FORMAL ALGEBRAIC STACKS
MATTHEW EMERTON

Abstract. We develop the basic theory of formal algebraic stacks, closely
following the development of the theory of formal algebraic spaces in the Stacks
Project [Stacks].

Contents
1. Introduction
2. 2-categorical background
3. Stack-theoretic background
4. Ind-algebraic stacks
5. Formal algebraic stacks
6. Ind-algebraic stacks and formal algebraic stacks
7. Adic morphisms
8. Properties of formal algebraic stacks
9. Functorial closed substacks of formal algebraic stacks
10. Morphisms to affine formal algebraic spaces
11. A criterion for a formal algebraic stack to be locally Noetherian
References

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1. Introduction
Our goal in this paper is to introduce some basic definitions and results related
to the notion of a formal algebraic stack over a scheme S.
1.1. Overview. In the theory of formal schemes, one finds a tension between two
points of view. Firstly, a formal scheme admits an open cover by affine formal
schemes, and an affine formal scheme may be regarded as a geometric object (theformal spectrum) associated to a certain kind of complete ring. In particular, we canultimately refer back to these complete rings for defining various local properties ofa formal scheme, or for making geometric constructions involving formal schemes.
As simple examples: the notion of a locally Noetherian formal scheme rests on
having such ring-based models for the local structure of formal schemes, as does
the notion of a p-adic formal scheme being flat over Zp , and the construction of theflat part of a p-adic formal scheme which is not itself flat over Zp .
On the other hand, formal schemes are also often profitably regarded as certain
kinds of Ind-schemes, namely Ind-schemes with respect to transition morphismsthat are thickenings (in the sense that they induce isomorphisms on the underlying
The author is supported in part by the NSF grants DMS-1303450 and DMS-1601871.
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reduced subschemes). This point of view fits well into the functor-of-points perspective, and in practice, formal schemes often arise as Ind-schemes in this way.
However, certain properties of formal schemes are difficult to access from this viewpoint: consider the difficulty of deciding whether or not a certain p-adic formal
scheme is flat over Zp , when it is given as a direct limit of schemes, on each ofwhich p is locally nilpotent! (Even less obvious is how to construct the Zp -flat partof such a formal scheme, when it is described as an Ind-scheme in this way.)
In the Stacks Project [Stacks] (whose overall framework we follow) the theoryof formal schemes is incorporated into the more general theory of formal algebraic
spaces [Stacks, Tag 0AHW], and the latter theory is developed in a manner thattakes into account both of the viewpoints described above. Formal algebraic spaces
are always regarded as sheaves on the big fppf site of some base scheme S (so thefunctor-of-points viewpoint is built in to the foundations), and affine formal algebraic spaces are defined as Ind-affine schemes [Stacks, Tag 0AI7] with thickenings
as transition morphisms (the inductive limit being formed in the category of fppf
sheaves), while general formal algebraic spaces are defined as sheaves that admit anétale cover by a disjoint union of affine formal algebraic spaces [Stacks, Tag 0AIL].
However, under mild hypotheses (quasi-compactness and quasi-separatedness),one can show that a formal algebraic space admits a description as an Ind-algebraic
space with thickenings as transition morphisms [Stacks, Tag 0AJE] (see also [Yas09,
Prop. 3.32], where this result is originally proved, although in a somewhat different
foundational set-up). Conversely, such an Ind-algebraic space is in fact a formal
algebraic space [Stacks, Tag 0AIU]. This latter result, while not difficult to prove(the key input is the invariance of the étale site with respect to thickenings), is
crucial. It shows that the Ind-object under consideration may be described (étale
locally) in terms of certain complete rings, which, as already noted above, provides
a key input to many of the basic definitions and constructions in the theory offormal schemes and formal algebraic spaces.
In this paper we develop the theory of formal algebraic stacks along similar lines
to the Stacks Project’s development of the theory of formal algebraic spaces. In
particular, while our definition of a formal algebraic stack in Section 5 will be interms of having local models that are affine formal algebraic spaces (local now in thesmooth topology), in Section 6 we show that (under the hypotheses of being quasicompact and quasi-separated) they admit a description as Ind-algebraic stacks, and
we also show that certain Ind-algebraic stacks are formal algebraic stacks. (Themain obstacle to be overcome in the context of stacks is that the smooth site is notinvariant under thickenings, so that the construction of smooth local affine formal
models for the Ind-algebraic stacks under consideration is more involved in generalthan in the case of formal algebraic spaces.)
In Section 7 we define the notion of adic morphisms, which provides the relative
version of the notion of an adic formal algebraic stack. (For example, if A is a
Noetherian ring complete with respect to an ideal I, then an I-adic formal algebraic
stack consists of a formal algebraic stack equipped with an adic morphism to Spf A.)
In Section 8 we develop a general framework for defining various properties of formal
algebraic stacks in terms of properties of affine formal algebraic spaces, and thenapply it to define a number of such properties.
The remaining sections are somewhat more specialized to the applications that
we have in mind. In Section 9, we develop a framework for defining closed substacks

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of formal algebraic stacks in a functorial manner. For example, this allows us todefine the flat part of a locally Noetherian formal algebraic stack over Spf Zp . In
Section 10, we prove a technical result which, in applications, will allow us to provethat, in certain situations, the “scheme-theoretic image” of a p-adic formal algebraic
stack is again a p-adic formal algebraic stack. Finally, in Section 11, we provide
a criterion (phrased in terms of the existence of a certain obstruction theory) for
a formal algebraic stack over a locally Noetherian base scheme to itself be locally
Noetherian.
The earlier sections develop necessary preliminary material. Section 2 gives a
rapid development of the theory of colimits in 2-categories, which forms the technical basis for the theory of Ind-algebraic stacks. We follow the very concrete
view-point on 2-categories adopted in [Stacks], and also in [GR04]. We recommend that the reader skip or at best skim this section, and rely instead on theirintuition regarding the formation of Ind-stacks. (We also note that in this draft,some arguments in Section 2 are incomplete or missing. We hope to fill them in
eventually.)
Section 3 discusses several other stack-theoretic notions that will be required in
our later development. The emphasis is on extending various geometric properties
and constructions (properties of morphisms, underlying reduced substacks, topological invariance of the small étale site) from the context of algebraic stacks tomore general contexts, so that we may apply them when studying formal algebraic
stacks or Ind-algebraic stacks. Ind-algebraic stacks themselves are formally definedin Section 4.
We anticipate that much of the material in this paper will ultimately be incorporated into the Stacks Project, and indeed, some of it already has been, sometimes
undergoing improvements in the process (e.g. Lemma 6.4 below).

1.2. Conventions. We will work in the overall framework of the Stacks Project[Stacks], and so will work with stacks on the fppf site of S, although we could
equally well have worked with stacks on the étale site.
Throughout the paper, all stacks under consideration will be stacks in groupoids,
and so we usually write simply stack rather than the more cumbersome stack ingroupoids. (We make an exception in the foundational Section 2, where we will
write stacks in groupoids, just to make the context of our discussion clear.)

1.3. Acknowledgements. The writing of this paper was prompted by the requirements of the papers [Car+19] and [EG19a], and I would like to thank my coauthors
Ana Caraiani, Toby Gee, and David Savitt for a fruitful collaboration, as well as fortheir helpful comments on the present paper. Toby Gee in particular has stewarded
the development of this paper since its inception, and I am indebted to him for his
encouragement, and for his careful reading of its many iterations.
I would also like to thank Bao V. Le Hung, whose suggestions led to my original
proof of Lemma 6.4, as well as Brian Conrad, who read an earlier version of themanuscript closely and made many helpful suggestions for improvement. Finally,
I owe a special thanks to Johan de Jong, for his support and encouragement, andfor his work on incorporating the material of this paper into the Stacks Project.

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2. 2-categorical backgroundIn this section, we recall and develop various 2-categorical notions that we will
require in the sequel. The most important of these is the notion of 2-colimits ina 2-category, which we will use in Section 4 to define the notion of Ind-algebraic
stacks. We also recall the notion of a strict pseudo-functor between 2-categories,
which will be useful for describing the functorial properties of certain constructions
we make later on (such as underlying reduced substacks), as well as some related
notions.
The discussion that follows would perhaps be better expressed in a more flexible
higher categorical language. However, we follow the more concrete approach of thestacks project [Stacks] (which is also the approach adopted in [GR04]) by working
explicitly with 2-categories in the sense defined there; this explicit approach also
accords well with our own mathematical limitations.
We first remind the reader that the notion of 2-category is defined in [Stacks,
Tag 003H] (the notion defined there is also sometimes referred to as a strict 2category), while the notion of pseudo-functor from a category to a 2-category isdefined in [Stacks, Tag 003N] (and see also Definition 2.1 below).
The main examples of 2-categories that we will consider are the 2-category of
all categories lying over some base category D, the 2-category of categories fibred
over D, the 2-category of categories fibred in groupoids over D, and (assuming that
D is equipped with the structure of a site) the 2-category of stacks in groupoids
over D.
As already remarked, our first goal is to define the notion of a 2-diagram ina 2-category, and the related notion of 2-colimit. In fact, just as a diagram in a
category C indexed by a category I is simply a functor from I to C, a 2-diagram isin a 2-category C indexed by a category I is simply a pseudo-functor from I to C.
We formalize this in the following definition (which may then be regarded simply
as a recollection of the notion of pseudo-functor).
Definition 2.1. If I is a category and C is a 2-category, then a 2-diagram in
C indexed by I is a pseudo-functor (ϕ, α) from I to C. Explicitly, then, such a
2-diagram consists of the following data:(1) For each x ∈ I, an object ϕ(x) of C.
(2) For each morphism f : x → y in I, a 1-morphism ϕ(f ) : ϕ(x) → ϕ(y) in C.
(3) For every object x of I, a 2-morphism αx : idϕ(x) → ϕ(idx ) in C.
(4) For every composable pair of morphisms f : x → y, g : y → z in I, a2-morphism αg,f : ϕ(g ◦ f ) → ϕ(g) ◦ ϕ(f ) in C.
This data is required to satisfy the following conditions:(4) The various 2-morphisms αx and αg,f are isomorphisms.
(5) For any morphism f : x → y in I, we have
αidy ,f = αy ? idϕ(f )andαf,idx = idϕ(f ) ? αx .
(6) For any composable triple f : w → x, g : x → y, h : y → z of morphisms
in I, we have(idϕ(h) ? αg,f ) ◦ αh,g◦f = (αh,g ? idϕ(f ) ) ◦ αh◦g,f .

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Remark 2.2. In this section we typically denote a 2-diagram in C indexed by I by
the pair (ϕ, α); this is understood to be short-hand for the data described in thepreceding definition. In later sections (when we apply the general theory here in thecase of Ind-algebraic stacks), the indexing category I will typically be a partiallyordered set, the 2-category C will be the category of stacks in groupoids over somesite, and we will often denote a 2-diagram in this category simply by {Xi }, where
i runs over the elements of I. In the more precise notation introduced above, wehave Xi := ϕ(i), while the 2-morphisms α are left implicit (and are presumed to beevident from a consideration of the situation at hand).
Remark 2.3. For the brave reader who attempts to compare our discussion with thatin [GR04], we note the the direction of the coherence constraint αg,f in Definition 2.1(which follows the conventions of [Stacks]) is opposite to that of [GR04, Def. 2.1.14].
Since this coherence constraint is required to be an isomorphism, this doesn’t haveany affect on the subsequent development of the theory, but should be taken into
account when comparing formulas here or in [Stacks] to those of [GR04].
A key fact is that the 2-diagrams indexed by I with values in C may naturally
be regarded as the objects of a 2-category (see e.g. [GR04, Def. 2.2.12 (ii)]). The
next several definitions and remarks explicate this additional structure.
We begin by defining the notion of a 1-morphism between two 2-diagrams.
Definition 2.4. Let I be a category, let C be a 2-category, and let (ϕ1 , α1 ) and(ϕ2 , α2 ) be 2-diagrams indexed by I with values in C.
A 1-morphism (F, ψ) from (ϕ1 , α1 ) to (ϕ2 , α2 ) consists of the following data:(1) For each object x of I, a 1-morphism Fx : ϕ1 (x) → ϕ2 (x) in C.
(2) For each morphism f : x → y in I, a 2-morphism ψf : ϕ2 (f ) ◦ Fx →Fy ◦ ϕ1 (f ) in C.
This data is required to satisfy the following conditions:(3) The various 2-morphisms ψf are isomorphisms.
(4) For any object x of I, the two 2-morphisms
idFx ?α1,x : Fx = Fx ◦ idϕ1 (x) → Fx ◦ ϕ1 (idx )andψidx ◦ (α2,x ? idFx ) : Fx = idϕ2 (x) ◦Fx → ϕ2 (idx ) ◦ Fx → Fx ◦ ϕ1 (x)coincide.
(5) For any pair f : x → y, g : y → z of composable morphisms in I, the2-morphisms(idFz ? α1,g,f ) ◦ ψg◦f : ϕ2 (g ◦ f ) ◦ Fx → Fz ◦ ϕ1 (g ◦ f ) → Fz ◦ ϕ1 (g) ◦ ϕ1 (f )and(ψg ? idϕ1 (f ) ) ◦ (idϕ2 (g) ?ψf ) ◦ (α2,g,f ? idFx ) : ϕ2 (g ◦ f ) ◦ Fx → ϕ2 (g) ◦ ϕ2 (f ) ◦ Fx
→ ϕ2 (g) ◦ Fy ◦ ϕ1 (f ) → Fz ◦ ϕ1 (g) ◦ ϕ1 (f )coincide.
Remark 2.5. A 1-morphism from (ϕ1 , α1 ) to (ϕ2 , α2 ), as in Definition 2.4, is thesame as a pseudo-natural transformation (in the sense of [GR04, Def. 2.2.1]) between
the pseudo-functors (ϕ1 , α) and (ϕ2 , α2 ).

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Definition 2.6. Suppose that I is a category and C is a 2-category, that (ϕ1 , α1 ),(ϕ2 , α2 ), and (ϕ3 , α3 ) are 2-diagrams indexed by I with values in C, that (F1 , ψ1 ) :(ϕ1 , α1 ) and (F2 , ψ2 ) : (ϕ2 , α2 ) → (ϕ3 , α)3 are 1-morphisms of 2-diagrams, in thesense of Definition 2.4. Then we define the composite 1-morphism(F3 , ψ3 ) := (F2 , ψ2 ) ◦ (F1 , ψ1 ) : (ϕ1 , α1 ) → (ϕ3 , α3 )as follows:
For an object x of I, the 1-morphism F3,x is defined via
F3,x := F2,x ◦ F1,x : ϕ1 (x) → ϕ3 (x),while for a morphism f : x → y of I, the 2-morphism ψ3,f is defined via
ψ3,f := (ψ2,f ? idF1,x ) ◦ (idF2,y ?ψ1,2 ) : ϕ3 (x) ◦ F3,x = ϕ3 (x) ◦ F2,x ◦ F1,x
→ F2,y ◦ ϕ2 (f ) ◦ F1,x → F2,y ◦ F1,y ◦ ϕ1 (f ) = F3,y ◦ ϕ1 (f ).
We leave it to the reader to check that, defined in this manner, the composite isindeed a 1-morphism of 2-diagrams. (See also [GR04, Rem. 2.2.5 (iii)].)We next define the notion of a 2-morphism between two 1-morphisms of 2diagrams.
Definition 2.7. If (ϕ1 , α1 ) and (ϕ2 , α2 ) are two 2-diagrams in the 2-category C
indexed by the category I, and if (F1 , ψ1 ) and (F2 , ψ2 ) are two 1-morphisms from
(ϕ1 , α1 ) to (ϕ2 , α2 ), then a 2-morphism ξ from (F1 , ψ1 ) to (F2 , ψ2 ) consists of a2-morphism ξx : F1,x → F2,x for each object x of I, such that for each morphismf : x → y in I, we have(ξy ? idϕ1 (f ) ) ◦ ψ1,f = ψ2,f ◦ (idϕ2 (f ) ?ξx ).
Remark 2.8. If we interpret 1-morphisms from (ϕ1 , α1 ) to (ϕ2 , α2 ) as pseudonatural transformations, as in Remark 2.5, then we may interpret 2-morphisms
between 1-morphisms as modifications between those pseudo-natural transformations, in the sense of [GR04, Def. 2.2.9].
We now define the various forms of composition for 2-morphisms between 1morphisms of diagrams.
Definition 2.9. (1) If (ϕ1 , α1 ) and (ϕ2 , α2 ) are two 2-diagrams in the 2-category C
indexed by the category I, if (F1 , ψ1 ), (F2 , ψ2 ), and (F3 , ψ3 ) are three 1-morphisms
from (ϕ1 , α1 ) to (ϕ2 , α2 ), and if ξ1 and ξ2 are 2-morphisms from from (F1 , ψ1 ) to(F2 , ψ2 ) and from (F2 , ψ2 ) to (F3 , ψ3 ) respectively, then we define the composite
ξ3 := ξ2 ◦ ξ1 from (F1 , ψ1 ) to (F3 , ψ3 ) via
ξ3,x := ξ2,x ◦ ξ1,x : F1,x → F2,x → F3,x ,for each object x of I. (The reader may easily check that ξ3 satisfies the requirements to be a 2-morphism.)(2) If (ϕ1 , α1 ), (ϕ2 , α2 ), and (ϕ3 , α3 ) are three 2-diagrams in the 2-category
C indexed by the category I, if (F1 , ψ1 ) and (G1 , η1 ) are two 1-morphisms from
(ϕ1 , α1 ) to (ϕ2 , α2 ), if (F2 , ψ2 ) and (G2 , η2 ) are two 1-morphisms from (ϕ2 , α2 ) to
(ϕ3 , α3 ), if ξ1 is a 2-morphism from (F1 , ψ1 ) to (G1 , η1 ), and if ξ2 is a 2-morphismfrom (F2 , ψ2 ) to (G2 , η2 ), then we define ξ3 := ξ2 ? ξ1 to be the 2-morphism from
(F2 , ψ2 ) ◦ (F1 , ψ1 ) to (G2 , η2 ) ◦ (G1 , η1 ) defined via
ξ3,x := ξ2,x ? ξ1,x

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for each object x ∈ I.
We can now explain how the 2-diagrams indexed by the category I, with values
in the 2-category C, themselves form a 2-category.
Definition 2.10. Fix a category I and a 2-category C.
(1) If (ϕ1 , α1 ) and (ϕ2 , α2 ) are two 2-diagrams indexed by I with values in C,
then we let MorC (ϕ1 , α1 ), (ϕ2 , α2 ) denote the category whose objects are the 1morphisms from (ϕ1 , α1 ) to (ϕ2 , α2 ), and whose morphisms are the 2-morphisms
between 1-morphisms. The composition of morphisms is given by Definition 2.9 (1).
(The reader may easily verify that with composition defined in this manner, we doindeed obtain a category.)(2) If (ϕ1 , α1 ), (ϕ2 , α2 ), and (ϕ3 , α3 ) are three 2-diagrams in the 2-category C
indexed by the category I, then we define the composition bifunctorMorC (ϕ1 , α1 ), (ϕ2 , α2 ) ×MorC (ϕ2 , α3 ), (ϕ3 , α3 ) → MorC (ϕ1 , α1 ), (ϕ3 , α3 )via Definition 2.6 (for objects) and Definition 2.9 (2) (for morphisms). (The reader
may verify that with these definitions we do obtain a bifunctor.)(3) We let 2 - Diag(I, C) denote the 2-category whose objects are the 2-diagrams
indexed by I with values in C, whose morphisms categories are defined as in part (1),and whose composition bifunctors are defined as in part (2). (The reader may verify
that with these definitions, we do indeed obtain a 2-category.)
Remark 2.11. The reader may immediately verify that in the context of Definition 2.10, if C is a (2, 1)-category, then the same is true of 2 - Diag(I, C).
Remark 2.12. The 2-category structure on 2 - Diag(I, C) is one aspect of the 3category structure on the collection of all 2-categories, which we don’t discuss here.
(See also [GR04, Rem. 2.2.5 (vi)].)
We wish to explain how the 2-categories of 2-diagrams behave with respect to achange of indexing category. Since these are genuine 2-categories, we will require
an appropriate notion of pseudo-functor between 2-categories, neither of which a
1-category. The stacks project doesn’t discuss the general notion of a pseudofunctor between 2-categories (the reader can consult [GR04] for a discussion of thisnotion in the same spirit as that of the Stacks Project), and we won’t require thisnotion either. However, we will require the following very special case of the generaldefinition. (See e.g. [GR04, Def. 2.1.14 (ii)].)Definition 2.13. If C and C 0 are both 2-categories, then a strict pseudo-functor
F : C → C 0 consists of a function mapping the set of objects of C to the set ofobjects of C 0 (denoted X 7→
 F (X )), and for eachpair of objects X and Y of C, afunctor FX ,Y : MorC X , Y → MorC F (X ), F (Y) , which takes the unit object tothe unit object in the case X = Y, and such that these functors are compatible
with composition in C and C 0 .
Remark 2.14. The strictness in the preceding definition is the requirement thatthe functor F respects unit objects and composition precisely, rather than up tovarious 2-isomorphisms in C 0 . Again, we refer to [GR04] for the general definitionof a pseudo-functor (when the source as well as the target is allowed to be a general
2-category, rather than just a 1-category, so that [Stacks, Tag 003N] no longer
applies).

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Example 2.15. The 2-fibre product provides an example of a pseudo-functor between two 2-categories that is not strict in general. More precisely, if C is a 2category that admits 2-fibre products, if Y is an object of C, and if Z → Y is
a 1-morphism in C, then the map which on objects is defined via X 7→ X ×Y Z
extends to a pseudo-functor C/Y → C/Z which is not strict in general (and hence
is outside the scope of our discussion). In Remark 3.25, however, we consider aparticular case in which the 2-fibre product can be constructed in such a way that
this pseudo-functor does become strict.
For later use, we also recall what it means for a strict pseudo-functor to be fully
faithful, or to be a 2-equivalence. (The definitions actually make sense for a general
pseudo-functor; see e.g. [GR04, Def. 2.4.9 (ii), (iv)].)Definition 2.16. Let F : C → C 0 be a strict pseudo-functor between a pair of two
categories.
(1) We say that F is fully faithful (resp. strongly faithful) if for each pair of objects
X , Y ∈ C, the functor FX ,Y is an equivalence of categories (resp. an isomorphismof categories).
(2) We say that F is a 2-equivalence (resp. a strong 2-equivalence) is F is fully
faithful (resp. strongly faithful) and if each object of C 0 is equivalent (resp. 1isomorphic) to an object in the image of F .
Remark 2.17. One can define the notion of strict pseudo-natural transformation between strict pseudo-functors in a fairly evident sense (see e.g. [GR04, Def. 2.2.1 (ii)]),and one may form a 2-category whose objects are all 2-categories,1 whose 1-morphisms
are the strict pseudo-functor, and whose 2-morphisms are the strict pseudo-naturaltransformations. One can then show that a strict pseudo-functor F : C → C 0 is astrong 2-equivalence if and only if it is an equivalence in this 2-category; in particular, we may find a strict pseudo-functor G : C 0 → C which is quasi-inverse to F ina fairly evident sense. (The reader can see [GR04, Rem. 2.4.29] for more details.)
If a strict pseudo-functor F : C → C 0 is a 2-equivalence, then it also admits
a certain kind of quasi-inverse pseudo-functor G; see [GR04, Cor. 2.4.30]. However, the pseudo-functor G will not be strict in general (and the expression of thequasi-inverse property involves the notion of modification between pseudo-naturaltransformations), and so we don’t discuss this point any further here.
The next definition makes precise the sense in which we can precompose a 2diagram with a functor between index categories.
Definition 2.18. Let C be a 2-category, and let G : Ie → I be a functor between
categories. We define a strict pseudo-functor (in the sense of Definition 2.13)
e C)
G∗ : 2 - Diag(I, C) → 2 - Diag(I,as follows:(1) If (ϕ, α) is a 2-diagram with values in the 2-category C, indexed by thecategory I, then we let G∗ (ϕ, α) ◦ G denote the 2-diagram (ϕ̃, α̃) with values in C,e whose values on objects x̃ of Ie are defined by ϕ̃(x̃) := ϕ G(x̃) , whose
indexed by I,
values on morphisms f˜ : x̃ → ỹ in Ie are defined by ϕ̃(f˜) := ϕ G(f˜) , for whiche are defined by α̃x̃ := αG(x̃) ,the natural transformations α̃x̃ , for an object x̃ of I,
1We ignore set-theoretic issues here.

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and for which the natural transformations α̃f˜,g̃ , for composable pairs of morphisms
e are defined by α̃ ˜ := αG(g),G(g) . One immediately verifies (using the(f˜, g̃) in I,g̃,ffunctorial properties of G) that (ϕ̃, α̃) is a 2-diagram.
(2) If (F, ψ) is a 1-morphism between the objects (ϕ1 , α1 ) and (ϕ2 , α2 ) of 2 - Diag(I, C),
then we let G∗ (F, ψ) denote the 1-morphism (Fe, ψ̃) from G∗ (ϕ1 , α1 ) to G∗ (ϕ2 , α2 ),whose values on objects x̃ of Ie are defined by Fex̃ = FG(x̃) , and whose values onmorphisms f˜ : x̃ → ỹ in Ie are defined by ψ̃f˜ := ψG(f˜) . One immediately verifies(using the functorial properties of G) that (Fe, ψ̃) is a 1-morphism.
(3) If (ϕ1 , α1 ) and (ϕ2 , α2 ) are objects of 2 - Diag(I, C), if (F1 , ψ1 ) and (F2 , ψ2 )are two 1-morphisms between these objects, and if ξ is a 2-morphism between
these 1-morphisms, then we define the 2-morphism G∗ ξ between G∗ (F1 , ψ1 ) andG∗ (F2 , ψ2 ) to be the 2-morphism ξ˜ whose values on objects x̃ of Ie are defined byξ˜x̃ := ξG(x̃) . One immediately verifies (using the functorial properties of G) that ξ˜
is a 2-morphism.
The reader may similarly verify that the above definitions, taken together, define
e C).
a strict pseudo-functor G∗ : 2 - Diag(I, C) → 2 - Diag(I,Our aim, now, is to define the notion of a 2-colimit of a 2-diagram. In order todo this, we have to define the notation of a constant 2-diagram. Before doing this,we recall the notion of a sub-2-category of a 2-category, as well as some related
notions. (See e.g. [Stacks, Tag 02X7] and [GR04, Def. 2.4.9 (iii)].)Definition 2.19. If C is a 2-category, then a sub-2-category C 0 of C consists of asubset of the set of objects of C, and for each pair of objects X and Y of C, a subcategory MorC 0 (X , Y) of MorC (X , Y), which contains idX when X and Y coincide,and such that composition of morphisms in C restricts to induce a composition ofmorphisms in C 0 .
We say that C 0 is a full sub-2-category if the inclusion MorC 0 (X , Y) ,→ MorC (X , Y)
is an equivalence of categories, for each pair of objects X , Y of C 0 . We say that C 0 is
a strong sub-2-category of C if MorC 0 (X , Y) = MorC (X , Y) for each pair of objects
X , Y of C 0 . (Thus, choosing a strong sub-2-category of C amounts to choosing a
subset of the set of objects of C.)
Remark 2.20. Evidently, if C is a (2, 1)-category, so is any full sub-2-category of C.
Example 2.21. The image of a fully faithful strict pseudo-functor is a full sub-2category of its target, while the image of a strongly faithful strict pseudo-functor
is a strong sub-2-category of its target.
We now explain how the 2-category C may be embedded as a sub-2-category of
2 - Diag(I, C) (for any index category I).
Definition 2.22. If C is a 2-category and I is an index category, then for any object
X of C, we define the constant 2-diagram ΦX indexed by I to be the object (ϕ, α)of 2 - Diag(I, C) defined by setting ϕ(x) := X for every object x of I, ϕ(f ) = idX
for every morphism f of I, and by defining all the 2-morphisms α to be ididX .
Remark 2.23. In the context of Definition 2.22, let I gp denote the groupoid completion of I, i.e. the localization of I at all its morphisms. The reader may easily verify
that if X and Y are two objects of C, then Mor
 C (ΦX , ΦY ) is naturally isomorphic to
the functor category Fun I gp , MorC (X , Y) (whose objects are functors from I gp

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to MorC (X , Y), and whose morphisms are natural transformations between thesefunctors).
In particular, by mapping each object of MorC (X , Y) to the corresponding constant functor (and mapping morphisms to the corresponding natural transformation), we obtain an inclusion of categories MorC (X , Y) ⊆ MorC (ΦX , ΦY ). In thisway we regard C as a sub-2-category of 2 - Diag(I, C).
If F : X → Y is a 1-morphism in C, we will let ΦF : ΦX → ΦY denote thecorresponding constant 1-morphism in 2 - Diag(I, C), and we will use analogous
notation for constant 2-morphisms, if necessary.
Because of its importance in the definition of 2-colimits, we spell out the notionof 1-morphism of 2-diagrams in the particular case when the target is a constant2-diagram.
Definition 2.24. If (ϕ, α) is a 2-diagram in the 2-category C, indexed by thecategory I, and if X is an object of C, then a 1-morphism (F, ψ) from (ϕ, α) to theconstant 2-diagram ΦX consists of the the following data:(1) For each object x of I, a 1-morphism Fx : ϕ(x) → X in C.
(2) For each morphism f : x → y in I, a 2-morphism ψf : Fx → Fy ◦ ϕ(f ) in C.
This data is required to satisfy the following conditions:(3) The various 2-morphisms ψf are isomorphisms.
(4) For any object x of I, we have ψidx = idFx ?αx .
(5) For any pair f : x → y, g : y → z of composable morphisms in I, we have(idFz ? αg,f ) ◦ ψg◦f = (ψg ? idϕ(f ) ) ◦ ψf .
We similarly spell out the definition of 2-morphisms between 1-morphisms inthis special case.
Definition 2.25. If (ϕ, α) is a 2-diagram in the 2-category C indexed by the category I, if X is an object of C, and if (F1 , ψ1 ) and (F2 , ψ2 ) are two 1-morphisms
from (ϕ, α) to ΦX , then a 2-morphism ξ from (F1 , ψ1 ) to (F2 , ψ2 ) consists of a 2morphism ξx : F1,x → F2,x for each object x of I, and such that for each morphismf : x → y in I, we have(ξy ? idϕ(f ) ) ◦ ψ1,f = ψ2,f ◦ ξx .
We may now define the notion of 2-colimit (see e.g. [GR04, Def. 2.5.1], but notethat they work in the more general context in which I is also allowed to be a
2-category). To do this, we note that if (ϕ, α) is an object of 2 - Diag(I, C), if X
and Y are two objects of C, and if (F, ψ) is an object of MorC (ϕ, α), ΦX , then byevaluating the composition bifunctorMorC (ϕ, α), ΦX × MorC (ΦX , ΦY ) → MorC (ϕ, α), Yof 2 - Diag(I, C) on (F, ψ), we obtain a functor MorC (ΦX , ΦY ) → MorC (ϕ, α), ΦY ),which we may restrict to the constant subcategory MorC (X , Y) of MorC (ΦX , ΦY )(see Remark 2.23) so as to obtain a functor
(2.26)
MorC (X , Y) → MorC (ϕ, α), Y .
Definition 2.27. If (ϕ, α) is a 2-diagram in the 2-category C, then we say that a
1-morphism (F, ψ) : (ϕ, α) → ΦX , for some object X of C, realizes X as a 2-colimit

FORMAL ALGEBRAIC STACKS

11

of the 2-diagram (ϕ, α), if for every object Y of C, the induced functor (2.26) is anequivalence of categories.
If X is a 2-colimit of (ϕ, α), then we will use the notation∼

2 - colimI (ϕ, α) −→ X .
(The notation on the arrow should be understood as indicating an equivalence, andis justified by the following remark.)
Remark 2.28. Standard arguments show that if X0 and X1 are two objects of C,
equipped with 1-morphisms (F0 , ψ0 ) : (ϕ, α) → ΦX0 and (F1 , ψ1 ) : (ϕ, α) → ΦX1
realizing each of them as the 2-colimit of (ϕ, α) in C, then there is a 1-morphism
F : X0 → X1 which is an equivalence in C, andfor which ΦF ◦ (F0 , ψ0 ) is isomorphic
to (F1 , ψ1 ) (in the category MorC (ϕ, α), ΦX1 ).
Before making our next remark, we recall the notion of an initial object in a
2-category.
Definition 2.29. If C is a 2-category, an object of x of C is said to be an initial
object of C if for each object y of C, the category MorC (x, y) is a contractiblegroupoid, or equivalently a connected setoid2, i.e. a category equivalent to the onepoint category.
Remark 2.30. As noted in [Stacks, Tag 003O], where the analogous definition of a
final object is given, it is perhaps best to restrict this definition to the situation inwhich C is actually a (2, 1)-category, and indeed, this is the only case in which wewill apply it.
Remark 2.31. In the case when C is a (2, 1)-category, there is an alternate characterization of 2-colimits, as initial objects in a certain 2-category. Namely, if (ϕ, α) is a
2-diagram in the 2-category C, we define a new 2-category MorC (ϕ, α) as follows:
its set of objects is the disjoint
union, indexed by the objects
X of C, of the set of ob
jects of MorC (ϕ, α), X ; a 1-morphism in MorC (ϕ, α) between two 1-morphisms(F, ψ) : (ϕ, α) → X and (G, η) : (ϕ, α) → Y consists ofa 1-morphism H : X → Y
in C, together with a 2-morphism ξ in MorC (ϕ, α), Y between ΦH ◦ (F, ψ) and(G, η); a 2-morphism in MorC (ϕ, α) between 1-morphisms (H, ξ) and (H 0 , ξ 0 ) consists of a 2-morphism θ : H → H 0 in C such that ξx0 ◦ (θ ? idFx ) = ξx for each object
x of the index category I. We leave it to the reader to deduce the definitionsof the various
 compositions in MorC (ϕ, α) . If C is a (2, 1)-category, then so is
MorC (ϕ, α) .
We furthermore leave it to the reader to then verify, in the case when C is a (2, 1)category, that a morphism (F, ψ) : (ϕ, α) → X realizes the object X as the 2-colimitof (ϕ, α) if and only if this morphism, thought of as an object of MorC (ϕ, α) , isan initial object in this 2-category, in the sense of Definition 2.29. (The following
lemma, whose proof we leave to the reader, aids in this.)Lemma 2.32. If F : A → B is a morphism of groupoids, then F is an equivalence if
and only if, for every object b of B, there exists an object a ∈ A and an isomorphism∼
ξ : F (a) −→ b, and if, furthermore, for any other object a0 ∈ A endowed with an
2We follow the terminology of [Stacks], and refer to a groupoid, each of whose connectedcomponents is contractible, as a setoid. Such a groupoid is equivalent to a discrete set, hence the
name.

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M. EMERTON∼∼

isomorphism ξ 0 : F (a0 ) −→ b, there is a unique isomorphism θ : a −→ a0 such thatξ = ξ 0 ◦ F (θ).
Remark 2.33. There is a completely analogous theory of 2-limits in a 2-category C,
which admit a characterization analogous to the characterization of 2-colimits given
in Definition 2.27, and also, when C is actually a (2, 1)-category, a characterizationin terms of final objects in a 2-category, as in Remark 2.31.
As one example, if the index category I is the partially ordered set
•

•


•



then forming the 2-limit in C of a diagram indexed by I amounts to forming the
2-fibre product of objects in C. (The treatment of 2-fibre products in [Stacks, Tag
003O] follows the “final object in a 2-category” approach.)
We next establish some basic functorial property of 2-colimits.
Lemma 2.34. Let (ϕ, α) be a 2-diagram indexed by the category I, with values in
the 2-category C, and let G : Ie → I be a functor between categories. Suppose that(F, ψ) : (ϕ, α) → ΦX is a 1-morphism realizing the object X of C as a 2-colimit ofe Y is a 1-morphism realizing the object Y as(ϕ, α), and that (Fe, ψ̃) : G∗ (ϕ, α) → Φ
a 2-colimit of G∗ (ϕ, α). (Here we use Φ, with an appropriate subscript, to denotee with an appropriate subscript, to denotea constant object indexed by I, and use Φ,ea constant object indexed by I.) Then there is a 1-morphism H : Y → X in C for∼e H ◦ (Fe, ψ̃) −→which there exists a 2-isomorphism of 1-morphisms ΦG∗ (F, ψ).
e X , i.e. is an
Proof. The pull-back G∗ (F, ψ) is a 1-morphism from G∗ (ϕ, α) to Φ∗ee
object of the category MorC G (ϕ, α), ΦX . Since (F , ψ̃) realizes Y as a 2-colimitof (ϕ, α), composition with (Fe, ψ̃) induces an equivalence of categories
∼eX .
MorC (Y, X ) −→ MorX G∗ (ϕ, α), ΦIn particular, there is an object H of the domain of this equivalence, i.e. a 1e H ◦ (Fe, ψ̃) (which is its image under the equivmorphism H : Y → X , such that Φ∗e X to the object G∗ (F, ψ) of the target.
alence) is isomorphic in MorX G (ϕ, α), ΦThis proves the lemma.
In the particular case of the preceding lemma when G : Ie → I is cofinal (in thesense of [Stacks, Tag 04E6]), we can say more.
Since we will need to apply it, we recall the definition of cofinality.
Definition 2.35. We say that a functor G : Ie → I is cofinal if the following
conditions hold:(1) For any object x of I, there is an object x̃ of Ie and a morphism x → G(x̃)
in I.
e and any pair of mor(2) For any object x of I, any pair of objects x̃, x̃0 of I,
phisms x → G(x̃), x → G(x̃0 ), there exists a sequence of morphismsx̃ = x̃0 ← x̃1 → x̃2 ← x̃3 → · · · ← x̃2n−3 → x̃2n−2 ← x̃2n−1 → x̃2n = x̃0

FORMAL ALGEBRAIC STACKS

13

e and morphisms x → G(x̃i ) for each i = 0, . . . , 2n, such that thein I,morphisms for i = 0 and 2n coincide with the given morphisms, and suchthat each diagramxx
G(x̃2i−2 ) oG(x̃2i−1 )&/ G(x̃2i )commutes.
Lemma 2.36. If, in the context of Lemma 2.34, the functor G is cofinal, for any
object Z of C, the pull-back functor
eZ(2.37)G∗ : MorC (ϕ, α), ΦZ → MorC G∗ (ϕ, α), Φ
is an equivalence of categories. In particular, the 2-diagram (ϕ, α) admits a 2colimit if and only if G∗ (ϕ, α) does, and furthermore, the 1-morphism H (given by
Lemma 2.34) between these 2-colimits is an equivalence.
Proof. The claim regarding 2-colimits follows directly from the definition, together
with the claimed equivalence of categories. Thus, for any object Z of C, we mustshow that (2.37) is an equivalence. To do this, we will show in turn that G∗ isfaithful, full, and essentially surjective.
To begin with, suppose that (F1 , ψ1 ) and (F2 , ψ2 ) are two 1-morphisms from(ϕ, α) to ΦX , that ξ1 and ξ2 are two 2-morphisms between them, and that G∗ ξ1 =G∗ ξ2 ; by definition, this means that
(2.38)

ξ1,G(x̃) = ξ2,G(tx)efor any object x̃ of I.
e and choose (as we may) an object x̃ of I,e and aNow let x be an object of I,
morphism f : x → G(x̃). Then for each i = 1, 2, we have an equality(2.39)(ξi,G(x̃) ? idϕ(f ) ) ◦ ψ1,f = ψ2,f ◦ ξi,x .

Since the 2-morphisms ψ1,f and ψ2,f are isomorphisms, we deduce from (2.38)
and (2.39) that ξ1,x = ξ2,x , as required.
Next, we suppose given a 2-morphism ξ˜ : G∗ (F1 , ψ1 ) → G∗ (F2 , ψ2 ); we mustshow that ξ˜ = G∗ ξ for some 2-morphism ξ : (F1 , ψ1 ) → (F2 , ψ2 ). By definition, theedata of ξ˜ amounts to a 2-morphism ξ˜x̃ : F1,G(x̃) → F2,G(x̃) , for each object x̃ of I.
These 2-morphisms satisfy the identity(2.40)(ξ˜ỹ ? idϕ(G(f˜)) ) ◦ ψ1,G(f˜) = ψ2,G(f˜) ◦ ξ˜x̃efor any morphism f˜ : x̃ → ỹ in I.
By the assumption of cofinality, given any object x of I, we may find an object
e and a morphism f : x → G(x̃). We now use (2.39) to define ξx , namely wex̃ of I,set−1
ξx := ψ2,f
◦ (ξ˜x̃ ? idϕ(f ) ) ◦ ψ1,f .
Of course, we need to check that this is well-defined. A consideration of property (2)ein Definition 2.35 shows that it suffices to show that if f˜ : x̃ → ỹ is a morphism in I,
then the 2-morphism ξx obtained by using the preceding formula coincides with the

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M. EMERTON

2-morphism obtained by replacing x̃ with ỹ, and replacing f with the composite
x → G(x̃) → G(ỹ). In other words, we have to show that
−1
−1
ψ2,f
◦ (ξ˜x̃ ? idϕ(f ) ) ◦ ψ1,f = ψ2,G(◦ (ξ˜ỹ ? idϕ(G(f˜)◦f ) ) ◦ ψ1,G(f˜)◦f .
f˜)◦fThis follows from a consideration of the commutative diagramψ1,G(f˜)◦fF1,x/ F1,G(ỹ) ◦ ϕ G(f˜) ◦ f



ψ1,f

y
F1,G(x̃) ◦ ϕ(f )ψ1,G(f˜) ?idϕ(f )idF1,G(ỹ) ?αG(f˜),f t/ F1,G(ỹ) ◦ ϕ G(f˜) ◦ ϕ(f )ξ̃ỹ ?idϕ(f˜)◦ϕ(f )F2,xξ̃x̃ ?idϕ(f )ψ2,f ◦G(f˜)ξ̃ỹ ?idϕ(G(f˜)◦f )/ F2,G(ỹ) ◦ ϕ G(f˜) ◦ f

ψ2,f



y
F2,G(x̃) ◦ ϕ(f )idF2,G(ỹ) ?αG(f˜),f t˜/ F2,G(ỹ) ◦ ϕ G(f ) ◦ ϕ(f )ψ2,G(f˜) ?idϕ(f )(The commutativity of the top and bottom faces follows from the fact that
(F1 , ψ1 ) and (F2 , ψ2 ) are 1-morphisms, the commutativity of the front face follows
from (2.40), and the commutativity of the remaining face is clear.) In carrying out
this verification, it is helpful to remember that all the 2-morphisms ψ and α are infact 2-isomorphisms.
Now that the 2-morphisms ξx are defined, we have to verify that they satisfy thenecessary conditions to define a 2-morphism between (F1 , ψ1 ) and (F2 , ψ2 ). Thisis achieved by a consideration of various commutative diagrams similar to the one
appearing above. (Details will appear in a future draft.)
Lemma 2.41. Let I be a category, let C be a 2-category, and let (ϕ1 , α1 ) and(ϕ2 , α2 ) be 2-diagrams with values in C, indexed by I. Suppose that (F1 , ψ1 ) :(ϕ1 , α1 ) → ΦX1 and (F2 , ψ2 ) : (ϕ2 , α2 ) → ΦX2 are 1-morphisms realizing X1(resp. X2 ) as the 2-colimit of (ϕ1 , α1 ) (resp. (ϕ2 , α2 )). Then any 1-morphism of
2-diagrams (F, ψ) : (ϕ1 , ψ1 ) → (ϕ2 , ψ2 ) induces a 1-morphism H : X1 → X2 in C,
such that the 1-morphisms of 2-diagrams ΦH ◦ (F1 , ψ1 ) and (F2 , ψ2 ) ◦ (F, ψ) are
2-isomorphic.
Proof. Since (F2 , ψ2 ) ◦ (F, ψ) is an object of MorC (ϕ1 , α1 ), ΦX2 , this follows fromthe defining property of X1 as a 2-colimit of (ϕ1 , α1 ).
Example 2.42. If the index category I is filtered and admits a countable skeleton,∼
then we may find an equivalence N −→ I. In this case any 2-colimit computed along
I can be replaced by an equivalent 2-colimit computed along N. Furthermore, any
2-diagram (ϕ, α) indexed by N may be replaced by a strictly commutative diagram(ϕ,e id): for each n ∈ N, we define ϕ(n)e:= ϕ(n), define ϕ
en,n := id, and defineϕen,n+1 := ϕn,n+1 . But in general, given any pair m < n, we defineϕem,n := ϕn−1,n ◦ ϕn−2,n−1 ◦ · · · ◦ ϕm+1,m+2 ◦ ϕm,m+1 .
The data α may be used to construct an equivalence between (ϕ, α) and (ϕ,e id).

FORMAL ALGEBRAIC STACKS

15

We turn to proving that 2-colimits exist in various situations, beginning with thecase of the (2, 1)-category of categories fibred in groupoids over some base category.
In order to do this, we will use the notion of relative groupoid completion.
Lemma 2.43. If X is a category fibred over the base category D, then there isa category X gp fibred in groupoids over D, which we call the relative groupoid
completion of X over D, and a 1-morphism F : X → X gp of categories fibred overD, with the property that for any category Y fibred in groupoids over D, the functor
F ∗ : FunD (X gp , Y) → FunD (X , Y) is an equivalence of categories.
Proof. Our construction of X gp , which is somewhat clumsy, proceeds by strictification. Namely, we know from [Stacks, Tag 004A] that X is equivalent to a presheafof categories on D, and so we may assume that X is in fact a presheaf of categories. Now given any category, we may localize it at all its morphisms, to obtaina groupoid which receives a functor from the given category, and is universal for
functors from the category to groupoids. (This is the usual groupoid completion.)
Applying this construction to the values of the given presheaf of categories, we
obtain a presheaf of groupoids, which then serves as a relative groupoid completionof X over D.
Lemma 2.44. If (ϕ, α) is a 2-diagram in the (2, 1)-category C of categories fiberedin groupoids over a base category D, then (ϕ, α) admits a 2-colimit in C.
Proof. We begin by constructing a category X fibred over D, which almost receives a morphism from (ϕ, α), More precisely, we will construct the data (F, ψ)
attached to such a morphism, but the various 2-morphisms ψf will not necessarily
be isomorphisms. Modulo this difficulty, and the fact that X will not be fibred ingroupoids, we will see that X satisfies the universal property to be a 2-colimit of(ϕ, α). We will then pass to a groupoid completion X gp of C. Doing this will force
the 2-morphisms ψf to become isomorphisms, so that obtain a genuine morphismfrom (ϕ, α) to X gp . The already-proved universal property of X , together withthe universal property of X gp , when taken together, will then show that X gp is a
2-colimit of (ϕ, α).
We give an explicit description of X , and of the functors Fx : ϕ(x) → X andnatural transformations ψf . We define the set of objects of X to be the disjoint
union of the set of objects of each of the categories ϕ(x), as x ranges over theobjects of the index category I. For any two objects x and y of I, if a is an objectof ϕ(x) and if b is an object of ϕ(y), then we also define MorX (a, b) as a disjoint
union:aMorX (a, b) :=Morϕ(y) ϕ(f )(a), b .
f ∈MorI (x,y)

For notational purpose, we also write, for any f ∈ MorI (x, y),MorX (a, b)[f ] := Modϕ(y) ϕ(f )(a), b ,so that we can rewrite the preceding disjoint union asaMorX (a, b) :=MorX (a, b)[f ] ;f ∈MorI (x,y)

and if l ∈ Morϕ(y) ϕ(f )(a), b) , then we write l[f ] to denote l when thought of anan element of MorX (a, b)[f ] ⊆ MorX (a, b).

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M. EMERTON

The composition of morphisms in X is defined as follows: for a pair of composable
morphisms f : x → y and g : y → z, and objects a of ϕ(x), b of ϕ(y), and c of ϕ(z),we define the pairingpg,f : MorX (b, c)[g] × MorX (a, b)[f ] → MorX (a, c)[gf ]via(m[g] , l[f ] ) 7→ m ◦ ϕ(g)(l) ◦ (αg,f )a[gf ];

the composition
MorX (b, c) × MorX (a, b) → MorX (a, c)is then defined as the disjoint union of the various pairings pg,f . We leave theverification that X forms a category to the reader.
If x ∈ I, then we let πx denote the functor ϕ(x) → D realizing ϕ(x) as a category
over D. We define a functor π : X → D as follows:on objects it is defined to be thedisjoint union of the given maps Ob (ϕ(x) → Ob(D); on morphisms, it is defined
as follows: if f : x → y is a morphism in I, if a and b are objects of ϕ(x) and ϕ(y)respectively, and if l : ϕ(f )(a) → b is a morphismin ϕ(y), thought of as a morphism

in X , then (noting that πx (a) = πy ϕ(f )(a) because ϕ(f ) is a 1-morphism between
categories lying over D) we define π(l) : πx (a) = πy ϕ(f )(a) → πy (b) to be themorphism πy (l). Again, we leave to the reader the verification that π is in fact afunctor from X to D. (When making this verification, it helps to remember that,by the definition of the 2-category of categories fibred over D [Stacks, Tag 02XP],
all the 2-morphisms in this category are required to project to identity morphisms
in D, so that if x is any object of I, then πx (αx )a = idπx (a) for all objects a offgϕ(x); and similarly, if x −→ y −→ z are morphisms in I, thenπz (αg,f )a = idπx (a) = id
πz (g◦f )(a)for all objects a of ϕ(x).)
The functor π : X → D realizes X as a category fibred over D. To see this, wehave to show that for any object b ∈ X , and any morphism l : u → π(b) in D,
there is a morphism ˜l : a → b in X lying over l, and which is strongly Cartesian.
Suppose that b is an object of ϕ(x). Then π(b) = πx (b) by definition, and by theassumption that ϕ(x) is fibred over D, there exists a morphism m : a → b in ϕ(x)such that πx (m) = l, and which is strongly Cartesian. Now define˜l := (m ◦ α−1 )[idx ] ∈ MorX (a, b)[idx ] ⊆ MorX (a, b).
x,aUsing the fact that m is strongly Cartesian, the reader can verify that ˜l is strongly
Cartesian, as required.
We define what we might call a proto-morphism (F, ψ) : (ϕ, α) → X . Namely, forany x ∈ I, we define a morphism Fx : ϕ(x) → X , which we take to be the obvious
inclusion on objects, while on morphisms, we define Fx as follows: if l : a → b is a−1morphism in ϕ(x), then we define Fx (l) to be the morphism l ◦ αx,a: ϕ(idx )(a) →[idx ]b ∈ Morϕ(x) ϕ(idx )(a), b (interpreted as the subset MorX (a, b)
of MorX (a, b)).
We leave the verification that Fx is a functor to the reader. Note, though, thatby construction (and recalling the point already remarked upon above, that 2morphisms in C project to identity morphisms in D), we have that π ◦ Fx = πx .
Also, the very construction of strongly Cartesian morphisms in X given above

FORMAL ALGEBRAIC STACKS

17

shows that Fx takes strongly Cartesian morphisms in ϕ(x) to strongly Cartesianmorphisms in X . Thus Fx is in fact a 1-morphism in C.
∼
For a morphism f : x → y in I, a natural transformation ψf : Fx −→ Fy ◦ ϕ(f )is defined as follows: for an object a of ϕ(x), the morphism (ψf )a : a → ϕ(f )(a) isdefined to be the element[f ]
(idϕ(f )(a) )[f ] ∈ MorX a, ϕ(f )(a)⊆ ModX a, ϕ(f )(a) .
The natural transformations ψf project under π to identity natural transformations
in D, and thus are indeed 2-morphisms in C.
As the reader can verify, the couple (F, ψ) satisfies all the conditions of Definition 2.24 other than condition (3): in general, the natural transformations ψf will
not be isomorphisms.
Now suppose that Y is a category fibred in groupoids over D, and that Φ : X → Y
is a morphism of categories fibred over D. Composing (F, ψ) with Φ in the evident way, we obtain a genuine morphism Φ ◦ (F, ψ) : (ϕ, α) → Y, since all the
2-morphisms in Y are necessarily isomorphisms (so that condition (3) of Definition 2.24 is automatically satisfied, while the remaining conditions are evidently
satisfied, since they are satisfied by (F, ψ)). In this way we obtain a functor
(2.45)FunD (X , Y) → MorC (ϕ, α), Y
whose source is the category of morphisms between X and Y (thought of as categories fibred over D) and whose target is the category of morphisms from (ϕ, α)
to Y. We will show that this functor is an isomorphism of categories.
To this end, we first re-express some of the preceding constructions in a simple
formula. Namely, if f : x → y is a morphism in I, if a is an object of ϕ(x), if b isan object of ϕ(y), then we then have the formula(2.46)

l[f ] = Fy (l) ◦ (ψf )a .

Thus, if Φ : X → Y is a 1-morphism, and if we define (G, η) := Φ ◦ (F, ψ), thenwe see from this formula that
(2.47)

Φ(l[f ] ) = Gy (l) ◦ (ηf )a ,for each morphism l[f ] ∈ MorX (a, b)[f ] . It is now straightforward to verify that,conversely, if one is given a morphism (G, η) : (ϕ, α) → Y to a category Y fibred ingroupoids over D, and if one defines the functor Φ : X → Y on objects via mapping
an object a ∈ ϕ(x) (thought of as an object of X ) to Gx (a), and on morphisms
via the formula (2.47), then Φ is indeed a 1-morphism from X to Y, and that thefunctor (2.45) induces a bijection on objects. It is similarly verified to induce abijection on morphisms, and thus is indeed an isomorphism of categories.
Now let X → X gp be a groupoid completion of X . By the preceding discussion,
the proto-morphism (F, ψ) induces a genuine morphism (F gp , ψ gp ) : (ϕ, α) → X gp .
Furthermore, for any category fibred in groupoids Y over D, the induced functor
FunD (X gp , Y) → FunD (X , Y) is an equivalence of categories. We thus see that
(F gp , ψ gp ) realizes X gp as a 2-colimit of (ϕ, α).
Lemma 2.48. If (ϕ, α) is a 2-diagram in the (2, 1)-category C of stacks in groupoids
over a site D, then (ϕ, α) admits a 2-colimit in C.
Proof. By Lemma 2.44, we may form a 2-colimit X of (ϕ, α) in the (2, 1)-categoryof categories fibred in groupoids over D. If we let Xe denote the stackification of X

18

M. EMERTON

over D (in the sense, e.g., of [Stacks, Tag 02ZO]), then the 2-categorical universal
property of Xe (see [Stacks, Tag 0436] and also [Stacks, Tag 04W9]) shows that Xe
forms a 2-colimit of (ϕ, α) in the (2, 1)-category of stacks in groupoids over D. 
Lemma 2.49. Let C be a (2, 1)-category of categories fibred in groupoids over thebase category D, let (ϕ, α) be a 2-diagram in C, indexed by the filtered category I,
and let (F, ψ) : (ϕ, α) → X be a 2-colimit of (ϕ, α). If each of the functors ϕ(f )(as f runs over morphisms in I) is faithful (resp. fully faithful), then each of thefunctors Fx : ϕ(x) → X (as x runs over the objects of I) is faithful (resp. fully
faithful).
Proof. The proof will appear in a subsequent draft.



The following result concerning the values of a 2-colimit of stacks is often useful.
Lemma 2.50. If {Xi }i∈I is a 2-diagram in the (2, 1)-category of stacks in groupoids
over the site D, and if T is an object D, then there is a natural morphism ofgroupoids 2 − colimi Xi (T ) → (2 − colimi Xi )(T ). If I is furthermore filtered, and if
T is quasi-compact then this morphism is fully faithful. Continuing to assume that
I is filtered, if in addition either T is quasi-separated or the transition morphisms
in the 2-diagram {Xi } are monomorphisms, then this morphism is an equivalence.
Proof. This is standard; a proof should appear in a future draft.



We introduce one more 2-categorical notion, the 2-categorical analogue of theusual categorical construction C/y (which for any object y of a category C, is thecategory whose objects are C-morphisms f : x → y, with a morphism between
f : x → y and f 0 : x0 → y consisting of a C-morphism g : x → x0 such thatf = f 0 ◦ g).
Definition 2.51. If C is a 2-category, and Y is an object of C, then we let C/Ydenote the 2-category whose objects consist of 1-morphisms F : X → Y in C.
If F : X → Y and F 0 : X 0 → Y are two objects of C/Y, then the objects of
MorC/Y (F, F 0 ) (which are 1-morphisms from F to F 0 in C/Y) consist of pairs (G, α),where G : X → X 0 is a 1-morphism and α : F → F 0 ◦ G is a 2-morphism, while
the morphisms (G, α) → (H, β) (which are the 2-morphisms from (G, α) to (H, β)
in C/X ) consist of of 2-morphisms γ : G → H such that (idF 0 ?γ) ◦ α = β. (See
[GR04, Ex. 2.1.11 (ii)].)3. Stack-theoretic backgroundIn this section we introduce and study various stack-theoretic notions that will
be needed in our developing the theory of formal algebraic stacks. One key pointis that in this theory, one studies many stacks (in groupoids) that are not algebraic
stacks (e.g. formal algebraic stacks themselves, as well as Ind-algebraic stacks), andso it is important to have a strong and flexible theory of the geometric propertiesof morphisms between such stacks. We develop some of that theory here, with anemphasis on so-called morphisms that are representable by algebraic stacks.
Another important construction in the context of formal algebraic stacks is theformation of the underlying reduced algebraic stack. We develop a general version ofthat theory in this section (after introducing our conventions related to substacks),
which is flexible enough to recover the existing theory for algebraic stacks, butwhich also applies to formal algebraic stacks and Ind-algebraic stacks.

FORMAL ALGEBRAIC STACKS

19

Related to this, we also prove a form of topological invariance of the small étalesite which applies to not-necessarily-algebraic stacks.
Throughout this section, we fix a scheme S, and by a stack over S, we mean acategory fibred in groupoids over the category Sch /S,3 which is a stack with respect
to the fppf topology on Sch /S.
We begin by recalling the following definition, which generalizes the notion of amorphism between stacks being representable by algebraic spaces.
Definition 3.1. (1) We say that a morphism X → Y of stacks over S is representable by Deligne–Mumford stacks, or DM for short, if for any morphism of stacks
Z → Y whose source is a Deligne–Mumford stack, the fibre product X ×Y Z is alsoa Deligne–Mumford stack.
(2) We say that a morphism X → Y of stacks over S is representable by algebraic
stacks, or algebraic,4 if for any morphism of stacks Z → Y whose source is analgebraic stack, the fibre product X ×Y Z is again an algebraic stack.
We will be primarily interested in the second of these two definitions, but the firstis also useful. (In particular, it appears naturally in the context of étale morphismsof algebraic stacks.)
Remark 3.2. It is easy to check, in the context of the preceding definition, thatin verifying either condition (1) or (2), it suffices to restrict to the case when Z
is actually a scheme. Indeed, suppose that condition (1) is satisfied in that case,and then consider the case of a morphism Z → Y whose source is a Deligne–
Mumford stack. We may choose an étale surjection U → Z whose source is ascheme (and which is representable by algebraic spaces, since Z is an algebraic
stack), and consider the base-changed morphism X ×Y U → X ×Y Z; this is again
representable by algebraic spaces, étale, and surjective. By assumption, the source
is a Deligne–Mumford stack, and thus we may find an étale surjection V → X ×Y U
whose source is a scheme (and which again is representable by algebraic spaces).
The composite V → X ×Y U → X ×Y Z is then a morphism that is representableby algebraic spaces, étale, and surjective, from a scheme to the stack X ×Y Z. Thus
this fibre product is itself a Deligne–Mumford stack (since it is an algebraic stack,by [Stacks, Tag 05UL], which admits a surjective étale morphism from a scheme).
Similarly, if condition (2) is satisfied when Z is a scheme, then consider the caseof a morphism Z → Y whose source is an algebraic stack. We choose a smoothsurjection U → Z whose source is a scheme (and which is representable by algebraic
spaces, since Z is an algebraic stack), and consider the base-changed morphism
X ×Y U → X ×Y Z; this is again representable by algebraic spaces, smooth, andsurjective. By assumption, the source is an algebraic stack, and thus we may finda smooth surjection V → X ×Y U whose source is a scheme (and which again isrepresentable by algebraic spaces). The composite V → X ×Y U → X ×Y Z isthen a morphism that is representable by algebraic spaces, smooth, and surjective,
from a scheme to the stack X ×Y Z. Thus this fibre product is itself an algebraic
stack [Stacks, Tag 05UL].
In particular (since schemes are DM algebraic stacks), we see that a DM morphism is representable by algebraic stacks.
3We ignore all set-theoretic issues throughout this write-up. The concerned reader should add
their preferred flavour of foundations.
4This latter term is the one used in the Stacks Project [Stacks, Tag 06CF].

20

M. EMERTON

Finally, we remark that since any scheme is locally affine, and since the property
of a stack over a base scheme T being algebraic, or Deligne–Mumford, may bechecked Zariski locally on T , we see that in order to verify either condition (1)or (2), it even suffices to consider the case when Z is an affine scheme.
The following lemma is sometimes useful for verifying the representability properties of a morphism.
Lemma 3.3. If X → Y → Z are morphisms of stacks over S, then the first arrow is representable (i.e. representable by schemes, resp. representable by algebraic
spaces, resp. representable by Deligne–Mumford stacks, resp. representable by algebraic stacks) if and only if, for each morphism T → Z whose source is a scheme,
the induced morphism T ×Z X → T ×Z Y is representable (resp. representable byalgebraic spaces, resp. representable by Deligne–Mumford stacks, resp. representableby algebraic stacks).
Proof. If we are given a morphism f : T → Y, whose source is a scheme, then wecan also consider the composite g : T → Y → Z, and the morphism f induces asection s : T → T ×g,Z Y. The projection T ×f,Y X → T can then be obtained as thepull-back under s of the morphism T ×g,Z X → T ×g,Z Y. If this latter morphism
has a given representability property, so does its pull-back under s. (Here we arealso taking into account Remark 3.2, of course.)Lemma 3.4. If f : X → Y is a morphism of stacks over S that is representable byalgebraic stacks, then the diagonal morphism ∆f : X → X ×Y X is representable byalgebraic spaces. In addition, the morphism f is DM (resp. representable by algebraic spaces) if and only if ∆f is furthermore unramified (resp. a monomorphism).
Proof. We apply the criterion of Lemma 3.3 to the sequence of morphisms X →X ×Y X → Y. Let Z → Y be a morphism whose source is a scheme, and write
W := X ×Y Z. Then W is an algebraic stack over Z, and the morphism ∆f pullsback to the diagonal morphism ∆W/Z . This morphism is representable by algebraic
spaces, and so (since Z → Y was arbitrary) so is the morphism ∆f .
The morphism ∆f is furthermore unramified (resp. a monomorphism) if andonly if each of the base-changed morphisms ∆W/Z is unramified (resp. a monomorphism). Each of these morphism is unramified (resp. a monomorphism), in turn, if
and only if each of the algebraic stacks W is in fact a Deligne–Mumford stack [Stacks,
Tag 06N3] (resp. an algebraic space [Stacks, Tag 04SZ]). This completes the proofof the lemma.
Lemma 3.5. A morphism X → Y from a sheaf over S to a stack over S isrepresentable by algebraic stacks if and only if it is representable by algebraic spaces.
Proof. The if direction is clear. For the only if direction, suppose that X → Y isrepresentable by algebraic stacks, and consider a morphism T → Y whose source
is a scheme. Then the fibre product X ×Y T is an algebraic stack, by assumption,
but it is also a sheaf. Thus it is in fact an algebraic space [Stacks, Tag 04SZ]. 
The following lemmas give additional criteria for deducing that certain morphisms of stacks are representable (in various senses).
Lemma 3.6. Suppose that X → Y → Z are morphisms of stacks over S, suchthat the second morphism, and also the composite morphism, are both representable

FORMAL ALGEBRAIC STACKS

21(i.e. representable by schemes, resp. representable by algebraic spaces, resp. representable by Deligne–Mumford stacks, resp. representable by algebraic stacks). Thenthe morphism X → Y is representable (resp. representable by algebraic spaces, resp.
representable by Deligne–Mumford stacks, resp. representable by algebraic stacks).
Proof. We apply the criterion of Lemma 3.3, and so find that it suffices to show that
if T → Z is any morphism whose source is a scheme, the base-changed morphism
T ×Z X → T ×Z Y is representable in the appropriate way. This follows from thefact that a morphism of schemes (resp. algebraic spaces, resp. Deligne–Mumford
stacks, resp. algebraic stacks) is representable by schemes (resp. algebraic spaces,
resp. Deligne–Mumford stacks, resp. algebraic stacks).
Lemma 3.7. If X → Y and Y → Z are morphisms of stacks over S for which
the composite X → Y → Z is representable by algebraic spaces (resp. Deligne–
Mumford stacks, resp. algebraic stacks) and the diagonal morphism Y → Y ×Z Y isrepresentable by algebraic spaces,5 then X → Y is representable by algebraic spaces(resp. Deligne–Mumford stacks, resp. algebraic stacks).
Proof. This follows from the usual graph argument: we factor X → Y as thecomposite of the graph X → X ×Z Y and the projection from X ×Z Y to Y. The
first morphism is representable by algebraic spaces (being a base-change of the
diagonal morphism Y → Y ×Z Y), and the second morphism is representable byalgebraic spaces (resp. Deligne–Mumford stacks, resp. algebraic stacks), being thebase-change of the morphism X → Z. Thus their composite is representable byalgebraic spaces (resp. Deligne–Mumford stacks, resp. algebraic stacks), as claimed.

Recall the following definition [Stacks, Tag 07Y8].
Definition 3.8. We say that a category fibred in groupoids X over a locally Noetherian scheme S satisfies the strong Rim–Schlessinger criterion, denoted (RS*), if,
given an affine open subset Spec Λ ⊆ S, and a fibre product diagram of Λ-algebras
BO 0

/B
O

A0 = A × B B 0/A

in which B 0 → B surjective with square zero kernel, the induced functor on fibre
categories
XSpec(A0 ) −→ XSpec(A) ×XSpec(B) XSpec(B 0 )is an equivalence of categories.
We now show that the strong Rim–Schlessinger condition can be pulled backalong morphisms representable by algebraic stacks.
Lemma 3.9. If X → Y is a morphism of stacks over a locally Noetherian scheme S
which is representable by algebraic stacks, and if Y satisfies (RS*), then so does X .
5Such a diagonal morphism is always representable by sheaves, and hence if it representableby algebraic stacks, it is in fact representable by algebraic spaces; this explains our apparently
restrictive hypothesis.

22

M. EMERTON

Proof. By [Stacks, Tag 06N7] (and [Stacks, Tag 042X]), we may (and do) replace
X by an isomorphic stack such that the morphism X → Y realizes X as a category
fibred in groupoids over Y. Let
X = Spec B/ X 0 = Spec B 0
Y = Spec A

/ Y 0 = Spec A0

be a pushout diagram in the category of schemes obtained by taking Spec of aCartesian square of rings as in Definition 3.8. We must show that the naturalfunctor between groupoids(3.10)X (Y 0 ) → X (X 0 ) ×X (X) X (Y )is an equivalence. Since Y satisfies (RS*), the functor
Y(Y 0 ) → Y(X 0 ) ×Y(X) Y(Y )is an equivalence, and so we may place the functor (3.10) into a 2-commutative
diagram of groupoids
X (Y 0 )/ X (X 0 ) ×X (X) X (Y )w
Y(Y 0 )
Since X → Y realizes the source as being fibred in groupoids over the target, thesame is true of the vertical arrow in this diagram. Thus, in order to check that thehorizontal functor is an equivalence, it suffices to check that it induces an equivalence on the fibre categories over each object in the base. (It is straightforward
to verify directly that this suffices. Alternatively, note that the diagonal arrow iseasily verified to also realize its source as a category fibred in groupoids over its
target, and then apply part (1) of [Stacks, Tag 003Z].)
This reduces our problem to verifying that the functor analogous to (3.10), obtained by replacing X by X ×Y Y 0 for some given morphism Y 0 → Y, is an equivalence. Since X → Y is representable by algebraic stacks, this fibre product isan algebraic stack. We are therefore done, since algebraic stacks satisfy (RS*),by [Stacks, Tag 0CXP].
Following [Stacks, Tag 03YK, Tag 04XB], we can define properties of morphisms
representable by algebraic stacks in the following way.
Definition 3.11. If P is a property of morphisms of algebraic stacks which isfppf local on the target, and preserved by arbitrary base-change, then we say thata morphism f : X → Y of stacks which is representable by algebraic stacks hasproperty P if and only if for every algebraic stack Z and morphism Z → Y, thebase-changed morphism of algebraic stacks Z ×Y X → Z has property P .
Remark 3.12. When applying Definition 3.11, it suffices to consider the case when
Z is actually a scheme, or even an affine scheme (since any algebraic stack admits asmooth, and so in particular fppf, cover by a scheme, and hence even by a disjoint
union of affine schemes, and by assumption the property P may be tested for fppf
locally).

FORMAL ALGEBRAIC STACKS

23

Remark 3.13. Some of the properties to which we will apply Definition 3.11 are flat,
locally of finite presentation, locally of finite type, proper, smooth, and surjective.
Remark 3.14. We may also apply Definition 3.11 to the property of being étale.
Recall that a morphism X → Y of algebraic stacks is said to be étale if it isunramified, flat, and locally of finite presentation, or equivalently, if it is a DM
morphism, with the property that for any morphism Z → Y whose source is ascheme, the base-changed morphism X ×Y Z → Z (whose source is a Deligne–
Mumford stack, by assumption) is étale, in the sense that for some (or, equivalently,any) surjective étale morphism U → X ×Y Z whose source is a scheme, the composite
morphism U → X ×Y Z → Z is étale.
We then find that a morphism X → Y of stacks over S which is representable byalgebraic stacks is étale (in the sense given by Definition 3.11 and the discussion of
the preceding paragraph) if and only if it is a DM morphism, with the additional
property that for any morphism Z → Y whose source is a scheme, the base-changed
morphism X ×Y Z → Z (whose source is a Deligne–Mumford stack, by assumption)is an étale morphism in the sense described in the preceding paragraph.
Remark 3.15. If we apply Definition 3.11 to the property of being a monomorphism,
then we find that a morphism f : X → Y, which is representable by algebraic
stacks, is a monomorphism if and only if ∆f : X → X ×Y X is an isomorphism(since this may be checked after pulling back over morphisms T → Y whose source
is a scheme), and thus if and only if f is fully faithful as a functor.
Furthermore, Lemma 3.4 then shows that if a morphism representable by algebraic stacks is a monomorphism, then it is actually representable by algebraic
spaces (since if the associated diagonal morphism is an isomorphism, it is certainly
a monomorphism). It then follows from [Stacks, Tag 040X] that such a morphismis also universally injective (where this latter property is again interpreted via Definition 3.11).
Example 3.16. If Y is a stack whose diagonal is representable by algebraic spaces,
then any morphism X → Y whose source is an algebraic stack is representableby algebraic stacks [EG19b, Lem. 2.3.1], and so Definition 3.11 applies in suchcases. Many such properties are discussed in detail in [EG19b, § 2.3]. (Note that
Definition 3.11 is compatible with [EG19b, Def. 2.3.4], although the latter definition
is more expansive than the former, in that it does not require that the property P
be fppf local on the target. In practice, all the example properties considered indetail in [EG19b, § 2.3] are in fact fppf local on the target.)
It is sometimes useful to describe a given stack as a quotient stack, and westate and prove a lemma that gives a general criterion for this, together with somerelated results. We begin by establishing a general context in which to consider
such a result. To this end, suppose that X is a stack over S, and that U → X
is a morphism whose source is an fppf sheaf on Sch /S, and which is representableby algebraic spaces and locally of finite presentation. We may then form the fibre
product R := U ×X U which is again an fppf sheaf on Sch /S, having the structure ofa groupoid in sheaves over U . Suppose now that R0 is another groupoid in sheaves
over U , for which each of the projections R0 ⇒ U are representable by algebraic
spaces, flat, and locally of finite presentation, and that we are given a morphism
R0 → R of groupoids in sheaves over U .

24

M. EMERTON

Following the discussion of [Stacks, Tag 0440], we may form the quotient stackX 0 := [U/R0 ]. (In the cited discussion, the quotient stack is defined when U and
R are in fact algebraic spaces, but the discussion there — and in particular, thedefinition of the quotient stack given in [Stacks, Tag 044Q] — makes sense whenever
U is a sheaf and R0 is a groupoid in sheaves over U whose projections to U are
representable by algebraic spaces, flat, and locally of finite presentation.) Note∼
that, by construction, there is a canonical isomorphism R0 −→ U ×X 0 U.
In the preceding situation, we have the following lemma relating the various
objects involved.
Lemma 3.17. We put ourselves in the context of the preceding discussion.
(1) The morphism U → X factors (up to 2-isomorphism) through a morphism
X 0 → X , which is furthermore representable by algebraic stacks.
(2) There is a 2-Cartesian square
/R

R0

X0

∆X 0 /X/ X 0 ×X X 0

in which the horizontal arrows are representable by algebraic spaces, and in which
the vertical arrows are representable by algebraic spaces, flat, locally of finite presentation, and surjective.
Proof. The stack [U/R0 ] is defined as the stackification of the presheaf in groupoids
over S whose objects are morphisms T → U , and whose morphisms are morphisms
T → R0 projecting to a given pair of morphisms T ⇒ U . The given morphism
R0 → R = U ×X U allows us construct a morphism from this presheaf to X , andhence a morphism from its stackification X 0 to X .
Consider the composite U → X 0 → X (which by assumption is representable byalgebraic spaces). If we base-change this along a morphism T → X whose source
is a scheme, we obtain morphisms
T ×X U → T ×X X 0 → T.
The source of the first arrow is an algebraic space, and first arrow itself is representable by algebraic spaces, flat, and surjective (since it a base-change of themorphism U → X 0 , which has these properties). Thus T ×X X 0 is indeed an algebraic stack [Stacks, Tag 05UL]; this proves (1).
It follows from (1) together with Lemma 3.4 that ∆X 0 /X is representable byalgebraic spaces, proving the first claim of (2). Consider now the morphism R =U ×X U → X 0 ×X X 0 . Since U → X 0 is representable by algebraic spaces, flat,
locally of finite presentation, and surjective, the same is true of this morphism. It
provides the right-hand arrow of the diagram in (2). If we pull this map back along∆X 0 /X , we obtain the 2-Cartesian square
U ×X 0 U

/R


X0


/ X 0 ×X X 0∆X 0 /X∼

Taking into account the isomorphism R0 −→ U ×X 0 U, we obtain the 2-Cartesiansquare of (2).


FORMAL ALGEBRAIC STACKS25

The preceding lemma has various applications, such as to the following criterion
for presenting a stack as a quotient.
Lemma 3.18. Suppose that X is a stack over S, and that U → X is a morphismfrom a sheaf to X which is representable by algebraic spaces, flat, surjective, andlocally of finite presentation. Then, if we write R := U ×X U, the morphism U → X∼
induces an isomorphism [U/R] −→ X .
Proof. We apply Lemma 3.17, taking R0 = R (as the hypotheses of the present
lemma allow us to do). Since the upper right arrow in the diagram of part (2) ofthat lemma is then an isomorphism, we conclude that ∆X 0 /X is also an isomorphism (as the property of being an isomorphism may be checked fppf locally on thetarget), and thus that X 0 → X is a monomorphism. Since an isomorphism is inparticular a monomorphism, and since X 0 → X is representable by algebraic stacks,
by Lemma 3.17 (1), it is in fact representable by algebraic spaces, by Lemma 3.4.
Consider the sequence of morphisms U → X 0 → X , each of which we know to
be representable by algebraic spaces. Both the first arrow and the composite arefurthermore flat, locally of finite presentation, and surjective. Since each of theseproperties may be checked fppf locally on the source, we find that the second arrowalso has these properties. In conclusion, the morphism X 0 → X is representable byalgebraic spaces, a monomorphism, flat, locally of finite presentation, and surjective.
It is thus an isomorphism [Stacks, Tag 05W5].

We also have the following variant of the preceding result.
Lemma 3.19. Suppose that X is a stack over S, that U → X is a morphism froma sheaf to X which is representable by algebraic spaces, and that the projections
R := U ×X U ⇒ U (which are again representable by algebraic spaces, being thebase-change of morphisms which are so representable) are flat and locally of finite
presentation. Then the morphism U → X induces a monomorphism [U/R] → X .
Proof. As in the proof of Lemma 3.18, we may apply Lemma 3.17 (2), taking
R0 = R. We find that ∆X 0 /X is an isomorphism, as required.

Remark 3.20. In the particular case when X itself is an algebraic stack, results along
the lines of the preceding lemmas are presented in [Stacks, Tag 04T4] and [Stacks,
Tag 04T5].
If X is an algebraic stack and U = Spec k, and the morphism Spec k → X is
locally of finite presentation, then the projections R := U ×X U ⇒ U are automatically flat, as well as being locally of finite presentation, and the preceding lemmathen gives a construction of the residual gerbe [Stacks, Tag 06ML] at the imagex ∈ |X | of Spec k.
In what follows, we will be given a stack X , and will want to talk about various
substacks of X . We briefly digress to explain our conventions regarding substacks.
At the least, a substack of X should be a full subcategory of X that is fibredin groupoids over S, and satisfies an appropriate descent condition on objects (seee.g. [Stacks, Tag 04TU]). However, in order to avoid complications with the possibility of two substacks Y and Z of X being isomorphic as substacks of X (i.e.
admitting an isomorphism of stacks, i.e. an equivalence of categories in groupoids

26

M. EMERTON

fibred over S, in a way that the diagram
Y

/X
?

∼


Z

2-commutes; here the the horizontal and diagonal arrows are the inclusions of Y
and Z as subcategories of X ) without being actually equal, we adopt the following
convention. (Compare [Stacks, Tag 0AIN].)
Convention 3.21. If X is a stack (in groupoids, as always) over S, then a substackof X consists of a strictly full subcategory of Y of X which also forms a stack ingroupoids over S.
Remark 3.22. This rigidification of the notion of substack is slightly unnatural froma 2-categorical view-point, since it places an emphasis on equality and isomorphism,
rather than on equivalence. However, it will simplify some later constructions, by
enforcing a certain rigidity (which, for example, will ensure the strictness of various
pseudo-functors that we construct).
Remark 3.23. Any monomorphism Z → X of stacks over S factors uniquely as
Z → Y ,→ X , where the first arrow is an isomorphism, and the second arrow is
the inclusion of a substack Y into X . (A monomorphism of stacks is just a fully
faithful functor of the underlying categories, compatible with their structure of
categories fibred over S. This factorization is then just the usual factorization of afully faithful functor into an equivalence onto its essential image, followed by the
inclusion of the essential image into the target category of the functor.)
Remark 3.24. If X is a stack over S, then we can speak of one substack of X
containing another; this holds provided the set of objects of one is contained in
the set of objects of the other. If {Y}i∈I is a collection of substacks of S, thenTi∈I Yi (the full subcategory of X whose set of objects is the intersection of theset of objects of each of the Yi ) is again a substack of X ; this substack is maximal
with respect to being contained in each of the Yi .
If C is a subcategory of X , then weTit makes sense to talk of the substack of X
generated by C: it is the intersection i∈I Yi , where Yi runs over all substacks of
X containing C.
In particular, if Y and Z are two substacks of X , then we may form the substackY ∩ Z, which realizes the 2-fibre product Y ×X Z as a substack of X .
Remark 3.25. Related to the final paragraph of the preceding remark, we make thefollowing observation: In general, if X → Y and Z → Y are morphisms of stacks,then the fibre product (which of course means the 2-fibre product) X ×Y Z is only
defined up to isomorphism of stacks, i.e. up to equivalence of categories fibred ingroupoids over S. However, if Z → Y is the inclusion of a substack (which is inparticular a monomorphism of stacks, i.e. a fully faithful functor), then (following
the prescription of Remark 3.23), we may consider the essential image of the induced
monomorphism X ×Y Z → X , which is a substack of X . This substack then gives
a canonical choice of model of X ×Y Z; concretely, it is the full subcategory of X
that contains precisely those objects whose images in Y are objects of Z.
If we interpret X ×Y Z in this way, then X 7→ X ×Y Z becomes a strict pseudofunctor (in the sense of Definition 2.13) from St/Y → St/Z. (Here we are denoting

FORMAL ALGEBRAIC STACKS

27

by St the 2-category of stacks fibred in groupoids over Sch /S with the fppf topology,
and then following Definition 2.51 to define the 2-categories St/X and St/Y of stacks
lying over Y and Z respectively.) We apply this observation in various contexts
below.
Definition 3.26. If X is a stack over S, then we say that a substack Y of X is
open (resp. closed, resp. locally closed) if the inclusion Y ,→ X is representable byalgebraic spaces, and induces an open (resp. closed, resp. locally closed) immersion
when pulled-back along any morphism T → X whose source is a scheme. (Notethat this inclusion is then in fact representable (by schemes), since a locally closed
sub-algebraic space of a scheme is in fact a scheme.)
We now turn to defining, and proving the basic properties of, the underlying
reduced substack of a stack over S. To this end, we let (Sch /S)red denote the fullsubcategory of Sch /S consisting of reduced schemes.
Definition 3.27. Let X be a stack (fibered in groupoids, as always) over S, and
let F : X → Sch /S be the functor which realizes X as a stack over S. We thenlet (Xred )0 denote the full subcategory of X whose set of objects consists of thoseobjects x of X for which F (x) is an object of (Sch /S)red , and let Xred denote thesubstack of X generated by (Xred )0 (in the sense discussed in Remark 3.24). We
refer to Xred as the underlying reduced substack of X .
Remark 3.28. If Y → X is a morphism of stacks over S (i.e. a functor of categories
fibred in groupoids over S), then this morphism evidently restricts to a functor(Yred )0 → (Xred )0 of categories over Sch /S, and hence also restricts to a functor
Yred → Xred of the corresponding reduced substacks. Thus the formation of Xredis a strict pseudo-functor (in the sense of Definition (2.13)) from the 2-category ofstacks over S to itself.
Remark 3.29. If X is an algebraic stack, then Xred is precisely the underlying
reduced algebraic substack of X . (To see this, let Z denote the underlying reduced
algebraic substack of X , and first note that if T → X is a morphism whose sourceis reduced, then this morphism factors through Z; thus (Xred )0 is contained in
the substack Z of X , and hence so is Xred . Conversely, choose a smooth surjective
morphism U → X whose source is a scheme, which then induces a smooth surjective
morphism Ured → Z. The composite morphism Ured → Z ,→ X then gives anobject of (Xred )0 , and hence of Xred . The stack property of Xred then shows thatthe morphism Z ,→ X factors through Xred .)Remark 3.30. If X is a formal algebraic space over S, regarded as a stack in setoids
over S, then an argument essentially identical to that of Remark 3.29 shows thatthe substack Xred , which corresponds to a subfunctor of X, is equal to Xred , theunderlying reduced algebraic space of X, as constructed in [Stacks, Tag 0AIN].
Remark 3.31. If X and Z are stacks over S, equipped with a morphism to the stackY over S, then the inclusions Xred ,→ X and Zred ,→ Z induce an isomorphism∼(Xred ×Yred Zred )red −→ (X ×Y Z)red(as one easily checks using the characterization of the various reduced substacks
involved; note also that we are using Remark 3.28 to see that the given morphisms
X , Z → Y restrict to morphisms Xred , Zred → Yred ).

28

M. EMERTON

We have the following more explicit description of the substack Xred of X .
Lemma 3.32. If X is a stack over S, then a morphism T → X whose source
is a scheme factors through the underlying reduced substack Xred if and only ifthere exists a morphism of schemes T 0 → T which is surjective, flat, and locally offinite presentation, such that the composite T 0 → T → X factors through a reduced
scheme.
Proof. Let Z denote the full subcategory of X whose objects consist of all morphisms T → X satisfying the criterion of the lemma, i.e. for which there a morphismof schemes that is surjective, flat, and locally of finite presentation, such that thecomposite T 0 → X factors as T 0 → T 00 → X , where T 00 is reduced.
We first note that Z certainly contains (Xred )0 (which classifies the morphisms
T → X for which T itself is reduced). We next show that Z is contained in Xred .
Indeed, if T → X is a morphism which is an object of Z, and T 0 and T 00 are above,
then certainly the morphism T 00 → X lies in Xred (since it lies in (Xred )0 ), and thus
so does its pull-back T 0 → X (since Xred is a category fibred in groupoids over S).
The stack property of Xred then implies that the originally given morphism T → X
must be an object of Xred .
Finally, we show that Z is itself a substack of X . Since it contains (Xred )0 andis contained in Xred , this will show that Z = Xred , and so prove the lemma.
Suppose, then, that T → X is an object of Z. If U → T is a morphism ofschemes, then the projection U 0 := T 0 ×T U is surjective, flat, and locally of finite
presentation, and the composite morphism U 0 → U → T → X admits the alternate
factorization U 0 → T 0 → T → X , and thus factors through a reduced scheme (since
T 0 → T → X does). Thus Z is a subcategory fibred in groupoids of X .
Lastly, suppose given a morphism T → X , and a morphism T 0 → T which issurjective, flat, and locally of finite presentation, such that the composite T 0 →T → X is an object of Z. By definition, then, there exists a morphism T 00 → T 0
that is surjective, flat, and locally of finite presentation, such that the composite
T 00 → T 0 → T → X factors through a reduced scheme. Since the composite
T 00 → T 0 → T is surjective, flat, and locally of finite presentation, we see that infact the morphism T → X is an object of Z, and so Z is indeed a substack of X . 
We now prove some lemmas concerning underlying reduced substacks which holdin the level of generality of Definition 3.27.
Lemma 3.33. If X is a stack over S, then (Xred )red = Xred .
Proof. This follows immediately from the explicit description of Xred given by
Lemma 3.32.
Lemma 3.34. If Y → X is a morphism of stacks over S that is representable byalgebraic stacks, and for which the induced morphism Yred → Xred is also representable by algebraic stacks, then Y → X is surjective if and only if Yred → Xred issurjective.
Proof. Write Y 0 := Xred ×X Y, regarded as a substack as Y (following the prescription of Remark 3.25); we then have the commutative diagram of morphisms ofFORMAL ALGEBRAIC STACKS

stacks

/ Y0

/Y

" 
Xred


/X

Yred

29

in which the horizontal arrows are inclusions of substacks, while the diagonal andvertical arrows are representable by algebraic stacks. Passing to underlying reduced
substacks of the various stacks appearing in the top row, and taking into accountLemma 3.33, we find that (Y 0 )red = Yred .
Let T → X be a morphism whose source is a scheme, and let Tred → Xred be theinduced morphism. Since Tred ,→ T is a thickening of schemes,∼

Tred ×Xred Y 0 −→ Tred ×X Y ,→ T ×X Y
is a thickening of algebraic stacks, and so T ×X Y → T is surjective if and only
Tred ×Xred Y 0 → Tred is surjective. Replacing X by Xred and Y by Y 0 (as noted
above, these latter two stacks have the same underlying reduced substack), we may
thus assume that X = Xred (so that also Y = Y 0 ), and it suffices to show that if T
is a scheme, then the projection(3.35)T ×X Yred → T

is surjective if and only if the projection(3.36)T ×X Y → T

is surjective.
Let V → T ×X Y be a surjective morphism whose source is a scheme (such amorphism exists, since the target is an algebraic stack). Then the composite Vred →V → T ×X Y → Y factors through Yred . Thus the composite Vred → V → T ×X Y
factors through a morphism Vred → T ×X Yred .
Since T ×X Yred ,→ T ×X Y is a monomorphism of algebraic stacks (and thus
universally injective; this is a special case of Remark 3.15), while the composite
Vred → T ×X Yred ,→ T ×X Y
is surjective, we conclude that the second arrow in this composite is in fact universally bijective. Thus either of the projections (3.35) or (3.36) is surjective if andonly if the other is. This completes the proof of the lemma.
Lemma 3.37. If U → X is a morphism of stacks over S which is representable byalgebraic stacks and smooth, then the induced morphism Ured → U ×X Xred is an
isomorphism.
Proof. Following the prescription of Remark 3.25, we regard U ×X Xred as a substackof U. As already observed in the statement of the lemma, the fact that the given
morphism U → X induces a morphism Ured → Xred shows that this substack
contains the substack Ured of U. To prove the lemma, we have to show the reverse
inclusion.
Thus suppose T → U is an object of the fibre product, which is to say that thecomposite T → U → X factors through Xred . Lemma 3.32 shows that we mayfind a morphism of schemes T 0 → T which is surjective, flat, and locally of finite
presentation, for which the composite T 0 → T → U → X factors as T 0 → T 00 → X ,with T 00 is reduced. The morphisms T 0 → T 00 and T 0 → T → U induce a morphism

30

M. EMERTON

T 0 → T 00 ×X U, whose target (by virtue of our assumptions on the morphism
U → X ) is an algebraic stack for which the projection to T 00 is smooth. Since T 00 is
reduced, so is the fibre product T 00 ×X U, and so we may choose a surjective smoothmorphism V → T 00 ×X U whose source is a reduced scheme.
Since the diagonal of the algebraic stack T 00 ×X U is representable by algebraic
spaces, the fibre product TT0 00 ×X U V is an algebraic space, and so we may findan étale surjective morphism V 0 → TT0 00 ×X U V whose source is a scheme. The
projection V 0 → T 0 is surjective and smooth, and thus the composite V 0 → T 0 → T
is surjective, flat, and locally of finite presentation. Furthermore, the composite
V 0 → T 0 → T → U admits the alternative factorization V 0 → V → T 00 ×X U → U.
Since V is reduced, we find (again applying Lemma 3.32) that the original morphism
T → U is an object of Ured , as required.

We next prove a general version of the topological invariance of the small Zariski
and étale sites. We begin by stating a general form of the infinitesimal lifting
property for an étale morphism of stacks.
Lemma 3.38. If U → X is a morphism of stacks which is DM and étale, if Z0 ,→ Z
is a closed immersion of algebraic stacks that is a thickening, and if we are givena morphism Z → X (which induces the composite morphism Z0 ,→ Z → X ), thenthe morphism of groupoids MorSt/X (Z, U) → MorSt/X (Z0 , U) is an equivalence.
∼

Proof. We have canonical equivalences MorSt/X (Z0 , U) −→ MorSt/Z (Z0 , U ×X Z)∼
and MorSt/X (Z, U) −→ MorSt/Z (Z, U ×X Z). Thus we may assume that Z equals X ,and so in particular reduce to the case when U → X is an étale morphism ofalgebraic stacks. Thus we suppose from now that we are in this situation (but wedon’t restrict to the case when Z equal X ).
We first suppose that Z = T is an affine scheme, so that T0 is a closed subscheme,with T0 ,→ T being a thickening. In this case the statement of the lemma follows
from the infinitesimal criterion as stated in [Ryd11, Cor. B.9].
Using the stack property of U to glue morphisms, one then extends successively
to the case when Z is an arbitrary scheme, to the case when when Z is an algebraic
space, and finally to the case when Z is an algebraic stack. (We leave the details
to the reader.)
We next define the small étale and small Zariski 2-categories attached to a stack
X over S.
Definition 3.39. If X is a stack over S, and if St/X denotes the 2-category ofstacks lying over X (following the notation of Definition 2.51), then we let EtX
denote the strong subcategory (in the sense of Definition 2.19) of St/X whose
objects are the 1-morphisms U → X of stacks which are DM and étale.
We let ZarX denote the strong subcategory of St/X consisting of morphisms
U → X which are monomorphisms.
Remark 3.40. Recall that open immersions of schemes coincide with étale monomorphisms [Stacks, Tag 025G]. This statement generalizes to morphisms U → X ofstacks which are representable by algebraic stacks and étale — such a morphism isan open immersion if and only if its a monomorphism. (The only direction is clear.
Conversely, if U → X is a monomorphism, then ∆U /X is an isomorphism, andhence in particular a monomorphism, and so U → X is representable by algebraic

FORMAL ALGEBRAIC STACKS

31

spaces. If T → X is a morphism from a scheme, then T ×X U → T is an étale
monomorphism from an algebraic space to a scheme, hence an étale monomorphismfrom a scheme to a scheme [Stacks, Tag 0418], and hence an open immersion.) This
explains our choice of notation ZarX .
We can now prove our result on topological invariance.
Lemma 3.41. If X0 → X is the inclusion of a closed substack into a stack over S,
which is furthermore a thickening, then the strict pseudo-functor U 7→ U0 := U ×X
X0 (see Remark 3.25 for an explanation of the strictness) induces 2-equivalences (inthe sense of Definition 2.16 (2)) of 2-categories EtX → EtX0 and ZarX → ZarX0 .
Proof. We begin by proving that the strict pseudo-functor EtX → EtX0 is fully
faithful. If U → X and V → X are objects of EtX , then we must show that thefunctorMorX (U, V) → MorX (U0 , V0 )is an equivalence of categories. The fact that X0 ,→ X is a monomorphism, andthe descriptions of U0 and V0 as fibre products, taken together show that
MorX0 (U0 , V0 ) → MorX (U0 , V)
is an equivalence. Thus it suffices to show that the composite
MorX (U, V) → MorX (U0 , V0 ) → MorX (U0 , V)
is an equivalence. This follows from Lemma 3.38.
To complete the proof that EtX → EtX0 is a 2-equivalence, it remains to showthat any object of EtX0 is equivalent to a morphism U0 → X0 , for some object
U → X of EtX .
If T → X is a morphism with T a scheme, write T0 := T ×X X0 , so that T0 is a
closed subscheme of T , with T0 ,→ T being a thickening. Given U0 → X0 as above,
we then define, for any morphism of schemes T → S, a groupoid U(T ) via
U(T ) := X (T ) ×X0 (T0 ) U0 (T0 ).
In this manner we obtain a category fibred in groupoids U over S, which is in facta stack, since each of X , X0 , and U0 is a stack. By construction, we also havea morphism U → X , and a morphism U ×X X0 → U0 , which one verifies is an
isomorphism. It remains to show that this morphism is DM and étale.
If we fix T → X , then we see that T ×X U is equivalent to the stack overT which associates, to a morphism of schemes T 0 → T , the groupoid (T ×X0
U0 )(T00 ). To simplify notation, then, we replace X by T , and thus assume givena thickening T0 ,→ T and a Deligne–Mumford stack U0 equipped with an étalemorphism U0 → T0 ; our goal is to show that U(T 0 ) := U0 (T00 ) defines a Deligne–
Mumford stack which is étale over T . We may write U0 := [U0 /R0 ] for somescheme étale over T0 , and some étale groupoid in algebraic spaces R0 over U0 . The
topological invariance of the étale site for algebraic spaces [Stacks, Tag 05ZG] thenlifts U0 to an étale morphism U → T , and R0 to an étale groupoid in algebraic
spaces over U . The tautological morphism U0 → [U0 /R0 ] := U0 induces a morphism
U → U (by definition of U), which is easily verified to induce an isomorphism∼[U/R] −→ U. Thus U is a Deligne–Mumford stack which is étale over T , as required.
The statement concerning Zariski topologies follows from the fact that, since
X0 ,→ X is a thickening, an étale morphism U → X is a monomorphism if and only
if Ured → Xred is (by [Stacks, Tag 0CJC]).


32

M. EMERTON

4. Ind-algebraic stacks
In this section, we use the theory of 2-colimits presented in Section 2 to define
Ind-algebraic stacks, and establish some simple results concerning them.
We first recall the notion of an Ind-algebraic space.
Definition 4.1. If S is a scheme, then a sheaf X on the fppf site of S is called
an Ind-algebraic space if there exists a directed system {Xi }i∈I of algebraic spaces
over S and an isomorphism limi Xi ∼= X, where the inductive limit is computed in−→
the category of sheaves on the fppf site of S. We say that X is an Ind-scheme if
the Xi can be taken to be schemes.
We wish to extend this notion so as to define a notion of Ind-algebraic stack. Outof habit and old-fashioned convention, we will usually use the notation lim rather−→
than 2 − colim. We also frequently write “2-directed system” to mean a 2-diagramwith a filtered index category.
Definition 4.2. A stack in groupoids X on the fppf site of S is called an Indalgebraic stack if there exists a 2-directed system {Xi }i∈I of algebraic stacks over
S and an isomorphism of stacks limi Xi ∼= X , where the left-hand side denotes the−→
2-colimit computed in the 2-category of stacks in groupoids on the fppf site of S.
Example 4.3. We give an illustrative example of an Ind-algebraic stack. Fix abase algebraic space B (lying over the base scheme S), and let {Xi }i∈I denote a
directed system of algebraic spaces that are of finite presentation and separated
over B. For each Xi , we have the stack CohXi /B of coherent sheaves on Xi whose
support is proper over B. This is an algebraic stack [Stacks, Tag 08WB]. If i ≤ i0 ,then pushforward induces a morphism CohXi /B → CohXi0 /B [Stacks, Tag 08DS].
Since the pushforward along a composite is naturally isomorphic to the compositeof the corresponding pushforwards, in a manner that is compatible with triplecomposites, we see that {CohXi /B }i∈I forms a 2-directed system of algebraic stacks.
We may thus form the 2-colimit limi CohXi /B . This is an Ind-algebraic stack, which−→
morally may be regarded as the stack of coherent sheaves on the Ind-algebraic space
X := limi Xi whose support is proper over B.
−→
We now establish some simple results related to Ind-algebraic stacks and morphisms between them.
∼

Lemma 4.4. If X is an Ind-algebraic stack, and if limi Xi −→ X is an iso−→morphism, where {Xi }i∈I is a 2-directed system of algebraic stacks, then there isa natural morphism of groupoids 2 − colimi Xi (T ) → X (T ). If T is quasi-compact,
then this morphism is fully faithful. If in addition either T is quasi-separated or
the transition morphisms in the 2-directed system {Xi } are monomorphisms, thenthis morphism is an equivalence.
Proof. This follows from Lemma 2.50.

Lemma 4.5. Let {Xi }i∈I be a 2-directed system of algebraic stacks, all of whose∼transition morphisms are monomorphisms (i.e. fully faithful). If limi Xi −→ X ,−→then each of the induced morphisms(4.6)

Xi → X

is a monomorphism, and is representable by algebraic stacks.

FORMAL ALGEBRAIC STACKS

33

Furthermore, if P is a property of morphisms of algebraic stacks that is preserved under base-change and fppf local on the target, then each of the transition
morphisms in the 2-directed system {Xi } has property P if and only if each of theinduced morphisms (4.6) has property P (in the sense of Definition 3.11).
Proof. To show that the induced morphism (4.6) is a monomorphism, it suffices toevaluate it on morphisms T → X , with T an affine (and hence quasi-compact andquasi-separated) scheme. By Lemma 4.4, it then suffices to show that the naturalmorphism Xi (T ) → 2−colimi Xi (T ) is fully faithful, given that each of the transition
morphisms in the 2-colimit is fully faithful. This follows from Lemma 2.49.
To prove that (4.6) is representable by algebraic spaces, we again consider morphisms T → X with T being affine. As already noted, such a morphism factors
through Xi0 for some i0 ≥ i, and since (by the result of the preceding paragraph)the morphism Xı0 → X is a monomorphism, the canonical morphism
T ×Xi0 Xi → T ×X Xi
is an isomorphism. The source of this morphism is an algebraic stack, since it is
the 2-fibre product of algebraic stacks. Thus the target is also an algebraic stack,which (taking into account Remark 3.2) is what we had to show.
To prove the final claim of the lemma, suppose first that each transition morphism has property P . From the argument of the preceding paragraph, we obtain∼an isomorphism T ×Xi0 Xi −→ T ×X Xi of algebraic stacks over T . By assumption the projection of the source onto T has property P , and hence so does the
projection of the target onto T . Thus (taking into account Remark 3.12), we findthat (4.6) has property P , as required. Conversely, suppose that each of these lattermorphisms has property P , and let Xi → Xi0 be a transition morphism. Since (4.6)
is a monomorphism, in the 2-commutative diagram
Xi ×X Xi0

Xi

$
/ Xi0

we find that vertical morphism is an isomorphism. The diagonal morphism hasproperty P , being the base-change to Xi0 of (4.6) (which has property P by assumption), and thus so does the horizontal arrow.

We refer to [Stacks, Tag 07XL], for the definition of a stack over S being limit
preserving.
Lemma 4.7. Suppose that {Xi }i∈I is a 2-directed system of algebraic stacks, each∼
locally of finite presentation over S. If X −→ limi Xi , then X is limit preserving.
−→
Proof. For each of the algebraic stacks Xi over S, being locally of finite presentation
is equivalent to being limit preserving [EG19b, Lem. 2.1.9]. Since the property of being limit preserving is tested on affine S-schemes (by definition), and affine schemes
are in particular quasi-compact, the present lemma follows from Lemma 4.4.
Lemma 4.8. Suppose that we are given stacks X , Y, and Z over S, and morphisms
X → Y and Z → Y, latter being representable by algebraic stacks. Suppose further
that X is an Ind-algebraic stack, and let {Xi }i∈I be a 2-directed system of algebraic

34

M. EMERTON∼

stacks for which there is an isomorphism limi Xi −→ X . Then {Xi ×Y Z}i∈I again−→
forms a 2-directed system of algebraic stacks, and there is an isomorphism∼

lim Xi ×Y Z −→ X ×Y Z.
−→i

In particular, X ×Y Z is again an Ind-algebraic stack.
Proof. Since Z → Y is representable by algebraic stacks (by assumption), the 2fibre products Xi ×Y Z are indeed algebraic stacks. Each of the morphisms Xi → X
induces a morphism Xi ×Y Z → X ×Y Z, which are compatible (in a 2-categoricalsense) as i varies. Thus we obtain a morphism limi Xi ×Y Z → X ×Y Z, which we−→must show is an isomorphism of stacks. To check this, it suffices to show that it
induces an equivalence of groupoids after evaluating on any affine S-scheme. Thisis easily verified using Lemma 2.50

Remark 4.9. A useful special case of the preceding lemma comes by setting X
equal to Y; namely, we see that if Z → Y is a morphism of stacks over S that isrepresentable by algebraic stacks, and if Y is an Ind-algebraic stack, then so is Z.
More precisely, if {Yi }i∈I is a 2-directed system of algebraic stacks for which there∼
is an isomorphism lim Yi −→ Y, then each of 2-fibre products Zi := Yi ×Y Z is an−→algebraic stack, which fit into a 2-directed system {Zi }, and there is an isomorphism∼
lim Zi −→ Z.
−→
We now state and prove a lemma that provides a supplement to the preceding
remark, but before doing so, we introduce some notation and conventions. Namely,
suppose that {Xi }i∈I and {Yi }i∈I are two 2-directed systems of algebraic stacks,
with the same indexing category I. We note that a 1-morphism between these
2-directed systems (in the sense of Definition 2.4) consists of giving a morphism ofstacks fi : Xi → Yi ,for each object i of I, such that for each i0 ≥ i the square
/ Xi0

Xi

(4.10)

fi0

fi


/ Yi0


Yi

(the horizontal arrows being the transition morphisms in the respective 2-directed
systems) is 2-commutative. In fact, the 2-commutativity constraints of these diagrams are part of the data of the 1-morphism, and there is an additional conditionon the 2-commutativity constraints (conditions (4) and (5) of Definition 2.4); however, in practice we won’t introduce notation for these constraints, and will denotethe 1-morphism simply by {fi }i∈I .
Lemma 2.41 shows that the 1-morphism {fi } induces a limiting morphism
lim fi : lim Xi → lim Yi .
−→
−→
−→
i

i

i

Lemma 4.11. Suppose that {fi } : {Xi }i∈I → {Yi } is a 1-morphism between two
2-directed systems of algebraic stacks (with the same indexing category I). Let
f : X := limi Xi → limi Yi =: Y denote the induced morphism limi fi .
−→−→−→(1) If each of the morphisms fi is a monomorphism (i.e. fully faithful), then themorphism f is a monomorphism.

FORMAL ALGEBRAIC STACKS

35(2) If each of the squares (4.10) is 2-Cartesian, then the morphism f is representable by algebraic stacks, and for any i ∈ I, the canonical morphism Xi →X ×Y Yi is an isomorphism.
(3) Suppose that each of the squares (4.10) is 2-Cartesian, and let P be a property
of morphisms of algebraic stacks which is preserved by base-change and fppf local
on the target. Then f has property P (in the sense of Definition 3.11) if and only
if each fi has property P .
Proof. The claim of (1) may be verified by evaluating on affine schemes (which
are in particular quasi-compact and quasi-separated); its proof is then a routine
application of Lemma 4.4.
We turn to proving (2), and thus assume that each of the squares (4.10) is
2-Cartesian. Let(4.12)T → X ×Y Yi

be a morphism whose source is an affine scheme. By Lemma 4.4, we may factor the
induced morphism T → X (up to 2-isomorphism) through a morphism T → Xi0 ,for some i0 ≥ i. The induced morphism T → Y then factors (up to 2-isomorphism)
through a morphism T → Yi0 . Since the composites T → Yi0 → Y and T → Yi → Y
coincide (up to 2-isomorphism), we find, again by Lemma 4.4, that the composites
T → Yi → Yi00 and T → Yi0 → Yi00 are 2-isomorphic for some i00 ≥ i. Thus thegiven morphism (4.12) factors (up to 2-isomorphism) through the 2-fibre productXi00 ×Yi00 Yi . By assumption the canonical map Xi → Xi00 ×Yi00 Yi in an isomorphism,
which is to say, an equivalence of categories, and thus the given morphism factors(up to 2-isomorphism) through Xi . Thus the canonical morphism of groupoids
Xi (T ) → (X ×Y Yi )(T ) is essentially surjective. Similar arguments show that it isfull and faithful, and hence an equivalence. This proves the claimed isomorphismof (2).
Now suppose that T → Y is a morphism whose source is an affine scheme.
By Lemma 4.4, we may factor this morphism (up to 2-isomorphism) through amorphism T → Yi for some i. We then find that
X ×Y T ∼= (X ×Y Yi ) ×Yi T ∼= Xi ×Yi T,
so that X ×Y T is an algebraic stack (being the 2-fibre product of algebraic stacks
over an algebraic stack). Taking into account Remark 3.2, this shows that themorphism X → Y of (1) is representable by algebraic stacks, and completes theproof of (2).
Finally, suppose given a property P as in (3). If f has property P , then, taking
into account (2), we find that each of the morphisms fi : Xi → Yi has property P .
Conversely, suppose that each fi does have property P . If T → Y is morphism
whose source is an affine scheme, then Lemma 4.4 again shows that this morphism
may be factored (up to 2-isomorphism) through Yi for some i ∈ I. Then we computethat
X ×Y T ∼= (X ×Y Yi ) ×Y T ∼= Xi ×Y T,iithe last isomorphism holding by (2). Since Xi → Yi has property P by assumption,and since P is preserved by base-change, we see that X ×Y T also has property P ,as required. (Here we are taking into account Remark 3.12 to restrict to affine
schemes T as test objects.)
36

M. EMERTON

The preceding lemma can be applied to deduce properties of the diagonal of an
Ind-algebraic stack in certain contexts.
Lemma 4.13. Let P be a property of morphisms of algebraic spaces which is preserved by base-change and fppf local on the target. Let {Xi }i∈I be a 2-directed
system of algebraic stacks, in which the transition morphisms are monomorphisms,∼
write limi Xi −→ X , and suppose given a morphism X → Y, where Y is a stack over−→S whose diagonal morphism Y → Y ×S Y is representable by algebraic spaces, andfor which each of the diagonal morphisms ∆Xi /Y : Xi → Xi ×Y Xi have property P .
Then the diagonal morphism ∆X /Y : X → X ×Y X is representable by algebraic
spaces, and has property P .
Proof. Form the 2-directed system {Xi ×Y Xi }. If i ≤ i0 , then since Xi → Xi0 is amonomorphism, the diagram
/ Xi0
Xi

Xi ×Y Xi
/ Xi0 ×Y Xi0

is 2-Cartesian. The present lemma thus follows from Lemma 4.11 (2) and (3). (Notethat the diagonal morphism ∆X /Y is always representable by sheaves, and hence isrepresentable by algebraic spaces if it is representable by algebraic stacks.)
Remark 4.14. Taking Y = S in the preceding lemma, we find in particular that
if X is an Ind-algebraic stack obtained by taking the 2-colimit of a 2-directed
system of algebraic stacks whose transition morphisms are monomorphisms, then
X → X ×S X is representable by algebraic spaces.
Example 4.15. Some properties P to which the preceding lemma applies are thatof being a monomorphism, in which case the lemma shows that if each of themorphisms Xi → Y is representable by algebraic spaces, then so is the morphism
X → Y, and that of being quasi-compact and quasi-separated, in which case thelemma shows that if each of the morphisms Xi → Y is quasi-separated, then so is
the morphism X → Y.6∼

Lemma 4.16. Suppose that X is an Ind-algebraic stack, and write limi Xi −→ X ,−→for some 2-directed system {Xi }i∈I of algebraic stacks. The induced morphism
limi (Xi )red → Xred is then an isomorphism, and so in particular Xred is again an−→
Ind-algebraic stack.
Proof. Lemma 4.11 (2) shows that there is an induced monomorphism∼lim(Xi )red ,→ lim Xi −→ X ,
−→
−→
i

i

whose essential image is then a substack of the target. We claim that this substack is
precisely Xred . Remark 3.28 shows that this substack is contained in Xred . To provethe claim, then, it suffices to show that if T → X is a morphism from a reduced
affine scheme to X , then we may factor this morphism (up to a 2-isomorphism)
through (Xi )red , for some i. This follows from Lemma 4.4, which shows that the
6Note that the morphism X → Y need not be representable by algebraic stacks, but the lemmashows that the diagonal X → X ×Y X is representable by algebraic spaces, so that the notion of
X → Y being quasi-separated makes sense.

FORMAL ALGEBRAIC STACKS

37

morphism T → X factors (up to a 2-isomorphism) through Xi , for some i, and thus
through (Xi )red , since T is reduced.

Remark 4.17. In general, in the context of the preceding lemma, the monomorphism of Ind-algebraic stacks Xred ,→ X need not be representable by algebraic
stacks/spaces.7 Indeed, this is already the case with Ind-schemes, as the following
example shows.
Example 4.18. Consider the Ind-scheme X := limn Xn , where−→
Xn := Spec C[x, y] x(x − 1) · · · (x − n)y 2 .
Applying Lemma 4.16, one finds that

Xred = lim(Xn )red = lim Spec C[x, y]/ x(x − 1) · · · (x − n)y .
−→
−→
n
n
Applying Lemma 4.8, we then find that
∼
X0 ×X Xred −→ lim X0 ×X (Xn )red−→n∼−→ lim Spec C[x, y]/ xy 2 , x(x − 1) · · · (x − n)y
−→



n

which is an affine formal algebraic space (in the sense of [Stacks, Tag 0AI7]; seealso Definition 5.1 below), but not an algebraic space.
There is one context, closely related to that of formal algebraic stacks (as we
will see in the discussion of Section 6) in which we can obtain more precise control
over the morphism Xred → X .
Lemma 4.19. Suppose that X is an Ind-algebraic stack, and that we can write∼
limi Xi −→ X , for some 2-directed system {Xi }i∈I of algebraic stacks for which−→
the transition morphisms are thickenings. Then Xred is an algebraic stack, as well
as a closed substack of X , each of the induced morphisms (Xi )red → Xred is an
isomorphism, and the morphism Xred ,→ X is a thickening.
Proof. Since the transition morphisms in the 2-directed system {Xi } are thickenings, they induce isomorphisms on underlying reduced substacks, i.e. the transition
morphisms in the 2-directed system {(Xi )red } are isomorphisms. Taking into account Lemma 4.16, we find that indeed each of the morphisms (Xi )red → Xred is an
isomorphism, and in particular, that Xred is an algebraic stack.
If i0 ≥ i, then the transition morphism Xi → Xi0 is a thickening, and so inparticular is a monomorphism. It thus follows from Lemma 4.5 that each of themorphisms Xi → X is a monomorphism. This explains why the second arrow in
the following sequence of morphisms is an isomorphism:∼∼∼(Xi )red −→ Xi ×Xi (Xi )red −→ Xi ×X (Xi )red −→ Xi ×X Xred .
Clearly the first arrow is also an isomorphism, and in the preceding paragraph weshowed that the third arrow is an isomorphism as well. Consequently, the composite(Xi )red → Xi ×X Xred is an isomorphism, and thus the morphism(4.20)

Xi ×X Xred → Xi

7Recall that for monomorphism of stacks, being representable by algebraic stacks is equivalent
to being representable by algebraic spaces.

38

M. EMERTON

is a thickening.
Lemma 4.4 shows that if T → X is an morphism whose source is an affine
scheme, then this morphism factors (up to 2-isomorphism) through some Xi . Thus
the morphism T ×X Xred → T is obtained (up to 2-isomorphism) as the base-changeof the morphism (4.20), and so is a thickening. This completes the proof of thelemma.

Lemma 4.21. (1) If X is an Ind-algebraic stack, and if U → X is an object of
EtX , then U is again an Ind-algebraic stack.
∼(2) Suppose furthermore that there is an isomorphism limi Xi −→ X , where−→{Xi }i∈I is a 2-directed system of algebraic stacks for which the transition morphisms
are thickenings. Then the strict pseudo-functor U 7→ Ured induces 2-equivalences(in the sense of Definition 2.16 (2)) ZarX → ZarXred and EtX → EtXred .
Proof. If U → X is DM, then it is in particular representable by algebraic stacks.
Thus if X is an Ind-algebraic stack, then so is U, by Remark 4.9. This proves (1).
∼Suppose now that we are in the context of (2). Lemma 3.37 shows that Ured −→U ×X Xred , while Lemma 4.19 shows that Xred ,→ X is a thickening. The claimed
2-equivalences thus follow from Lemma 3.41.
Lemma 4.22. If X is an Ind-algebraic stack over a locally Noetherian scheme S,
then X satisfies the condition (RS*).
Proof. Since the diagram appearing in the definition of (RS*) (i.e. Definition 3.8)
involves only affine schemes, which are in particular quasi-separated, we see that
this follows from Lemma 4.4, together with [Stacks, Tag 0CXP], which shows thatalgebraic stacks satisfy (RS*).

5. Formal algebraic stacks
In this section, we begin by briefly recalling the definition of a formal algebraic
space from [Stacks], and then proceed to give the definition of, and establish somebasic results regarding, formal algebraic stacks.
We fix a base scheme S. The following definitions follow [Stacks, Tag 0AI7] (foraffine formal algebraic spaces) and [Stacks, Tag 0AIL] (for formal algebraic spaces
in general).
Definition 5.1. (1) An affine formal algebraic space over S is a sheaf X on the fppf∼site of S which admits a description as an Ind-scheme X −→ limi Xi , where the−→Xi are affine schemes and the transition morphisms are thickenings (in the senseof [Stacks, Tag 04EX]).
(2) A formal algebraic` space over S is a sheaf X on the fppf site of S which
receives a morphism
Ui → X which is representable (by schemes), étale, andsurjective, and whose source is a disjoint union of affine formal algebraic spaces Ui .
Remark 5.2. We remark that the notion of affine formal algebraic space used in[Stacks], and recalled above, is more general than the notion of affine formal scheme
defined in [EGA1, §10] or other references. Similarly, the notion of formal algebraic
space used in [Stacks] is more general than the notion developed in e.g. Knutson’s
original treatment [Knu71, Ch. 5], where only the separated Noetherian case istreated. (And in our intended applications, it is actually crucial that we work in a

FORMAL ALGEBRAIC STACKS

39

framework that allows formal algebraic spaces and formal algebraic stacks that are
not a priori Noetherian.)
The following definition is the evident analogue of Definition 5.1 in the context
of stacks.
Definition 5.3. Let X be a stack in groupoids8 on the fppf site of a scheme S.
We say that X is a formal algebraic stack if there is a morphism U → X , whose
domain U is a formal algebraic space, and which is representable by algebraic spaces,smooth and surjective.
Remark 5.4. This definition should be compared with the characterization of algebraic stacks as stacks which receive a morphism whose source is an algebraic space,
and which is representable, smooth, and surjective [Stacks, Tag 05UL].
Remark 5.5. If we take into account the definition of a formal algebraic space, wefind that we may equivalently
define a formal algebraic stack to be a stack X which`
receives a morphism Xi → X which is representable by algebraic spaces, smooth,and surjective, and whose source is a disjoint union of affine formal algebraic spaces.
Remark 5.6. If, in Definition 5.3, we ask that the morphism U → X be étale,rather than merely smooth, we obtain the notion of a formal Deligne–Mumford
stack. Since our interest is primarily in the case of formal algebraic stacks that are
not necessarily Deligne–Mumford, we won’t say much about this special case of thedefinition in these notes.
Example 5.7. A formal algebraic space over S may be regarded as a formal algebraic
stack in a natural way, by reinterpreting the underlying sheaf as a category fibred
in setoidsover the fppf site of S. We will prove in Lemma 6.11 that, conversely,
if X is a formal algebraic stack which has trivial inertia stack (i.e. corresponds
to a category fibred in setoids, rather than just in groupoids), then X is in fact
(isomorphic to) a formal algebraic space. (In the same result, we also show that it
suffices to check the triviality of the inertia stack after pulling back to geometricpoints.)Example 5.8. Since any algebraic space is also a formal algebraic space, an algebraic
stack over S is also a formal algebraic stack over S.
Example 5.9. Suppose that X is an algebraic stack over S, and that T ⊆ |X | is alocally closed subset of the topological space underlying X . We may then define
Xb|T , the completion of X along T , to be the substack of X defined as
Xb|T (U ) := {f : U → X | f (|U |) ⊆ T }.
(For the analogous construction in the case of algebraic spaces, see [Stacks, Tag
0AIX].9)
To see that Xb|T is a formal algebraic stack, choose a smooth surjection V → X ,where V is an algebraic space. If we let T 0 denote the preimage of T in |V |, then
8Since from now on we only consider stacks in groupoids, we will typically write “stack” rather
than “stack in groupoids”.
9In that discussion, the subset T is required to be closed. But note that if T is locally closed
in |X |, then there is an open substack U of X so that T is closed in |U |. One sees immediately∼from the definition that the open immersion U → X induces an isomorphism Ub|T −→ Xb|T , andso, after replacing X by U , we may reduce ourselves to the case when T is closed.

40

M. EMERTON

the fibre product Xb|T ×X V is naturally isomorphic to Vb|T 0 , the completion of V
along T 0 . Since Vb|T 0 is a formal algebraic space, by [Stacks, Tag 0AIZ], and since
the projection Xb|T ×X V → Xb|T is representable by algebraic spaces, smooth, andsurjective (being the base-change of a morphism with these properties), we see that
Xb|T is indeed a formal algebraic stack.
Remark 5.10. In some other treatments of formal algebraic spaces and stacks, suchas that of [Har05, App. A], these notions are defined by considering sheaves orstacks on a site NilpX , where X is a formal scheme. However, in the frameworkof the Stacks Project, a formal scheme X (over some base scheme S) is simply asheaf on the fppf site of S, and NilpX can then be thought of as the category of
S-morphisms T → X (with T an S-scheme). Giving a sheaf, or stack, on NilpX ,is then equivalent to giving a sheaf, or stack, on the fppf site of S, equipped witha morphism (of sheaves, or stacks) to X. In particular, giving a formal algebraic
space, or formal algebraic stack, on NilpX , say in the sense of [Har05, App. A], willthen amount to giving a formal algebraic space, in the sense of the Stacks Project,
or a formal algebraic stack, in the sense defined above, equipped with a morphism
to X; thus our approach to the subject, while differing in its foundational set-up
from that of [Har05, App. A], is nevertheless compatible with that approach. (Seee.g. [Stacks, Tag 00XZ] and [Stacks, Tag 04WV] for a discussion of the relevant sheaf
and stack-theoretic background, as well as [Stacks, Tag 03I3] for an explanation of
how it applies in the context of algebraic spaces, and [Stacks, Tag 04X4] for thecorresponding explanation in the context of algebraic stacks.)(In fact, there are some other minor differences between our treatment and thatof [Har05, App. A], which we will briefly discuss. One is that, as already noted, wework with the fppf topology, rather than the étale topology. This is harmless inpractice. Another is that the definition of a formal algebraic space in the Stacks
Project is more general than that of [Har05, App. A]: for example, non-quasiseparated objects are allowed, and the underlying notion of formal scheme is also
more general than that of [EGA1, §10] or other references. This means that ournotion of formal algebraic stack is correspondingly more general.)
We begin our theoretical development by proving the representability by algebraic spaces of the diagonal morphism of a formal algebraic stack. (The corresponding result for formal algebraic spaces is [Stacks, Tag 0AIP].) Before giving theproof, we recall a lemma which provides a general strategy for studying propertiesof diagonal morphisms (which is useful in part for its notational convenience).
Lemma 5.11. If X is a category fibred in groupoids over the scheme S, and if P isa property of morphisms of sheaves (on the fppf site of S) which is preserved underarbitrary base-change, then the following are equivalent:(1) If T → X ×S X is any morphism whose source is a scheme, then the basechanged morphism
T ×X ×S X X → T
has the property P .
(2) If T1 → X and T2 → X are arbitrary morphisms to X whose sources areschemes, then the base-changed morphism
T1 ×X T2 → T1 ×S T2
has the property P .

FORMAL ALGEBRAIC STACKS

41

Proof. This is well-known, but we briefly recall the proof. The key point is that∼
we there is an isomorphism T1 ×X T2 −→ (T1 ×S T2 ) ×X ×S X X , which allows us torewrite the morphism of (2) in the form
(T1 ×S T2 ) ×X ×S X X → T1 ×S T2 .
Taking this into account, we see that if (1) holds, then we may deduce (2) byapplying (1) to the morphism T := T1 ×S T2 → X ×S X . Conversely, if (2) holds,
and if we are in the context of (1), the given morphism T → X ×S X corresponds
to a pair of morphisms T ⇒ X . Applying (2) to the product of these morphisms,
which is a morphism T ×S T → X ×S X , we find that the base-changed morphism(T ×S T ) ×X ×S X X → T ×S T has the property P . Pulling back this morphism
along the diagonal morphism T → T ×S T shows that (1) holds.
Lemma 5.12. The diagonal of a formal algebraic stack is representable by algebraic
spaces, and locally of finite type.
Proof. Taking into account Lemma 5.11 (and taking P to be the property of being
representable by algebraic spaces), we consider a pair of morphisms T1 , T2 → X ,each of whose source is an S-scheme. If we choose a morphism U → X as in thedefinition of a formal algebraic stack (so U is a formal algebraic space, and themorphism is representable by algebraic spaces, smooth, and surjective), then we
may pull back this morphism along the morphism of sheaves T1 ×X T2 → X toobtain a morphism of sheaves(T1 ×X U ) ×U (T2 ×X U ) → T1 ×X T2
which is representable by algebraic spaces, smooth, and surjective. Each fibre
product Ti ×X U is an algebraic space (since U → X is representable by algebraic
spaces), and so their fibre product over the formal algebraic space U is again an
algebraic space (since the diagonal of U is representable). Thus T1 ×X T2 is a sheaf
receiving a morphism whose source is an algebraic space, and which is representable,smooth, and surjective. It follows that T1 ×X T2 is itself an algebraic space [Stacks,
Tag 04S6]. This shows that the diagonal of X is representable by algebraic spaces.
To see that the diagonal is furthermore locally of finite type, we must show thatthe natural morphism T1 ×X T2 → T1 ×S T2 is locally of finite type. This may bechecked smooth locally on the source and target, and so it suffices to show that thenatural morphism(T1 ×X U ) ×U (T2 ×X U ) → (T1 ×X U ) ×S (T2 ×X U )is locally of finite type. This is a base-change of the diagonal morphism U → U ×S U ,which is indeed locally of finite type [Stacks, Tag 0AIP].

We recall another general result related to diagonals (see also [EG19b, Prop. 2.3.16,
2.3.18], where essentially the same result is proved).
Lemma 5.13. (1) If X is a category fibred in groupoids over S whose diagonal∆ : X → X ×S X is representable by algebraic spaces, then the double diagonal∆∆ : X → X ×X ×X X is also representable by algebraic spaces. If the diagonal ∆is furthermore locally of finite type, then the double diagonal ∆∆ is locally of finite
presentation.
(2) If f : X → Y is a morphism of categories fibred in groupoids over S which isrepresentable by algebraic spaces, and if the diagonal morphism ∆X : X → X ×S X

42

M. EMERTON

and the double diagonal morphism ∆∆Y : Y → Y ×Y×S Y Y are each also representable by algebraic spaces (by (1), the last hypothesis holds if ∆Y is representableby algebraic spaces), then the diagonal morphism ∆f : X → X ×Y X is also representable by algebraic spaces. If furthermore each of ∆X and ∆∆Y are locally offinite type, then so is ∆f . (By (1), this hypothesis on ∆∆Y holds if ∆Y is locally offinite type.)
Proof. We begin by proving (1). If T is an S-scheme, then the groupoid of T -valued
points of X ×X ×S X X is equivalent to the groupoid classifying pairs (ξ ∈ X (T ), α ∈Aut(ξ)), and given an object (ξ, α) of this groupoid, the groupoid (which is in fact asetoid) of T 0 -valued points of the corresponding fibre product T ×X ,∆∆ X ×X ×S X X
is equivalent to the sheaf that classifies morphisms T 0 → T such that αT 0 = idξT 0 .
This sheaf can be described as the fibre productT ×α×id,Aut(ξ)×T Aut(ξ),∆ Aut(ξ).
If ∆ is representable by algebraic spaces, then Aut(ξ) is an algebraic space, andhence so is this fibre product. If ∆ is furthermore locally of finite type, then Aut(ξ)
is locally of finite type over T , and hence its diagonal is locally of finite presentation [Stacks, Tag 084P]. Thus this fibre product is locally of finite presentation
over T .
Claim (2) follows by the usual graph argument: we factor the diagonal morphism∆X as the composite∆fX −→ X ×Y X → X ×S X .
This composite is representable by algebraic spaces by assumption, and since thesecond morphism is a base-change of the diagonal ∆Y of Y, its diagonal is representable by algebraic spaces by assumption. Factoring ∆f asΓ∆f

X −→ X ×X ×S X (X ×Y X ) → X ×Y X ,we see that it is a composite of morphisms that are representable by algebraic
spaces, and so is representable by algebraic spaces itself. The claim regarding ∆fbeing locally of finite type (under the appropriate hypotheses) follows in the sameway.
Lemma 5.14. (1) If X is a formal algebraic stack, then the double diagonal∆∆ : X → XX ×S X X is representable by algebraic spaces, and locally of finite presentation.
(2) If f : X → Y is a morphism of formal algebraic stacks, then ∆f : X →X ×Y X is representable by algebraic spaces, and locally of finite type.
Proof. This follows immediately from Lemmas 5.12 and 5.13.

Lemma 5.12 shows that the discussion of Example 3.16 applies to morphisms
from algebraic stacks to formal algebraic stacks. For example, it makes sense to
speak of a morphism from an algebraic stack to a formal algebraic stack being flatand surjective, and so the statement of the following lemma makes sense. (Thelemma may be thought of as showing that the property of a formal algebraic stackactually being an algebraic stack can be checked flat locally.)
Lemma 5.15. If X → Y is a morphism from an algebraic stack to a formal
algebraic stack which is flat and surjective (in the sense explained in Example 3.16),
then Y is itself an algebraic stack.

FORMAL ALGEBRAIC STACKS43

Proof. It suffices to show that if W → Y is a morphism from an affine formal algebraic space to Y which is representable by algebraic spaces, smooth, and surjective,then W is actually an affine scheme. (Indeed, from this, and the assumption
that`
Y is a formal algebraic stack, it follows that Y admits a morphism
Wi → Y
whose source is a disjoint union of affine schemes which is representable by algebraic spaces, smooth, and surjective, and thus — by Remark 5.4 — that Y is analgebraic stack.) Given such a morphism, we find that X ×Y W → W is flat andsurjective, with its source being an algebraic stack. We may then find a morphism
U → X ×Y W whose source is a scheme, and which is smooth and surjective, andthe composite morphism U → W is then representable by algebraic stacks, flat,and surjective. Since its source is a scheme and its target has a diagonal whichis representable (by schemes) [Stacks, Tag 0AIP]), it is in fact representable, flat,and surjective. Since W is an affine formal algebraic space, we may find an affine
scheme U 0 , and a morphism U 0 → U which is locally on the source an open immersion, such that the composite U 0 → U → W is surjective. Replacing U by U 0 ,we may thus assume that U and W are both affine. The morphism U → W thenfactors as U → V ,→ W, where V is an affine scheme and V ,→ W is a thickening.
Thus if T → V is any morphism whose source is a scheme, we obtain morphisms
T ×W U → T ×W V → T,with the composite being faithfully flat, and the second arrow being a thickening.
Since faithfully flat morphisms are scheme-theoretically dominant, we find thatthe second arrow is necessarily an isomorphism, and thus that V → W is itself
an isomorphism (since T → W was arbitrary). Thus W is an affine scheme, as
claimed.

We recall that in the Stacks Project, a formal algebraic space is defined to
be locally countably indexed, resp. locally adic*, if it admits an étale surjection
from a union of affine formal algebraic spaces that are countably indexed, resp.
adic* [Stacks, Tag 0AKY]. (An affine formal algebraic space is countably indexed
if it may be written as an inductive limit of a sequence of thickenings of affine
schemes [Stacks, Tag 0AII]; an affine formal algebraic space is adic* if it is of theform Spf A for an adic* topological ring A, i.e. a topological ring which is complete,10 and which admits a finitely generated ideal — called an ideal of definitionof A — whose powers are all open, and form a basis of open neighbourhoods ofzero [Stacks, Tag 0AID].) We now extend these notions to the context of formal
algebraic stacks.
Definition 5.16. We say that a formal algebraic stack X is countably indexed, resp.
locally adic*, if we may find a representable morphism U → X which is smoothand surjective, and whose source is a countably indexed, resp. locally adic*, formal
algebraic space.
Remark 5.17. Note that it follows from [Stacks, Tag 0AKS], resp. [Stacks, Tag
0AKT], that if a formal algebraic space (thought of as a formal algebraic stack) is
locally countably indexed, resp. locally adic*, in the sense of this definition, then it
is in fact locally countably indexed, resp. locally adic*, in the sense already definedin the Stacks Project.
10By convention, complete here means complete and separated.

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M. EMERTON

Example 5.18. If X is an algebraic stack over S, and if T is a closed subset of |X |for which the open immersion X \ T → X is quasi-compact, then the completion
Xb|T is locally adic*. To see this, we use a fact which was noted in the discussionof Example 5.9, namely that if V → X is a smooth surjection whose source isan algebraic space, and if T 0 denotes the preimage of T in |V |, then Vb|T 0 admits amorphism to Xb|T which is representable by algebraic spaces, smooth, and surjective.
If X \ T → X is quasi-compact, then so is the base-changed morphism V \ T 0 → V ,and so it follows from [Stacks, Tag 0AQ1] that Vb|T 0 is locally adic*.
We next discuss various base-change results in the context of formal algebraic
stacks.
Lemma 5.19. Let X → Y be a morphism of stacks over S which is representableby algebraic spaces.
(1) If T → Y is a morphism from a formal algebraic space to Y, then X ×Y T
is again a formal algebraic space over S. If T is locally countably indexed,
resp. locally adic*, then so is X ×Y T .
(2) If Z → Y is a morphism from a formal algebraic stack to Y, then X ×Y Z
is again a formal algebraic stack over S. If Z is locally countably indexed,
resp. locally adic*, then so is X ×Y Z.
Proof. We begin with (1). By definition,
we may find a family {Ti } of affine formal`
algebraic spaces and a morphism Ti → T which `is representable (by schemes),étale and surjective. The base-changed morphism
X ×Y Ti → X ×Y T is thenrepresentable, étale and surjective. Thus it suffices to show that each of the products
X ×Y Ti is a formal algebraic space.
Since Ti is an affine formal algebraic space, we may write Ti ∼
T
as
= lim
−→λ i,λan inductive limit of schemes, with the transition morphisms being thickenings.
Lemma 4.8 shows that we then have an induced isomorphism
X ×Y Ti,λ ,X ×Y Ti ∼
= lim
−→
λ

expressing X ×Y Ti as an inductive limit of algebraic spaces, with the transition
morphisms being thickenings. It follows from [Stacks, Tag 0AIU] that X ×Y Ti is
indeed a formal algebraic space.
Now suppose that T is furthermore locally countably indexed, resp. locally adic*.
The projection X ×Y T → T is a morphism representable by algebraic spaces, whose
source is a formal algebraic space (by what we have already proved). To simplify
notation, denote the source of this projection by X; we wish to show that X is
locally countably indexed, resp. locally adic*. Since` T is locally countably indexed,
resp. locally adic*, we may find a morphism i Ti → T which is representable,étale, and surjective, with each Ti being a countably indexed, resp.
` adic*, affine
formal algebraic space. Base-changing over X yields a morphism X ×T Ti → X
which is again representable, étale, and surjective.
In order to conclude that X is locally countably indexed, resp. locally adic*, itsuffices to prove that each of the fibre products X ×T Ti , which a priori is a formal
algebraic space, is furthermore locally countably indexed, resp. locally adic*. Let
U → X ×T Ti be an étale representable morphism whose source is an affine formal
algebraic space. Since the composite U → X ×T Ti → Ti is representable by

FORMAL ALGEBRAIC STACKS

45

algebraic spaces and Ti is countably indexed, resp. adic*, it follows from [Stacks,
Tag 0AKQ] that U is countably indexed, resp. adic*, as required.
We now turn to proving (2). Choose a morphism T → Z which is representableby algebraic spaces, smooth, and surjective, and whose source T is a formal algebraic
space; furthermore, choose T to be locally countably indexed or locally adic* if Z
is. Taking the fibre product of this morphism with X over Y yields a morphism
X ×Y T → X ×Y Z which is again representable by algebraic spaces, smooth, andsurjective. By (1), the source of this morphism is a formal algebraic space, which
is locally countably indexed or locally adic* if T is, and thus the target is indeed aformal algebraic stack, which is locally countably index or locally adic* if T is, andthus (by our choice of T ), if Z is.

Remark 5.20. Let us note some particular consequences of the preceding result,namely: if X → Y is a morphism of sheaves over S which is representable byalgebraic spaces, and if Y is furthermore a formal algebraic space (resp. a locallycountably indexed formal algebraic space, resp. a locally adic* formal algebraic
space), then the same is true of X (set T = Y = Y in (1)); and similarly, if X → Y
is a morphism of stacks over S which is representable by algebraic spaces, and if
Y is furthermore a formal algebraic stack (resp. a locally countably indexed formal
algebraic stack, resp. a locally adic* formal algebraic stack), then the same is trueof X (set Z = Y in (2)).
Lemma 5.21. If Y is a stack over S whose diagonal is representable by algebraic
spaces, and if U → Y and V → Y are morphisms each of whose domains is a formal
algebraic space, then U ×Y V is again a formal algebraic space.
Proof. If we interpret U ×Y V as the pull-back of U ×S V over the diagonal of Y,
then this follows from Lemma 5.19 (1).
Lemma 5.22. If X → Y and Z → Y are morphism of stacks over S, with X
and Z being formal algebraic stacks, and Y having its diagonal being representableby algebraic spaces (e.g., by Lemma 5.12, one could take Y itself to be a formal
algebraic stack), then X ×Y Z is again a formal algebraic stack.
Proof. Choose morphisms U → X and V → Z that are representable by algebraic
spaces, surjective, and smooth, whose domains are formal algebraic spaces. ThenU ×Y V → X ×Y Z is again representable by algebraic spaces, surjective andsmooth, and its source is again a formal algebraic space, by Lemma 5.21 (since∆Y/S is representable by algebraic spaces by assumption). Thus X ×Y Z is indeed
a formal algebraic stack.

We next establish an analogue of Lemma 5.19 for morphisms representable byalgebraic stacks. Because we don’t have a complete analogue of [Stacks, Tag 0AIU](to which we appealed in the proof of that lemma) in the context of formal algebraic
stacks, we are not able to prove a complete analogue of Lemma 5.19, but instead
make do with the following slightly weaker variant.
Lemma 5.23. Let X → Y be a morphism of stacks over S which is representableby algebraic stacks, and consider a morphism Z → Y whose source is a formal
algebraic stack.
(1) If X → Y is DM, then X ×Y Z is a formal algebraic stack.

46

M. EMERTON(2) If X is a formal algebraic stack, then the fibre product X ×Y Z is a formal
algebraic stack.
(3) If Z is locally countably indexed, resp. locally adic*, then X ×Y Z is a locallycountably indexed, resp. locally adic*, formal algebraic stack.
`
Proof. Choose (as we may) a morphism i Ti → Z which is representable byalgebraic spaces, smooth, and surjective, and whose source `is the disjoint union ofaffine formal algebraic spaces. The base-changed morphism i X ×Y Ti → X ×Y Z
is again representable by algebraic spaces, smooth, and surjective, and so if we canshow that each of the fibre products X ×Y Ti appearing in its source is a formal
algebraic stack, the same is true of its target. In summary, we are reduced to thecase when Z is in fact an affine formal algebraic space T (we omit the subscripts ifrom now on, since they play no further role). If Z is furthermore locally countableindexed, resp. locally adic*, then we may also choose T to be countably indexed,
resp. adic*.
We begin by proving (1). For this, we follow the proof of Lemma 5.19 (1).
Namely, since T is an affine formal algebraic space, we may write T ∼
T as
= lim
−→λ λ
an inductive limit of schemes, with the transition morphisms being thickenings, andso by Lemma 4.8 we have an induced isomorphism
X ×Y T ∼X ×Y Tλ ,
= lim
−→
λ

expressing X ×Y T as a 2-colimit limit of a 2-directed system of Deligne–Mumford
stacks, with the transition morphisms being thickenings. It follows from Lemma 6.1
below11 that X ×Y T is indeed a formal algebraic stack.
In the context of (2), we may additionally choose a formal algebraic space U
and a morphism U → X which is representable by algebraic spaces, smooth, andsurjective. The base-changed morphism U ×Y T → X ×Y T is then representable byalgebraic spaces, smooth, and surjective, and so to prove that X ×Y T is a formal
algebraic stack, it suffices to prove that U ×Y T is a formal algebraic space. The
composite U → X → Y is representable by algebraic stacks, and so by Lemma 3.5 itis in fact representable by algebraic spaces; thus Lemma 5.19 (1) shows that indeed
U ×Y T is a formal algebraic space.
Suppose now that we are in the context of (3). As we already noted above, since
Z is locally countably indexed, we may assume that T is countably indexed (andeven adic*, if Z is locally adic*). Thus we may write T ∼T , where {Tn }
= lim
−→ n
is a directed sequence of affine schemes related by thickenings. This induces anisomorphism
X ×Y T ∼X ×Y Tn ,
= lim
−→
n
realizing X ×Y T as the 2-colimit of a directed sequence of algebraic stacks related bythickenings. (Here we have used the fact that X → Y is representable by algebraic
stacks.) Lemma 6.3 below12 then shows that X ×Y T is a countably indexed formal
algebraic stack, as required.
11The reader will verify that the proof of this lemma doesn’t require any of the results fromthe theory of formal algebraic stacks that are developed in this note; it is a direct consequence ofthe topological invariance of the small étale site for Deligne–Mumford stacks. Thus although its
proof is postponed until the next section, there is no circularity in invoking it here.
12The reader can easily verify that the proof of that result is independent of present lemma.

FORMAL ALGEBRAIC STACKS

47

Choose a morphism U → X ×Y T which is representable by algebraic spaces,smooth, and surjective. Arguing as in the proof of (2), we find that the composite
U → X ×Y T → T is representable by algebraic spaces, and thus Lemma 5.19 (1)shows that U is locally countably indexed, resp. locally adic*, if T is. Thus, bydefinition, the same is true of X ×Y T .

Remark 5.24. As a useful consequence of the preceding lemma, we note that if
X → Y is a morphism of stacks over S which is representable by algebraic stacks,
with Y furthermore being a formal algebraic stack, then if either the morphism isin fact DM, or if Y is locally countably (resp. locally adic*), then X is a formal
algebraic stack, which is locally countable (resp. locally adic*) if Y is. (To see this,set Z = Y in (1) or (3), as appropriate).
It can be useful on occasion to view a formal algebraic stack as a quotient
stack. Suppose, then, that X is a formal algebraic stack over S, and that U → X
is a morphism whose source is a formal algebraic space, and which satisfies theconditions of Definition 5.3. As in the discussion preceding Lemma 3.18, we maythen form the fibre product R := U ×X U which is a formal algebraic space (byLemmas 5.12 and 5.21) having the structure of a groupoid in sheaves over U , andfor which each of the projections are representable by algebraic spaces and smooth.
Lemma 5.25. In the context of the preceding discussion, the morphism U → X
induces an isomorphism of stacks [U/R] → X .
Proof. This follows from Lemma 3.18.



We now discuss the properties of the underlying reduced substack of a formal
algebraic stack (the definition of which is provided by Definition 3.27).
Lemma 5.26. If X is is a formal algebraic stack over S, then Xred is a closed
and reduced algebraic substack of X , and the inclusion Xred ,→ X is a thickening(in the sense that its base-change over any algebraic space induces a thickening ofalgebraic spaces). Furthermore, any morphism Y → X with Y a reduced algebraic
stack factors through Xred .
Remark 5.27. Although the proof of this lemma that follows is slightly involved inits details, it can be summarized as follows: we choose a morphism U → X whose
source is a formal algebraic space, and which is representable by algebraic spaces,smooth, and surjective, and then, setting R := U ×X U , and using Lemma 5.25 to∼∼write [U/R] −→ X , we show that there is an induced isomorphism [Ured /Rred ] −→Xred .
Proof of Lemma 5.26. Choose a morphism U → X whose source is a formal algebraic space, and which is representable by algebraic spaces, smooth, and surjective.
Let Ured denote the underlying reduced algebraic space of U , in the sense of [Stacks,
Tag 0AIN]. The natural morphism Ured ,→ U is a thickening (in the sense that its
base-change over any morphism T → U , whose source is a scheme, is a thickening),
and thus so is its base-change(5.28)

Ured ×X Ured ,→ Ured ×X U.

We first claim that this base-changed thickening is an isomorphism. First, we notethat source and target are algebraic spaces, since Ured is, and since U → X isrepresentable by algebraic spaces, as is the composite Ured → U → X . Secondly,

48

M. EMERTON

since U → X is smooth, so is the base-changed morphism of algebraic spaces
Ured ×X U → Ured . Since Ured is reduced, we deduce that Ured ×X U is also
reduced, and hence the projection onto U factors through Ured . Thus we obtain amorphism
Ured ×X U → Ured ×X Ured ,which is easily verified to be an inverse to (5.28). Thus (5.28) is indeed an isomorphism.
We write Rred := Ured ×X Ured ⇒ Ured . (The motivation for this notation isthat, if we write R := U ×X U, then arguments similar to those of the preceding
paragraph show that Rred is the underlying reduced algebraic space of the formal
algebraic space R.) Identifying Rred with Ured ×X U (as the preceding paragraphallows us to do), and then also with U ×X Ured (by symmetry), we find that eachof the projections Rred ⇒ Ured is a smooth morphism of algebraic spaces. Since
Rred is thus smooth over Ured (by either projection), we see that Rred ×Ured Rred(the fibre product being formed with respect to any of the projections to Ured ) is
reduced, and hence is the underlying reduced algebraic space of R ×U R. Thus thegroupoid structure on R over U (it is a groupoid in formal algebraic spaces over theformal algebraic space U ) induces a groupoid structure on Rred over Ured , realizing
it as a smooth groupoid in algebraic spaces over Ured . The definition of Rred as afibre product over X shows that the composite
Ured ,→ U → X
induces a morphism(5.29)[Ured /Rred ] → X .

We claim that this latter morphism is representable by algebraic spaces, and is athickening (and so in particular a monomorphism), whose essential image is equal to
Xred . Note that this description of Xred (namely that it is isomorphic to [Ured /Rred ])shows that Xred is a reduced algebraic stack.
Since the properties of being representable by algebraic spaces and a thickening can be checked smooth locally on the target, in order to show that (5.29) isrepresentable by algebraic spaces, and a thickening, it suffices to verify this after
pulling back along the morphism U → X . The morphisms Ured → [Ured /Rred ] andUred ,→ U induce a morphism
Ured → [Ured /Rred ] ×X U,
which is in fact an isomorphism: indeed, to check that this morphism is an isomorphism, we can do so after pulling back over the smooth surjective morphism
Ured ×X U → [Ured /Rred ] ×X U ; but the pulled back morphism is the morphism
Rred → Ured ×X U,
which we have already shown to be an isomorphism. We conclude that (5.29) is
indeed representable by algebraic spaces and a thickening, since Ured ,→ U is.
The final claim of the lemma follows from Remarks 3.28 and 3.29, and so tocomplete the proof of the lemma, it suffices to show that the essential image of (5.29)
is equal to Xred . Choose a scheme V equipped with a surjective étale morphism
V → Ured . (This is possible, since Ured is an algebraic space.) Since Ured is reduced,
so is V . Thus the morphism V → X is in fact an object of (Xred )0 , and hence ofXred . Since V → Ured → [Ured /Rred ] is a smooth (and so fppf) cover of the target,

FORMAL ALGEBRAIC STACKS

49

we then find that the thickening (5.29) factors through Xred . To show that Xred isequal to the essential image of this thickening, it suffices to show that any object of(Xred )0 lies in the essential image of this thickening. This amounts to showing that
if T → X is a morphism whose source is a reduced scheme, then this morphism
factors through [Ured /Rred ]. Since [Ured /Rred ] is an fppf stack, it suffices to checkthis fppf locally on T . We may thus replace T by T ×X U , a reduced algebraic
space admitting a smooth surjection onto T . The morphism of this algebraic space
to X then factors through U by construction, and hence through Ured , and hence
through [Ured /Rred ], as required.

Example 5.30. If X is an algebraic stack and T is a locally closed subset of |X |,then the reduced algebraic stack underlying the completion Xb|T is the locally closed
substack of X obtained by equipping T with its reduced induced structure [Stacks,
Tag 050C].
We note the following lemma.
Lemma 5.31. If Y → X is a morphism of formal algebraic stacks that is representable by algebraic stacks, then the induced morphism Yred → Xred is also representable by algebraic spaces, and is surjective if and only if the original morphismis.
Proof. The morphism Yred → Y is the inclusion of a closed substack, and hence evenrepresentable by schemes, so that the composite Yred → Y → X is representableby algebraic stacks. Since Xred → X is a monomorphism, we see that the same istrue of the induced morphism Yred → Xred . The second claim of the lemma follows
from the first, together with Lemma 3.34.

We now prove a version of the topological invariance of the small Zariski andétale sites for a formal algebraic stack. (We also state the case of algebraic stacks,
since the argument applies just as well in that case.)
Lemma 5.32. If X is an algebraic stack (resp. a formal algebraic stack), and if
U → X is an object of EtX , then U is again an algebraic stack (resp. a formal
algebraic stack). Furthermore, the strict pseudo-functor U 7→ Ured induces an isomorphism of categories ZarX → ZarXred , as well as a 2-equivalence (in the sense of
Definition 2.16 (2)) of 2-categories EtX → EtXred .
Proof. If U → X is DM, then it is in particular representable by algebraic stacks.
Thus if X is an algebraic stack, then U certainly is, while if X is a formal algebraic
stack, then so is U, by Lemma 5.23 (1).13 (See also Remark 5.24.) Lemma 3.37∼
shows that Ured −→ U ×X Xred , while Lemma 5.26 shows that Xred ,→ X is a thickening. The claimed equivalence and 2-equivalence thus follow from Lemma 3.41. 
We briefly develop the notion of quasi-compact (and also of quasi-separated)
formal algebraic stacks, and of quasi-compact morphisms of formal algebraic stacks.
We refer to [Stacks, Tag 0AJ8] for the corresponding definitions in the context offormal algebraic spaces.
Lemma 5.33. If X is a formal algebraic stack, then the following are equivalent:
13The proof of this latter result relies on Lemma 6.1, whose proof in turn relies on the topological invariance of the small étale site for algebraic stacks. Since that case of the present lemma
does not involve any application of Lemmas 5.23 or 6.1, there is no circularity in the argument.

50

M. EMERTON(1) Xred is a quasi-compact algebraic stack.
(2) There exists a quasi-compact formal algebraic space U and a morphism
U → X which is representable by algebraic spaces, smooth, and surjective.
(3) There exists an affine formal algebraic space U and a morphism U → X
which is representable by algebraic spaces, smooth, and surjective.
Proof. Clearly condition (3) implies condition (2). Conversely, condition (2) implies
condition (3), by [Stacks, Tag 0AJ9]. If U → X is a morphism as in condition (2),then Ured → Xred (which Lemma 3.37 shows is obtained by pulling back the given
morphism U → X along the thickening Xred ,→ X ) is a smooth surjective morphismfrom an algebraic space to an algebraic stack whose source is quasi-compact; itfollows that the target is also quasi-compact, and so (2) implies (1).
Conversely, suppose that (1) holds, and choose a morphism U → X which isrepresentable by algebraic spaces, smooth, and surjective, and whose source is aformal algebraic space. Invoking Lemma 3.37 again, we see that Ured := U ×X Xredis the underlying reduced subspace of U , and that the morphism Ured → Xred issmooth and surjective. Since Xred is quasi-compact by assumption, we may alsofind a quasi-compact scheme V and a morphism V → Xred which is smooth andsurjective. If we form the fibre product Ured ×Xred V, then its projection onto eachof Ured and V is smooth, and hence open, as well as surjective. Since V is quasicompact, we may then find a quasi-compact open subspace W of Ured ×Xred V
which surjects onto V . The image W 0 of W under the projection to Ured is thena quasi-compact open subspace subspace of Ured which surjects onto Xred . If we
let W 00 denote the corresponding open subspace of U , then W 00 is quasi-compact,
and W 00 → X is representable by algebraic spaces, smooth, and surjective (this00last property following from Lemma 5.31 and the fact that Wred
= W 0 → Xred issurjective). Thus (2) holds.
Definition 5.34. We say that a formal algebraic stack is quasi-compact if it satisfies
the equivalent conditions of Lemma 5.33.
We say that a formal algebraic stack is quasi-separated if its diagonal morphism(which Lemma 5.12 shows is representable by algebraic spaces) is quasi-compact
and quasi-separated.
Lemma 5.35. For a morphism X → Y of formal algebraic stacks, the following
are equivalent.
(1) The induced morphism Xred → Yred is a quasi-compact morphism of algebraic stacks.
(2) For every morphism Z → Y whose source is a quasi-compact formal algebraic stack, the fibre product formal algebraic stack X ×Y Z is quasi-compact.
(2”) For every morphism Z → Y whose source is a quasi-compact formal algebraic space, the fibre product formal algebraic stack X ×Y Z is quasi-compact.
(2”) For every morphism Z → Y whose source is an affine formal algebraic
space, the fibre product formal algebraic stack X ×Y Z is quasi-compact.
(3) For every morphism Z → Y whose source is a quasi-compact algebraic
stack, the fibre product formal algebraic stack X ×Y Z is quasi-compact.
(3’) For every morphism Z → Y whose source is a quasi-compact scheme, thefibre product formal algebraic stack X ×Y Z is quasi-compact.
(3”) For every morphism Z → Y whose source is an affine scheme, the fibre
product formal algebraic stack X ×Y Z is quasi-compact.

FORMAL ALGEBRAIC STACKS

51`(4) There exists a morphism
Yi → Y whose source is a disjoint union ofaffine formal algebraic spaces, and which is representable by algebraic spaces,smooth, and surjective, such that each of the fibre product formal algebraic
stacks X ×Y Yi are quasi-compact.
Proof. The argument is routine, and we leave it to the reader.



Definition 5.36. We say that a morphism of formal algebraic stacks is quasicompact if it satisfies the equivalent conditions of Lemma 5.35.
6. Ind-algebraic stacks and formal algebraic stacks
As recalled in Definition 5.1, an affine formal algebraic space X is, by definition [Stacks, Tag 0AI7], a sheaf on the fppf site of S that can be written as an
Ind-scheme
X∼
X,
= lim
−→ i
i

where the Xi are affine schemes and the transition morphisms are thickenings.
More generally, if X is a formal algebraic space which is quasi-compact and quasiseparated, then [Stacks, Tag 0AJE] shows that it may written as an Ind-algebraic
space
X∼
X,
= lim
−→ i
i

where the Xi are algebraic spaces and the transition morphisms are thickenings.
Conversely, any Ind-algebraic space of this form is a formal algebraic space [Stacks,
Tag 0AIU]. The proof of this latter result depends on the topological invarianceof the étale site. Thus, it carries over directly to the context of Deligne–Mumford
stacks: any 2-colimit of a 2-directed system of Deligne–Mumford stacks with respect to transition morphisms that are thickenings is necessarily a formal Deligne–
Mumford stack.
Here is a formal statement and proof of this result.
Lemma 6.1. If {Xi }i is a 2-directed system of Deligne–Mumford stacks, with eachof the transition morphisms being a thickening, then X := limi Xi is a formal−→
algebraic stack, and there is a morphism U → X whose source is a formal algebraic
space, and which is representable by algebraic spaces, étale, and surjective.
Proof. Fix an index i0 , and choose an étale morphism Ui0 → Xi0 whose source
is a scheme. For each i ≥ i0 , it follows from Lemma 3.41 that we may find anétale morphism Ui → Xi which is a thickening of Ui0 → Xi0 . (We note thatUi is indeed a scheme, as the notation suggests, by [Stacks, Tag 0BPW].) The
transition morphisms Xi → Xi0 induce compatible transition morphisms Ui → Ui0 ,which make {Ui } into a directed system of schemes, whose transition morphisms are
thickenings. If we set U := limi≥i Ui , then U is a formal algebraic space, by [Stacks,−→ 0
Tag 0AIU].) By Lemma 2.34, the morphisms Ui → Xi induce a morphism∼

U → lim Xi −→ lim Xi := X ,
−→
−→
i≥i0

i

which, by Lemma 4.11, is representable by algebraic spaces, étale, and surjective,
as required; note that the claimed isomorphism is provided by Lemma 2.36.
52

M. EMERTONIn the general case of formal algebraic stacks that are not necessarily Deligne–
Mumford, less seems to be known about the possibility of relating them to 2-colimitsof thickenings. However, we do have the following results, the first of which is adirect analogue for formal algebraic stacks of [Stacks, Tag 0AJE] (and whose proof
follows closely the proof of that result).
Lemma 6.2. If X is a quasi-compact and quasi-separated formal algebraic stack,
X for a 2-directed system {Xi }i∈I of algebraic stacks in which thethen X ∼
= lim
−→i i
transition morphisms are thickenings. If X is furthermore locally countably indexed,
then we may choose the indexing category to be countable, and so write X ∼
X
= lim
−→n n
as the 2-colimit of a directed sequence of algebraic stacks with respect to transition
morphisms that are thickenings.
Proof. Choose a morphism V → X which is representable by algebraic spaces,smooth, and surjective, and whose source is an affine formal algebraic space (as wemay, since X is quasi-compact), and write V = limi∈I Vi as the inductive limit of−→affine schemes via thickenings.
If i ∈ I, then V ×X Vi is an algebraic space, which is quasi-compact, since Vi and
V are quasi-compact (being an affine scheme and a formal affine algebraic space,respectively), and X is quasi-separated. Thus the projection V ×X Vi → V factors
through Vj , for some j ≥ i. If we let Zi denote the scheme-theoretic image in Vj
of this projection, then Zi (thought of as a closed subsheaf of V ) is independent of
the choice of j, and so we may regard it as being the scheme-theoretic image of themorphism V ×X Vi → V. Since the projections V ×X Vi → V and Vi ×X V → V
are isomorphic, we may equally well regard Zi as the scheme-theoretic image of thelatter morphism.
We now compute that Zi ×X V and V ×X Zi coincide as subsheaves of V ×X V ;indeed, since the formation of scheme-theoretic images of quasi-compact morphisms
is compatible with flat base-change, the first may be described, using the firstdescription of Zi , as the scheme-theoretic image of the morphism V ×X Vi ×X V →V ×X V (projection onto the first and third factors), while the second may bedescribed, using the second description of Zi , as the scheme-theoretic image of themorphism V ×X Vi ×X V → V ×X V (where the morphism is again projection
onto the first and third factors). Thus each of these fibre products also coincides
with Zi ×X Zi . We denote these three (equal) fibre products by Ri . The lastdescription of Ri gives it a natural structure of a groupoid in algebraic spaces overZi , while the first two descriptions show that it is a smooth groupoid (because theprojections Zi ×X Zi ⇒ Zi are obtained by base-change from the smooth morphism
V → X ). Thus we may form the algebraic stack Xi := [Zi /Ri ], and the composite
Zi → V → X factors through Xi , i.e. induces a morphism Xi → X . Essentiallyby definition, the base-change of this morphism to V over X recovers the closed
immersion Zi → X , and thus this morphism itself is a closed immersion.
Since Zi contains Vi (when both are thought of as closed subsheaves of V ), we seethat limi Zi ∼X → X is also an isomorphism,= V. Thus the natural morphism lim−→−→i iand we have proved the first statement of the lemma.
If X is furthermore locally countably indexed, then V is countably indexed, andso the indexing category I can be taken to be countable. This proves the second
statement of the lemma.
FORMAL ALGEBRAIC STACKS53

The next result gives a partial analogue of [Stacks, Tag 0AIU], or, in other words,
a partial converse to the preceding result.
Lemma 6.3. If X1 ,→ X2 ,→ · · · ,→ Xn ,→ · · · is a sequence of finite orderthickenings of algebraic stacks, then limn Xn is a locally countably indexed formal−→
algebraic stack.
`
Proof. Choose (as we may) a morphism U1,i → X1 which is smooth and surjective, and whose source is a disjoint union of affine schemes. Lemma 6.4 below allows
us to successively lift each of the morphisms U1,i → X1 to a morphism Un,i → Xn ,so that Un−1,i is identified with the fibre product Xn−1 ×Xn Un,i . If we write
Ui := limn Ui,n , then Ui is an inductive limit of a sequence of affine schemes with−→respect to thickenings, and so is a countably indexed affine formal algebraic space.
Furthermore, there is an evident morphism Ui → limn Xn , which by construction is−→`representable by algebraic spaces and smooth. The induced morphism Ui → X
is furthermore surjective, and thus realizes X as a locally countably indexed formal
algebraic stack.

Lemma 6.4. If X ,→ X 0 is a finite order thickening of algebraic stacks, and if
U → X is a smooth morphism whose source is an affine scheme, then we may finda smooth morphism U 0 → X 0 whose source is an affine scheme, and which fits intoa Cartesian diagram
/ U0
U

X


/ X0

Proof. This is proved in the stated generality as [Stacks, Tag 0CKI].
Here we present our original argument, which requires the addition assumptionthat each of X and X 0 has a quasi-compact diagonal.
An evident induction reduces us to the case when X ,→ X 0 is a first order thickening. In this case we argue using the deformation theoretic formalism developedin [Ols06] (and we require the assumptions on the diagonals of X and X 0 that we
do because it is a running hypothesis in that paper). Namely, Theorem 1.4 of that
paper shows that the obstruction to deforming U → X to a flat morphism U 0 → X 0
is given by an element of Ext2OU (LU/X , I), where LU/X denotes the cotangent complex of U over X , and I denotes the pull-back to U of the ideal sheaf on X 0 thatcuts out X . Since U → X is smooth, the cotangent complex LU/X is precisely the
space of differentials Ω1U/X , which is a locally free sheaf of finite rank on U . (Werefer to [LMB00; Ols07] for the general theory of the cotangent complex on algebraic stacks, and to [LMB00, Lem. 17.5.8] for this particular fact. More precisely,
it is proved there that the cotangent complex of a morphism that is representableby algebraic spaces and smooth collapses to the sheaf Ω1 supported in degree zero.
The formation of Ω1 is compatible with smooth base-change, and so is easily seen
to be locally free, by reduction to the case of a smooth morphismof schemes.) Thus
the Ext2 above may be rewritten as H 2 U, I ⊗ (Ω1U/X )∨ , which vanishes, since U
is affine.
Finally, the following lemma (Lemma 6.5) shows that the flat morphism U 0 → X 0
is in fact smooth, since the base-changed morphism U → X is.
54

M. EMERTON

Lemma 6.5. Suppose that
X

/ X0



/ Y0
Y
is a Cartesian diagram of morphisms of algebraic stacks, in which the horizontal
arrows are finite order thickenings, and the vertical arrows are flat.
(1) If the left-hand vertical arrow is locally of finite presentation, so is theright-hand vertical arrow.
(2) If the left-hand vertical arrow is smooth, so is the right-hand vertical arrow.
Proof. The properties of being locally of finite presentation, or of being smooth,may be checked smooth locally. Thus if we choose a smooth surjective morphism
V 0 → Y 0 whose source is a scheme and pull-back the diagram over V 0 , we may
assume that Y = V is a scheme. If we then choose a smooth surjective morphism
U 0 → X 0 , and write U = X ×X 0 U 0 , then in order to prove the lemma, it suffices toprove the corresponding statement for the Cartesian diagram
U

/ U0


V


/ V0

in which all the objects appearing are schemes. Since the properties to be checked
are local on the source, we reduce to the case where all the schemes involved areaffine, in which the case the problem may be rephrased as follows: if A → A/I isa surjection of rings with I a nilpotent ideal, if B is a flat A-algebra, and if B/IB
is of finite presentation (resp. smooth) over A/I, then B is of finite presentation(resp. smooth) over A.
If f : A[x1 , . . . , xn ] → B is a morphism for which the induced morphism f :(A/I)[x1 , . . . , xn ] → B/I is surjective, then Nakayama’s Lemma shows that themorphism f itself is surjective. If J denotes its kernel, then since B is flat over A,
we see that J/IJ is identified with the kernel of f . Nakayama’s Lemma again showsthat if J/IJ is finitely generated over (A/I)[x1 , . . . , xn ], then J is finitely generated
over A[x1 , . . . , xn ]. Thus we have shown that if B/IB is finitely presented over A/I,
then B is finitely presented over A.
If furthermore B/IB is smooth over A/I, then we find that B is a finitely
presented A-algebra, for which the induced morphism Spec B → Spec A has smoothfibres. The fibrewise criterion for smoothness [Stacks, Tag 00TF] then shows that
B is a smooth A-algebra, as required.

We have the following corollary to Lemma 6.3.
Corollary 6.6. Suppose that X is an Ind-algebraic stack that can be written as∼
the 2-colimit X −→ lim Xn of a directed sequence (Xn )n≥1 in which the Xn are−→
algebraic stacks, and the transition morphisms are closed immersions. If Xred is aquasi-compact algebraic stack, then X is a locally countably indexed formal algebraic
stack.
Proof. By assumption, Xred is a quasi-compact algebraic stack, and Lemma 4.4∼
shows that the induced isomorphism Xred −→ limn Yn,red (given by Lemma 4.16)−→

FORMAL ALGEBRAIC STACKS

55

necessarily factors through Yn,red for some sufficiently large value of n. This showsthat the sequence (Yn,red )n≥1 stabilizes when n becomes sufficiently large, and thusthat the sequence (Yn )n≥1 becomes a sequence of thickenings for sufficiently large
values of n. Thus Lemma 6.3 shows that X is indeed a locally countably indexed
formal algebraic stack.

We now turn to establishing the characterization of formal algebraic spaces discussed in Example 5.7 above. Lemma 6.2 plays a crucial role in the argument.
Lemma 6.7. If X is a formal algebraic stack, then the following are equivalent:(1) X is an affine formal algebraic space.
(2) Xred is an affine scheme.
Proof. That (1) implies (2) is clear, and we focus on proving the converse implication. Thus we assume that Xred is an affine scheme. Then Xred is quasi-compact
and quasi-separated, and hence the same is true of X . Lemma 6.2 allows us towrite X ∼X as the 2-colimit of a 2-directed system of algebraic stacks having
= lim
−→i i
thickenings for transition morphisms. Since Xred ∼= (Xi )red for each i, we conclude
from [Stacks, Tag 0BPW] that each Xi is an affine scheme. Thus X is indeed anaffine formal algebraic space.

We can now show that formal algebraic spaces are precisely the formal algebraic
stacks that have trivial inertia stacks. This is the analogue, for formal algebraic
stacks, of [Stacks, Tag 04SZ], and we begin by recalling that statement, as well asa useful variant statement [Con07, Thm. 2.5.1 (1)], which reduces one to checking
triviality of the inertia stack over field-valued, or even geometric, points. (Since
the paper [Con07] imposes they blanket hypothesis that algebraic stacks be quasiseparated, in the sense of [LMB00], we give a proof here of its Thm. 2.2.5 (1) whichapplies without any such assumption.)
Lemma 6.8. If X is an algebraic stack, then the following are equivalent:(1) X is a stack in setoids.
(2) The canonical morphism IX → X is an equivalence (where IX denotes theinertia stack of X ).
(3) X is an algebraic space.
(4) For any morphism x : Spec k → X with k a field, the automorphism groupalgebraic space of x is trivial.
(5) For any morphism x : Spec k → X with k an algebraically closed field, theautomorphism group algebraic space of x is trivial.
Proof. The equivalence of (1), (2), and (3) is precisely the statement of [Stacks,
Tag 04SZ]. Clearly (4) implies (5), and since the formation of the automorphismgroup algebraic space of an object of X is compatible with the formation of flatbase-change, we see that in fact (4) and (5) are equivalent, and are implied by (1).
Now (1) is equivalent to the statement that for any morphism T → X withsource a scheme, the automorphism group algebraic space of this T -valued point of
X is trivial. If we assume (5), then we find that the fibre of this group algebraic
space over each geometric point of T is trivial. It follows from Lemma 6.9 that thisgroup algebraic space itself is trivial (note that this group algebraic space admits
the identity section, so that all the hypotheses of that lemma are satisfied); thus (5)implies (1).


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The lemma that follows isolates the key argument from the proof of [Con07,
Thm. 2.2.5 (1)].
Lemma 6.9. Let X → Y be a locally finite type morphism of algebraic spaces withthe property that, for any morphism Spec k → Y , with k being an algebraicallyclosed field, the base-changed morphism X ×Y Spec k → Spec k is an isomorphism.
If the morphism X → Y admits a section, then it itself is an isomorphism.
Proof. The property of being an isomorphism may be checked étale locally on thetarget, and so, pulling back over an étale surjection V → Y with V a scheme, we
may assume that Y is a scheme.
We first claim that the morphism X → Y is unramified. This may be checkedétale locally on the source, and so composing with an étale surjection U → X
with U a scheme, we are reduced to showing that a morphism of schemes U → Y
which is locally of finite type, and which satisfies the additional property that foreach morphism Spec k → Y with k an algebraically closed field, the base-changed
morphism U ×Y Spec k → Spec k is étale, is unramified. Since the property of amorphism being étale is fppf local on the target, we see that in fact Uy → Spec κ(y)is étale for each y ∈ Y . The fact that U → Y is unramified now follows from [Stacks,
Tag 02G8].
Since X → Y is unramified, the diagonal ∆X/Y is an open immersion [Stacks,
Tag 05W1]. Thus the section Y → X (which we assumed to exist) is an open
immersion. It is also surjective (by virtue of our assumption regarding geometricpoints), and thus is an isomorphism. This completes the proof of the lemma.
Remark 6.10. In the preceding lemma, the hypothesis that X → Y admits a section
is crucial. For example, if Z is a closed subscheme of the scheme X, and U := X \Z,then the natural morphism Z q U → X satisfies the hypothesis on geometric points
from the lemma, but typically is not an isomorphism.
We now return to the context of formal algebraic stacks.
Lemma 6.11. Let X be a formal algebraic stack. The following are equivalent:(1) X is a stack in setoids.
(2) The canonical morphism IX → X is an equivalence (where IX denotes theinertia stack of X ).
(3) X is a formal algebraic space.
(4) Xred is an algebraic space.
(5) For any morphism x : Spec k → X with k a field, the automorphism groupof x is trivial.
(6) For any morphism x : Spec k → X with k an algebraically closed field, theautomorphism group of x is trivial.
Proof. The equivalence of (1) and (2) is a general fact about stacks; see e.g. [Stacks,
Tag 04ZM], and the fact that (3) implies (2) was observed in Example 5.7. Suppose
now that (1) holds. Since Xred is a substack of X , it is also a category fibred insetoids (rather than just in groupoids), and so it follows from [Stacks, Tag 04SZ]
that Xred is an algebraic space, so (1) implies (4).
Since Spec k is reduced for any field k, the groupoid of morphisms from Spec k toeach of X and Xred coincide. Furthermore, if x : Spec k → Xred is such a morphism,
then because Xred → X is a monomorphism, the automorphisms of x, thought ofeither as a Spec k-valued point of either Xred or of X , coincide [Stacks, Tag 06R5].

FORMAL ALGEBRAIC STACKS

57

Thus, conditions (5) and (6) are equivalent to the analogous conditions for Xred ,which in turn are equivalent to (4), by Lemma 6.8.
It remains to show that (4) implies (3). In fact, we first show that (4) implies (1)and (2); equivalently, we show that if Xred is a formal algebraic space, then the
diagonal morphism X → X ×S X is a monomorphism. Since Xred → X is amonomorphism, the diagram
Xred/ Xred ×S XredX
/ X ×S X

whose horizontal arrows are the evident diagonal morphisms, is 2-Cartesian. Its
horizontal arrows are representable by algebraic spaces, and locally of finite type(Lemma 5.12), while its vertical arrows are representable by algebraic spaces, andthickenings. Since Xred is an algebraic space, by assumption, the upper horizontal
arrow is furthermore a monomorphism. It follows from [Stacks, Tag 0BPJ] that thelower horizontal arrow is also a monomorphism, as required.
Thus, assuming that Xred is an algebraic space, we have shown that X is astack in setoids. We now show that it `is in fact a formal algebraic space. To thisend, choose (as we may) a morphism
Ui,red → Xred which is representable byalgebraic spaces and étale, and whose source is a disjoint union of affine schemes
Ui,red . Lemmas 5.32 and 6.7 allow us to promote each Ui,red to an affine formal
algebraic space Ui , equipped with a morphism Ui `→ X which is representable byalgebraic spaces and étale. The induced morphism i Ui → X is then representableby algebraic spaces, étale, and surjective, so realizes X as a formal algebraic space.
This completes the proof that (4) implies (3), and so also completes the proof ofthe lemma.
Remark 6.12. One can establish a characterization of formal Deligne–Mumford
stacks analogous to the characterization of formal algebraic spaces given by thepreceding lemma, using very similar arguments. Namely, a formal algebraic stack
X is formal Deligne–Mumford if and only if Xred is a Deligne–Mumford stack, if
and only if ∆X is unramified, if and only if the automorphism group schemes ofpoints of X are étale, if and only if this last condition holds for field-valued, or evenmerely algebraically closed field-valued, points of X .
We may use the results developed in this section to define the notion of a schemetheoretically dominant morphism between formal algebraic stacks that are quasicompact and quasi-separated. Namely, suppose that X → Y is a morphism of
formal algebraic stacks, each of which is quasi-compact and quasi-separated. The
usual diagonal argument shows that X → Y is then also quasi-compact and quasiseparated. Lemma 6.2 allows us to write X ∼X , with the transition morphisms
= lim
−→ λ
being thickenings of algebraic stacks. Similarly, we may write Y ∼Y . For each
= lim
−→ µ
index λ, the composite morphism Xλ → X → Y factors through one of the Yµ (sinceXλ is quasi-compact and quasi-separated). The resulting morphism Xλ → Yµ is
quasi-compact, and so has a scheme-theoretic image (see e.g. [EG19b, §3.1], orExample 9.9 below), which we denote by Zλ . One easily checks that Zλ , thoughtof as a closed substack of Y, is independent of the particular choice of the indexµ used in its definition, and we may regard Zλ as the scheme-theoretic image of58

M. EMERTON

Xλ in Y. Evidently Zλ is a closed substack of Zλ0 if λ ≤ λ0 , with the resulting
morphism Zλ ,→ Zλ0 being a thickening. In particular, we may form the 2-colimit
lim Zλ , which is an Ind-algebraic stack.There is a natural morphism lim Zλ → Y.
−→−→
Definition 6.13. A morphism X → Y of quasi-compact and quasi-separated formal algebraic stacks is called scheme-theoretically dominant if, in the notation of
the preceding discussion, the morphism lim Zλ → Y is an isomorphism.
−→
Remark 6.14. One easily verifies that the condition of the preceding definition is
independent of the particular representation of X as an inductive limit X ∼
X .
= lim
−→ λ
The property of being scheme-theoretically dominant is preserved by a suitableflat base-change.
Lemma 6.15. If X → Y is a scheme-theoretically dominant morphism between
quasi-compact and quasi-separated formal algebraic stacks, and if Y 0 → Y is a morphism whose source is again a quasi-compact and quasi-separated algebraic stack,and which is furthermore representable by algebraic stacks and flat, then the basechanged morphism X ×Y Y 0 → Y 0 is scheme-theoretically dominant.
X ; since X →Proof. As in the discussion preceding Definition 6.13, write X ∼
= lim
−→ λ
Y is scheme-theoretically dominant by assumption, we obtain the corresponding
description Y ∼Y , where the composite Xλ → X → Y factors through a
= lim
−→ λ
scheme-theoretically dominant morphism Xλ → Yλ , for each index λ. Let X 0 :=X ×Y Y 0 . If we write Xλ0 := Xλ ×Y Y 0 , then Xλ0 is an algebraic stack, the morphism
Xλ0 ,→ X 0 is a closed immersion, the Xλ0 form a 2-directed system of algebraic stacks
with thickenings as transition morphisms, and the natural morphism lim Xλ0 → X 0 is−→
an isomorphism. Similarly, if we write Yλ0 := Yλ ×Y Y 0 , then Yλ0 is an algebraic stack,000
the morphism Yλ ,→ Y is a closed immersion, the Yλ form a 2-directed systemhaving thickenings as transition morphisms, and the natural morphism lim Yλ0 → Y 0−→
is an isomorphism.
Since Xλ → Yλ is a scheme-theoretically dominant quasi-compact morphism ofalgebraic stacks, since the formation of scheme-theoretic images of quasi-compact
morphisms is compatible with flat base-change, and since Yλ0 → Yλ is flat, we findthat Xλ0 → Yλ0 is again scheme-theoretically dominant. Thus by definition X 0 → Y 0
is scheme-theoretically dominant, as claimed.

7. Adic morphisms
We now discuss the concept of an adic morphism, beginning by recalling thisnotion in the context of formal algebraic spaces.
Definition 7.1. If f : X → Y is a morphism of sheaves over S with Y being a
locally adic* formal algebraic space, then we say that f is adic if and only if it
representable by algebraic spaces.
Remark 7.2. This is the definition of [Stacks, Tag 0AQ3]. (Note that in thatdefinition, it is required that additionally X be a locally adic* formal algebraic
space; however, Remark 5.20 shows that this holds automatically if the morphism
is representable by algebraic spaces.)
We extend this definition to the context of formal algebraic stacks as follows.

FORMAL ALGEBRAIC STACKS59

Definition 7.3. We say that a morphism X → Y of stacks over S, whose target
Y is a locally adic* formal algebraic stack, is adic if and only if it is representableby algebraic stacks.
Remark 7.4. In the context of Definition 7.3, Remark 5.24 shows that the source
X of an adic morphism is necessarily a locally adic* formal algebraic stack.
Remark 7.5. It follows from Lemma 3.5 that a morphism of sheaves whose target
is a locally adic* algebraic space, thought of as a morphism of stacks, is adic in thesense of Definition 7.3 if and only if it is adic in the sense of Definition 7.1.
Suppose now that A is an adic* topological ring. with finitely generated ideal ofdefinition I. As usual, write Spf A := limn Spec A/I n ; then the natural morphism−→
Spf A → Spec A is a monomorphism. In particular, a stack over Spec A admits at
most one morphism to Spf A.
Definition 7.6. We say that a formal algebraic stack X over Spec A is an I-adic
formal algebraic stack if it admits a morphism X → Spf A which is furthermore
adic.
Example 7.7. If X is an algebraic stack over Spec A, then the I-adic completion Xbof X gives an example of an I-adic formal algebraic stack. We may define the I-adic
completion in two ways: as the fibre product Spf A ×Spec A X , or as the completionof X along its closed substack Spec A/I ×Spec A X . (The first definition makes it
clear that Xb is I-adic, since its structure morphism to Spf A is the base-change ofthe morphism X → Spec A, which is representable by algebraic stacks since X itself
is an algebraic stack.)
Example 7.8. Suppose that {Xn }n≥1 is a directed sequence of algebraic stacks overSpec A, for which each Xn in fact lies over Spec A/I n , such that each of the induced
morphisms Xn → Spec A/I n ×Spec A/I n+1 Xn+1 is an isomorphism. We may thenform the Ind-algebraic stack X := limn Xn , and the obvious morphism X → Spf A−→
is representable by algebraic stacks, or, equivalently, adic, so that X is an I-adic
formal algebraic stack.
We have the following simple lemmas.
Lemma 7.9. If A is an adic* topological ring, with finitely generated ideal ofdefinition I, if X → Y is a morphism of stacks over Spec A which is representableby algebraic stacks, and if Y is furthermore an I-adic formal algebraic stack, then
X is also an I-adic formal algebraic stack.
Proof. Remark 7.4 shows that X is again a formal algebraic stack over Spec A. The
fact that it is I-adic follows from the fact that a composite of morphisms that are
representable by algebraic stacks is again representable by algebraic stacks.
Lemma 7.10. If A is an adic* topological ring, with finitely generated ideal ofdefinition I, then any morphism X → Y between I-adic formal algebraic stacks isnecessarily adic.
Proof. Since Spf A → Spec A is a monomorphism, we have that Y ×Spf A Y ∼=Y ×Spec A Y, and hence Lemma 5.12 shows that the diagonal morphism Y → Y ×Spf A
Y is representable by algebraic spaces. The claim of the lemma then follows from
Lemma 3.7.


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8. Properties of formal algebraic stacks
We fix a base-scheme S, and throughout, we work with objects (schemes, sheaves,
stacks, etc.) over S. We will define certain properties of formal algebraic stacks(over S)14 in terms of properties of formal algebraic spaces that are of a sufficiently
local nature.
Definition 8.1. We say that a property P of formal algebraic spaces is smooth
local (resp. fppf local) if it satisfies the following three conditions:(1) If X → Y is a morphism of formal algebraic spaces which is representableby algebraic spaces and smooth (resp. flat and locally of finite presentation),and if Y satisfies P , then so does X.
(2) If X → Y is a morphism of formal algebraic spaces which is representableby algebraic spaces, smooth (resp. flat and locally of finite presentation),and surjective, and if X satisfies P , then so does Y .
(3) If {Xi }i∈I is a collection `of formal algebraic spaces, each of which satisfies P ,then the disjoint union i Xi also satisfies P .
Remark 8.2. Since smooth morphisms are in particular flat and locally of finite
presentation, we see that an fppf local property is in particular smooth local.
Lemma 8.3. If X is a formal algebraic stack over S, and if P is a property offormal algebraic spaces that is smooth local (resp. fppf local) then the following are
equivalent:(1) For every morphism U → X whose domain is a formal algebraic space, andwhich is representable by algebraic spaces and smooth (resp. flat and locallyof finite presentation), the formal algebraic space U has property P .
(2) For every morphism U → X whose domain is an affine formal algebraic
space, and which is representable by algebraic spaces and smooth (resp. flatand locally of finite presentation), the affine formal algebraic space U has
property P .
(3) There exists a formal algebraic space V having the property P , and a morphism V → X which is representable by algebraic spaces, smooth (resp. flatand locally of finite presentation), and surjective.
(4) There exists a family {Ui } of affine
formal algebraic spaces, each having the`property P , and a morphism Ui → X which is representable by algebraic
spaces, smooth (resp. flat and locally of finite presentation), and surjective.
Proof. Clearly condition (1) implies condition (2), and almost as clearly, condition (4) implies condition (3) (take the formal algebraic space V of (3) to be thedisjoint union of the affine formal algebraic`spaces Ui of (4)). Since X is a formal
algebraic stack, there exists a morphism
Ui → X , whose source is a disjoint
union of affine formal algebraic spaces, which is representable by algebraic spaces,smooth, and surjective. Since smooth morphisms are in particular flat and locallyof finite presentation, we see that condition (2) implies condition (4).
Finally, suppose that (3) holds, and choose a morphism V → X as in the statement of (3). Now consider a morphism U → X as in the statement of (1). The
projection U ×X V → U is then representable by algebraic spaces, smooth (resp. flat
14As already indicated, we will work with formal algebraic spaces over S throughout, and so
we won’t continue to emphasize that all our objects and morphisms are over S.

FORMAL ALGEBRAIC STACKS

61

and locally of finite presentation), and surjective, while the projection U ×X V → V
is representable by algebraic spaces, and smooth (resp. flat and locally of finite presentation). Since V has property P , by assumption, we conclude first that the fibre
product U ×X V has property P , and then that U has property P , as required. 
Definition 8.4. If P is a smooth local property of formal algebraic spaces, thenwe say that a formal algebraic stack X is P if X satisfies the equivalent conditionsof Lemma 8.3.
Remark 8.5. Since fppf local properties are in particular smooth local, this definition
applies to such properties. Also, note that since formal algebraic spaces are particular types of formal algebraic stacks, this definition encompasses formal algebraic
spaces. Similarly, algebraic stacks are particular types of formal algebraic stacks,
and so this definition also encompasses algebraic stacks. In this case it coincides
with the notion of an algebraic stack being P in the sense of [Stacks, Tag 04YG](provided that the property P , as understood for formal algebraic spaces, coincides
with the property P for schemes, when the latter are thought of as particular casesof the former).
Having made this definition, Lemma 8.3 admits the following extension.
Lemma 8.6. If X is a formal algebraic stack over S, and if P is a property offormal algebraic spaces that is smooth local (resp. fppf local) then the following are
equivalent:(1) X has property P .
(2) For every morphism U → X whose domain is a formal algebraic stack, andwhich is representable by algebraic stacks and smooth (resp. flat and locallyof finite presentation), the formal algebraic stack U has property P .
(3) There exists a formal algebraic stack U that has property P , and a morphism
U → X which is representable by algebraic stacks, smooth (resp. flat andlocally of finite presentation), and surjective.
Proof. Clearly (2) implies (1), while (1) implies (3).
We next show that (1) implies (2). To this end, suppose that X has property P ,and that we are given a morphism`
U → X as in (2). Since U is a formal algebraic
stack,
we
may
find
a
disjoint
union
Ui of formal algebraic spaces, and a morphism`
Ui → U whichisrepresentablebyalgebraic
spaces, smooth, and surjective. The`
composite Ui → U → X is then representable by algebraic spaces (here we areapplying Lemma 3.5) and smooth (resp. flat and locally of finite presentation), and(by definition of X satisfying P and Lemma 8.3) Ui satisfies P . Thus U satisfies P(again by definition), as required.
Finally, we show that (3) implies (1). Thus, we suppose now that X satisfies (3),and let U → X be a morphism as in the` statement of (3). As in the preceding
paragraph, we`may find a disjoint union Ui of affine formal algebraic spaces, anda morphism
Ui → U which is representable by algebraic spaces, smooth, andsurjective.
Since
U furthermore satisfies P , each Ui satisfies P . The composite`
Ui → U → X is then representable by algebraic spaces (here we are applying
Lemma 3.5), smooth (resp. flat and locally of finite presentation), and surjective,and (by definition of X satisfying P , and Lemma 8.3) we see that X satisfies P , asclaimed.


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M. EMERTON

Remark 8.7. It is shown in [Stacks, Tag 0AKS], resp. [Stacks, Tag 0AKT], thatthe property of being countably indexed, resp. adic*, is an fppf local property
of affine formal algebraic spaces. It follows immediately that the property of aformal algebraic space being locally countably indexed or locally adic* (in the senseof [Stacks, Tag 0AKY]) is again fppf local. If we apply the preceding framework
to the property P of being locally countably indexed, resp. locally adic*, thenwe recover the notion of a locally countably indexed, resp. locally adic*, formal
algebraic stack which we introduced above.
The following notion provides another instance of the preceding framework.
Maybe there is better
terminology, or establishedterminology, for thisnotion?

Definition 8.8. We say that a formal algebraic stack X is residually Jacobson if
Xred is a Jacobson algebraic stack, in the sense of [Stacks, Tag 04YH].
Remark 8.9. Since the property of a scheme being Jacobson depends only on its
underlying topological space, and hence only on its underlying reduced subscheme,
and since this property is also local for the fppf topology on schemes [Stacks, Tag0368], we see that the property of an affine formal algebraic space being residually
Jacobson (in the sense of Definition 8.8) is fppf local. Thus we may apply thepreceding framework to the property P of being residually Jacobson, and we recover
the notion of Definition 8.8.
Remark 8.10. An affine formal algebraic space is defined to be Noetherian if it is of
the form Spf A, where A is a Noetherian adic topological ring [Stacks, Tag 0AID].
It is shown in [Stacks, Tag 0AKW] that the property of being Noetherian is fppf
local. It follows immediately that the property of an algebraic space being locally
Noetherian (in the sense of [Stacks, Tag 0AKY]) is again fppf local. Thus the above
framework allows us to define the notion of a locally Noetherian formal algebraic
stack. This definition is sufficiently important that we record it separately (and also
take the opportunity to define the notion of Noetherian formal algebraic stacks).
Definition 8.11. Let X be a formal algebraic stack.
(1) We say that X is locally Noetherian if it so in the sense of Definition 8.4(taking P to be the property of a formal algebraic space being locally
Noetherian).
(2) We say that X is Noetherian if it is locally Noetherian, quasi-compact, andquasi-separated.
Remark 8.12. The definition of Noetherian formal algebraic stacks as being locally Noetherian algebraic stacks that are furthermore quasi-compact and quasiseparated is modelled on the analogous definition for algebraic stacks [Stacks, Tag0510] (and coincides with that definition when applied to an algebraic stack, thoughtof as a formal algebraic stack).
When we apply Definition 8.11 (1) to formal algebraic spaces (thought of as particular types of formal algebraic stacks), we recover the notion of locally Noetherian
formal algebraic spaces, as defined in [Stacks, Tag 0AKY]. A consideration of thedefinitions also shows that a formal algebraic stack X is locally Noetherian if andonly if it admits a morphism U → X which is representable by algebraic spaces,smooth, and surjective, and whose source is a locally Noetherian formal algebraic
space.

FORMAL ALGEBRAIC STACKS

63

Lemma 8.13. If Y is a locally Noetherian formal algebraic stack, and if X → Y isa morphism of formal algebraic stacks that is representable by algebraic stacks andlocally of finite type, then X is also locally Noetherian.
Proof. If V → Y is representable by algebraic spaces, smooth, and surjective, with
V being a locally Noetherian algebraic space, then the fibre product X ×Y V is
a formal algebraic stack, by Lemma 5.22, and the projection X ×Y V → X isagain representable by algebraic spaces, smooth, and surjective. Thus it suffices
to show that X ×Y V is locally Noetherian. The projection X ×Y V → V isrepresentable by algebraic stacks and locally of finite type. If we let U → X ×Y V bea morphism that is representable by algebraic spaces, smooth, and surjective, thenthe composite U → X ×Y V → V is representable by algebraic spaces (here we haveused Lemma 3.5) and locally of finite type. Thus its source is locally Noetherian,since its target is [Stacks, Tag 0AQ7]. Thus X ×Y V is locally Noetherian, andhence so is X , as claimed.

Recall that a Noetherian ring is said to be Cohen–Macaulay if its localization
at each prime ideal is a Cohen–Macaulay local ring [Stacks, Tag 00NC]. It infact suffices to check this property for the localizations at maximal ideals [Stacks,
Tag 00NB], and it is equivalent to ask that the completions at maximal ideals are
Cohen–Macaulay [Stacks, Tag 07NX].
Similarly, a Noetherian ring is said to be regular if its localization at each prime
ideal is a regular local ring [Stacks, Tag 00OD]. It in fact suffices to check this property for the localizations at maximal ideals [Stacks, Tag 0AFS], and it is equivalent
to ask that the completions at maximal ideals are regular [Stacks, Tag 07NY].
A ring is said to be normal if its localization at each prime ideal is a normal
local domain [Stacks, Tag 00GV]. It in fact suffices to check this property forthe localizations at maximal ideals [Stacks, Tag 030B]. Unfortunately (and as
is well-known), even in the Noetherian case, it is not equivalent to require thatthe completions at maximal ideals are normal domains; this latter condition is astronger condition in general.
The property of being reduced behaves similarly to that of being normal. Namely,a ring is reduced if and only if all its localizations are reduced if and only if all its
localizations at maximal ideals are reduced. In the Noetherian case, since completions are then faithfully flat, we also see that it suffices for the completion at every
maximal ideal to be reduced. However, in general this last condition is stronger
than that of being reduced.
The preceding discussion of normality and reducedness prompts us to make thefollowing definition.
Definition 8.13.1. A Noetherian ring A is defined to be analytically unramified if
the completion of A at each maximal ideal of A is reduced, and to be analyticallynormal if the completion of A at each maximal ideal of A is a normal local domain.
As remarked on above, we have the following lemma.
Lemma 8.14. If A is an analytically unramified (resp. analytically normal) Noetherian ring, then A is reduced (resp. normal).
Proof. As noted above, it suffices to show that the localization Am of A at eachof its maximal ideals m is reduced (resp. a normal domain). This follows from the
cm from this local ring to its completion is faithfully
fact that the morphism Am → A

Is this terminology
standard, or in conflictwith some other standard
notion? Note thatanalytically unramified isused in the Stacks Project,but only for local rings (asfar as I could tell).

64

M. EMERTON

cm is reduced (resp. normal).
flat (as A is Noetherian), and the assumption that A(See [Stacks, Tag 033F, Tag 033G].)

The following technical lemma will be useful below.
Lemma 8.15. If A → C is a smooth morphism of Noetherian rings, if n is acm is reduced (resp.
maximal ideal of C lying over a maximal ideal m of A, and if Acnormal), then the same is true of Cn .
cm ⊗A C, then the base-changed morphism A
cm → C 0 is
Proof. If we write C 0 := Aagain smooth, and the maximal ideal n of C corresponds to a maximal ideal n0 of0d
cn ∼C 0 for which there is an isomorphism C=C
n0 . Thus we may assume that A is areduced (resp. normal) complete Noetherian local ring, and in particular that A is
excellent [Stacks, Tag 07QW].
Since C is smooth over the reduced (resp. normal) and excellent ring A, it isagain reduced (resp. normal) and excellent, as is its localization Cn . (See [Stacks,
Tag 033C] and [Stacks, Tag 07QU].) An excellent reduced (resp. normal) local ringis analytically unramified (resp. analytically normal), as required (see e.g. [Stacks,
Tag 0C21, Tag 0C22]).
The next lemma records the fact that the properties of being analytically normal,
Cohen–Macaulay, or regular are preserved under completion.
Lemma 8.16. Let A be a Noetherian ring, and let I be an ideal of A. If A is
analytically normal, analytically unramified, Cohen–Macaulay, or regular, then sob of A.
is the I-adic completion A
Proof. By definition in the analytically normal and analytically unramified cases,
and as discussed above in the case of Cohen–Macaulayness or regularity, it sufficesb at any of its maximal ideals is normal, reduced,
to show that the completion of A
Cohen–Macaulay, or regular, as the case may be. This follows from the fact thatb necessarily contain I A,
b and so are in bijection with thethe maximal ideals of A
maximal ideals of A containing I, and that if m is such a maximal ideal, then Amb m have naturally isomorphic completions.
and (A)
Definition 8.17. We say that a Noetherian affine formal algebraic space is analytically normal, analytically unramified, Cohen–Macaulay, or regular, if, writing it in
the form Spf A for a Noetherian adic topological ring A, the ring A is analyticallynormal, Cohen–Macaulay, or regular.
We would like to apply the framework of Definition 8.4 to the properties ofDefinition 8.17, which would require showing that they are smooth local properties
of affine formal algebraic spaces. We are able to do this in the case of Cohen–
Macaulayness and regularity. In the case of analytically normal and analyticallyunramified rings, we are able to do this after imposing the additional hypothesis ofbeing residually Jacobson.
We begin by recalling that if Spf B → Spf A is a morphism of Noetherian affine
formal algebraic spaces, then there is a corresponding morphism A → B inducing
the given morphism on formal spectra [Stacks, Tag 0AN0]. If this morphism isrepresentable by algebraic spaces (or, equivalently, adic), and if I is an ideal ofdefinition of A, then the topology on B is again the I-adic topology (which is theFORMAL ALGEBRAIC STACKS

65

reason for introducing the terminology adic to describe this type of morphism);see [Stacks, Tag 0AN5] and [Stacks, Tag 0APU].
We next note the following lemma.
Lemma 8.18. A morphism of Noetherian affine formal algebraic spaces Spf B →Spf A which is representable by algebraic spaces is (faithfully) flat if and only if thecorresponding morphism A → B is (faithfully) flat.
Proof. If A is equipped with I-adic topology, then the condition that Spf B →Spf A be representable by algebraic spaces is equivalent to the condition that B be
equipped with the IB-adic topology. The condition that this morphism furthermore
be (faithfully) flat is equivalent to the condition that the morphism A/I i → B/I ibe (faithfully) flat for each i ≥ 0. If A → B is (faithfully) flat, then certainly eachof these morphisms is (faithfully) flat, and so the if the direction of the lemma
follows.
Suppose, conversely, that each of the morphisms A/I i → B/I i is flat. Let J → A
be an inclusion of an ideal into A. Write Ji := J ∩ I i ; the Artin–Rees lemmashows that the filtration {Ji } of J is cofinal with the I-adic filtration. Since B is
Noetherian, the Artin–Rees lemma also shows that J ⊗A B is I-adically complete,and so the morphism J ⊗A B → lim(J/Ji ) ⊗A/I i B/I i is an isomorphism. Since←−
B/I i is flat over A/I i , and since each of the morphisms J/Ji → A/I i is injective,
we see that (J/Ji ) ⊗A/I i B/I i → B/I i is injective, and so, passing to the projective
limit, we find that J ⊗A B → B is injective. As J was arbitrary, we conclude that
A → B is flat, as claimed.
Suppose in addition that A/I → B/I is faithfully flat, and let M be a non-zero
finitely generated A-module. Artin–Rees implies that M is I-adically complete,and hence that M/IM 6= 0. Thus (M/IM ) ⊗A/I B/I 6= 0, and so in particular
M ⊗A B 6= 0. Thus in this case A → B is faithfully flat.
Lemma 8.19. Suppose that X → Y is a morphism of Noetherian affine formal
algebraic spaces which is representable by algebraic spaces, flat, and surjective. If
X is furthermore analytically normal, analytically unramified, Cohen–Macaulay, or
regular, then the same is true of Y .
Proof. If we write X = Spf B and Y = Spf A, then Lemma 8.18 shows that A → B
is faithfully flat, and so the Cohen–Macaulayness or regularity of B implies thatof A [Stacks, Tag 0352, Tag 0353 ].
Suppose then that B is analytically normal (resp. analytically unramified). Since
A is I-adically complete, any maximal ideal m of A contains I. Since the morphism
A/IA → B/IB is faithfully flat, we may then find a maximal ideal n of B which
cm → Bcn is flat and local, and hence faithfully flat.
pulls back to m. The morphism A
Since the target is normal (resp. reduced), so is the source [Stacks, Tag 033G, Tag
033F]. Thus A is analytically normal (resp. analytically unramified), as claimed.
Lemma 8.20. If X → Y is a morphism of affine formal algebraic spaces whichis representable by algeraic spaces and smooth, and if Y is Noetherian and either
residually Jacobson and analytically normal, residually Jacobson and analyticallyunramified, Cohen–Macaulay, or regular, then the same is true of X.
Proof. Recall Remark 8.9, which shows that the property of being residually Jacobson is fppf local. Write X = Spf B and Y = Spf A; Lemma 8.18 shows that the66

M. EMERTON

morphism A → B is flat. Since A is I-adically complete, a collection of elements
f1 , . . . , fr ∈ A generates the unit ideal in A if and only if it generates the unit ideal
in A/I i for each i > 0. If these equivalent conditions hold, then each Spec A/I i
is the union of the affine open subsets Spec(A/I i )fj , and Spf A/I i is the union ofddthe open formal subspaces Spf A
fj (where Afj denotes the I-adic completion of thelocalization Afj ). Similarly, we may write Spf B as the union of the open formaldsubspaces Spf B
fj .
If A satisfies one of the properties under consideration, then so does each of its
dlocalizations Afj , and hence so does each of the completions A
fj (by Lemma 8.16).
d
If we can show that Bfj satisfies the same property, then by applying Lemma 8.19`
dto the morphism Spf B
fj → Spf B, we find that Spf B also satisfies this property.
Thus, we are free to prove the lemma after localizing on Spf A, in the above sense.
By assumption the morphism Spec B/I → Spec A/IS is smooth. Thus we mayfind a cover by distinguished affine opens Spec A/I = Spec(A/I)gj so that eachof the morphisms Spec(B/I)gj → Spec(A/I)gj is in fact standard smooth, in thedsense of [Stacks, Tag 00T6]. (See [Stacks, Tag 00TA].) Thus, replacing A by A
gjd
and B by Bgj , we may assume that B/I admits a presentation as an (A/I)-algebraof the form
B/I = (A/I)[x1 , . . . , xn ]/(f 1 , . . . , f c ),∂f i 
with the determinant ∆ := detbeing invertible. If we choose lifts of∂xj 1≤i,j≤c
the f j to elements fj ∈ A[x1 , . . . , xn ], and write C := A[x1 , . . . , xn ]/(f1 , . . . , fc )∂fi and ∆ := det, then C[1/∆] is a smooth A-algebra. Since ∆ is invert∂xj 1≤i,j≤cible in C/I, we note that C/I i = C[1/∆]/I i for each i, and thus that the I-adic
completions of C and of C[1/∆] coincide; we denote this common I-adic compleb Repeated application of the infinitesimal lifting property for smoothtion by C.
morphisms gives a morphism of A-algebras C[1/∆] → B, inducing an isomorphism
C/I ∼= B/I, and hence a surjection C/I i → B/I i for each i. Let Ji denote thekernel of this surjection. Since B/I i is flat over A/I i , we find that Ji /IJi ∼= J1 = 0,ib∼and thus that Ji = IJi = · · · = I Ji = 0. Thus we obtain an isomorphism C= B.
Since C[1/∆] is smooth over A, it inherits Cohen–Macaulayness or regularity
from A (see [Stacks, Tag 0339] and [Stacks, Tag 033A]). Thus, by Lemma 8.16, sobdoes its I-adic completion C.
It remains to consider the case when A is residually Jacobson and either anb thenalytically normal or analytically unramified. If n is a maximal ideal of C
it contains I, and so corresponds to a maximal ideal of C/IC. The morphism
A/IA → C/IC is then a finite type morphism of Jacobson rings, and so we see
cm is either normal orthat n lies over a maximal ideal m of A. By assumption A
reduced. Since C[1/∆] is smooth over A, we may apply Lemma 8.15 to find thatcbn (which is naturally isomorphic to the completion of C at n ∩ C) is either normalCb is analytically normal, or analytically unramified, as required.
or reduced. Thus C(We have already noted that it is residually Jacobson.)
Lemma 8.21. If X is a formal algebraic space, then the following properties areequivalent:FORMAL ALGEBRAIC STACKS

67(1) If U → X is a morphism whose source is an affine formal algebraic space
and which is representable by algebraic spaces and smooth, then U is Noetherian and furthermore is residually Jacobson and analytically normal(resp. residually Jacobonson and analytically unramified, resp. Cohen–Macaulay,
resp. regular).
`(2) There exists a morphism i Ui → X, which is representable by algebraic
spaces and smooth, with each Ui being an affine formal algebraic space which
is Noetherian and is residually Jacobson and analytically normal (resp.
residually Jacobonson and analytically unramified, resp. Cohen–Macaulay,
resp. regular).
Proof. This follows from Lemmas 8.19 and 8.20 by standard manipulations.



Definition 8.22. We say that a formal algebraic space is residually Jacobson
and analytically normal, residually Jacobson and analytically unramified, Cohen–
Macaulay, or regular, if it satisfies the equivalent conditions of Lemma 8.21 forwhichever of these four conditions is under consideration.
Remark 8.23. Lemma 8.21 allows us to follow the prescription of Definition 8.4
and then define the notion of a formal algebraic stack being residually Jacobson
and analytically normal, residually Jacobson and analytically unramified, Cohen–
Macaulay, or regular.
We recall the following theorem of Gabber [RL14, Thm. 9.2].
Theorem 8.24. If A is a quasi-excellent Noetherian ring, then the completion of
A at any ideal is again quasi-excellent.
Corollary 8.25. Suppose that A is a quasi-excellent Noetherian ring which is Iadically complete for some ideal I. If Spf B → Spf A is a finite type adic morphismof affine formal algebraic spaces, then B is reduced if and only if B is analyticallyunramified.
Proof. Since Spf B → Spf A is adic, the topology on B is the I-adic topology, andso Spf B := lim Spec B/I i B. Since Spf B is furthermore of finite type over Spf A,−→
we see that B/I is of finite type over A/I. Choose a surjection A[x1 , . . . , xn ] →B/I (for some sufficiently large value of n). We may lift this to a morphismA[x1 , . . . , xn ] → B. If C denotes the I-adic completion of the domain of thismorphism, then there is an induced morphism C → B. The image of this morphism is I-adically dense in B, and also I-adically complete. Thus this morphism
is surjective, and so B is a quotient of C. Theorem 8.24 shows that C is quasiexcellent, and thus so is its quotient B. We have already observed that analyticallyunramified implies reduced in general; on the other hand, since B is quasi-excellent,
it is analytically unramified if it is reduced.

We now discuss a slightly different class of examples of smooth local properties.
Definition 8.26. If P is a property of S-algebraic stacks that is inherited by closed
substacks, then we say that a formal algebraic stack X is Ind-P if we may write∼
X −→ limi Xi as an inductive limit of algebraic spaces having property P with−→respect to transition morphisms that are thickenings.
Lemma 8.27. If P is a property of S-algebraic stacks that is inherited by closed
substacks, if X is an Ind-P formal algebraic stack, and if Y ,→ X is a closed
immersion, then Y is also Ind-P .

68

M. EMERTON∼

Proof. If we write X −→ lim Xi as in Definition 8.26, thesn taking the fibre product−→∼with Y over X yields an isomorphism Y −→ lim Xi ×X Y. Each morphism Yi :=−→Xi ×X Y → Xi is a closed immersion, and since Xi has property P by assumption,
we find that the same is true of Yi . Thus Y also satisfies the requirement of
Definition 8.26, and hence is Ind-P .
Lemma 8.28. If P is a property of S-algebraic stacks that is inherited by closed
substacks, and if X is an Ind-P formal algebraic stack, then Xred is an algebraic
stack satisfying P .
∼∼Proof. If X −→ lim Xi as in Definition 8.26, then Xred −→ (Xi )red for any choice−→of i. Since (Xi )red is a closed substack of Xi , which satisfies P by assumption, ourhypothesis on P implies that (Xi )red , and hence also Xred , satisfies P .
Lemma 8.29. If X is a quasi-compact and quasi-separated formal algebraic stack,then the following are equivalent:(1) X is Ind-P ;∼(1’) We may write X −→ limi Xi as an inductive limit of algebraic stacks having−→
property P with respect to transition morphisms that are closed immersions;(2) If Z ,→ X is a closed immersion whose source is an algebraic stack then Z
satisfies P ;(3) Any morphism T → X , whose source is an affine scheme, may be factored
as T → Z ,→ X , where Z is an algebraic stack satisfying P and the second arrowis a closed immersion;(3’) Any morphism T → X , whose source is a quasi-compact algebraic stack,may be factored as T → Z ,→ X , where Z is an algebraic stack satisfying P , andthe second arrow is a closed immersion.
Proof. It is clear that (3’) implies (3). Suppose that (3) holds, and consider aclosed immersion Z ,→ X whose source is an algebraic stack. Since X is quasicompact, so is Z, and so we may find a morphism T → Z whose source is an affine
scheme, and which is smooth and surjective. By assumption this morphism factors
as T → Z 0 ,→ X , with the second morphism being a closed immersion whose source
Z 0 is an algebraic stack satisying P . Thus we find that Z is a closed substack of Z 0 ,and so our assumption on P implies that Z also satisfies P . Hence (3) implies (2).
∼Suppose now that (2) holds. By Lemma 6.2 we may write limi Xi −→ X for−→some directed system of algebraic spaces {Xi }, with the transition morphisms being
thickenings. Applying (2) to each of the various closed embedding Xi ,→ X showsthat each Xi then satisfies P , so that (1) holds.
Obviously (1) implies (1’). Finally, suppose that (1’) holds, and write X ∼=lim Xi with the Xi satisfying P , and the morphisms being closed immersions. Any−→morphism T → X as in (3’) then factors through one of the Xi and so (3’) holds,
as claimed.
Lemma 8.30. If P is a property of S-algebraic stacks that is inherited by closed
substacks, and if X is a quasi-compact and quasi-separated algebraic stack that isInd-P when regarded as a formal algebraic stack, then X satisfies P .
Proof. This follows by applying condition (2) of Lemma 8.29 (which by that lemma∼
follows from the fact that X is Ind-P ) to the identity morphism X −→ X .
FORMAL ALGEBRAIC STACKS

69

Lemma 8.31. Let P be a property of S-algebraic stacks that is inherited by closed
subschemes, and which is also smooth local (respectively fppf local). If X → Y is amorphism of formal algebraic stacks over S that is representable by algebraic spaces
and smooth (respectively fppf), and if Y is Ind-P , then X is Ind-P . If the given
morphism is furthermore surjective, and if Y is quasi-compact and quasi-separated,then the converse also holds: namely, if X is Ind-P then so is Y.
Proof. If Z ,→ Y is a closed immersion whose source is an algebraic stack, then X ×Y
Z ,→ X is a closed immersion whose source is an algebraic space, and X ×Y Z → Z
is smooth (respectively flat and locally of finite presentation), and is furthermore∼
surjective if X → Y is surjective. Thus if Y is Ind-P , and if we write limi Yi −→ Y,−→
where each Yi is an algebraic stack satisfying P and the transition morphisms are∼
thickenings, then we obtain an isomorphism limi X ×Y Yi −→ X . Since P is smooth−→(respectively fppf) local, we find that each X ×Y Yi satisfies P , and thus that X is
Ind-P .
Conversely, if X is Ind-P — so that it satisfies condition (2) of 8.29 — and if
X → Y is surjective, then we find that Y also satisfies condition (2) of Lemma 8.29,
and thus is Ind-P , provided that it is also quasi-compact and quasi-separated (sothat that lemma applies to Y).
Lemma 8.32. Let P be a property of S-algebraic stacks that is inherited by closed
substacks, and which is also smooth local (respectively fppf local).
If X is a formal algebraic stack over S, then the following properties are equivalent:(1) If U is a quasi-compact and quasi-separated formal algebraic stack equippedwith a morphism U → X which is representable by algebraic spaces andsmooth (respectively fppf), then
` U is Ind-P .
(2) We may find a morphism
Ui → X which is representable by algebraic
spaces, smooth (respectively fppf), and surjective, with each Ui being a formal affine algebraic space which` is Ind-P .
(3) We may find a morphism
Ui → X which is representable by algebraic
spaces, smooth (respectively fppf), and surjective, with each Ui being a formal algebraic stack which is Ind-P .
Proof.
Since X is a formal algebraic space, by definition we may find a morphism`
Ui → X which is representable by algebraic spaces, smooth, and surjective. Thus
if (1) holds, each of the Ui is Ind-P , and so (2) holds as well. Clearly` (2) implies (3).
Suppose then that condition (3) holds, and fix a morphism` Ui → X as inits statement. Let U → X be a morphism as in (1). ThenU ×X Ui → U isrepresentable by algebraic spaces, smooth (respectively fppf), and surjective. Eachfibre product` U ×X Ui is again a formal algebraic stack, and so we may find amorphism Vi,j → U ×X Ui which is representable by algebraic spaces, smooth,and surjective, with each Vi,j an affine formal algebraic space. Then the compositeaa(8.33)
Vi,j →U ×X Ui → U
i,ji

is representable by algebraic spaces, smooth (respectively fppf), and surjective.
Lemma 8.31, applied first to the various morphisms Vi,j → Ui , and then to themorphism (8.33) (taking into account that U is quasi-compact and quasi-separated
by assumption), implies that U is Ind-P .
70

M. EMERTON

Definition 8.34. Let P be a property of S-algebraic stacks that is inherited byclosed substacks, and which is also smooth local. We say that a formal algebraic
stack X over S is locally Ind-P if it satisfies the equivalent conditions of Lemma 8.32.
It is evident from this lemma that this property that the property of being locally
Ind-P , when restricted to formal algebraic spaces, is smooth local in the sense of
Definition 8.1, so that we may also regard ourselves as being in the context ofDefinition 8.4.
Lemma 8.35. Let P be a property of S-algebraic stacks that is inherited by closed
substacks, and which is also smooth local (so that Definition 8.34 applies), and let
X be a locally Ind-P formal algebraic stack. If Z → X is a morphism whose source
Z is an algebraic stack and which is unramified,15 then Z satisfies P .
`Proof. By condition (2) of Lemma 8.32, we may choose a morphism
Ui → X
whose source is a disjoint union of affine formal algebraic spaces, and which isrepresentable by algebraic spaces, smooth, and surjective,
for which each Ui is Ind`P . Pulling-back over Z, we obtain a morphism Ui ×X Z → Z, which is again
representable by algebraic spaces, smooth, and surjective. Thus, to conclude that
Z satisfies P , it suffices to show (by Lemma 8.6) that each of the fibre products
Ui ×X Z is an algebraic stack satisfying P .
Since Z is an algebraic stack, and since Ui ×X Z → Z is representable byalgebraic spaces, we see that the fibre product Ui ×X Z is itself an algebraic stack.
The morphism Ui ×X Z → Ui is furthermore unramified, being the base-change of anunramified morphism. In order to show that Ui ×X Z satisfies P , it suffices to showthat for any smooth morphism V → Ui ×X Z whose source is an affine scheme, theaffine scheme V satisfies P , and this is equivalent to showing the image of V under
this morphism, which is an open substack V of Ui ×X Z, satisfies P . Consider
such a morphism; since Ui is Ind-P by assumption, the composite V → Ui ×X
Z → Ui factors through a closed affine subscheme Ti of Ui which satisfies P (byLemma 8.29). Also, the morphism V → Ti is unramified (since the composite V →Ti → Ui is, being the composite of an unramified morphism and an open immersion,and since the diagonal of Ui is also unramified, being a monomorphism that is locallyof finite type [Stacks, Tag 0AIP]). Thus it has étale, and hence unramified, diagonal,
and so is a DM morphism [Stacks, Tag 04YW]. As is explained in that reference,since Ti is a scheme, the algebraic stack V is in fact a Deligne–Mumford stack, andso we may find a surjective étale morphism V 0 → V with V 0 a scheme. Consequently
V 0 → V → Ti is unramified, and thus, taking into account our assumptions on P
and the étale local structure of unramified morphisms [Stacks, Tag 04HH], we findthat each affine open subset of V 0 satisfies P . Thus V 0 satisfies P , and hence sodoes V. This completes the proof.
Lemma 8.36. Let P be a property of S-algebraic stacks that is inherited by closed
substacks, and which is also smooth local. If X is a locally Ind-P formal algebraic
stack, then Xred is an algebraic stack satisfying P .
Proof. This follows from Lemma 8.35, applied to the closed immersion Xred ,→ X .
15The notion of a morphism Z → X being unramified is defined, thanks to Lemma 5.12,
Definition 3.11 and Example 3.16.

FORMAL ALGEBRAIC STACKS

71

Lemma 8.37. Let P be a property of S-algebraic stacks that is inherited by closed
substacks, and which is also smooth local. If X is an algebraic stack which is locally
Ind-P when regarded as an algebraic stack, then X in fact satisfies P .
∼Proof. This follows from Lemma 8.35, applied to the identity morphism X −→ X .
The following lemma shows that being locally Ind-P is not so far from being
Ind-P itself.
Lemma 8.38. Let P be a property of S-algebraic stacks that is inherited by closed
substacks, and which is also smooth local. If X is a formal algebraic stack thatis Ind-P , then X is locally Ind-P . Conversely, if X is locally Ind-P and is also
quasi-compact and quasi-separated, then X is Ind-P .
Proof. If X is Ind-P , then the identity morphism X → X satisfies condition (3)of Lemma 8.32, showing that X is locally Ind-P . Conversely, suppose that X is
locally Ind-P and also quasi-compact and quasi-separated. If Z ,→ X is a closed
immersion whose source is an algebraic stack, then Lemma 8.35 implies that Z
satisfies P . Thus X satisfies condition (2) of Lemma 8.29, and so X is Ind-P bythat same lemma.
Remark 8.39. Two examples of a property P to which we may apply Definitions 8.26
and 8.34 are being locally of finite type over S, and being locally Noetherian. When
S is locally Noetherian, the former condition implies the latter, so we see thata formal algebraic stack that is (locally) Ind-locally of finite type over a locally
Noetherian scheme S is in particular (locally) Ind-locally Noetherian.
We also emphasize that while any (locally) Noetherian formal algebraic stack is(locally) Ind-locally Noetherian, the condition of being (locally) Ind-locally Noetherian is considerably weaker.
Example 8.40. A typical example of an Ind-locally Noetherian formal algebraic
space X which is not Noetherian is as follows: for each n ≥ 0, set
An := C[x, y]/(x(x − 1) · · · (x − n)y, y 2 ),and set X := lim Spec An , with respect to the evident transition morphisms.
−→
In Section 11 below, we will give a criterion for a locally Ind-locally of finite
type formal algebraic stack over a locally Noetherian base scheme S to be locally
Noetherian.
We close this section by mentioning that it would be natural, and desirable, todevelop a framework for defining properties of morphisms between formal algebraic
stacks, analogous to the corresponding framework for morphisms between formal
algebraic spaces that is developed in [Stacks, Tag 0ANA]; namely, based on properties of continuous ring maps. In particular, this is a natural framework for defining
properties of morphisms that are not representable by algebraic spaces. We don’t
do this in general here, but give one example, as an illustration. As in [Stacks, Tag
0ANA], it will be necessary to restrict the class of formal algebraic stacks for which
we make the definition, ultimately because the theory of continuous ring maps is not
terribly robust, and often requires auxiliary hypotheses to yield reasonable results.
Lemma 8.41. If f : X → Y is a morphism between locally Noetherian formal
algebraic stacks, then the following properties are equivalent:

72

M. EMERTON(1) For every commutative diagram of morphisms
/V

U

X

f


/Y

in which U ∼= Spf A and V ∼= Spf B are affine formal algebraic spaces and in which
the vertical arrows are representable by algebraic spaces and flat (so that U and
V are necessarily Noetherian affine formal algebraic spaces, by Remark 8.10 andLemma 8.3), the ring morphism B → A induced by the upper horizontal arrow isflat.
`(2) There exists a morphism j Vj → Y, where each Vj ∼= Spf Bj is an affine
formal algebraic space and which is `representable by algebraic spaces, flat, and surjective, and for each j a morphism i Uj,i → X ×Y Vj , where each Uj,i ∼= Spf Aj,iis an affine formal algebraic space and which is representable by algebraic spaces,
flat, and surjective, such that each of the induced morphisms Bj → Ai,j is flat. (Inorder to make sense of this induced morphism, we again note that each Uj,i and
Vj is necessarily a Noetherian affine formal algebraic space, by Remark 8.10 andLemma 8.3).
`(3) There exists a morphism i Ui → X , where each Ui ∼= Spf Ai is an affine
formal algebraic space which is representable by algebraic spaces, flat, and surjective,
and a factorization of the induced morphism Ui → Y as Ui → Vi → Y, whereVi ∼= Spf Bi is an affine formal algebraic space, such that the morphism Vi → Y isrepresentable by algebraic spaces and flat, and such the induced morphism Bi → Ai
is flat. (Again, in order to make sense of this induced morphism, we note that eachUj,i and Vj is necessarily a Noetherian affine formal algebraic space, by Remark 8.10
and Lemma 8.3).
If f : X → Y is furthermore adic, i.e. representable by algebraic stacks, thenthese conditions are equivalent to f being flat in the sense of Definition 3.11.
Proof. It is clear that (1) implies (2), and that (2) implies (3). By standard manipulations (see e.g. the proof of [Stacks, Tag 0ANG], on which the present argument
is modelled), to see that (3) implies (1), it suffices to show the following: if
U0 ∼= Spf A0

/ V0 ∼= Spf B 0U∼
= Spf B/V ∼= Spf Ais a commutative diagram of morphisms of Noetherian affine formal algebraic
spaces, in which the vertical arrows are representable by algebraic spaces and flat,with the left-hand arrow being furthermore surjective, and if the upper horizontal
arrow induces a flat morphism B 0 → A0 , then the lower horizontal morphism induces a flat morphism B → A. To see this, consider the corresponding diagram of
induced ring morphisms
/ A0
BO 0
O
A

/B

FORMAL ALGEBRAIC STACKS73

The left and right-hand arrows are flat, with the right-hand arrow even being
faithfully flat, by Lemma 8.18. The upper horizontal arrow is flat by assumption.
Since the composite of flat morphisms are flat, we find that the morphism B → A
becomes flat after composition with the faithfully flat morphism A0 → A0 . It follows
that B → A is itself flat, as required.
It remains to be shown that if X → Y is representable by algebraic stacks, thenconditions (1), (2), and (3) are equivalent to X → Y being flat in the previously
defined sense, namely that for any morphism Z → Y whose source is an algebraic
stack, or even simply a scheme, that the induced morphism X ×Y Z → Z is flat.
We begin `with some preliminary constructions. Namely, we choose (as we may)a morphism Vj → Y whose source is a disjoint union of Noetherian affine formal
algebraic spaces, which is representable by algebraic
spaces, smooth, and surjective.
`
Then we obtain a base-changed morphism
X ×Y Vj → X , whose source is adisjoint union of formal algebraic stacks (by Lemma 5.22) Lemma 8.13), which isrepresentable by algebraic spaces, smooth, and surjective, and such that each of
the projections X ×Y Vj → Vj is representable by algebraic stacks. Lemma 8.13
shows that each of the fibre products X ×Y Vj is furthermore locally
Noetherian,`and so we may (and do) choose, for each of them, a morphism Uj,i → X ×Y Vj
whose source is a disjoint union of Noetherian affine formal algebraic stacks, whichis representable by algebraic spaces, smooth, and surjective. Each of the composite
morphisms Uj,i → X ×Y Vj → Vj is then representable by algebraic stacks, andhence by algebraic spaces (by Lemma 3.5).
Now suppose that X` → Y is flat, in the sense of Definition 3.11. Then each of thecomposite morphism Uj,i → X ×Y Vj → Vj is a composite of morphisms that are
representable by algebraic stacks and flat in this sense, and hence this morphism(which we have seen is representable by algebraic spaces) is flat. Lemma 8.18
then implies that each of these morphisms is induced by a flat morphism of thecorresponding topological rings. Thus (2) holds (and hence so do (1) and (3)).
Suppose instead now that condition (1) holds. Then we find that each of themorphisms Uj,i → Vj is induced by a flat morphism of topological rings, and hence
by Lemma 8.18,`is flat (when regarded
as a morphism representable by algebraic`
spaces). Since
Uj,i → X are
Vj → Y are each morphisms representable byalgebraic spaces, smooth, and surjective, standard arguments then show that X →Y is flat in the sense of Definition 3.11.

Definition 8.42. We say that a morphism X → Y of locally Noetherian formal
algebraic stacks is flat if it satisfies the equivalent conditions (1), (2), and (3) ofLemma 8.41. (The final statement of the lemma ensures that this definition iscompatible with the existing definition in the case of adic morphisms.)
Our final result proves that the regularity properties discussed in Remark 8.23
are flat local in the sense of Definition 8.42.
Lemma 8.43. If X → Y is a flat and surjective morphism of locally Noetherian
formal algebraic stacks, and if X satisfies any of the conditions of Remark 8.23,
then the same is true of Y.
Proof. This amounts to showing that if A → B is a faithfully flat morphism ofadically complete Noetherian local rings, and if B is residually Jacobson and analytically normal, residually Jacobson and analytically unramified, Cohen-Macaulay,

74

M. EMERTON

or regular, then the same is true of A. This follows by an argument analogous tothat used to prove Lemma 8.19.

9. Functorial closed substacks of formal algebraic stacks
In this section we explain how to extend various constructions of closed substacks
from the context of affine formal algebraic spaces to the context of formal algebraic
stacks. In order to maximize the flexibility of our construction, we put ourselves in
the following situation: we fix a base formal algebraic stack S, we let D denote the2-category of formal algebraic stacks lying over S, and then we fix a strong sub-2category (in the sense of Definition 2.19) C of D, which we assume has the following
property: if U → X is a morphism that is representable by algebraic spaces andsmooth, and if we are given a morphism X → S making X an object of C, then U(equipped with the composite morphism U → X → S) is also an object of C.
We let Aff C denote the full subcategory of C whose objects are the affine formal
algebraic spaces belonging to C. (We note that since any object X of C receives
a morphism U → X which is representable by algebraic spaces, smooth, and surjective, and whose source is a disjoint union of affine formal algebraic spaces, Aff C
contains plenty of objects. We note also that Aff C is (equivalent to) a usual category (i.e. a 1-category), because its objects are formal algebraic spaces rather thanmore general formal algebraic stacks.) Let i : Aff C → D denote the inclusion, andsuppose that we are given a functor F : Aff C → D, equipped with a natural transformation F ,→ i which is a closed immersion, in the sense that for any object U of
Aff C , the corresponding morphism F (U ) ,→ U is representable by algebraic spaces,
and a closed immersion (and so is in fact representable by schemes).
We assume that F satisfies the following axiom:
Axiom 9.1. If U → V is a morphism in Aff C which is representable by algebraic
spaces and smooth, then the natural morphism F (U ) → U ×V F (V ) is an isomorphism.
Lemma 9.2. Let X be an object of C, and let T → X be a morphism whose source
is a scheme. The following properties
are equivalent:`(1) For some morphism
Vi → X whose source is a disjoint union of affine
formal algebraic spaces, and which is representable by algebraic spaces, smooth, andsurjective, each of the induced morphisms T ×X Vi → Vi factors through F (Vi ).
(2) For any morphism U → X which is representable by algebraic spaces andsmooth, and whose source is an affine formal algebraic space, the base-changed
morphism T ×X U → U factors through F (U ).
Furthermore, there is a closed substack Z ,→ X , characterized by the property
that a morphism T → X , whose source is a scheme, factors through Z if and only
if it satisfies the preceding two equivalent conditions.
Proof. Let T → X be a morphism whose source is a scheme. Since X is a formal
algebraic stack, we may choose a morphism V → X which is representable byalgebraic spaces, smooth,and surjective, and whose source may be written as a`disjoint union V = Vi of affine formal algebraic schemes. It is clear that if (1)holds, then (2) holds for any such choice. Suppose conversely that (2) holds forsome such choice; we will show that (1) holds, i.e. that T ×X U → U factors throughF (U ) for every U → X as in the statement of (1).

FORMAL ALGEBRAIC STACKS

75`
Consider the base-changed morphism
U ×X Vi → U ; its source is a formal
algebraic space, its target is the affine formal algebraic space U , and it is representable by algebraic spaces, smooth, and surjective. Since U is quasi-compact, we
may find a finite sequence W1 , . . . , Wn of affine algebraic spaces, such that each Wj
is equipped with a morphism Wj → U ×X Vi(j) (where the index i(j) depends on j)which is representable by algebraic spaces and étale, and such that the composite
n
a
j=1

Wj →

n
a

U ×X Vi(j) → U

j=1

is (representable by algebraic spaces, smooth, and) surjective.
We are assuming that T ×X Vi → Vi factors through F (Vi ) for each index i. In
particular, for each value of j, we have that T ×X Vi(j) → Vi(j) factors throughF (Vi(j) ). Since, for each value of j, the composite Wj → U ×X Vi(j) → Vi(j) isrepresentable by algebraic spaces and smooth, we find by Axiom 9.1 that
T ×X W j → W jfactors through F (Wj ) for each value of j.
`
We want to show that T ×X U → U factors through F (U ). Since Wj → U
is surjective, it suffices to show that T ×X Wj factors through F (U ) ×U Wj foreach value of j. Taking into account Axiom 9.1, this is equivalent to showing that
T ×X Wj factors through F (Wj ) for each value of j, which condition has alreadybeen shown to hold. Thus (1) does indeed hold.
We now show that the condition that T ×X Vi → Vi factors through F (Vi ) cuts
out a closed substack Z of X . Since the property of a morphism factoring througha closed subscheme of the target can be checked after passing to a faithfully flat
cover of the source, we see that Z is indeed a substack of X . By descent for closed
subschemes, it suffices to show that the base-changed morphism Z ×X Vi → Vi
is a closed immersion for each index i. Now Z ×X Vi classifies morphisms T →Vi for which T ×Vi (Vi ×X Vj ) → Vj factors through F (Vj ) for each index j, or
equivalently, for which T ×Vi (Vi ×X Vj ) → Vi ×X Vj factors through Vi ×X F (Vj )for each index j. We claim that F (Vi ) ×X Vj and Vi ×X F (Vj ) coincide, as closed
subsheaves of Vi ×X Vj . Given this, we see that Z ×X Vi classifies T → Vi forwhich
T ×Vi (Vi ×X Vj ) → Vi ×X Vj factors through F (Vi ) ×X Vj for each j. Since`
V×i
X Vj → Vi is representable by algebraic spaces, smooth, and surjective, wejsee that Z ×X Vi in fact classifies T → Vi for which T factors through F (Vi ). In
other words, the base-changed morphism Z ×X Vi → Vi coincides with the closed
immersion F (Vi ) → Vi ; thus Z → X is a closed substack, as claimed.
It remains to show that F (Vi ) ×X Vj and Vi ×X F (Vj ) coincide as subsheaves of
Vi ×X Vj . To this end, note that Vi ×X Vj is a formal algebraic space, and so we mayfind a collection of morphisms W → Vi ×X Vj which are representable and étale, andwhose coproduct surjects onto Vi ×X Vj . It suffices to check that F (Vi ) ×X Vj andVi ×X Vj pull back to the same subsheaf of W , for each such W . To see this, notethat since each of the composites W → Vi ×X Vj → Vi and W → Vi ×X Vj → Vj isrepresentable by algebraic spaces and smooth, Axiom 9.1 shows that both of thesepull-backs coincide with the subsheaf F (W ) of W .
Definition 9.3. If X is an object of C, then we define Fe(X ) to be the closed
substack of X constructed in Lemma 9.2

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M. EMERTON

Lemma 9.4. If f : X → Y is a morphism in C (so, more concretely, X → Y isa functor of categories fibred in groupoids over S), then f restricts to a morphism
Fe(X ) → Fe(Y).
Proof. Let T → Fe(X ) be a morphism whose source is a scheme, and let V →Y be a morphism whose source is an affine formal algebraic space, and which isrepresentable by algebraic spaces and smooth; we must show that the base-changed
morphism(9.5)

T ×Y V → V

factors through F (V ).
The morphism X ×Y V → V is representable by algebraic spaces and smooth, andLemma 5.22
` shows that its source is a formal algebraic stack. Thus we may find amorphism Ui → X ×Y V whose source is a disjoint union of affine formal algebraic
spaces, and which is representable by algebraic spaces, smooth, and surjective. The
induced morphisma∼T ×X Ui → T ×X (X ×Y V ) −→ T ×Y V
is then also representable by algebraic spaces, smooth, and surjective, and so toshow that (9.5) factors through F (V ), it suffices to show that the same is true ofeach of the induced morphisms(9.6)

T ×X Ui → V.

But by assumption, the morphism T → X factors through Fe(X ), and hence theprojection T ×X Ui → Ui factors through F (Ui ). Since the morphism Ui → V
restricts to a morphism F (Ui ) → F (V ) by assumption, we see that indeed each of
the morphisms (9.6) factors through F (V ), as required.
Proposition 9.7. Let j : C → D denote the inclusion. Then Fe induces a strict
functor (in the sense of Definition 2.13) Fe : C → D, equipped with a natural trans∼
formation Fe ,→ j, together with a natural isomorphism F −→ Fe| Aff C , which satisfies
the following conditions:(1) For each object X of C, the morphism Fe(X ) ,→ X is representable by algebraic spaces and a closed immersion (and thus is also representable byschemes).
(2) If X → Y is a morphism in C which is representable by algebraic stacks
and smooth, then the natural transformation Fe(X ) ,→ X ×Y Fe(Y) is an
isomorphism.
Proof. By construction Fe(X ) is a full subcategory fibred in groupoids of X , andLemma 9.4 shows that any morphism X → Y in C restricts to morphism Fe(X ) →Fe(Y). Thus Fe indeed induces a strict pseudo-functor C → D. The natural transformation Fe(X ) ,→ X is then given simply by the inclusion Z ,→ X , while if U is∼an object of Aff C , the natural isomorphism F (U ) −→ Fe(U ) is given by the isomorphism which identifies F (U ) with its image (thought of as a subsheaf of U ) under
the given natural transformation F (U ) ,→ U .
Condition (1) is now satisfied by construction, and we turn to verifying condition (2). Lemma 9.4 shows that there is an inclusion of closed substacks Fe(X ) →X ×Y Fe(Y), which we must show is an isomorphism, i.e. we must show that (theFORMAL ALGEBRAIC STACKS

77

essential image in X of) the 2-fibre product X ×Y Fe(Y) is contained in Fe(X ). Thatis, we need to prove the following: (a) if T → X is any morphism whose source isa scheme, and if for every morphism U → X which is representable by algebraic
spaces and smooth, and whose source is an affine formal stack, the base-changed
morphism T ×X U → U factors through F (U ), then it factors through this fibre
product. Applying the corresponding characterization of Fe(Y), what we know is:(b) the morphism T → X factors through the fibre product X ×Y Fe(Y) if andonly if, for any morphism V → Y which is representable by algebraic spaces andsmooth, and whose source is an affine formal algebraic stack, the base-changed
morphism T ×Y V → V factors through F (V ). If we rewrite this last fibre productas T ×X (X ×Y V ), then we see that this is equivalent to asking that the projection
T ×X (X ×Y V ) → X ×Y V factor through X ×Y F (V ). Our goal is to deduce (a)from (b).
In the context of (b), the fibre product X ×`
Y V is a formal algebraic stack(Lemma 5.23), and so we may find a morphism
Ui → X ×Y V which is representable by algebraic spaces, smooth, and surjective, with each Ui being an affine
formal algebraic space. Since the property of factoring through a closed substack
can be checked smooth locally, we find that T ×X (X ×Y V ) → X ×Y V factors
through X ×Y F (V ) if and only if T ×X Ui → Ui factors through Ui ×V F (V ),or equivalently (by Axiom 9.1; note that Lemma 3.5 ensures that the morphism
Ui → V , which a priori is representable by algebraic stacks, is in fact representable by algebraic spaces), through F (Ui ). Thus we are indeed able to deduce (a)from (b).

Lemma 9.8. Suppose that F is compatible with flat base-change, in the sense that
if U → V is a morphism in Aff C which is representable by algebraic spaces and flat,then the natural morphism F (U ) → U ×V F (V ) is an isomorphism. Then the sameis true of Fe: i.e. if X → Y is a morphism in C representable by algebraic stacks,then the natural morphism Fe(X ) → X ×Y Fe(Y) is an isomorphism.
Proof. The proof is identical to the proof of part (2) of Proposition 9.7, using the
hypothesis on F in place of Axiom 9.1.

We now present various applications of the preceding framework.
Example 9.9. We may use the preceding framework as a foundation for the theoryof scheme-theoretic images for quasi-compact morphisms of algebraic stacks. (Seee.g. [EG19b, §3.1] for a more ad hoc development of this theory.)
Suppose that T → S is a quasi-compact morphism of algebraic stacks, and take
C to be the category of algebraic stacks lying over S. The full subcategory Aff C
then consists of the category of affine schemes equipped with a morphism to S.
If Spec A → S is an object of Aff C , define F (Spec A) to be the scheme-theoretic
image in A of the base-changed morphism Spec A ×S T → Spec A (defined in
an obvious way: consider a smooth surjection U → Spec A ×S T whose sourceis an affine scheme — which is possible, since Spec A ×S T is quasi-compact —and define F (Spec A) to the be the scheme-theoretic image of the composite U →Spec A ×S T → Spec A; well-definendness, i.e. independence of the choice of U ,is easily checked). Axiom 9.1 is easily verified in this context, using the known
base-change properties of scheme-theoretic images of quasi-compact morphisms ofschemes [Stacks, Tag 081I], and thus F extends to a functor Fe. For any morphism

78

M. EMERTON

of algebraic stacks X → S, we declare Fe(X ) to be the scheme-theoretic image of thebase-changed morphism T ×S X → X . In particular, Fe(S) is the scheme-theoretic
image of the morphism T → S itself.
Example 9.10. Suppose that A is a quasi-excellent Noetherian ring which is Iadically complete with respect to an ideal I of A. Let C denote the category offormal algebraic stacks equipped with a morphism X → Spf A which is adic andlocally of finite type. The full subcategory Aff C then consists of affine formal
algebraic spaces Spf B, endowed with a morphism Spf B → Spf A which is adic
and of finite type. This morphism in turn corresponds to a morphism A → B,with B being I-adically complete and B/I being finite type over A/I. Consider the
reduced quotient Bred of B. Since B is Noetherian (by [Stacks, Tag 0AKP]; notethat [Stacks, Tag 0AKN] ensures that for a morphism of affine formal algebraic
spaces, being representable by algebraic spaces is equivalent to being representableby schemes), the A-algebra Bred is again I-adically complete, and Bred /I is also offinite type over A/I; thus Spf Bred again lies in C. We define F (Spf B) := Spf Bred ,equipped with the morphism Spf Bred ,→ Spf B.
Corollary 8.25 shows that for any quotient of B, being reduced is equivalent tobeing analytically unramified. When combined with Lemma 8.20, this shows thatthe functor F satisfies Axiom 9.1. Thus we may extend F to a functor Fe on C,
which associates to any I-adic formal algebraic stack, locally of finite type over
Spf A, its associated reduced formal algebraic substack.16
Example 9.11. Suppose that A is a complete DVR, with maximal ideal I. Let C
denote the category of locally Noetherian formal algebraic stacks equipped with amorphism X → Spf A. The full subcategory Aff C then consists of Noetherian affine
formal algebraic spaces Spf B endowed with a morphism Spf B → Spf A, which inturn corresponds to a morphism A → B, with B being a Noetherian A-algebra,
complete with respect to an ideal J that contains IB. Let Bfl denote the maximal
A-flat quotient of B, i.e. the quotient of B by its ideal of I-power torsion elements.
Since B is Noetherian, the quotient Bfl is again J-adically complete, and we define
F (Spf B) := Spf Bfl , equipped with its canonical morphism Spf Bfl ,→ Spf B.
Since smooth morphisms are in particular flat, it follows easily from Lemma 8.18
that the functor F satisfies Axiom 9.1, and so F extends to a functor Fe on C, whichassociates to any locally Noetherian formal algebraic stack X lying over Spf A its
flat part Xfl . By construction, the morphism Xfl → Spf A is flat, in the sense of
Definition 8.42, and it is easily seen to be the maximal substack of X with respect
to this property, in that if Y → X is a morphism of locally Noetherian formal
algebraic stacks for which the composite Y → Spf A is flat, then Y → X factors
through Xfl .
The preceding construction applies in particular to X which are I-adic and locallyof finite type over Spf A (since such formal algebraic stacks are in particular locally
Noetherian). In this case, the morphism Xfl → Spf A is again adic and locally offinite type (being the composite of such a morphism with a closed immersion).
Remark 9.12. Lemma 9.8 ensures that the constructions described in Examples 9.9
and 9.11 are compatible with base-change under morphisms in their respective
categories C that are representable by algebraic spaces and flat. (In the case of
16We use this terminology to distinguish this construction from that of the underlying reduced
algebraic substack of Lemma 5.26.

FORMAL ALGEBRAIC STACKS79

Example 9.9, the compatibility of the formation of scheme-theoretic images of quasicompact morphisms with respect to flat base-change [Stacks, Tag 081I] shows thatthe hypothesis of Lemma 9.8 is satisfied. In the case of Example 9.11, the hypothesisof Lemma 9.8 is easily verified, using Lemma 8.18.)
10. Morphisms to affine formal algebraic spaces
Recall that if X is an algebraic stack, then we have a structure sheaf OX on X .
(We refer to [Stacks, Tag 06TF] for the foundations of sheaves on stacks; see [Stacks,
Tag 06TU] in particular for a discussion of the structure sheaf.) If B = Γ(X , OX ),then there is a canonical morphism X → Spec B, which is initial in the categoryof morphisms from X to affine schemes. We now explain how to extend this construction to the context of (certain) formal algebraic stacks.
To this end let X be a quasi-compact and quasi-separated formal algebraic stack,which is furthermore locally countably indexed. By Lemma 6.2 we may write
X ∼
X ,
= lim
−→ n
n≥1

where each Xn is an algebraic stack, and the transition morphisms Xn ,→ Xn+1
are thickenings. For each n ≥ 1, write Bn0 := Γ(Xn , OXn ). Each of the natural
0morphisms Bn+1
→ Bn0 has locally nilpotent kernel. We may form the projective
0
limit B := limn Bn ; if we equip each Bn0 with its discrete topology, and equip B with←−the projective limit topology, then B is a complete topological ring, which is easily
seen to be independent (up to canonical isomorphism) of the choice of descriptionof X as a 2-colimit. If we let Jn denote the kernel of the surjection B → Bn0 , then
Jn is open in B. If Bn denotes the image of B in Bn0 (so that Bn ∼= B/Jn ), then
the transition morphisms Bn+1 → Bn are surjective with locally nilpotent kernels(so that the corresponding morphisms Spec Bn → Spec Bn+1 are thickenings), and∼we have an induced isomorphism of topological rings B −→ limn Bn . Each of the←−ideals Jn is a weak ideal of definition in B, realizing B as a weakly admissible
complete topological ring (in the sense of [Stacks, Tag 0AMV]).
Definition 10.1. If X is a quasi-compact, quasi-separated, locally countably indexed formal algebraic stack, then we write Γ(X , OX ) := B, where B is the weakly
admissible linearly topologized ring attached to X as in the preceding discussion.
Lemma 10.2. If X is a quasi-compact, quasi-separated, locally countably indexed
formal algebraic stack, then there is a natural morphism of formal algebraic stacks
X → Spf Γ(X , OX ), which is initial in the category of morphisms from X to weakly
admissible affine formal algebraic spaces.
Proof. Let Y be a weakly admissible affine formal algebraic space, and write Y ∼=limI Spec A/I, where A is a weakly admissible topological ring and I runs over−→
the weak ideals of definition of A. If X → Y is a morphism, then the composite
Xn → X → Y factors through Spec A/I for some I. Since Spec A/I is affine, we
obtain a further factorization of the form
Xn → Spec Bn0 → Spec A/I → Spf A.
(Here we are using the notation introduced in the discussion preceding Definition 10.1.) The induced morphism A → Bn0 (which is continuous when Bn0 is

80

M. EMERTON

given its discrete topology) is independent of the particular weak ideal of definition I considered in this factorization, and we obtain a continuous morphism
A → limn Bn0 = B, inducing the desired morphism Spf B → Spf A ∼
=Y.
←−Suppose now that R is a Noetherian ring, complete with respect to a principal
ideal I = tR. We regard R as a topological ring by equipping it with the I-adic, or
equivalently t-adic, topology. We will refer to an I-adic formal algebraic stack (inthe sense of Definition 7.6) as a t-adic formal algebraic stack. By definition, to givea formal algebraic stack X the structure of a t-adic formal algebraic stack structure
is to equip it with a morphism X → Spf R which is representable by algebraic
spaces. We then say that X is a t-adic formal algebraic stack of finite presentation
if this morphism is of finite presentation.
If X is a t-adic formal algebraic stack of finite presentation, then it is bothquasi-compact and quasi-separated, as well as being locally countably indexed.
Thus Definition 10.1 applies, and allows us to define the weakly admissible linearly
topologized ring Γ(X , OX ). Lemma 10.2 then shows that the structure morphism
X → Spf R induces a natural morphism Spf Γ(X , OX ) → Spf R, which corresponds
in turn to a continuous ring morphism R → Γ(X , OX ).
Lemma 10.3. If X is a t-adic formal algebraic stack of finite presentation, then
the natural topology on Γ(X , OX ) coincides with the t-adic topology; equivalently,
the morphism R → Γ(X , OX ) is taut.
Remark 10.4. The notion of a taut morphism of linearly topologized rings is definedin [Stacks, Tag 0AMX]. The equivalence claimed in the statement of the lemma
follows from [Stacks, Tag 0APU].
Proof of Lemma 10.3. Since X is quasi-compact, we may choose a morphism U →X whose source is an affine formal algebraic space, and which is representable byalgebraic spaces, smooth, and surjective. Since X is furthermore quasi-separated,the fibre product U ×X U is again quasi-compact, and so we may similarly find amorphism V → U ×X U whose source is an affine formal algebraic space, and whichis representable by algebraic spaces, smooth, and surjective.
Write Xn := Spec R/tn ×Spf R X , and similarly write Un := Spec R/tn ×Spf R U
and Vn := Spec R/tn ×Spf R V . We then have exact sequences of R-modules
0 → Γ(Xn , OXn ) → Γ(Un , OUn ) → Γ(Vn , OVn )(the first arrow being given by pull-back, and the second by the difference of thetwo pull-back maps), and (passing to the projective limit in n)
0 → Γ(X , OX ) → Γ(U, OU ) → Γ(V, OV ).
We also have isomorphisms
Γ(U, OU )/tn Γ(U, OU ) ∼= Γ(Un , OUn )andΓ(V, OV )/tn Γ(V, OV ) ∼= Γ(Vn , OVn ).
We let Jn ⊆ Γ(X , OX ) denote the kernel of the morphism Γ(X , OX ) → Γ(Xn , OXn );since Γ(Xn , OXn ) embeds into Γ(Un , OUn ), we see that
Jn = tn Γ(U, OU ) ∩ Γ(Xn , OXn ).
Since each of the morphisms U → Spf R and V → Spf R is representable by algebraic spaces and of finite type, each of U and V is Noetherian [Stacks, Tag 0AQ7].

FORMAL ALGEBRAIC STACKS81

In particular, the ideal in Γ(V, OV ) consisting of the t-power torsion elements isfinitely generated, and hence is annihilated by some uniformly bounded power of t,
say tN . It follows thattn Γ(X , OX ) ⊆ Jn = tn Γ(U, OU ) ∩ Γ(X , OX ) ⊆ tn−N Γ(X , OX )if n ≥ N. Thus the topology on Γ(X , OX ) induced by the ideals Jn , which is its
natural topology, coincides with its t-adic topology, as claimed.
We give an application of the preceding construction and results which will be
used in [Car+19] and [EG19a]. We continue to assume that R is a Noetherian ring,
complete with respect to the t-adic topology, for some element t ∈ R. We furthermore assume given quasi-compact and quasi-separated formal algebraic stacks X
and Y, and a commutative diagram
X

/Y
" 
Spf R

in which the diagonal arrow makes X into a t-adic formal algebraic stack of finite
presentation. The usual graph argument shows that the horizontal morphism X →Y is necessarily quasi-compact, quasi-separated, and representable by algebraic
stacks. Suppose in addition that this morphism is proper, and scheme-theoreticallydominant (in the sense of Definition 6.13). Finally, suppose that Y is Ind-locally offinite type over Spec R (in the sense of Remark 8.39).
Proposition 10.5. In the preceding situation, and under the preceding hypotheses,
the morphism Y → Spf R is adic, and hence realizes Y as a t-adic formal algebraic
stack of finite presentation.
Proof. Since Y is quasi-compact and quasi-separated by assumption, it suffices toshow that it is a t-adic formal algebraic stack that is locally of finite presentationin the evident sense. Since R is Noetherian, this is equivalent to being locally offinite type, and it is this latter property that we will establish.
Since Y is quasi-compact, we may find a morphism V → Y whose source isan affine formal algebraic space, and which is representable by algebraic spaces,smooth, and surjective. In order to show that Y → Spf R is adic (i.e. is representable by algebraic stacks) and (locally) of finite type, it suffices to show that thecomposite V → Y → Spf R is representable by algebraic stacks and of finite type, or
equivalently (by Lemma 3.5), representable by algebraic spaces and of finite type,or equivalently (by definition), adic and of finite type.
If we let Xn := Spec R/tn ×Spf R X , then X ∼X . Since X → Y is scheme= lim−→ ntheoretically dominant, we may write Y ∼Yn as the 2-colimit of a directed= lim−→
sequence of algebraic stacks with thickenings as transition morphisms, such that themorphism X → Y induces a scheme-theoretically dominant morphism of algebraic
stacks Xn → Yn for each n. If we write Vn := Yn ×Y V , then each Vn is an algebraic
space admitting a closed immersion Vn ,→ V , and hence is in fact an affine scheme,
say Vn ∼lim Vn → V is an isomorphism, and= Spec An . The natural morphism −→so we have V ∼An . Our goal, then, is to show that the= Spf A, where A ∼= lim←−projective limit topology on A coincides with the t-adic topology on A.

82

M. EMERTON

Since Y is Ind-locally of finite type over Spf R, as well as being quasi-compact
and quasi-separated, each of the closed algebraic substacks Yn is locally of finite
type over Spec R/tn (by Lemma 8.29 (2)). Since Y is quasi-compact and quasiseparated, so is its closed substack Yn ; thus in fact each Yn is of finite type overSpec R/tn , and so Vn is also of finite type over Spec R/tn . Equivalently, the ring
An is of finite type over R/tn .
Write B := Γ(X ×Y V, OX ×Y V ), and let Bn denote the image of the naturalmorphism B → Γ(Xn ×Y V, OXn ×Y V ). Lemma 10.2 shows that the morphism
X ×Y V → Spf A induces a morphism A → B. As in the proof of Lemma 6.15,
we find that each of induced morphisms Xn ×Y V → Vn is scheme-theoreticallydominant, which implies (indeed, is equivalent to) the fact that each An is equal
to the image of the composite morphism A → B → Bn . Since Xn ×Y V → Vn isfurthermore proper, we see that Bn is a finite An -algebra for each n. (See e.g. [Ols05]
for a proof of this finiteness statement for proper morphisms in the stacky context.)
As already noted, we have an isomorphism of topological rings A ∼A , as= lim←−n nwell as an isomorphism of topological rings B ∼
B
.
Since
A
embeds
into
Bn
= lim
n
n
←−n
for each n, we see that the injection A ,→ B is a topological embedding.
Since A is complete, and since the t-adic topology on A is finer than its given
topology, we see that A is t-adically complete. Thus, if we fix a value of n, thensince An if of finite type over R/tn , we may find a morphism from R{x1 , . . . , xr }(restricted power series over R in some number of variables x1 , . . . , xr ) to A whose
composite with the surjection A → An is again surjective. Since Bn is finite over An ,it is thus also finite over R{x1 , . . . , xr }.
Lemma 10.3 shows that the topology on B induced by writing it as the projective
limit B ∼B coincides with the t-adic topology. Thus B/tB is a quotient of
= lim
←−n n
Bn for some sufficiently large value of n, and so, applying the construction of
the preceding paragraph with this value of n, we find that B/tB is finite as an
R{x1 , . . . , xr }-algebra. The topological version of Nakayama’s Lemma then showsthat B is finite as an R{x1 , . . . , xr }-algebra.
Now R{x1 , . . . , xn } is Noetherian, and so the Artin–Rees Lemma shows that thet-adic topology on B induces the t-adic topology on its R{x1 , . . . , xr }-submodule A.
Taking into account Lemma 10.3 again, as well as the fact (noted above) that thegiven topology on B induces the given topology on A, we find that the given
topology on A is the t-adic topology, as required. To see that Spf A → Spf R is offinite type, note that A/tm A is now known to be a discrete quotient of A, for any
value of m, so that we have a closed immersion Spec A/tm A ,→ Spec An for somesufficiently large value of n. Since the target of this closed immersion is of finite
type over R/tn , so is the source.

11. A criterion for a formal algebraic stack to be locally
Noetherian
Our main objective in this section is to provide a criterion for a locally Ind-locallyof finite type formal algebraic stack X (in the sense discussed in Remark 8.39) lying
over a locally Noetherian scheme S to in fact be locally Noetherian. Our argument,
which can be thought of as a generalization of the proof that a pro-representing
object for a formal deformation problem whose tangent space is finite dimensional
is necessarily a complete Noetherian local ring, will be based on the existenceof an obstruction theory satisfying certain properties, similar to those consideredFORMAL ALGEBRAIC STACKS

83

in [Stacks, Tag 07YV]. Before introducing the relevant notions, though, we firstestablish some simple criteria for affine formal algebraic spaces to be Noetherian.
Lemma 11.1. Let Spf A = limn Spec An be a countably indexed affine formal al−→gebraic space of Ind-locally finite type over a Noetherian ring Λ; so each An isa finite type Λ-algebra, and the transition morphisms Spec An ,→ Spec An+1 arethickenings of Λ-schemes. Let In denote the nilradical of An , and suppose further
that for each N ≥ 1, the limit limn Spec An /InN eventually stabilizes, i.e. that the−→Nmorphism Spec An /InN → Spec An+1 /In+1
is an isomorphism for each fixed valueof N and for sufficiently large values of n. Then the topological ring A = limn An←−
is a Noetherian Λ-algebra, endowed with the I-adic topology, for some ideal I ⊆ Awith respect to which A is complete.
Proof. For each value of N ≥ 0, choose n(N ) large enough that each of the tranN +1
N +1→ An(N ) /In(Nsition morphisms Am /Im) is an isomorphism if m ≥ n(N ), andN +1
write BN := An(N ) /In(N) . We may, and do, choose n(N ) so that it is a monotone increasing function of n, in which case there are natural transition morphisms
BN +1 → BN , which are thickenings. Also, since each An is Noetherian, each of
the ideals In is nilpotent, and so if N is chosen so large that InN +1 = 0, then there
is a natural surjection BN → An /InN +1 = An . Thus {BN } is an inverse system ofthickenings that is cofinal with the sequence {An }, and hence gives rise to the sameaffine formal algebraic space. Replacing the sequence {An } by the sequence {BN },we may then assume that Inn+1 = 0 for each n (so in particular, A0 is reduced),∼
and that An /Inn −→ An−1 .
Since A0 is a finite type Λ-algebra, we may choose a surjection Λ[x1 , . . . , xr ] →A0 , and lift this morphism to a morphism of Λ-algebras Λ[x1 , . . . , xr ] → A. Also,∼
if n ≥ 1, then In /In2 −→ I1 /I12 is a finitely generated A0 -module, and thus we may
choose compatible surjections of Λ[x1 , . . . , xr ]-modules(11.2)

Λ[x1 , . . . , xr ]s → In /In2

for some value of s.
Since each In is a nilpotent ideal in An , the morphisms (11.2) can be used toinduce a compatible sequence of surjections Λ[x1 , . . . , xr ][[y1 , . . . , ys ]] → An , which
in turn give rise to a morphismΛ[x1 , . . . , xr ][[y1 , . . . , ys ]] → A.
If we equip the source with its (y1 , . . . , ys )-adic topology, then this morphism istaut (in the sense of from [Stacks, Tag 0AMX]) with dense image, and so it follows
from [Stacks, Tag 0APT] that this morphism is itself surjective and open, so that
A is Noetherian and adic.
Lemma 11.3. If X is a countably indexed affine formal algebraic space of Indlocally finite type over a Noetherian ring Λ, then the following conditions are equivalent:(1) X is Noetherian.
(2) Let Spf A := limn Spec An be any countably indexed affine formal algebraic−→
space of Ind-locally finite type over Λ (so {An } is an inverse system of finite typeΛ-algebras with the transition morphisms Spec An ,→ Spec An+1 being thickenings),
let In denote the nilradical of An , and suppose that there exists some N ≥ 0 suchthat InN = 0 for all n. Then any morphism x
b : Spf A → X of affine formal algebraic

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M. EMERTON

spaces (or equivalently, any compatible family of morphisms xn : Spec An → X) is
effective, i.e. there exists a morphism x : Spec A → X inducing xb (or equivalently,
inducing each of the xn ).
(3) Let A be any finite type Λ-algebra, and let {A0n } be an inverse system ofsquare-zero thickenings 0 → Jn → A0n → A → 0, with the property that each Jn is a
finite type A-module that is furthermore annihilated by the nil-radical of A. Then,if A0 := limn Spec A0n , any morphism x
b : Spf A0 → X is effective.
−→
Proof. Suppose first that X is Noetherian, so that we may find an isomorphism
X ∼= Spf B, where B is a Noetherian ring, endowed with its J-adic topology forsome ideal J with respect to which B is complete. Let xb : Spf A → Spf B bea morphism as in (2). Then the induced morphism B → An maps J into In ,and hence contains J N in its kernel. Thus we may pass to the inverse limit andobtain a morphism B → B/J N → A, which in turn induces the desired morphismx : Spec A → Spf B. Thus (1) implies (2).
Conversely, suppose that condition (2) holds. Choose an isomorphism Spf B ∼= X,
where B = limn Bn is an inverse limit of finite type Λ-algebras Bn with respect to←−transition maps that are thickenings, and let Jn be the nil-radical of Bn . Choose
N ≥ 0, and for each value of n, let An := Bn /JnN . By assumption the morphism
limn Spec An → Spf B is effective, and thus factors through Spec Bn for some suf−→
ficiently large value of n. This implies that the inverse system Bn /JnN eventuallystabilizes, and it follows from Lemma 11.1 that X is Noetherian, i.e. that (1) holds.
Condition (3) is a special case of condition (2). Indeed, the assumptions imply
that each A0n is of finite type over Λ. Let I denote the nilradical of A, and choose
N so that I N = 0. Then, if In0 denotes the nilradical of A0n , we see that (In0 )N ⊆ Jn ,and thus that (In0 )N +1 = 0.
Suppose now that (3) holds, and let x
b : Spf A → X be a morphism as in thesetting of (2). We will prove that x
b is effective by induction on N . Consider the
inverse system of short exact sequences (indexed by n)(11.4)

0 → InN −1 → An → An /InN −1 → 0.

By induction, the morphism limn Spec An /InN −1 → X is effective. Since X is Ind−→
locally of finite type, it factors through a morphism y : Spec C → X, where C
is of finite type over Λ. Pulling back each of the short exact sequences (11.4) by
the corresponding morphism C → An /InN −1 , we obtain an inverse system of short
exact sequences(11.5)

0 → InN −1 → A0n → C → 0.

We are now in the situation of (3), and thus the morphism Spf A0 := lim Spec A0n →−→X is effective. The natural morphism A0 → A induces a corresponding morphism
Spec A → Spec A0 , which, composed with the just-constructed morphism Spec A0 →X, yields the desired morphism x : Spec A → X.

We now introduce the obstruction theoretic notion that will be applied in our
criterion for Noetherianness. We begin by recalling some material from [Stacks,
Tag 07Y6].
Let X be a category fibred in groupoids over a locally Noetherian scheme S which
satisfies the condition (RS*) of Definition 3.8. If x : Spec A → X is a morphism forwhich the composite morphism Spec A → S factors through an affine open subset
of S, then in [Stacks, Tag 07Y9] there is defined a functor Tx from the category

FORMAL ALGEBRAIC STACKS

85

of A-module to itself, whose formation is also functorial in the pair (x, A) [Stacks,
Tag 07YA], and such that for any A-module M , there is a natural identification ofTx (M ) with the set of lifts of x to morphisms x0 : Spec A[M ] → X (where A[M ]denotes the square zero extension of A by M ).
We now make the following key definition.
Definition 11.6. Let X be a category fibred in groupoids over a locally Noetherian
scheme S, which satisfies (RS*). We say that X admits a nice obstruction theory
if, for each finite type S-algebra A which lies over an affine open subscheme of S,
equipped with a morphism x : Spec A → X , there exists a complex of A-modules•
K(x,A), as well as an auxiliary A-module C(x,A) , such that the following conditions
are satisfied:•(1) The complex K(x,A)
is bounded above and has finitely generated cohomology modules (or, equivalently, is isomorphic in the derived category to abounded above complex of finitely generated A-modules), while C(x,A) is
finitely generated and locally free.
•(2) The formation of K(x,A)
is compatible (in the derived category) with pullback, as is the formation of C(x,A) . More precisely, if f : Spec B → Spec A,
inducing the morphism y : Spec B → Spec A → X , then there is a natural∼••
isomorphism in the derived category f ∗ K(x,A)−→ K(y,B), as well as a∼natural isomorphism f ∗ C(x,A) −→ C(y,B) .
(3) For any pair (x, A) as above, and for any A-module M , we have a short
exact sequence•
0 → C(x,A) ⊗ M → Tx (M ) → H 1 (K(x,A)⊗L M ) → 0

whose formation is functorial in M , as well as in the pair (x, A), in the sensethat if we have a homomorphism A → B, and let y denote the composite
xSpec B → Spec A −→ X ,and if N is a B-module (regarded also as an A-module via the given morphism A → B), then the diagram0/ C(x,A) ⊗ N/ Tx (N )/ H 1 (K •

L

N)

/0

0


/ C(y,B) ⊗ N/ Ty (N )/ H 1 (K •(y,B) ⊗

N)

/0(x,A) ⊗

Lcommutes; here the left-most and right-most vertical arrows are induced by
the pull-back isomorphisms of condition (2) above, while the middle verticalarrow is the natural map arising from the functoriality of T , as discussed
in [Stacks, Tag 07YA].
(4) For any pair (x, A) as above, and for any A-module M , the cohomology•
module H 2 (K(x,A)⊗L M ) serves as an obstruction module, so that for any
deformation situation (x, A0 → A), we have a functorial obstruction element•
ox (A0 ) ∈ H 2 (K(x,A)⊗L I) (where I denotes the square-zero kernel of thesurjection A0 → A).
Remark 11.7. In Definition 11.6, it might be more natural to dispense with themodules C(x,A) , and in condition (3), to simply require the existence of a natural

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M. EMERTON•
isomorphism between Tx (M ) and H 1 (K(x,A)⊗L M ). However the slightly morerelaxed condition that we have imposed gives technical flexibility; for example, it
makes it easy to prove Lemma 11.12 below.

Remark 11.8. We give an alternative formulation of condition (3) of Definition 11.6.
•
Namely, suppose given a natural surjection Tx (M ) → H 1 (K(x,A)⊗L M ), and let
Kx (M ) denote the kernel; the formation of Kx (M ) is then functorial in M , and in
the pair (A, x).
Recall that a functor on A-modules is of the form N ⊗ – if and only if it isA-linear, right exact, and commutes with inductive limits. The functor Kx (–) is•certainly A-linear (since Tx (–) is [Stacks, Tag 07Y9], as is H 1 (K(x,A)⊗L –)). Thus∼

asking for a natural isomorphism Kx (M ) −→ C(x,A) ⊗ M , with C(x,A) finitely
generated and locally free, is equivalent to asking that Kx (–) be exact and commute
with inductive limits, and that Kx (A) be finitely generated.
Remark 11.9. We suppose given a nice obstruction theory in the sense of Definition 11.6. Let A be a finite type S-algebra that lies over an affine open subschemeof S, and let x : Spec A → X be a morphism. If M is an A-module, then there is∼a natural isomorphism limi Mi −→ M, where Mi runs over the finitely generated−→
submodules of M , inducing natural isomorphisms∼••
lim H • (K(x,A)⊗L Mi ) −→ H • (K(x,A)⊗L M ).
−→i

Since S is locally Noetherian, the finite type S-algebra A is Noetherian. If∼
0 → I → A0 → A → 0 is a square zero thickening, then we may write limi A0i −→ A0 ,−→where A0i runs over the finite type S-subalgebras of A0 that surject onto A. If we
set Ii := A0i ∩ I, then Ii is a finite A-module, and there is a natural isomorphism∼
limi Ii −→ I, and so a corresponding natural isomorphism−→(11.10)∼••lim H • (K(x,A)⊗L Ii ) −→ H • (K(x,A)⊗L I).
−→i

Since the formation of obstruction elements is functorial, we find that ox (A0 ) (whichlies in the target of this isomorphism), may be identified with the image of ox (A0i )(for any choice of i; the images for the various different choices of i coincide) under
this isomorphism. The isomorphism (11.10) then shows that ox (A0 ) = 0 if and only
if ox (A0i ) = 0 for some value (and hence for all sufficiently large values) of i.
If X is limit preserving, then there is an equivalence of groupoids∼

lim X (Ai ) −→ X (A)−→i(the left-hand side being the 2-colimit). Using this, we find that the morphism x
lifts to A0 if and only if it lifts to A0i for some value of i. Similarly, we see that theformation of Tx (M ) is compatible with the formation of inductive limits.
In conclusion, we find that if X is limit preserving, then, in order to construct
a nice obstruction theory, it suffices to consider the case of finitely generated Amodules M , and square zero extensions
0 → I → A0 → A → 0
for which A0 is of finite type over S (or, equivalently, for which I is a finite Amodule).

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87

Remark 11.11. For later reference, we briefly discuss a certain construction related
to obstruction theory computations.
Suppose that
0 → I → A0 → A → 0
is a square zero thickening, and that ϕ : I → J is a morphism of A-modules. We
may pushforward the extension by ϕ to obtain a morphism of short exact sequences0/JO/ ϕ∗ A0
O

/A

/0

0

/I

/ A0

/A

/0

We note that ϕ∗ A0 admits a natural ring structure, so that the preceding diagram
becomes a morphism of square zero extensions. Namely, we may construct ϕ∗ A0
as the quotient of the product A0 × J by an anti-diagonally embedded copy of I.
If we regard A0 × J as the trivial square zero extension A0 [J], then the diagonallyembedded copy of I is an ideal, and so the quotient ϕ∗ A0 becomes a square zeroextension of A by J, and the above diagram becomes a morphism of square zerothickenings.
If we are now given a morphism x : Spec A → X as in Definition 11.6, the functo•
riality of the formation of obstruction elements shows that ox (ϕ∗ A0 ) ∈ H 2 (K(x,A)
⊗L J)
0
2
•
L
is the image of ox (A ) ∈ H (K(x,A) ⊗ I) under the morphism induced by ϕ.
Lemma 11.12. If U → X is a morphism of categories fibred in groupoids over a
locally Noetherian scheme S that is representable by algebraic spaces and smooth,and if X satisfies (RS*) and admits a nice obstruction theory (in the sense of
Definition 11.6), then the same is true of U.
Proof. It follows from Lemma 3.9 that U satisfies (RS*).
Suppose given a morphism u : Spec A → U, where A is a finite type S-algebralying over an affine open subscheme of S, and let x : Spec A → X denote the••composite of f with the given morphism U → X . We define K(u,A):= K(x,A).
The infinitesimal lifting criterion for smooth morphisms (see e.g. [Ryd11, Appendix B]) shows that the natural morphism Tu (M ) → Tx (M ) is surjective. The
composite•
Tu (M ) → Tx (M ) → H 1 (K(x,A)⊗L M )(the latter map being the surjection provided by the good obstruction theory for X ,which is assumed to exist) is then also surjective; we next compute its kernel. To thisend, write U := Spec A ×X U; then U is an algebraic space smooth over Spec A, andthe morphism u corresponds to a section of U over Spec A. The conormal bundleof this section to U is then a finitely generated locally free sheaf on Spec A, whose
global sections form a finitely generated locally free A-module P , and we have anatural short exact sequence
0 → P ⊗ M → Tu (M ) → Tx (M ) → 0.
•
The kernel Ku (M ) of the surjection Tu (M ) → H 1 (K(x,A)⊗L M ) then sits in ashort exact sequence

0 → P ⊗ M → Ku (M ) → C(x,A) ⊗ M → 0,

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M. EMERTON

whose formation is functorial in M . The five-lemma then shows that Ku (M ) is anexact functor of M , and evidently Ku (A) is finitely generated (being an extensionof finitely generated A-modules). Remark 11.8 then shows that there is a natural∼
isomorphism Ku (M ) −→ C(u,A) ⊗ M , for some finitely generated and locally freeA-module C(u,A) .
Suppose now that 0 → I → A0 → A → 0 is a square-zero thickening. We claimthat if we define•⊗L I),ou (A0 ) := ox (A0 ) ∈ H 2 (K(x,A)
then we obtain functorial obstruction elements for extending morphism into U oversquare-zero thickenings.
Certainly if u extends to a morphism u0 : Spec A0 → U, then the composite
0x : Spec A0 → U → X is an extension of x. Conversely, if such an extension
x0 of x exists, then the infinitesimal lifting criterion shows that we may lift x0 toa morphism u0 : Spec A0 → U which extends x. This verifies the final condition•
for K(u,A)
to give a good obstruction theory, and so completes the proof of thelemma.

Theorem 11.13. If X is a locally countably indexed and locally Ind-locally of finite
type formal algebraic stack over the locally Noetherian scheme S which admits a niceobstruction theory, then X is locally Noetherian.
Remark 11.14. Recall that, by Lemma 4.22, the Ind-algebraic stack X satisfies(RS*), so that it makes sense to speak of it admitting a nice obstruction theory.
`
Proof of Theorem 11.13. We may find a morphism i Ui → X whose source isa disjoint union of affine formal algebraic spaces, and which is representable byalgebraic spaces, smooth, and surjective. We may furthermore assume that foreach i, the composite morphism Ui → X → S factors through an open affine subsetof S.
By assumption, each Ui is locally countably indexed and of Ind-locally finite
type over S, and in order to prove the theorem, we must show that each Ui is infact a Noetherian affine formal algebraic space. Lemma 11.12 shows that each Ui
admits a nice obstruction theory, and thus, replacing X by Ui , and replacing S
by an appropriately chosen affine open subscheme Spec Λ, it suffices to prove thetheorem in the case when X = X is a countably indexed affine formal algebraic
space of Ind-locally finite type over Spec Λ for a Noetherian ring Λ.
We need to show that X is Noetherian; to do this, it is enough, by Lemma 11.3,
to show that X satisfies condition (3) of that lemma. Thus we suppose given asequence of square-zero thickenings(11.15)0 → Jn → A0n → A → 0,

where A is a finite type Λ-algebra, and Jn is a finite type Ared -module, as well asa morphism x
b : Spf A0 → X , where
0 → J → A0 → A → 0
is the inverse limit of the thickenings (11.15). We must show that x
b is effective; todo this, we argue by induction on the dimension of Spec Ared .
Let y : Spec A → X be the morphism induced by xb, and let yred : Spec Ared → X•
denote the morphism induced by y. Each of the Ared -modules H i (K(y) isred ,Ared )finitely generated, and furthermore vanishes if i is sufficiently large. Thus we mayFORMAL ALGEBRAIC STACKS

89

find an element f ∈ A such that Spec A[1/f ] is a dense open subset of Spec A (or,equivalently, such that Spec Ared [1/f ] is a dense open subset of Spec Ared ), and•such that each of the localizations H i (K(y)[1/f ] is locally free for i ≥ 1.
red ,Ared )We recall two standard spectral sequences that we will require, namely(11.16)q•p+q•⊗L M )
E2p,q := Tor−p(K(yAred H (K(yred ,Ared ) ), M ) =⇒ H
red ,Ared )(for any Ared -module M ), and(11.17)••⊗L J).
⊗L Jn ) =⇒ H p+q (K(yE2p,q := Rp lim H q (K(yred ,Ared )red ,Ared )←−nFrom the first of these spectral sequences, we obtain morphisms(11.18)••H 1 (K(y) ⊗ Jn → H 1 (K(y⊗L Jn ).
red ,Ared )red ,Ared )If we let Cn denote the cokernels of these morphisms, then Lemma 11.22 below
shows that each Cn is annihilated by some power of f which is bounded independently of n. Replacing f by this power, we may and do assume that thesecokernels are in fact annihilated by f . Since the transition morphisms Jn+1 → Jn
are surjective, the same is true of the transition morphisms••
H 1 (K(y) ⊗ Jn+1 → H 1 (K(y) ⊗ Jn ,red ,Ared )red ,Ared )•) ⊗ Jn = 0. If we let Mn denote the image of (11.18),
so that R1 limn H 1 (K(yred ,Ared )←−then the right exactness of R1 limn shows that also R1 limn Mn = 0 (alternatively,←−←−
we could note that the transition morphisms Mn+1 → Mn are again surjective),and hence that∼•
R1 lim H 1 (K(y⊗L Jn ) −→ R1 lim Cn
red ,Ared )
←
−
←
−
n
n

is annihilated by f . A consideration of (11.17) then shows that there is a surjection(11.19)••H 2 (K(y⊗L J) → lim H 2 (K(y⊗L Jn ),
red ,Ared )red ,Ared )←−nwhose kernel is annihilated by f .
We are now ready to apply our obstruction theory. Note that since each of
the A-modules Jn , as well as their inverse limit J, is in fact an Ared -module, andsince the formation of the complex K • is compatible with base-change, we haveisomorphisms••
H i (K(y,A)⊗L M ) ∼⊗L M )= H i (K(yred ,Ared )when M is equal to one of the Jn or to J. In particular, we may view the ob•
struction classes oy (A0n ) as elements of H 2 (K(y⊗L Jn ), and similarly, we
red ,Ared )•may view the obstruction class oy (A0 ) as an element of H 2 (K(y⊗L J). Since
red ,Ared )the formation of obstruction classes is functorial, we see that the obstruction classoy (A0 ) maps to the compatible sequence of obstruction classes oy (A0n ) under thesurjection (11.19). However, the existence of the morphisms Spec A0n → X showsthat each of the obstruction classes oy (A0n ) vanishes, and hence that oy (A0 ) lies in
the kernel of (11.19). Thus this class is annihilated by f .
If A00 denotes the push-out of A0 along the morphism J → J given by multiplication by f , then we see that oy (A00 ) = 0. (See Remark 11.11 for a general discussionof pushouts and of the corresponding obstruction elements.) Thus there exists amorphism x00 : Spec A00 → X prolonging y. If we let A00n denote the pushout of A0n∼along the morphism Jn → Jn given by multiplication by f (so that A00 −→ limn A00n ),←−90

M. EMERTON

then the morphism x00n : Spec A00n → X obtained by restricting x00 may not coincidewith the composite(11.20)xnX,
Spec A00n → Spec A0n −→but differs from this composite by an element tn ∈ Tyred (Jn ). Since the morphisms
xn are compatible as n varies, so are the elements tn . The morphism••, J) → lim H 1 (K(y, Jn )H 1 (K(yred ,Ared )red ,Ared )←−nis surjective (as follows from a consideration of the spectral sequence (11.17)), andthe morphism
C(yred ,Ared ) ⊗ J → lim C(yred ,Ared ) ⊗ Jn←−nis even an isomorphism (since C(yred ,Ared ) is finitely generated and locally free).
Thus the morphism
Tyred (J) → lim Tyred (Jn )←−nis also surjective, and hence we can find an element t ∈ Tyred (J) which maps
to the element (tn ) ∈ limn Tyred (Jn ). Modifying x00 by t, we obtain a morphism←−(which we continue to denote by x00 ) whose restriction x00n to each Spec A00n coincides
with (11.20).
∼
Let us take a moment to take stock of our situation. If we write X −→ Spec B,0then the morphism x
b : Spf A → X corresponds to a continuous morphism B → A0 .
The fact that the composite Spf A00 → Spf A → X is induced by a morphism
Spec A00 → X is equivalent to the fact that the kernel of the composite(11.21)

B → A0 → A00

is an open ideal in B. The kernel of the morphism A0 → A00 is the ideal J[f ](the ideal of f -torsion elements in J), and the morphism A0 → A00 factors as thesurjection A0 → A0 /J[f ] followed by an embedding A0 /J[f ] ,→ A00 . Thus thekernel of the composite (11.21) and the kernel of the composite B → A0 → A0 /J[f ]coincide, and hence the composite
Spf(A0 /J[f ]) → Spf A0 → X
is effective, arising from a morphism z : Spec(A0 /J[f ]) → X.
We now consider the square zero thickening
0 → J[f ] → A0 → A0 /J[f ] → 0,
which may be written as an inverse limit of square zero thickenings
0 → Jn [f ] → Bn → A0 /J[f ] → 0.
This latter square zero thickening is obtained as the pull-back, via the morphism
A0 /J[f ] → A0n /Jn [f ], of the square zero thickening
0 → Jn [f ] → A0n → A0n /Jn [f ] → 0.
Since X satisfies (RS*), by Lemma 4.22, the compatible morphisms xn : Spec A0n →X and z : Spec(A0 /J[f ]) → X (compatible in the sense that they coincide whenrestricted to Spec A0n /Jn [f ]) may then be glued to yield a morphism
un : Spec Bn → X.

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91

These morphisms are compatible as n varies, and induce the given morphism xb:
A0 → X.
Since X is Ind-locally of finite type over Λ, we may find a finite type Λ-subalgebra
C of A0 /J[f ], such that the morphism z : Spec(A0 /J[f ]) → X factors throughSpec C. Since A is of finite type over Λ, we may enlarge C if necessary so that
C → A is surjective; we then let g denote an element of C which maps to the given∼
element f of A. Since (A0 /J[f ])red −→ Ared , the inclusion of C into A0 /J[f ] then∼induces an isomorphism of finite type Λ-algebras Cred −→ Ared , mapping the imageof g to the image of f . In particular, the Krull dimension of Cred /g Cred is less thanthe Krull dimension of Ared .
If we pull-back the square zero thickening A0 of A0 /J[f ] over C, we obtain asquare zero thickening
0 → J[f ] → C 0 → C → 0,
which is the inverse limit of square zero thickenings
0 → Jn [f ] → Cn0 → C → 0
obtained by pulling back each of the thickenings Bn . Each of the morphisms un :Spec Bn → X induces a morphism vn : Spec Cn0 → X. These morphisms are ofcourse compatible as n varies, and so give rise to a morphism vb : Spf C 0 → X. Since
the morphism x
b : Spf A0 → X factors as the compositevb

Spf A0 → Spf C 0 −→ X
by construction, we see that the effectivity of xb would follow from the effectivityof vb. It is this latter effectivity that we now prove.
Since C is of finite type over Λ, it is Noetherian, and so its ideal of g-power
torsion elements is annihilated by some fixed power of g, say g N . It follows thateach of the sequences
0 → Jn [f ] → Cn0 /g N +1 Cn0 → C/g N +1 C → 0
is short exact, as is the sequence
0 → J[f ] → C 0 /g N +1 C 0 → C/g N +1 C → 0
obtained as their inverse limit. Our inductive hypothesis then implies that themorphism Spf C 0 /g N +1 C 0 → X obtained by restricting vb is effective. Since C 0 canbe obtained as the fibre product∼

C 0 −→ (C 0 /g N +1 C 0 ) ×C/gN +1 C C,
another application of (RS*) yields a morphism v : Spec C → X, making vb effective,
and so completing the proof of the theorem.
The following result is presumably well-known, but for lack of a reference, we
include a proof.
Lemma 11.22. Let A be a Noetherian ring, let C • be a bounded above complexof A-modules all of whose cohomology modules are finitely generated, let f ∈ A,
and suppose that the cohomology modules of C • [1/f ] are furthermore locally freeA[1/f ]-modules. Then for each index i, there is a power f n of f such that thekernel and cokernel of the natural morphism(11.23)H i (C • ) ⊗ N → H i (C • ⊗L N )92

M. EMERTON

are annihilated by f n , for every A-module N (the power f n being independent of
the module N ).
Proof. Consider the spectral sequenceq•
p+q(C • ⊗L N ),E2p,q = TorA
−p H (C ), N ) =⇒ H

the edge map of which induces the morphism (11.23) in the statement of the lemma.
Since C • is bounded above, for a fixed value of i (and an arbitrary A-module N ),only finitely many terms of this spectral sequence contribute to the computationof the kernel and cokernel of (11.23). Thus, to prove the lemma, it suffices toshow that if M is a finitely generated A-module for which M [1/f ] is locally freeover A[1/f ], then for each positive degree i, there is a power f n of f such thatn
TorA(the power f n being independent of the choicei (M, N ) is annihilated by fof N ).
Let P• → M be a resolution of M by finitely generated projective A-modules.
Since M [1/f ] is locally free, the complex. . . → Pi+1 [1/f ] → Pi [1/f ] → . . . P1 [1/f ] → P0 [1/f ] → M [1/f ] → 0
is chain homotopic to zero. Write P−1 := M, and let s̃j : Pj [1/f ] → Pj+1 [1/f ] (forj ≥ −1) denote this chain homotopy, so that∂j+1 ◦ s̃j − s̃j−1 ∂j = idPj [1/f ] .
If we fix i > 0 and choose n large enough, we may then find morphisms sj : Pj →Pj+1 , for j = i and i − 1, such that sj [1/f ] = f n sj , and such that∂i+1 ◦ si − si−1 ∂i = f n idPi .
If we tj (for j = i and i − 1) denote the morphism Pj ⊗ N → Pj+1 ⊗ N inducedby sj , then we find that∂i+1 ◦ ti − ti−1 ∂i = f n idPi ⊗N .
The existence of such morphisms tj satisfying this identity evidently implies thatnTorAi (M, N ) := Hi (P• ⊗ N ) is annihilated by f , as required.
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Email address: emerton@math.uchicago.edu
Department of Mathematics, University of Chicago, 5734 S. University Ave., Chicago,
IL 60637, USA