,
then we could use the equivalence of categories between étale ϕ-modules and Galois
representations with Artinian coefficients to study the versal rings in which we areinterested using standard techniques from the formal deformation theory of Galois
representations; in particular, in the case of Breuil–Kisin modules, we could use theresults of [Kim11]. Even in the general framework that we have adopted, it seems
plausible that we could use the Artin–Schreier theory constructions that underliethat equivalence to replace the formal deformation theory of étale ϕ-modules by
the formal deformation theory of some more finitistic objects. However, we havefound it more direct, and interesting in its own right, to argue with the formal
deformation theory of étale ϕ-modules. We caution the reader that this leads us,
in what follows, to consider some rather large pro-Artinian rings!
5.3.1. Lifting rings. The main results of this subsection are Proposition 5.3.6 andTheorem 5.3.15.
We begin by studying the formal deformation theory of étale ϕ-modules.
Let E/Qp be a finite extension with ring of integers O, uniformiser $ and residue
field F, and let M be an étale ϕ-module with F-coefficients which is free of rank d.
Fix an integer a ≥ 1, and (following the notation of § 2.2) write CO/$a for the
category of Artinian local O/$a -algebras for which the structure map induces anisomorphism on residue fields.
Fix a choice of (ordered) OE,F -basis of M , or equivalently, an identification of
OE,F -modules(5.3.2)∼dM −→ OE,F.

5.3.3. Definition. A lifting of M to an object R of CO/$a is a triple consistingof an étale ϕ-module MR which is free of rank d, a choice of (ordered) OE,R -basisof MR , and an isomorphism MR ⊗R F ∼= M of étale ϕ-modules which takes thechosen basis of MR to the fixed basis of M . Equivalently, a lifting M consists of anétale ϕ-module MR endowed with an isomorphism of OE,R -modules(5.3.4)∼dMR −→ OE,Fdsuch that the étale ϕ-module structure on OE,F
which is obtained by reducing (5.3.4)
dinduced by themodulo mR coincides with the étale ϕ-module structure on OE,F
isomorphism (5.3.2).
By regarding the isomorphisms (5.3.2) and (5.3.4) as identifications, we see thatthe liftings of M admit yet another equivalent description: namely, the identification (5.3.2) allows us to regard M as simply a choice of matrix Φ ∈ GLd (OE,F ),108

M. EMERTON AND T. GEE

which describes the étale ϕ-module structure
d
d
d
OE,F
= ϕ∗ OE,F= ϕ∗ M → M = OE,F,

and the identification (5.3.4) allows us to regard the lifting MR as a matrix ΦR ∈GLd (OE,R ) lifting Φ; this is the matrix describing the étale ϕ-module structure
d
d
d
OE,R
= ϕ∗ OE,R= ϕ∗ MR → MR = OE,R.

We denote by D : CO/$a → Sets the functor taking R to the set of isomorphismclasses of liftings of M to R. (The functorial structure on D is the obvious one
induced by extension of scalars.)
5.3.5. Remark. There is a natural action of the subgroup 1 + mR Md (OE,R ) of
GLd (OE,R ) on D (R), given by change of basis. In terms of the descriptionof liftings via a choice of isomorphism (5.3.4), this is given by composing thedinduced by a matrix g ∈isomorphism (5.3.4) with the automorphism of OE,R
1 + mR Md (OE,R ). In terms of the description of liftings in terms of a matrix ΦR ,this is given by the twisted conjugation action ΦR 7→ gΦR ϕ(g)−1 .
We begin our study of liftings by establishing the pro-representability of D .
5.3.6. Proposition. The functor D is pro-representable by an object R ofpro -CO/$a .
Proof. We follow Dickinson’s appendix to [Gou01], which uses Grothendieck’s
representability theorem to prove the existence of universal deformation rings.
By [Gro95, Cor. to Prop. 3.1, §A], it is enough to show that D is left exact. Left
exactness is equivalent to preserving fibre products and terminal objects. Thereis a unique terminal object of CO/$a , namely F, and since D (F) consists of thetrivial lifting, it is also a terminal object. It remains to check that D preserves
fibre products in CO/$a ; but this is obvious if we think of liftings as being a choiceof ΦR lifting Φ.

5.3.7. Remark. If MR is a rank d free étale ϕ-module over the Artinian ring R
lifting M , then Lemma 5.2.15 shows that we may write MR = MR [1/u] for a free
finite height ϕ-module MR of some height F (depending on MR ). The unicity
statement of Lemma 5.2.9 (2) (with h = 1) then shows that if N is sufficiently large(depending on F ), the elements of 1 + uN mR Md (SR ) act freely (in the sense of theaction described in Remark 5.3.5) on the element of D (R) represented by M .
Thus, for a non-trivial thickening R of k, if the set D (R) is non-empty (which
e.g. necessarily will be the case if R is a k-algebra, since in that case we can simply
base-change M from k to R), then it is quite enormous! In particular, the prorepresenting ring R constructed in Proposition 5.3.6 does not admit a countablebasis of neighbourhoods of zero.
We now consider liftings of bounded height. If M is an étale ϕ-module with
A-coefficients, then we say that M admits a model of height F if there is a(not necessarily projective) ϕ-module M of height F with A-coefficients such thatM[1/u] = M .
5.3.8. Definition. Let DF be the subfunctor of D whose elements are the liftingsof M which admit a model of height F .
5.3.9. Proposition. The functor DF is pro-representable by a quotient RF of R .

SCHEME-THEORETIC IMAGES OF MORPHISMS OF STACKS

109

Proof. Let A be an Artinian quotient of R . It follows from Lemma 5.3.10 belowthat there is a unique maximal quotient AF of A for which the corresponding étale
ϕ-module admits a model of height F . We then take RF := limA AF .
←−
5.3.10. Lemma. Let A be Noetherian. The direct sum of two étale ϕ-modules with
A-coefficients which have models of height F also has a model of height F . Any
subquotient of an étale ϕ-module with A-coefficients with a model of height F also
has a model of height F .
Proof. The statement about direct sums is trivial. For subquotients, let 0 → M 0 →M → M 00 → 0 be a short exact sequence of étale ϕ-modules with A-coefficients,and suppose that M is a finite height ϕ-module of height F with M[1/u] = M .
Let M0 , M00 be respectively the kernel and image of the induced map M → M 00 ; thenit is easy to check that M0 , M00 are ϕ-modules of height F , and that M0 [1/u] = M 0 ,M00 [1/u] = M 00 .

5.3.11. Remark. As noted in Remark 5.3.7, there is an N (depending on F , andwhich we fix once and for all) such that for each object R of CO/$a , the elementsof 1 + uN mR Md (SR ) act freely on DF (R). This again implies that the rings RF
are huge.
However, since F is now fixed, we may also fix the integer N , and then systematically quotient out by the free action of 1 + uN mR Md (SR ). This leads to thefollowing definition.
5.3.12. Definition. We let DF : CO/$a → Sets denote the functor defined by
DF (R) := DF (R)/ 1 + uN mR Md (SR ) .
In the rest of this section we develop the theory needed to prove Theorem 5.3.15,
which shows that DF is pro-representable by a Noetherian ring.
5.3.13. Lemma. Let A be an Artinian O/$a -algebra. Suppose that M is an étale
ϕ-module with A-coefficients, and that M, M0 are two models of M of height F .
If j is minimal such that uj M ⊆ M0 , and k is such that uk ϕ∗ M ⊆ ϕ∗ M0 , thenk > (j − a)q.
Proof. By assumption we have uj−1 (M/M ∩ M0 ) 6= 0, so by Corollary 5.2.8 wehave u(j−a)q ϕ∗ (M/M ∩ M0 ) 6= 0. It is therefore enough to check that ϕ∗ M ∩
ϕM0 = ϕ∗ (M ∩ M0 ); but this follows from the flatness of ϕ (applied to the map
M ⊕ M0 → M , (x, y) 7→ x − y).

The following lemma is a generalisation of [CL09, Cor. 3.2.6], and the proof issimilar.
5.3.14. Lemma. Let A be an Artinian O/$a -algebra. If M is an étale ϕ-modulewith A-coefficients, and M has a model of height F , then it has both a minimalmodel Mmin of height F and a maximal model Mmax of height F . Furthermore,
Mmax /Mmin is an A-module of finite length.
Proof. Let M be a model of M of height F . We claim that there is an i ≥ 0 suchthat any other model M0 of height F satisfies ui M ⊆ M0 ⊆ u−i M.
To see this, we follow the proof of [Kis09b, Prop. 2.1.7], and choose r minimalsuch that ur M ⊆ ΦM (ϕ∗ M) ⊆ u−r M (note that r exists by Lemma 5.2.6), andchoose j minimal such that uj M ⊆ M0 , and l minimal such that M0 ⊆ u−l M. We

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must show that j, l are bounded independently of M0 . It follows from Corollary 5.2.8
that if uk ϕ∗ M ⊆ ϕ∗ M0 then k > (j − a)q. By Lemma 5.2.6 there is a constant n(a)such that F divides un(a) in SA , so thatΦM (ϕ∗ M) ⊆ u−r M ⊆ u−j−r M0 ⊆ u−n(a)−j−r ΦM0 (ϕ∗ M0 ).
It follows from Lemma 5.3.13 that n(a) + j + r > (j − a)q, so that j is bounded
independently of M0 .
Similarly, if we choose l minimal such that M0 ⊆ u−l M, thenΦM0 (ϕ∗ M0 ) ⊆ M0 ⊆ u−l M ⊆ u−r−l ΦM (ϕ∗ M),
so that by Lemma 5.3.13 again we have r + l > (l − a)q, and l is also bounded
independently of M0 , as required.
Since u−i M/ui M has finite length as an A-module, it follows that the categoryof models of M of height F is Artinian and Noetherian. To see that maximal andminimal models of height F exist, it is now enough to note that if M, M0 are
models of M of height F , then so are max(M, M0 ) := M + M0 and min(M, M0 ) :=M ∩ M0 .

5.3.15. Theorem. The functor DF is pro-representable by a Noetherian ring RF .
Furthermore, we may choose a natural transformation DF → DF which is a section
to the natural transformation DF → DF ; and thus for any R, we obtain a natural
isomorphism ∼DF (R) × 1 + uN mR Md (SR ) −→ DF (R).
Proof. Define the group-valued functor G on O/$a -algebras via G(A) := 1 +uN Md (SA ) (the group structure being given via multiplication of matrices). Then
G is in fact an affine group scheme, equal toSpec(O/$a )[{xi,j,k,n }1≤i≤f,1≤j,k≤d,N ≤n<∞ ],where we have written f := [k : Fp ] and have chosen a basis for W (k) over Zp ,f∼∼so that SA −→ A[[u]]f and 1 + uN Md (SA ) −→ 1 + uN Md (A[[u]]) ; the variables xi,j,k,n then parameterize the coefficient of un in the (j, k) entries of the ithsummand.
Now let S denote the pro-Artinian completion of(O/$a )[{xi,j,k,n }1≤i≤f,1≤j,k≤d,N ≤n<∞ ]at the closed point defined by setting $, as well as all the xi,j,k,n , equal to 0. (Thispoint corresponds to the identity 1 ∈ Md (SF ).) If we set G := Spf S, then for any
object R of CO/$a , we have G(R) = 1 + uN mR Md (SR ). The free action of G(R)
on DF (R), for each R, induces an equivalence relation
G × DF ⇒ DF ,the quotient of DF by which is of course just DF . The product G × DF is prob O/$a S. In order to proceed, we need to show that thisrepresentable by RF ⊗completed tensor product is topologically flat over RF , in the sense of [Gab65]. For
this, we have have to give a more explict description of S, and so we make a small
digression in order to do so.
Let V denote the free O/$a -module spanned by the variables{xi,j,k,n }1≤i≤f,1≤j,k≤d,N ≤n<∞ ,SCHEME-THEORETIC IMAGES OF MORPHISMS OF STACKS

111

endowed with its discrete topology. We let U 7→ U ∗ denote Pontrjagin duality on
topological O/$a -modules. If U is discrete, then U ∗ is profinite, and conversely.
In particular, if U is a discrete finite O/$a -module, then U ∗ is again both discreteand finite. Continuing to assume that U is discrete and finite, and writing Vδ∗ todenote V ∗ endowed with its discrete topology, we find that
HomO/$a (V, U ) = HomO/$a ,cont (U ∗ , V ∗ )= HomO/$a (U ∗ , V ∗ ) = HomO/$a ,cont (Vδ∗ )∗ , U .
Thus the canonical embedding V ,→ (Vδ∗ )∗ identifies (Vδ∗ )∗ with the pro-Artinian(equivalently, pro-finite) completion of V . Thus S = O/$a [[(Vδ∗ )∗ ]], in the notationb O/$a S =of [Gab65, §1.2.5]. In particular, S is topologically flat over O and RF ⊗∗RF [[(V )δ ]] is topologically flat over RF .
It follows from [Gab65, Thm. 1.4] that the kernel RF of the corresponding pairof morphisms
b O/$a S = RF [[(V ∗ )δ ]].
RF ⇒ RF ⊗pro-represents the quotient DF . The same result also shows that RF is topologicallyflat over RF .
b O/$a S is a power series ring over RF , the morphism G × DF → DF
Since RF ⊗satisfies an infinitesimal lifting criterion of the type considered in Definition 2.2.9
above. Thus this morphism is versal, in the terminology employed in that definition,or smooth, in the terminology of [Sta, Tag 06HG]. It is then no doubt a matter ofgeneral principles that DF → DF also satisfies this infinitesimal lifting property. In
our particular case we can see this directly: since any surjection R → R0 manifestly
induces a surjection G(R) → G(R0 ), one immediately confirms that DF → DF
satisfies the infinitesimal lifting property. An evident generalization of [Sta, Tag
06HL] to our pro-Artinian setting, a detailed proof of which may be be foundat [uh], then shows that RF ∼= RF [[{xi }i∈I ]], for some index set I. The morphism
RF → RF obtained by mapping each xi to 0 then determines a functorial section
DF → DF with the property stated in the theorem.
It remains to show that RF is Noetherian. By [Gro95, Prop. 5.1, §A], this isequivalent to showing that DF (F[]) is a finite-dimensional F-vector space. Elementsof DF (F[ε]) determine self-extensions 0 → M → M 0 → M → 0 with the property
that M 0 admits a model of height F . Let M0 be such a model; then the image of M0
in M is a model of M of height F , and thus contains Mmin , the minimal model ofheight F (whose existence is guaranteed by Lemma 5.3.14). We can replace M0 by
the preimage of Mmin , and accordingly we can assume that the image of M0 in M
is Mmin .
Similarly, the kernel of M0 → M is a model of M , and is therefore containedin Mmax ; replacing M0 by its sum with Mmax , we may assume that in fact M0 is
an extension of Mmin by Mmax . (Having made this replacement, M0 may only beof height F 2 , rather than of height F , but this does not matter for our argument.)
It suffices to show that the F-vector space of such extensions, considered up to theequivalence relation induced by (1 + uN εMd (Sk ))-conjugacy, is finite-dimensional.
After choosing bases for Mmax and Mmin , the possible matrices for ϕM0 are of
the form


Amax
B
0
Amin

112

M. EMERTON AND T. GEE

where Amin and Amax are respectively
the matrices of ϕMmin and ϕMmax . Conju
1 uN εXgating by matrices of the form
, we see that we are free to replace B
01
by B + uN XAmin − Amax ϕ(uN X). It therefore suffices to show that for somesufficiently large M , for every Y ∈ Md (Sk ) we can writeuM Y = uN XAmin − Amax ϕ(uN X)
for some X ∈ Md (Sk ).
Since Mmin has height F , it follows from Lemma 5.2.6 that for some t ≥ 0 andsome Zmin ∈ Md (Sk ) we can write Amin Zmin = ut Idd . It therefore suffices to showthat we can always solve the equation(5.3.16)

uM −N Y = ut X − u(q−1)N Amax ϕ(X)Zmin(as a solution to this equation with Y replaced by Y Zmin provides a solution to our
original equation).
For any V ∈ ut Md (Sk ), write δ(V ) := u(q−1)N −t Amax ϕ(V )Zmin ∈ Md (Sk ).
Note that if V ∈ us Md (Sk ) for some s ≥ t + 1, then δ(V ) ∈ uqs+(q−1)N −t Md (Sk ),and in particular δ(V ) ∈ us+1 Md (Sk ). It follows that the sum W := V + δ(V ) +δ 2 (V ) + · · · ∈ us Md (Sk ) converges, and W − δ(W ) = V . Therefore, if we set M :=N + 2t + 1, take V = uM −N −t Y , and write W = X, we have the required solution
to (5.3.16).
If E 0 /E is a finite extension with ring of integers O0 , then we have an obvious
map from ϕ-modules (and finite height ϕ-modules) with respect to O from thosewith respect to O0 , given by tensoring with O0 over O. We end this section withthe following reassuring results.
5.3.17. Lemma. If M is an étale ϕ-module with A-coefficients, then M admits amodel of height F if and only if M ⊗O O0 admits a model of height F .
Proof. Since the inclusion O ,→ O0 is split as an inclusion of O-modules (e.g.
because O0 is a faithfully flat and finite O-algebra), this is immediate from
Lemma 5.3.10.

If F0 is the residue field of O0 , then we have corresponding universal lifting


0
a
0
rings RO
0 , RO 0 ,F and RO 0 ,F for liftings to O /$ -algebras of MF ⊗F F .
 ∼ 
5.3.18. Corollary. We have natural isomorphisms of O0 /$a -algebras RO
0 = R ⊗O
0


0
0
∼
∼
0
O , RO0 ,F = RF ⊗O O , and RO ,F = RF ⊗O O

Proof. This follows easily from Lemma 2.2.13. Alternatively, we can argue slightly
more explicitly (but essentially equivalently) as follows. We give the argumentfor R , the other cases being essentially identical.
Tensoring over O0 with O and considering the universal property gives a natural
map RO⊗O O0 . Let RO
0 → R
0 ,F be the subring of RO 0 consisting of elements
whose reductions modulo the maximal ideal lie in F. Considering the matrix of theuniversal étale ϕ-module over RO
0 with respect to a basis which lifts (the extensionof scalars of) a basis of MF , we see that this universal étale ϕ-module is definedover ROIt follows from the universal property that there is a natural map
0 ,F .

R → RO0 ,F .

SCHEME-THEORETIC IMAGES OF MORPHISMS OF STACKS

113




0

Considering the composites RO⊗O O0 → RO
→ RO
0 → R
0 and
0 ,F ⊗O O

0R → RO
→
R
→
R
⊗
O
,
we
easilyobtainthefirstclaim.
The
second0 ,FO
O0claim then follows from Lemma 5.3.17.


5.4. Moduli of finite height ϕ-modules and of étale ϕ-modules. In thissubsection we will define the moduli stacks that we are interested in, and prove our
key results regarding them. We begin by establishing some terminology, which will
be important for all that follows.
We fix an integer a ≥ 1, and proceed to define various categories fibred ingroupoids (which will in fact be stacks, although some only in the Zariski topology)
over O/$a .
5.4.1. Definition. If a, d ≥ 1 are positive integers, then for any O/$a -algebra A,
we define Rad (A) to be the groupoid of étale ϕ-modules with A-coefficients which
are projective of rank d over OE,A . If A → B is a morphism of O/$a -algebras, and
if M is an object of Rad (A), then the pull-back of M to Ra (B) is defined to be thetensor product OE,B ⊗OE,A M .
The resulting category fibred in groupoids Rad is in fact a stack in groupoidsin the fpqc topology over O/$a , as follows from the results of [Dri06], and more
specifically from Theorem 5.1.18 above.
5.4.2. Definition. If a, d ≥ 1 are positive integers, and if F ∈ (W (k) ⊗Zp O)[u] isa polynomial that is congruent to a positive power of u modulo $, then for anyaO/$a -algebra A, we define Cd,F(A) to be the groupoid of ϕ-modules of height F
with A-coefficients which are projective of rank d over SA . If A → B is a morphismaof O/$a -algebras, and if M is an object of Cd,F(A), then the pull-back of M toa
Cd,F (B) is defined to be the tensor product SB ⊗SA M.
Again, it follows from Theorem 5.1.18 that the resulting category fibred ingroupoids Rad is in fact a stack in groupoids in the fpqc topology over O/$a . Thereais an obvious morphism Cd,F
→ Rad , defined by sending (M, ϕ) to (M[1/u], ϕ).
5.4.3. Remark. Our notation, and the entire set-up that we have just introduced, isvery much inspired by the work of Pappas and Rapoport [PR09]. Indeed, in the casewhen q = p, ϕ(u) = up , and F ∈ W (k)[u] is an Eisenstein polynomial, our stackaa
Cd,F
coincides with the stack Ch,W(k)[u]/F defined in [PR09, §3.b]. However, ourastack Rd is subtly different from the stack denoted in the same manner in [PR09].
In that reference, the étale ϕ-modules under consideration are not required to be
projective, but are required to be fpqc locally free. However, it seems to us that
it is necessary to impose this projectivity in order to obtain a stack, while thelocal freeness hypothesis seems unnatural from the point of view of our intended
applications (which is that Rad should provide models for moduli stacks of local
Galois representations — and a direct summand of a family of representations
should again form such a family); also, at a technical level, the effectivity resultof Theorem 5.4.19 (5) below depends on working with projective étale ϕ-modules
that are not necessarily locally free over the coefficient ring.
In spite of the difference between our definition of Rad and that of [PR09], wenevertheless rely on many of the arguments of that reference. In order to make theconnection between our set-up and that of [PR09], it is helpful to introduce thefollowing auxiliary objects, in which we require projectivity of our étale ϕ-modules,
but also impose freeness conditions, as in [PR09].

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M. EMERTON AND T. GEE

5.4.4. Definition. We define Rad,free to be the full subgroupoid of Rad classifying
free étale ϕ-modules of rank d.
If τ is any topology on the category of O/$a -modules lying between the Zariski
topology and the fpqc topology, then we define Rad,τ −free to be the full subgroupoidof Rad classifying projective étale ϕ-modules of rank d that are furthermore τ -locally
free over the ring of coefficients.
Taking into account the fact that Rad is an fpqc stack, the category fibred ingroupoids Rad,τ −free is evidently a stack in the topology τ . Indeed, one immediately
checks that it is the τ -stackification of Rad,free .
5.4.5. Remark. As already noted, the category fibred in groupoids Rad,free , as wellas the stacks Rad,τ −free , will play a purely auxiliary role. Furthermore, we need only
make one choice of topology τ and work with that particular choice throughout;e.g. we could simply take τ to be the Zariski topology.
From now on, for the duration of the paper, we fix a choice of a ≥ 1, and omit
it from the notation.
5.4.6. Lemma. The morphism Cd,F → Rd factors through Rd,τ −free (for any choiceof τ ).
Proof. This follows from Proposition 5.1.9 (1), which shows that a projective finite
height ϕ-module is actually Zariski locally free.

5.4.7. Proposition. If Spec A → Rd is a morphism with A a Noetherian O/$a algebra, then there exists a scheme-theoretically surjective morphism Spec B →Spec A such that the composite morphism Spec B → Rd factors through Rd,free .
Proof. Let M be the étale ϕ-module over A classified by the given A-valued point of
Rd , and let Mred denote the base-change of M over Ared . By Lemma 5.2.15, we mayfind a (not necessarily projective) finite height ϕ-module M ⊂ Mred with M[1/u] =Mred . The quotient M/uM is then a coherent sheaf on Spec Ared , and so we mayfind a dense open subset U ⊂ Spec Ared such that M/uM restricts to a free sheaf
over U . Thus, by Proposition 5.1.8, M restricts to a free finite height ϕ-module
over U , and so Mred restricts to a free étale ϕ-module over U (necessarily of rank d).
We may regard U as an open subset of Spec A, and without loss of generality we may in fact assume that U = Spec Af for some f ∈ A. Because Af is
Noetherian, the nilradical of Af is nilpotent, and so the kernel of the morphism
Af ((u)) → (Af )red ((u)) is also nilpotent. Thus the restriction of M to U is a projective Af ((u))-module which becomes free modulo a nilpotent ideal. By a standard
Nakayama-type argument, we see that this restriction itself is a free étale ϕ-module
over U .
We now note that we may choose a closed subscheme Z ,→ Spec A whose underlying
closed subset is equal to Spec A \ U, and for which the obvious morphism`
U Z → Spec A is scheme-theoretically dominant, in addition to being surjective.
(Since A is Noetherian, the kernel A[f ∞ ] of A → Af is equal to A[f n ] for somen ≥ 1, and we may take Z = Spec A/f n .) The proposition now follows by anevident Noetherian induction.

5.4.8. Proposition. Let A be an O/$a -algebra, and let M, N be projective étale ϕmodules of finite rank with A-coefficients. Then the functors on A-algebras taking B

SCHEME-THEORETIC IMAGES OF MORPHISMS OF STACKS

115

to Hom(MB , NB ) and Isom (MB , NB ) are both represented by affine schemes offinite presentation over A.
Proof. By Lemma 5.2.14, there are projective étale ϕ-modules P, Q of finite ranksuch that the étale ϕ-modules F := M ⊕ P and G := N ⊕ Q are both free of finite
rank. We now follow the proof of [PR09, Cor. 2.6(b)]. Choosing bases of F, G
as OE,A -modules, an element of Hom(FB , GB ) is given by a matrix g with coefficients in OE,B . If the matrices of ϕM , ϕN with respect to the chosen bases arerespectively X, Y then the condition that g respects ϕ is that ϕ(g) = Y −1 gX.
Choose an integer n ≥ 0 such that X, X −1 , Y, Y −1 all have entries with poles ofdegree at most n, and let s ≥ 0 be minimal such that g has poles of degree at most s.
By Corollary 5.2.8, ϕ(g) has poles of degree greater than (s − a)q. Since ϕ(g) =Y −1 gX, we see that we mustPhave (s − a)q < 2n + s, whence s < (2n + aq)/(q − 1).
∞
Writing the matrix g as i=−s gi ui , gi ∈ Md (W (k) ⊗Zp B), the equation g =−1
Y ϕ(g)X
and Lemma 5.2.9 show that the gi for i ≤ (2n + (a − 1)q)/(q − 1)determine all of the gi . It follows that Hom(FB , GB ) is represented by an affine
scheme of finite presentation over A.
Let e ∈ End(F ), f ∈ End(G) be the idempotents corresponding to M , N respectively. Since Hom(MB , NB ) ⊂ Hom(FB , GB ) is given by those g satisfyingg(1 − e) = 0 and (1 − f )g = 0, we see that it is represented by a closed subschemeof the scheme representing Hom(FB , GB ), and is therefore of finite presentation(for example by Lemma 2.6.3). Finally, the result for Isom (MB , NB ) follows by
regarding it as the subfunctor of pairs (α, β) ∈ Hom(MB , NB ) × Hom(NB , MB )satisfying αβ = IdNB , βα = IdMB .

The following theorem generalises some of the main results of [PR09] to our
setting. The proofs are almost identical, and we content ourselves with explaining
the changes that need to be made to the arguments of [PR09], rather than writing
them out in full.
5.4.9. Theorem. (1) The stack Cd,F is an algebraic stack of finite presentation overSpec O/$a , with affine diagonal.
(2) The morphism Cd,F → Rd,f pqc−free is representable by algebraic spaces,
proper, and of finite presentation.
(3) The diagonal morphism ∆ : Rd,f pqc−free → Rd,f pqc−free ×O/$a Rd,f pqc−freeis representable by algebraic spaces, affine, and of finite presentation.
Proof. In the case that O = Zp , q = p, and ϕ(u) = up , it follows from the main
results of [PR09] that Cd,F is an algebraic stack of finite type over Spec O/$a , andthat (2) holds. (Strictly speaking, [PR09] assume that F is an Eisenstein polynomial, but their arguments go through unchanged with our slightly more general
choice of F .) In the case of general O, q and ϕ, the arguments go over essentiallyunchanged provided that one replaces the use of [PR09, Prop. 2.2] with an appeal
to Lemma 5.2.9, and that in [PR09, §3] one replaces eah by the quantity n(a, h)
appearing in Lemma 5.2.6.
Part (3) is immediate from Proposition 5.4.8. To prove the remaining claimsof (1), we have to show that Cd,F is in fact of finite presentation over O/$a , withaffine diagonal. These facts are certainly implicit in the arguments of [PR09], but
for the reader’s convenience, we explain how they follow formally from the results
already established. Since O/$a is Noetherian, and since we know already that
Cd,F is finite type over O/$a , the diagonal morphism Cd,F → Cd,F ×O/πa Cd,F is116

M. EMERTON AND T. GEE

automatically quasi-separated (being a representable morphism between finite type
algebraic stacks over O/$a ), and so to show that Cd,F is of finite presentation over
O/$a , it suffices to show that this diagonal morphism is quasi-compact. Sinceaffine morphisms are quasi-compact, this will follow once we show that Cd,F has
affine diagonal. For this, we factor the diagonal of Cd,F as
Cd,F → Cd,F ×Rd,f pqc−free Cd,F → Cd,F ×O/$a Cd,F .
The first of these morphisms is a closed immersion, since Cd,F → Rd,f pqc−free isrepresentable and proper (by (2)), while the second is affine, being a base-changeof the diagonal morphism of Rd,f pqc−free → Rd,f pqc−free ×O/$a Rd,f pqc−free (whichis affine, by (3)). Their composite is thus an affine morphism, as claimed.
In Theorem 5.4.11 we prove the analogue of Theorem 5.4.9 for Rd . In order todo so we need the following Lemma.
5.4.10. Lemma. Let A be an O/$a -algebra, and let M be an étale ϕ-module with Acoefficients, which is free of rank d as an OE,A -module. Let T be an automorphismof M . Then the functor on A-algebras taking B to the set of T -invariant projective
ϕ-modules MB ⊂ MB of rank d and height F is representable by a projective Ascheme.
Proof. Let Gr denote the affine Grassmannian classifying projective SA -lattices
in M ; this is an Ind-projective A-scheme. We begin by showing that the subfunctorof Gr × Gr given by{(M, N) : M ⊆ N}
is a closed Ind-subscheme of Gr × Gr. To see this, we have to show that for any
A-algebra B, and any pair of lattices MB , NB ∈ Gr(B), the locus in Spec B overwhich MB ⊆ NB is closed. Equivalently, we need to show that the locus overwhich the morphism MB → MB /NB vanishes is closed. To see this, note that
we may factor this map as MB → P → Q → MB /NB , where P is a finite rank
(as a B-module) projective quotient (hence direct summand) of MB , and Q isa finite rank projective direct summand of MB /NB . Since the maps MB → P
and Q → MB /NB are both split, we are in fact considering the locus over which
the morphism P → Q vanishes, and this is obviously closed, as it is given by thevanishing of matrix entries.
The endomorphism T induces an automorphism T∗ of Gr (taking M to T (M)),
and we let ΓT := T∗ × id : Gr → Gr × Gr be the graph of T . Pulling back theclosed locus considered above by ΓT , we see that there is a closed Ind-subscheme
GrT of Gr, classifying the lattices M with T (M) ⊂ M.
The ϕ-module M corresponds to a morphism Spec A → Rd,f pqc−free , and foreach F , the fibre product GrF := Spec A ×Rd,f pqc−free Cd,F is a closed subschemeof Gr (it is a scheme by Theorem 5.4.9 (2)). Then the intersection of GrF and GrT
is closed in GrF , and is therefore projective over Spec A, as required.

We let End(Rn,free ) be the category fibred in groupoids over O/$a with
End(Rn,free )(A) = {(M, f )} where M ∈ Rn,free (A) and f ∈ End(M ). It contains a subcategory fibred in groupoids Projn,d , classifying those pairs (M, f )for which Im f is projective of rank d. There are natural morphisms Projn,d ⊆End(Rn,free ) → Rn,free and Projn,d → Rd , which respectively take (M, f ) to M

SCHEME-THEORETIC IMAGES OF MORPHISMS OF STACKS

117

and to Im f . These morphisms fit into the following commutative diagram.
Cn,F

Rn

Projn,dRn,freeEnd(Rn,free )Rd

5.4.11. Theorem. (1) The morphism Cd,F → Rd is representable by algebraic
spaces, proper, and of finite presentation.
(2) The diagonal morphism ∆ : Rd → Rd ×O/$a Rd is representable by algebraic
spaces, affine, and of finite presentation.
(3) Rd satisfies [1].
Proof. We begin with (1). Let B be an A-algebra, and let Spec B → Rd be
a morphism, corresponding to a projective étale ϕ-module MB of rank d. We
need to show that Spec B ×Rd Cd,F is representable by a proper algebraic spaceover Spec B of finite presentation. This can be checked étale locally, so in particular by Lemma 5.1.23 we can assume that MB is free over (W (k) ⊗O/$a B)((un ))for some n. The claim follows from Lemma 5.4.10, applied with u replaced by un ,and T being given by multiplication by u.
Part (2) is immediate from Proposition 5.4.8. For (3), by Proposition 2.3.19 andpart (2), it is enough to show that Rd → Spec O/$a is limit preserving on objects.
To this end, suppose that we have a morphism T → Rd , where T = lim Ti is a←−limit of affine schemes. By Lemma 5.2.14 we can lift the morphism T → Rd toa morphism T → Projn,d for some n. The composite morphism T → Projn,d →Rn,free → Rn lifts to a morphism T → Cn,F for some F (because every free étale
ϕ-module contains a free finite height ϕ-module, by Lemma 5.2.15) and since Cn,F
is locally of finite presentation, this morphism factors through Ti for some i.
Consequently, the composite T → Projn,d → Rn,free factors through Ti , and it
suffices to prove that the morphism Projn,d → Rn,free is limit preserving on objects.
This follows from Proposition 5.4.8, which shows that the morphism Projn,d →Rn,free is representable by schemes of finite presentation (note that the condition
that an endomorphism of a free module be idempotent is a closed condition, and istherefore of finite presentation by Lemma 2.6.3).

By Theorem 5.4.11, the running assumptions of Section 3.2 apply to the morphism Cd,F → Rd ; so we may use Definition 3.2.6 to define the scheme-theoretic
image of Cd,F → Rd , which we denote by Rd,F . The main result of this section is
Theorem 5.4.19 below, showing that Rd,F is an algebraic stack. Before proving it,
we study the versal rings of Rd and Rd,F .
Let F0 /F be a finite extension, and let MF0 be an étale ϕ-module with F0 coefficients, corresponding to a finite type point x : Spec F0 → Rd . Write O0
for the ring of integers in the compositum of E and W (F0 )[1/p], so that O0 has
residue field F0 .
As in Section 5.3, we fix a choice of (ordered) OE,F0 -basis of MF0 , and we let
D : CO0 /$a → Sets be the functor taking R to the set of isomorphism classes ofliftings of MF0 to R. By Remark 5.3.5, the group functor H defined via H(R) :=118

M. EMERTON AND T. GEE

R× + mR Md (OE,R ) acts on D via change of basis. By Proposition 5.3.6, D is
pro-representable by an object R of pro -CO0 /$a .
b d,x , defined by mapping any lift
5.4.12. Lemma. The natural morphism D → R
MR of MF0 over some test object R to the underlying étale ϕ-module (i.e. forgetting
the choice of basis of MR , as well as the chosen isomorphism between MR /mR
and MF0 ), is versal, and is also H-equivariant, for the change-of-basis action of Hb d,x .
on D and for the trivial action of H on R
Proof. The claimed equivariance is clear, since the morphism is defined in part byforgetting the chosen bases. To see the claimed versality, it suffices to show that if
MA is an étale ϕ-module with A-coefficients, where A is a finite Artinian O0 /$a algebra, and if MB is an étale ϕ-module with B-coefficients, with B a finite Artinian
O0 /$a -algebra admitting a surjection onto A, such that the base change (MB )Aof MB to A is isomorphic to MA , and we have compatible choices of basis for MA
and MB , then we may find an MB0 (together with such a basis) which lifts MA ,and is isomorphic to MB . The existence of such a lift amounts to showing that we
can lift the choice of basis, and this is clear from the surjectivity of GLd (OE,B ) →GLd (OE,A ).

The preceding lemma shows in particular that Rd admits versal rings at all finite
type points.
Let CSpf R denote the pull-back of Cd,F → Rd along the versal morphism

D = Spf R → Rd , and let R,C be the scheme-theoretic image (in the senseof Definition 3.2.15) of the morphism CSpf R → Spf R . By Lemmas 3.2.16
b d,F,x .
and 5.4.12, there is a versal morphism Spf R,C → R
5.4.13. Definition. We let D,C denote the subfunctor of D represented by
Spf R,C .
5.4.14. Remark. The equivariance statement of Lemma 5.4.12 implies that the Haction on D restricts to an H-action on D,C , and the the morphism D,C :=b d,F,x is H-equivariant, with respect to the induced H-action on D,C ,Spf R,C → R
b d,F,x .
and the trivial H-action on R
Our goal will be to show that an appropriately chosen subgroup functor of H
acts freely on D,C , and that the corresponding quotient DC of D,C also admits ab d,F,x , and is Noetherianly pro-representable. To this end, weversal morphism to R
will relate D,C to the subfunctor DF of D , where, as in Section 5.3, we let DF
be the subfunctor of D consisting of those M which have a model of height F . In
fact, we will show that D,C is a subfunctor of DF ; equivalently, we will show that
R,C , which is a priori a quotient of R , is actually a quotient of RF (the quotientof R that pro-represents DF , whose existence is proved in Proposition 5.3.9). We
begin with a useful general criterion for such a factorisation of the map R → R,C
to exist.
We momentarily place ourselves in the general deformation-theoretic context ofSubsection 2.2; that is, we fix a Noetherian ring Λ, and a finite ring map Λ → k,whose target is a field. We will also allow ourselves to use the language of formal
algebraic spaces from [Sta, Tag 0AHW], but not in a serious way. (If R is a local
ring with maximal ideal m, then Spf R is simply the Ind-scheme limi Spec R/mi ,−→SCHEME-THEORETIC IMAGES OF MORPHISMS OF STACKS

119

and giving a finite type morphism of formal algebraic spaces X → Spf R amounts
to giving a collection of compatible finite type morphisms of algebraic spaces Xi →Spec R/mi .)5.4.15. Lemma. Let R → S be a continuous surjection of objects in pro -CΛ , let
X → Spf R be a finite type morphism of formal algebraic spaces, and make thefollowing assumption: if A is any finite-type Artinian local R-algebra for which
the canonical morphism R → A factors through a discrete quotient of R, and forwhich the canonical morphism X ×Spf R Spec A → Spec A admits a section, thenthe canonical morphism R → A furthermore factors through S.
Then if A is any discrete Artinian quotient of R for which the base-changed morphism X ×Spf R Spec A → Spec A is scheme-theoretically dominant, the surjection
R → A factors through S.
Proof. The desired conclusion is equivalent to the claim that the closed immersion(5.4.16)

Spf S ×Spf R Spec A → Spec A

is an isomorphism. By Yoneda’s lemma it is enough to show that the following
condition (*) holds whenever B is a finite type A-algebra:(*) Any morphism Spec B → Spec A which can be factored through X ×Spf R
Spec A necessarily factors through the closed immersion (5.4.16).
(Indeed, if (*) holds, then, since the morphism X ×Spf R Spec A → Spec A is offinite type, by assumption, we see that it factors through (5.4.16). On the other
hand, this morphism is scheme-theoretically dominant, by assumption; thus (5.4.16)
is an isomorphism, as required.)
By considering the product of the localisations of a finite type A-algebra B at
all its maximal ideals, this will follow if we prove (*) when B is the localisationof a finite type A-algebra at one of its maximal ideals. Since such a localisationis Noetherian, this in turn will follow if we prove (*) when B is the completionof a finite type A-algebra at one of its maximal ideals. Considering the reductionof such a completion modulo the various powers of its maximal ideal, we thenreduce further to proving (*) in the the case when B is a finite type Artinian local
A-algebra. But in this case, condition (*) holds by assumption.

5.4.17. Proposition. D,C is a subfunctor of DF .
Proof. The claim of the proposition amounts to showing that the surjection R →R,C factors through RF . By Lemma 3.2.4, we may write R,C as the inverse limitof Artinian quotients A, for each of which the base-changed morphism Cd,F,a →Spec A is scheme-theoretically dominant. It suffices to show that each of the composite surjections R → R,C → A factors through RF . This will follow from
Lemma 5.4.15, taking R = R , S = RF , and X = Cd,F,Spf R , provided we showthat the hypotheses of that lemma hold.
To this end, let A be a finite type Artinian local R -algebra for which thecanonical morphism Cd,F,Spec A → Spec A admits a section, and let MA denote theétale ϕ-module corresponding to the induced morphism Spec A → Spf R . The
existence of the section to Cd,F,Spec A is, by definition, equivalent to the existenceof a projective ϕ-module MA of height F such that MA [1/u] = MA . Again bydefinition, we have MA ⊗A κ(A) = MF0 ⊗F0 κ(A). By Corollary 5.3.18, the functor
DF0 of liftings of MF0 ⊗F0 κ(A) which have a model of type F is pro-represented

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M. EMERTON AND T. GEE

by RF ⊗W (F0 ) W (κ(A)), so in particular the existence of MA implies that themorphism Spec A → Spf R factors through Spf RF , as required.

It follows from Proposition 5.4.17, together with Remark 5.3.11, that the actionof H(R) := R× +mR Md (OE,R ) on D,C (R) (for any test object R), whose existence
was noted in Remark 5.4.14, restricts to a free action of G(R) := 1+uN mR Md (SR ).
We then make the following definition (in analogy to Definition 5.3.12).
5.4.18. Definition. We let DC : CO/$a → Sets denote the functor defined by
DC (R) := D,C (R)/G(R).
By construction there is a Cartesian square
D,C

/ D


DC


/ DF

F

and the section DF → DF of Theorem 5.3.15 then restricts to a section DC → D,C .
An argument almost identical to that used in the proof of Theorem 5.3.15 showsthat DC (R) is pro-representable by some RC ; since DC is a subfunctor of DF , thispro-representing object is a quotient of the ring RF that pro-represents DF . As
the latter ring is Noetherian (by Theorem 5.3.15), so is RC .
With these various definitions and observations in place, we are now ready toprove our main theorem.
5.4.19. Theorem. The hypotheses of Theorem 1.1.1 hold for the morphism Cd,F →Rd . That is:(1) Cd,F is an algebraic stack, locally of finite presentation over Spec O/$a .
(2) Rd satisfies [3], and its diagonal is locally of finite presentation.
(3) Cd,F → Rd is a proper morphism.
(4) Rd admits versal rings at all finite type points.
(5) Rd,F satisfies [2].
Accordingly, Rd,F is an algebraic stack of finite presentation over Spec O/$a , andthe morphism Cd,F → Rd factors as Cd,F → Rd,F → Rd , with the first morphism
being a proper surjection, and the second a closed immersion.
Proof. Points (1), (2), (3) and (4) follow from Theorems 5.4.9 and 5.4.11 together
with Lemma 5.4.12, so it only remains to check (5). For this, it follows from
Lemma 2.2.5, Corollary 2.7.3, and Lemma 3.2.20 that we need only check thatthat Rd,F admits effective Noetherian versal rings at all finite type points.
The H-equivariance that was commented upon in Remark 5.4.14 implies thatb d,F,x factors through the quotient DC of D,C . The
the versal morphism D,C → RCb
induced morphism D → Rd,F,x is again versal (as one immediately checks, using
the chosen section DC → D,C ). As we have already observed, the functor DC is
pro-representable by a Noetherian ring RC .
To complete the verification of (5), we need to check that the morphism Spf RC =b d,F,x is effective. To this end, note that by Theorem 5.5.19 below, theDC → Rb d,x is induced by a morphism Spec RC → Rd . It remains tomorphism Spf RC → R
check that this morphism Spec RC → Rd factors through Rd,F . By Lemma 3.2.19,

SCHEME-THEORETIC IMAGES OF MORPHISMS OF STACKS

121

it suffices to show that the morphism Cd,F,RC → Spec RC is scheme-theoreticallydominant. This is precisely the statement of Lemma 5.5.4.
The theorem now follows from Theorem 1.1.1, except that we have only provedthat Rd,F is locally of finite presentation. In order to show that it is of finite
presentation over Spec O/$a , we must show that it is quasi-compact and quasiseparated. Since Cd,F is quasi-compact and the map Cd,F → Rd,F is surjective,
it follows from [Sta, Tag 050X] that Rd,F is quasi-compact. Since Rd has affinediagonal, by Theorem 5.4.9, and since Rd,F is a closed substack of Rd , the diagonalof Rd,F is also affine. Thus Rd,F is quasi-separated, as required.

We now describe Rd as an Ind-stack. Note that the inductive limit in the statement of the following theorem could equivalently be computed with respect to any
cofinal system of F s (see [Eme, §2]).
5.4.20. Theorem. If F |F 0 , then the natural morphism Rd,F → Rd,F 0 is a closed
immersion. Furthermore, the natural morphism limF Rd,F → Rd (where, following−→
Remark 4.2.7, the inductive limit is computed as a stack on any of the Zariski,étale, fppf, or fpqc sites) is an isomorphism of stacks. In particular, the stack Rd
is an Ind-algebraic stack which satisfies [1].
Proof. We first show that each of the morphisms Rd,F ,→ Rd,F 0 is a closed immersion; indeed, this follows from the fact that in the chain of monomorphisms
Rd,F ,→ Rd,F 0 ,→ Rdboth the composite and the second morphism are closed immersions.
It remains to be shown that the natural morphism limF Rd,F → Rd is an isomor−→phism. Since Rd satisfies [1], it suffices to show that if Spec A → Rd is a morphismwith A a Noetherian O/$a -algebra, then this morphism factors through the closed
substack Rd,F for some F . Equivalently, we must show that for some F , the closed
embedding Rd,F ×Rd Spec A ,→ Spec A is an isomorphism. It therefore suffices toshow that we may find a morphism Spec B → Spec A which is scheme-theoreticallydominant (equivalently, so that the corresponding morphism A → B is injective)
such that the induced morphism Spec B → Rd factors through Rd,F . By Proposition 5.4.7, we can find a scheme-theoretically dominant morphism Spec B → Spec A
such that Spec B → Rd factors through Rd,free . Since a free étale ϕ-module contains a free finite height ϕ-module, by Lemma 5.2.15, we see that the morphism
Spec B → Rd,free factors through Cd,F for some F , and thus through Rd,F , asrequired.

5.4.21. Remark. Rd is presumably not an algebraic stack. Indeed, since it is Indalgebraic and satisfies [1], if it were algebraic, it would be locally finite dimensional.
Since Rd is the inductive limit of its closed substacks Rd,F , it would follow that foreach finite type point x of Rd , there would be a uniform bound on the dimensionof Rd,F at x, independently of F . However, this dimension can be computed interms of the versal rings at x, and it is presumably straightforward to use thearguments of [Kim11] to compute the dimensions of the rings RF and RC andthereby obtain a contradiction (for example, in the case considered in Section 5.4.23
below, the results of [Kim11] directly imply that the algebraic stacks Rd,E h areequidimensional, with dimension growing linearly in h).

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M. EMERTON AND T. GEE

5.4.22. An alternative approach. As we now explain, by slightly altering the definitions of Cd,F and Rd , we could avoid appealing to the descent results of [Dri06],
without substantially altering our conclusions.

Namely, setting S := Spec O/$a , we define Ced,F := pro - (Cd,F )|Aff pf/S and
e d := pro - (Rd )|Aff
R. Without appealing to the results of [Dri06], we know that
pf/S

these are categories fibred in groupoids. Proposition 5.1.9 shows that (Cd,F )|Aff pf/S
is in fact an fppf stack. Using faithfully flat descent results from rigid analytic
geometry, one can similarly show that (Rd )|Aff pf/S is an fppf stack. Furthermore,
the arguments of [PR09], as adapted and modified in the present paper, show that
(Cd,F )|Aff pf/S is furthermore represented by an algebraic stack of finite type over S.
Lemma 2.5.4 then implies that this same algebraic stack represents Ced,F , whilee d is an fppf stack, which satisfies axiom [1] byLemma 2.5.5 (2) implies that R
Lemma 2.5.4. The arguments of [PR09] are easily adapted to prove the analogue ofe d . Furthermore, the proofs in the subsequent section
Theorem 5.4.9 for Ced,F and R
immediately adapt to establish the analogue of Theorem 5.4.20.
Appealing to the results of [Dri06], as we do, we in fact prove that Ced,F = Cd,F
e d = Rd . However, the primary appeal of this approach is aesthetic:
and that R
it allows us to give straightforward and natural definitions of the stacks that wewill study. In practice, and in applications, it seems that little would be lost byadopting the slightly weaker and more circumlocutious approach described here.
5.4.23. Galois representations. Let K/Qp be a finite extension with residue field k.
We now specialise to the case that q = p, ϕ(u) = up , and O = Zp . Let E be theminimal polynomial over W (k) of a fixed uniformiser π of K; then we refer to a
ϕ-module of height at most E h as a Breuil–Kisin module of height at most h. Fix ap= πn and π0 = π.
uniformiser π of K, and elements πn ∈ K, n ≥ 0, such that πn+1
Set K∞ = ∪n≥0 K(πn ), and GK∞ := Gal(K/K∞ ).
The connections between Breuil–Kisin modules, étale ϕ-modules, and Galois
representations are as follows. Let A be a finite8 Artinian Zp -algebra. By [Kis09b,
Lem. 1.2.7] (which is based on the results of [Fon94], and makes no use of therunning hypothesis in [Kis09b] that p 6= 2), there is an equivalence of abeliancategories between the category of continuous representations of GK∞ on finite Amodules, and the category of étale ϕ-modules with A-coefficients. Furthermore,
if we write T (M ) for the A-module with GK∞ -action corresponding to the étale
ϕ-module M , then M is a free SA [1/u]-module of rank d if and only if T (M ) is afree A-module of rank d.
If M is an étale ϕ-module, and V = T (M ) is the corresponding representationof GK∞ , then we say that M has height at most h if and only if it there is a Breuil–Kisin module M of height at most h with M[1/u] = M , and we say that V has
height at most h if and only if M has height at most h.
Suppose that p·A = 0. We say that a continuous representation of GK on a finite
free A-module is flat if it arises as the generic fibre of a finite flat group scheme overOK with an action of A. It follows from [Kis09b, Thm. 1.1.3, Lem. 1.2.5] (together
with the results of [Kim12] in the case that p = 2) that restriction to GK∞ induces
8Recall that an Artinian Z -algebra that is finitely generated as a Z -module is necessarilyppfinite as a set, so that the word “finite” here can be interpreted either in the commutative algebra
sense, or literally.

SCHEME-THEORETIC IMAGES OF MORPHISMS OF STACKS

123

an equivalence of categories between the category of flat representations of GK
on finite free A-modules and the category of representations of GK∞ of height at
most 1 on finite free A-modules.
The above discussion shows that in the preceding context, we may (somewhat informally) regard Ra as a moduli space of d-dimensional continuous representationsof GK∞ over Z/pa Z. Furthermore, if A is a reduced finite-Z/pa algebra, and thus a
product of finite fields, then any Breuil–Kisin module M over A is necessarily free.
Thus an A-valued point of Ra corresponds to a Galois representation of height h if
and only if it factors through Rad,E h . Indeed, we have the following result.
5.4.24. Theorem. The Fp -points of Rad,E h naturally biject with isomorphism classesof Galois representations GK∞ → GLd (Fp ) of height at most h.
Proof. Since Rd,E h is a finite type stack over Z/pa Z, any Fp -point comes froman F-point for some finite extension F/Fp , and by the definition of Rd (and thecorrespondence between étale ϕ-modules and continuous GK∞ -representations explained above), we see that we need to prove that a morphism Spec F → Rd factors
through Rd,E h if and only if the corresponding étale ϕ-module has height at most h(possibly after making a finite extension of scalars).
By the definition of Cd,E h , this latter condition is equivalent to the assertion
that the morphism Spec F → Rd factors through the morphism Cd,E h → Rd , while
by Lemma 3.2.14, the former condition is equivalent to the assertion that the fibreSpec F ×Rd Cd,E h be non-empty. Since this fibre product is a finite type F-algebraic
space (by Theorem 5.4.11 (1)), if it is non-empty it contains a point defined overa finite extension of F. Thus these conditions are indeed equivalent, if we allow
ourselves to replace F by an appropriate finite extension.

5.4.25. Corollary. There is an algebraic stack of finite type over Spec Fp , whose Fp points naturally biject with isomorphism classes of finite flat Galois representations
GK → GLd (Fp ).
Proof. In view of the equivalence between finite flat representations of GK and representations of GK∞ of height at most 1 explained above, this follows immediatelyfrom Theorem 5.4.24 (applied in the case a = h = 1).
5.5. Effectivity of projectivity. In this final subsection we prove Theorem 5.5.19,
which was used in the proof of Theorem 5.4.19 in order to show that Rd,F admits
effective Noetherian versal rings at all finite type points. The basic idea of theproof is to show that the universal family of finite height ϕ-modules on Cd,F canbe pushed forward to give a finite height ϕ-module on Spec RC , and the corresponding étale ϕ-module on Spec RC gives the required promotion of the morphism
Spf RC → Rd,F to a morphism Spec RC → Rd,F . It is, however, not obvious (atleast to us) that this construction gives a projective étale ϕ-module on Spec RC ,and that the resulting morphism Spec RC → Rd,F agrees with the given morphism
Spf RC → Rd,F , and we have to do some work to establish both of these claims.
5.5.1. Interpreting scheme-valued points of Cd,F . By definition, if A is an O/$a algebra, then giving an A-valued point of Cd,F is the same as giving a rank dprojective SA -module M equipped with a ϕ-module structure of height F . Supb denote the u-adic
pose more generally that T is a scheme over O/$a , and let Z
completion of T ×Spec Zp Spec S, i.e. the formal scheme obtained by completing this124

M. EMERTON AND T. GEE

fibre product along its closed subscheme cut out by u = 0. Lemma 5.2.4 showsthat the endomorphism ϕ of S induces a morphism ϕZb : Zb → Zb of formal schemes
over T .
Giving a T -valued point of Cd,F is then the same as giving a rank d locally free
c on Z,c → M
cb equipped with an injective morphism Φc : ϕ∗ M
coherent sheaf MbM
Zwhose cokernel is killed by F .
In our discussion below, we will be in the following situation: the scheme T will be
projective over a Noetherian affine scheme Spec A. The morphism T → Spec A theninduces a morphism Z := T ×Spec A Spec SA → Spec SA . The u-adic completionb introduced above (although theof the source is then equal to the formal scheme Z
map Z := T ×Spec A Spec SA → T ×Spec Zp Spec S need not be an isomorphism inc arises
general). In this situation, the Grothendieck existence theorem shows that M
as the u-adic completion of a rank d locally free sheaf M on Z. If π : Z → Spec SA
denotes the projection, then π∗ M is a coherent sheaf on Spec SA , which is to say, afinite type SA -module. Since multiplication by u on M is injective (as M is locallyfree), the same is true of multiplication by u on π∗ M (since π∗ is left exact).
b arises as the u-adic completion of an endomorphism
The endomorphism ϕZb of Z
ϕZ of Z; and in fact ϕZ is just the base-change to Z of the endomorphism ϕ of SA .
This latter endomorphism is finite and flat by Lemma 5.2.5, and thus so is ϕZ .
Consequently, if π : Z → Spec SA denotes the projection, we find that there is a∼natural isomorphism ϕ∗ π∗ M −→ π∗ ϕ∗Z M.
∗∗c
c
Now the morphism ΦM
c : ϕZb M → M algebraizes to a morphism ΦM : ϕZ M →M, and since the cokernel of ΦM
c is supported (set-theoretically) on the locus whereu = 0 (since it is annihilated by F ), it coincides with the cokernel of ΦM . Thus
this latter cokernel is also killed by F .
Since π∗ is left-exact, we may push forward the short exact sequenceΦM
0 → ϕ∗Z M −→M → M/ϕ∗Z M → 0

to obtain a left exact sequence
0 → π∗ ϕ∗Z M → π∗ M → π∗ (M/ϕ∗Z M),
which, using the previously noted isomorphism, we may rewrite as a left exact
sequence
0 → ϕ∗ π∗ M → π∗ M → π∗ (M/ϕ∗Z M).
Since F annihilates M/ϕ∗ M, it also annihilates π∗ (M/ϕ∗Z M), and thus we seethat π∗ M is naturally endowed with the structure of a ϕ-module of height F with
A-coefficients.
5.5.2. The pushforward of the universal finite height ϕ-module. To simplify notation, we write R for RC from now on. We also fix a descending sequence of ideals In
of R which is cofinal with the sequence mnR ; equivalently, which determines the mR adic topology on R. (For the moment, the reader can just imagine that In = mnR ;however, below we will make a different choice of In .) We write Rn := R/In , andb R [1/u] forlet Mn denote the universal étale ϕ-module over Spec Rn . We write Sccthe mR -adic completion of SR [1/u], and write M = limn Mn ; then M is a finite←−b R [1/u]-module, and we may recover the various Mn from M
c via theprojective S∼cnatural isomorphism Rn ⊗R M −→ Mn .

SCHEME-THEORETIC IMAGES OF MORPHISMS OF STACKS125

We aim to prove the following proposition.
5.5.3. Proposition. We may find a ϕ-module M0 of height F with R-coefficients0 [1/u] = M
0 [1/u] denotes m -adic completion).
c (where the hat on M\
such that M\
R
The construction of M0 involves the pushforward of the universal finite height
ϕ-module over R, and we begin by setting up some notation related to the studyof this pushforward. Note that each of the morphisms
Cd,F ×R Spec Rn → Spec Rn
is in fact a projective morphism of schemes πn : Xn → Spec Rn . These morphisms
are compatible as n varies, and by Grothendieck existence give an algebraization of
Cd,F ×Rb d,x Spf R → Spf R
to a projective morphism of schemes π : X → Spec R. (In the special case of
Breuil–Kisin modules of height 1, this is [Kis09b, Prop. 2.1.10], and the generalcase can be proved in exactly the same way (see e.g. [Kis10, Prop. 1.3] for the caseof Breuil–Kisin modules of arbitrary height); the key point is that Cd,F ×R
b d,x Spf R
inherits a natural very ample (formal) line bundle from the affine Grassmannian.)
5.5.4. Lemma. The morphism X → Spec R is scheme-theoretically dominant.
Proof. Recall that Spf R,C , which is a formal closed subscheme of Spf R , is defined to be the scheme-theoretic image of the morphism
CSpf R := Cd,F ×Rd Spf R → Spf R .
We may rephrase this as the conjunction of the following two statements: the fibre
product CSpf R,C := Cd,F ×Rd Spf R,C (which a priori is a closed formal subschemeof CSpf R ) is equal to CSpf R ; and the projection(5.5.5)

CSpf R,C → Spf R,C

is scheme-theoretically dominant, in the sense that the target of the projection isequal to its scheme-theoretic image. It is from this latter statement that we will
deduce the lemma.
As observed prior to the statement of Definition 5.4.18, there is a formal group
G over Spec O/$a acting freely on Spf R,C , and (remembering that R is ourabbreviated notation for RC ), the closed formal subscheme Spf R of Spf R,C isconstructed so that
(5.5.6)∼

Spf R ×Spec O/$a G −→ Spf R,C .

(Compare Theorem 5.3.15 and its proof.) We write CSpf R := Cd,F ×Rd Spf R. What
we will show is that
(5.5.7)

CSpf R → Spf R

is scheme-theoretically dominant; the lemma follows directly from this. (Indeed,by definition this implies that the scheme-theoretic image of X → Spec R contains
Spec Ri for any discrete quotient Ri of R, and thus is equal to Spec R; the reader
should compare the first paragraph of the proof of Lemma 3.2.4. Indeed, as one seesfrom a consideration of the remainder of that proof, the theorem on formal functions
implies that the scheme-theoretic dominance of the morphism X → Spec R ofschemes is equivalent to the scheme-theoretic dominance of the morphism (5.5.7)
of formal schemes.)126

M. EMERTON AND T. GEE

We first note that the G-action on Spf R,C lifts to a G-action on CSpf R,C .
Indeed, the morphism Spf R,C → Rd which is used in the definition of CSpf R,C asa fibre product is invariant under the action of G on its source, and so this G-action
indeed lifts to a G-action on the fibre product, i.e. on CSpf R,C . The isomorphism(5.5.6) then induces a corresponding isomorphism(5.5.8)∼

CSpf RC ×Spec O/$a G −→ CSpf R,C .

As we observed in the proof of Theorem 5.3.15, we have an isomorphism G =Spf O/$a [[(Vδ∗ )∗ ]] for a certain free O/$a -module V . From this, from (5.5.8), andfrom the scheme-theoretic dominance of (5.5.5), we deduce that (5.5.7) is schemetheoretically dominant, as required.

We set Sn := OXn (Xn ); since Xn is proper over the affine scheme Spec Rn , we
see that Sn is a finite Rn -algebra. Let Rn0 denote the image of Rn in Sn . By
the scheme-theoretic dominance of (5.5.7), the Rn0 are again a cofinal system ofArtinian quotients of R and so, replacing each Rn by Rn0 (which doesn’t change Xnor Sn ), we may, and indeed do, assume that each morphism Rn → Sn is injective.
We write Yn := Xn ×Spec Zp Spec S = Xn ×Spec Rn Spec SRn (the equality holding
because Rn is finite), and let Ybn denote the completion of Yn along the closed
subscheme “u = 0”. Equivalently, Ybn := Yn ×Spec SRn Spf SRn , where Spf SRn
is taken with respect to the u-adic topology on SRn . Each of the schemes Xn isequipped with a tautological projection to Cd,F , and we may apply the discussionof Subsection 5.5.1 to this morphism so as to obtain a locally free coherent sheaf
cn on the formal scheme Ybn , which by the Grothendieck existence theorem can beMpromoted to a locally free coherent sheaf Mn on Yn itself.
If πYn : Yn → Spec SRn denotes the projection, then the discussion of Subsection 5.5.1 shows that (πYn )∗ Mn is a ϕ-module of height F with coefficients in Rn .
Since, in particular, (π Yn )∗ Mn is u-torsion free, it naturally embeds into the étale
ϕ-module (πYn )∗ Mn [1/u]. Our next goal is to compute this étale ϕ-module, andto compare it to Mn ; we do this in (5.5.14) below.
Write
Un := Xn ×Spec Zp Spec OE = Xn ×Spec Rn Spec OE,Rn= Yn ×Spec SRn Spec OE,Rn .
The projection πn : Xn → Spec Rn induces a projection πU,n : Un → Spec OE,Rn .
The open immersion(5.5.9)Spec OE,Rn → Spec SRn

also induces an open immersion jn : Un → Yn .
In the statement of the following lemma, we regard the rank d projective OE,Rn module Mn as a locally free coherent sheaf on Spec OE,Rn .
∼∗5.5.10. Lemma. There are canonical isomorphisms jn∗ Mn −→ πU,n
Mn , compatible
with the actions of ϕ, and with varying n.

Proof. Choose an OE,Rn basis for Mn , and let N denote the free SRn -submodule∼of Mn generated by this basis. Then we have an isomorphism limi u1i N −→ Mn .
−→

SCHEME-THEORETIC IMAGES OF MORPHISMS OF STACKS

127

Pulling back via πn , we obtain an isomorphism of quasi-coherent sheaves
1∼(5.5.11)
lim i πn∗ N −→ πn∗ Mn−→ uion Yn .
∗
We pull the isomorphism (5.5.11) back to Ybn . On the one hand, if we let πdnN
1 d∗denote the u-adic completion of N, then the left hand side pulls back to limi ui πn N.
−→
On the other hand, the defining relationship between Mn and the u-adic complecn of Mn (namely, that Mn is defined by the morphism Rn → Rd that wastion M
itself used to construct the morphism Xn → Cd , which in turn defines the cohercn on Ybn ) shows that π ∗ Mn pulls back to the sheaf lim 1i M
c on Ybn .
ent sheaf Mn−→i u i
1 d
1cn )i are inductive systems of coherent sheaves on Ybn withSince ( ui πn∗ N)i and ( ui M
injective transition maps and having the same inductive limit on Ybn , we see thatthey are mutually cofinal. The equivalence of categories provided by Grothendieck
existence then shows that ( u1i πn∗ N)i and ( u1i Mn )i are mutually cofinal inductive
systems of coherent sheaves on Yn itself. Returning to (5.5.11), we find that1∼
lim i Mn −→ πn∗ Mn .
−→ ui
The left-hand side of this isomorphism is naturally identified with (jn )∗ jn∗ Mn . The
natural adjunction between jn∗ and (jn )∗ then shows that∼∗jn∗ Mn −→ jn∗ πn∗ Mn = πU,nMn ,as required.



Since jn is the base change of the open immersion (5.5.9), we find that
(5.5.12)(πU,n )∗ jn∗ Mn = OE,Rn ⊗SRn (πY,n )∗ Mn = (πY,n )∗ Mn [1/u].
Since OE,Rn is flat over Rn , we find that (πU,n )∗ OUn = Sn ⊗Rn OE,Rn = OE,Sn ,and so by the projection formula,∗(πU,n )∗ πU,n
Mn = OE,Sn ⊗OE,Rn Mn = Sn ⊗Rn Mn .

By Lemma 5.5.10, we may rewrite this as(5.5.13)(πU,n )∗ jn∗ Mn = Sn ⊗Rn Mn .

Comparing (5.5.12) and (5.5.13), we find that
(5.5.14)(πY,n )∗ Mn [1/u] = Sn ⊗Rn Mn .
Since Rn → Sn is injective, and since Mn is flat over Rn (being projective overthe flat Rn -algebra OE,Rn ), we find that the natural map Mn → Sn ⊗Rn Mn is
injective. By the left exactness of pushforward, we see that (πY,n )∗ Mn is u-torsion
free, so the natural map (πY,n )∗ Mn → (πY,n )∗ Mn [1/u] = Sn ⊗Rn Mn is also
injective. Consequently, we may define
M0n := (πY,n )∗ Mn ∩ Mn ,the intersection taking place in Sn ⊗Rn Mn . This is a ϕ-stable SRn -submoduleof Mn , whose formation is compatible with change in n.
5.5.15. Lemma. The inclusion M0n ,→ Mn induces an isomorphism M0n [1/u] =
Mn .

128

M. EMERTON AND T. GEE

Proof. By virtue of the definition of M0n , we find that
M0n [1/u] = (πY,n )∗ Mn [1/u] ∩ Mn = (Sn ⊗Rn Mn ) ∩ Mn = Mn ,the second-to-last equality following from (5.5.14).



We now define M0 := limn M0n ; this is an SR -module, which we endow with its←−
projective limit topology (in which each M0n is equipped with the discrete topology).
5.5.16. Lemma. The SR -module M0 is of finite type, and the projective limit topology on M0 coincides with the mR -adic topology on M0 .
In order to prove this lemma, we revisit some of the preceding constructions,with the various Xn replaced by the projective R-scheme X. To be precise, wewrite Y := X ×Spec R Spec SR , and write Yb to denote the mR -adic completion of Y .
(Note that this notation is not compatible with the analogous notation Ybn , whichb denote the mR -adic completion
denoted u-adic completion. If we similarly let X
b∼of X, then by the construction of X, there is an isomorphism X= Cd,F ×R
b d,x Spf R,∼band consequently, an isomorphism Y = Cd,F × b Spf SR , where Spf SR is formed
Rd,xwith respect to the mR -adic topology on SR .)We may regard Yb as the ind-scheme limn Yn , and the compatible (in n) collection−→
c on theof coherent sheaves Mn on the various Yn give rise to a coherent sheaf M
formal scheme Yb . By the Grothendieck existence theorem, this coherent sheaf arises
as the completion of a coherent sheaf M on the projective SR -scheme Y .
If we write πY for the projection Y → Spec SR , then the theorem on formal
functions shows that (πY )∗ M is a finite type SR -module, and that there is an
isomorphism(5.5.17)∼(πY )∗ M −→ lim(πY,n )∗ Mn ,←−nwhich furthermore identifies the mR -adic topology on the source with the projective
limit topology on the target.
Proof of Lemma 5.5.16. Since each M0n is an SRn -submodule of (πY,n )∗ Mn , it follows from (5.5.17) and the left exactness of inverse limits that M0 (= limn M0n )←−is an SR -submodule of (πY )∗ M. Since SR is Noetherian, and M0 is an SR submodule of the finite type SR -module (πY )∗ M, we see that M0 is of finite type.
The projective limit topology on M0 coincides with the topology induced by the
projective limit topology on (πY )∗ M induced by the right-hand side of the isomorphism (5.5.17). Thus it also coincides with the topology induced by the mR -adic
topology on (πY )∗ M, which by the Artin–Rees lemma is equal to the mR -adic
topology on M0 .
Proof of Proposition 5.5.3. We have already seen in Lemma 5.5.16 that M0 is a
finite type SR -module, and Lemmas 5.5.15 and 5.5.16 together show that
0 [1/u] = lim (M0 /mn M0 )[1/u] = lim M0 [1/u] = lim M = M
c.
M\
R
←
−
←
− n
←
− n
n
n
n

Since each M0n embeds into Mn = M0n [1/u], we see that each M0n is u-torsion free;thus M0 is also u-torsion free.

SCHEME-THEORETIC IMAGES OF MORPHISMS OF STACKS129

It remains to show that the ϕ-module structure on M0 makes it of height F .
The endomorphism ϕ of SR is finite flat, and thus ϕ∗ M0 = limn ϕ∗ M0n . It hence←−
suffices to show that each M0n is of height F .
We have already seen that each of the pushforwards (πYn )∗ Mn is of height F .
Now let x ∈ M0n = (πYn )∗ Mn ∩ Mn be arbitrary. Then F x ∈ F (πY,n )∗ Mn , sosince (πY,n )∗ Mn has height F , we see that there is some y ∈ ϕ∗ (πY,n )∗ Mn with
Φ(πY,n )∗ Mn (y) = F x. Since ΦMn : ϕ∗ Mn → Mn is a bijection, we have y ∈ ϕ∗ Mn ,so y ∈ ϕ∗ Mn ∩ϕ∗ (πY,n )∗ Mn . Since ϕ is flat, we have ϕ∗ Mn ∩ϕ∗ (πY,n )∗ Mn = ϕ∗ M0n ,so y ∈ ϕ∗ M0n , and M0n is of height F , as required.

5.5.18. Descent of projectivity. We now come to the main result of this subsection.
5.5.19. Theorem. There is a projective étale ϕ-module M over RC whose mRC -adic
completion is isomorphic to the base change to RC of the (formal) universal étale
ϕ-module over R .
Proof. We write M := M0 [1/u], where M0 is the ϕ-module of height F over RC
constructed in Proposition 5.5.3. By construction, M is an étale ϕ-module with
RC -coefficients, and the mRC -adic completion of M is a finite projective moduleb RC [1/u], and corresponds to the morphism Spf RC → Rd . Theorem 5.5.20
over S
below then shows that M itself is a projective étale ϕ-module, as required.

5.5.20. Theorem. Let R be a complete Noetherian local O/$a -algebra R with maximal ideal m, let M be an étale ϕ-module with R-coefficients, and suppose that the\ the m-adic
c is projective, or equivalently, free (over R((u)),
m-adic completion M
completion of R((u))). Then M itself is projective (over R((u))).
The remainder of this subsection is devoted to the proof of this theorem. However, before giving the proof in detail, we give an outline of it in the case that
R = K[[T1 , . . . , Tn ]], where K is a field of characteristic p. In this case, the ringR((u)) may be regarded as the ring of bounded holomorphic functions on the open
unit n-dimensional polydisk Dn over the complete discretely valued field K((u)),\ may be regarded as the formal completion of theand its m-adic completion R((u))\
local ring ODn ,0 , where 0 denotes the origin of Dn . The extension ODn ,0 → R((u))∼\
cis then faithfully flat, and since M −→ R((u)) ⊗R((u)) M is projective, the same istrue of ODn ,0 ⊗R((u)) M.
Writing the stalk as a direct limit of rings of holomorphic functions on a nested
sequence of polydisks centred at 0 of shrinking radius, we find that the restrictionof M to one of these polydisks is projective. The étale ϕ-module structure on M
then allows us to employ Frobenius amplification (“Dwork’s trick”) to show that M
itself gives rise to a projective module over the ring O(Dn ) of holomorphic functions
on Dn — i.e. we find that O(Dn ) ⊗R((u)) M is projective. Finally, the inclusion
R((u)) ,→ O(Dn ) of bounded holomorphic functions in all (i.e. not necessarily
bounded) holomorphic functions is faithfully flat, and so M itself is projective overR((u)).
The proof in the general case follows the same outline: we regard R((u)) as being
the ring of bounded holomorphic functions on a closed analytic subvariety of anopen unit polydisk and make the same Frobenius amplification argument. We passfrom the case of k-algebras to O/$a -algebras via the usual graded techniques.
We now fill in the details of the preceding sketch.

130

M. EMERTON AND T. GEE

Proof of Theorem 5.5.20. Assume to begin with that R is a complete Noetherian
local k-algebra; we may then write R = K[[T1 , . . . , Tn ]]/I, for some field extension
K of k, and some ideal I. For any integer m ≥ 0, we writeVmm

Am := R[x1 , . . . , xn , u]/(up x1 − T1 , . . . , up xn − Tn )[1/u],
and
V

m

m

Bm := R[x1 , . . . , xn , u]/(ux1 − T1p , . . . , uxn − Tnp )[1/u];
in both cases, the hat indicates u-adic completion. There are natural morphismsof R((u))-algebras(5.5.21)

Am ,→ Am+1 ,mdefined by mapping xi in the source to up (p−1) xi in the target, andBm+1 ,→ Bm ,defined by mapping xi in the source to up−1 xpi in the target; Lemmas 5.5.33
and 5.5.36 below (together with Remark 5.5.34) show that all these morphisms
are flat. There are also evident isomorphisms of R((u))-algebras(5.5.22)∼R((u)) ⊗ϕ,R((u)) Am −→ Am+1 ,defined by mapping 1 ⊗ xi in the source to xi in the target, as well as isomorphisms(5.5.23)∼R((u)) ⊗ϕ,R((u)) Bm+1 −→ Bm ,defined by mapping 1 ⊗ xi in the source to xpi in the target. (It is less evident
that the morphisms (5.5.23) are isomorphisms, but this follows from Lemma 5.5.35
below.) Note also that A0 = B0 .
We write A := limm Am . Being the direct limit of a sequence of Noetherian rings−→
with respect to flat transition morphisms, we see immediately that A is a coherent
ring. In fact, something stronger is true: A is a Noetherian local R((u))-algebra,
and there is an isomorphism
∼
\
b −→
A
R((u)),
where now the hats indicate m-adic completion. To see this, it suffices to consider
the case when the ideal I is zero. (Standard arguments with adic completions offinite type modules over Noetherian rings, and with direct limits, show that the\ are compatible with theb and R((u))
constructions of the various rings Am , A, A,
passage from R to R/J, for any ideal J of R. Now apply this observation to
the ideal I in the ring K[[T1 , . . . , Tn ]].) In this case, we see that A is the ring ofgerms of holomorphic functions at the origin of the unit n-dimensional polydisk
over the complete discretely valued field K((u)), and this ring is well-known to
be Noetherian, and to have the formal power series ring K((u))[[T1 , . . . , Tn ]] as its
m-adic completion (see [BGR84, Prop. 7, §7.3.2]).
The discussion of the preceding paragraph shows that the embedding A ,→\ is faithfully flat, and since M
c (which, by the Artin–Rees lemma, may beR((u))\ ⊗R((u)) M ) is projective, we conclude that
identified with R((u))
A ⊗R((u)) M

SCHEME-THEORETIC IMAGES OF MORPHISMS OF STACKS

131

is also projective over A. This in turn implies that
Am ⊗R((u)) M
is projective over Am , for some sufficiently large value of m. (Indeed, this projectivity is witnessed by a split surjection from a finite free A-module, and both thissurjection, and a splitting of the surjection, can be descended to some Am .)Composing the inclusion Am−1 → R((u)) ⊗ϕ,R((u)) Am−1 , defined via a 7→ 1 ⊗ a,with the isomorphism (5.5.22), we may regard Am as a faithfully flat Am−1 -algebra.
(Note that this is not the same as the R((u))-linear map Am−1 → Am consideredin (5.5.21).) We then compute that∼

Am ⊗Am−1 (Am−1 ⊗R((u)) M ) −→ Am ⊗ϕ,R((u)) M∼∼−→ Am ⊗R((u)) (R((u)) ⊗ϕ,R((u)) M ) −→ Am ⊗R((u)) M(the last isomorphism following from the étale ϕ-module structure on M ). Since
the property of being finitely generated projective can be detected after a faithfully flat base-change, we find that Am−1 ⊗R((u)) M is projective. Continuing via
descending induction, we conclude that A0 ⊗R((u)) M is a projective A0 -module,or equivalently, a projective B0 -module. A completely analogous argument, taking
into account (5.5.23), then shows that Bm ⊗R((u)) M is projective for each m ≥ 0.
Write B := limm Bm . Since the transition morphisms in this projective limit are←−flat morphisms of Noetherian Banach algebras over K((u)), we find that that B
is a Fréchet–Stein algebra, in the sense of [ST03]. As a consequence, any finitely
presented B-module N is coadmissible over B (by [ST03, Cor. 3.4]), so that thenatural morphism N → limm Bm ⊗B N is an isomorphism. In particular, if X is a←−
finitely generated (and hence finitely presented) R((u))-module, then B ⊗R((u)) X
is a finitely presented B-module, and hence there is an isomorphism(5.5.24)∼

B ⊗R((u)) X −→ lim Bm ⊗R((u)) X.
←−m

If we take X to be a finitely generated ideal I of R((u)), then (5.5.24) showsthat the morphism(5.5.25)

B ⊗R((u)) I → B

may be written as the projective limit of morphisms Bm ⊗R((u)) I → Bm . By
Lemma 5.5.32 below, each Bm is flat over R((u)), so we find that each of theselatter morphisms is injective. Thus so is the morphism (5.5.25). It follows that B
is flat over R((u)).
In fact B is faithfully flat over R((u)). To show this, it suffices (given the flatnessthat we have already proved, and the fact that flat maps satisfy going down) toshow that each maximal ideal of R((u)) is obtained via restriction from a maximal
ideal of B. Any such maximal ideal is the kernel of a surjection(5.5.26)R((u)) → L,

where L is a finite extension of K((u)) (see the proof of [dJ95, Lem. 7.1.9] for a
proof of this fact), and each Ti maps to an element ai ∈ L satisfying |ai | < 1 (i.e.
each ai lies in the maximal ideal of the ring of integers of L). If we choose m somthat |ai | ≤ |u|1/p for each ai , then the surjection (5.5.26) extends to a surjection
Bm → L. (This extension of the surjection R((u)) → L to a surjection Bm → L forsome sufficiently large value of m is also explained carefully in the proof of [dJ95,

132

M. EMERTON AND T. GEE

Lem. 7.1.9].) The kernel of the composite B → Bm → L is then a maximal idealof B that restricts to the kernel of (5.5.26).
Since M is finitely generated over R((u)), we obtain, from (5.5.24), an isomorphism∼

B ⊗R((u)) M −→ lim Bm ⊗R((u)) M ;←−msince each Bm ⊗R((u)) M is projective over Bm , Lemma 5.5.28 below shows that
B ⊗R((u)) M is projective over B. Since R((u)) → B is faithfully flat, we find that
M is projective, as required.
Finally, consider the general case of the theorem, in which R is assumed to
be a complete Noetherian local O/$a -algebra, without a necessarily equalling 1.
\ M , and M
c with its $-adic filtration, andEndow each of O/$a , R, R((u)), R((u)),•
alet Gr O/$ , etc., denote the corresponding associated graded object. We findthat Gr• O/$a = k[ε]/(εa ), that Gr• R is a complete local Noetherian algebra overk[ε]/(εa ), that Gr• R((u)) = (Gr• R)((u)), and that Gr• M is an étale ϕ-module overGr• R.
Since m-adic completion of finitely generated modules over a Noetherian local
ring (such as R((u))) is exact, we also see that the m-adic completion of Gr• R((u))\ and that the m-adic completion of Gr• M
is naturally isomorphic to Gr• R((u)),•cc is finitely generated
is naturally isomorphic to Gr M . The assumption that M••\\ then implies that Gr
c is finitely generated
and projective over R((u))
M = Gr M•• \\and projective over Gr R((u)) = Gr R((u)). (This is easily seen if one uses theequivalence between a module being finitely generated and projective and being a
direct summand of a finite rank free module.) The case of the theorem alreadyproved then shows that Gr• M is finitely generated and projective over Gr• R((u)),
or, equivalently, finitely generated and flat over Gr• R((u)). Lemma 5.5.37 (appliedwith R = O/$a and A = R((u))) shows that M is finitely generated and flat —and thus projective — over R((u)).
The following lemmas, which were used in the proof of the preceding theorem,
are presumably well-known to experts, but we include proofs, for lack of a reference.
5.5.27. Lemma. If B is a commutative Fréchet–Stein algebra (over a complete
discretely valued field K), if M is a finitely presented B-module, and if N is acoadmissible B-module, then M ⊗B N is again a coadmissible B-module.
Proof. If we choose a presentation B r → B s → M → 0, then we obtain a rightexact sequence
N r → N s → M ⊗B N → 0.
Since N is coadmissible, so are each of N r and N s , and thus so is M ⊗B N, being
the cokernel of a morphism between coadmissible B-modules.

5.5.28. Lemma. If B is a commutative Fréchet–Stein algebra over a complete dis∼cretely valued field K, say B −→ limn Bn , where each Bn is a Noetherian Banach←−
K-algebra, with the transition morphisms being flat, and if M is a finitely presented
B-module with the property that each tensor product Bn ⊗B M is a projective Bn module, then M is a projective B-module.

SCHEME-THEORETIC IMAGES OF MORPHISMS OF STACKS133

Proof. Since M is a finitely presented B-module, it is furthermore projective if andonly it is flat. To show that M is flat, it suffices to show that for each finitely
generated ideal I ⊆ B, the induced morphism M ⊗B I → M is injective.
Since I is finitely generated, it is the image of a morphism B r → B, for some r ≥0, and thus is coadmissible. Lemma 5.5.27 then shows that M ⊗B I is coadmissible.
The B-module M itself is also coadmissible (being finitely presented). Thus themorphism(5.5.29)

M ⊗B I → M

may be obtained as the projective limit of the morphisms
Bn ⊗B ⊗M ⊗B I → Bn ⊗B M.
We may rewrite each of these morphism as(5.5.30)(Bn ⊗B ⊗M ) ⊗Bn (Bn ⊗B I) → Bn ⊗B M.

Since Bn is flat over B, we see that the inclusion of I in B induces a sequenceof injections Bn ⊗B I ,→ Bn . Since Bn ⊗B M is projective, and thus flat, over
Bn , the morphisms (5.5.30) are then also injective. Hence so is their projective
limit (5.5.29).

5.5.31. Lemma. Let A be a ring, and a an element of A. If M → N is a morphismof A-modules whose kernel and cokernel are each annihilated by some power of a,c[1/a] → Nb [1/a] (where b denotes a-adic completion)then the induced morphism M
is an isomorphism.
Proof. This is standard, and follows easily from the definitions.



5.5.32. Lemma. Let A be a Noetherian ring, and let a, b1 , . . . , bm be elements of A.
If we write B := A[x1 , . . . , xm ]/(ax1 − b1 , . . . , axm − bm ), then the natural mapbb
A[1/a]→ B[1/a](where b denotes a-adic completion) is flat.
Proof. We begin with the case m = 1, where we write x, y for x1 , y1 , and b for b1 .
Note that each morphism in the sequence of natural morphisms
b → A[x]/(axbbA− b) → A[1/a]becomes an isomorphism after inverting a. Indeed, this is evidently the case for
their composite, and it is also evident that the first morphism becomes surjective
after inverting a. Similarly, each morphism in the sequence of natural morphismsb ⊗A B = A[x]/(axbbB := A[x]/(ax − b) → A− b) → B
becomes an isomorphism after passing to a-adic completions. Thus, in order toprove the lemma in the case m = 1, it suffices to note that the natural morphismbfrom A[x]/(ax− b) to its a-adic completion is flat, as follows from the Artin–Reesb is Noetherian, as A is). (Note that by the discussion
lemma (and the fact that Abbabout, the natural map A[1/a]→ B[1/a]is obtained from this map by inverting a.)
The general case follows by induction on m. Indeed, writing
C = A[x1 , . . . , xm−1 ]/(ax1 − b1 , . . . , axm−1 − bm−1 ),
b
b
b
b
b
we can factor A[1/a]→ B[1/a]as A[1/a]→ C[1/a]→ B[1/a],with the first map
being flat by the inductive hypothesis, and the second being flat by the case m = 1,
as B = C[xm ]/(axm − bm ).


134

M. EMERTON AND T. GEE

The following result is a somewhat technical modification of the precedinglemma.
5.5.33. Lemma. Let A be a Noetherian ring, and let a, a0 , b1 , . . . , bm , b01 , . . . , b0m
be elements of A. If we write B := A[x1 , . . . , xm ]/(ax1 − b1 b01 , . . . , axm − bm b0m )and C := A[y1 , . . . , ym ]/(aa0 y1 − b1 , . . . , aa0 ym − bm ), then there is a morphism of
A-algebras B → C defined by mapping each xi to a0 b0i yi , and the induced morphism
00bbB[1/aa] → C[1/aa] (where b denotes aa0 -adic completion) is flat.
Proof. We factor the morphism B → C as
B → B[y1 , . . . , ym ]/(aa0 y1 − b1 , . . . , aa0 ym − bm )→ B[y1 , . . . , ym ]/(aa0 y1 − b1 , x1 − a0 b01 y1 , . . . , aa0 ym − bm , xm − a0 b0m ym ) = C.
The second morphism is surjective, and its kernel, which is the ideal (x1 −a0 b01 y1 , . . . , xn − a0 b0m ym ), is annihilated by a. Lemma 5.5.31 shows that thissecond morphism becomes an isomorphism after aa0 -adically completing and theninverting aa0 . Thus it suffices to show that the first morphism becomes flat after
aa0 -adically completing and inverting aa0 ; this follows from Lemma 5.5.32.

5.5.34. Remark. We note, in the context of the preceding lemma, that if (a0 )r = asfor some r, s ≥ 1, then aa0 -adically completing is the same as a-adically completing,and inverting aa0 is the same as inverting a.
We also have the following variations on the preceding results.
5.5.35. Lemma. Let A be a Noetherian ring, and let a, b1 , . . . , bm be elements of A,
and let n be a positive integer. If we write B := A[x1 , . . . , xn ]/(an x1 −bn1 , . . . , an xn −bnm ) and C = A[y1 , . . . , yn ]/(ay1 −b1 , . . . ayn −bm ), then the morphism of A-algebras∼ bb
B → C defined by xi 7→ yin induces an isomorphism B[1/a]−→ C[1/a](where b
denotes a-adic completion).
Proof. The morphism B → C is finite: C is generated as a B-module by the variousem, for 1 ≤ ei ≤ n − 1. The image of yij in the cokernel of thismonomials y1e1 · · · ymmorphism is annihilated by aj , so the entire cokernel is annihilated by am(n−1) .
Note that the morphisms A[1/a] → B[1/a] → C[1/a] are all isomorphisms, sothat the kernel of the morphism B → C is contained in the kernel of the morphism
B → B[1/a]. Each element of this kernel is annihilated by some power of a. Since
this kernel is finitely generated (as B is Noetherian), we see that this entire kernel
is annihilated by some power of a. Combining this with the conclusion of thepreceding paragraph, and with Lemma 5.5.31, establishes the lemma.

5.5.36. Lemma. Let A be a Noetherian ring, let a, b1 , . . . , bm be elements of A, and
let n be a positive integer. If we write B := A[x1 , . . . , xm ]/ ax1 − bn1 , . . . , axm − bnm
and C = A[y1 , . . . , ym ]/(ay1 − b1 , . . . , aym − bm ), then the morphism of A-algebras∼ bb
B → C defined by xi 7→ an−1 yin induces a flat morphism B[1/a]−→ C[1/a](whereb denotes a-adic completion).
Proof. We factor the morphism B → C as
B = A[x1 , . . . , xm ]/ ax1 − bn1 , . . . , axm − bnm→ A[t1 , . . . , tm ]/(an t1 − bn1 , . . . , an tm − bnm )→ A[y1 , . . . , ym ]/(ay1 − b1 , . . . , aym − bm ) = C,SCHEME-THEORETIC IMAGES OF MORPHISMS OF STACKS

135

where the first morphism is defined by xi 7→ an−1 ti , and the second morphism isdefined by ti 7→ yin . The present lemma then follows from Lemmas 5.5.33 and 5.5.35(taking into account Remark 5.5.34).

Let R be an Artinian local ring, with maximal ideal I and residue field k. If M
is an R-module, then we let Gr• M denote the graded k-vector space associated to
the I-adic filtration on M (so Gri M := I i M/I i+1 M ).
5.5.37. Lemma. If A is an R-algebra, then an A-module M is (faithfully) flat over
A if and only if Gr• M is (faithfully) flat over Gr• A. Furthermore, if any of these
conditions holds, then the natural morphism Gr• A ⊗Gr0 A Gr0 M → Gr• M is an
isomorphism.
Proof. If M is flat over A, then considering the result of tensoring M by the various
short exact sequences
0 → I n A → I m A → I m A/I n A → 0
we find that the natural morphism Gri A ⊗Gr0 A Gr0 M → Gri M is an isomorphism,for each i. This proves the final assertion of the lemma. Furthermore, since (faithful) flatness is preserved under base-change, we see first that Gr0 M is flat over
Gr0 A (and faithfully flat if M is faithfully flat over A), and then (using the result
already proved) that Gr• M is flat over Gr• A (and faithfully flat if M is).
It is not quite as obvious that flatness of Gr• M over Gr• A implies the flatnessof M over A, but this is a standard fact in commutative algebra; e.g. it follows
from [Sta, Tag 0AS8]. (If i ≥ 0, then base-changing via the map Gr• A → Gr≤i A :=A/IA ⊕ I/I 2 ⊕ · · · ⊕ I i /I i+1 , we find that Gr≤i M := M/IM ⊕ · · · ⊕ I i M/I i+1 M isflat over Gr≤i A. In particular, the embedding I i /I i+1 A =: Gri A ,→ Gr≤i A induces
an embedding∼
I i /I i+1 ⊗A M −→ Gri A ⊗Gr≤i A Gr≤i M ;concretely, this means that the morphism
I i /I i+1 ⊗A M → I i M/I i+1 M
is an embedding. Letting i vary, and recalling that I is nilpotent, we deducefrom [Sta, Tag 0AS8] that M is flat over A.)
If Gr• M is furthermore faithfully flat over Gr• A, then (since faithful flatness ispreserved under base-change) we see that Gr0 M := M/IM is faithfully flat over
Gr0 A := A/IA. Because I is nilpotent, an A-module vanishes if and only if its
reduction mod I does, and we conclude that M is faithfully flat over A.
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Email address: emerton@math.uchicago.edu
Department of Mathematics, University of Chicago, 5734 S. University Ave., Chicago,
IL 60637
Email address: toby.gee@imperial.ac.uk
Department of Mathematics, Imperial College London, London SW7 2AZ, UK