Derived Algebraic Geometry XIII: Rational and p-adic Homotopy
TheoryDecember 15, 2011

Contents
1 Rational Homotopy Theory
1.1 Cohomological Eilenberg-Moore Spectral Sequences . . . . . . . . . . . . . . . . . . . . . . . .
1.2 k-Rational Homotopy Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3 Rational Homotopy Theory and E∞ -Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.4 Differential Graded Lie Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.5 Comparison with Formal Moduli Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4
5
9
15
18
24

2 E∞ -Algebras in Positive Characteristic27
2.1 Norm Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.2 Power Operations on E∞ -Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.3 Finitely Constructible Sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.4 A Universal Coefficient Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
2.5 Compactness of Relative Cochain Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
2.6 Affine Behavior of p-Constructible Morphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
3 p-Profinite Homotopy Theory
69
3.1 Pro-Constructible Sheaves and p-Profinite Spaces . . . . . . . . . . . . . . . . . . . . . . . . . 70
3.2 Whitehead’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
3.3 p-Profinite Spaces of Finite Type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
3.4 Connectivity in p-Profinite Homotopy Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
3.5 p-adic Homotopy Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
3.6 Étale Homotopy Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

1

Introduction
The principal goal of algebraic topology is to study topological spaces X by means of invariants associatedto X, such as the homology and cohomology groups H∗ (X; Z) and H∗ (X; Z). Ideally, we would like to devise
invariants X 7→ F (X) which satisfy the following requirements:(a) The invariant F is algebraic in nature: that is, it assigns to each space X something like a group or avector space, which is amenable to study using methods of abstract algebra.
(b) The invariant F is powerful: that is, F (X) contains a great deal of useful information about X.
There is a natural tension between these requirements: the simpler an object F (X) is, the less information
we should expect it to contain. Nevertheless, there are some invariants which do a good job of meeting bothobjectives. One example is provided by Sullivan’s approach to rational homotopy theory. To every topologicalspace X, Sullivan associates a polynomial deRham complex A(X), which is a commutative differential gradedalgebra over the field Q of rational numbers. This is an object of a reasonably algebraic nature (albeit notquite so simple as a group or a vector space), which at the same time captures the entire rational homotopy
type of X: for example, if X is a simply connected space whose homotopy groups {πn X}n≥2 are finitedimensional vector spaces over Q, then we can functorially recover X (up to homotopy equivalence) fromA(X).
As a chain complex, Sullivan’s polynomial deRham complex A(X) is quasi-isomorphic with the complex
of singular cochains C ∗ (X; Q). The advantage of A(X) of C ∗ (X; Q) is that A(X) is equipped with a multiplication which is commutative at the level of cochains, whereas the multiplication on C ∗ (X; Q) (givenby the Alexander-Whitney construction) is only commutative at the level of cohomology. We can therefore
think of A(X) as a remedy for the failure of the multiplication on C ∗ (X; Q) to be commutative at the level ofcochains. This is specific to the case of rational coefficients. If p is a prime number and Fp is the finite fieldwith p elements, then there is no way to replace the chain complex C ∗ (X; Fp ) by a commutative differential
graded algebra over Fp , which is functorially quasi-isomorphic to C ∗ (X; Fp ). However, a different remedy
is available over Fp : although the multiplication on C ∗ (X; Fp ) is not commutative, it is commutative up tocoherent homotopy. More precisely, C ∗ (X; Fp ) has the structure of an E∞ -algebra over Fp . Moreover, Mandell has used this observation to develop a “p-adic” counterpart of rational homotopy theory. For example,
he has shown that if X is a simply connected space whose homotopy groups are finitely generated modules
over Zp , then X can be functorially recovered from C ∗ (X; Fp ), together with its E∞ -algebra structure (see[54]).
Our goal in this paper is to give an exposition of rational and p-adic homotopy theory from the ∞categorical point of view, emphasizing connections with the earlier papers in this series. We will begin withthe case of rational homotopy theory. Sullivan’s work on the subject was preceded by Quillen, who showedthat the homotopy theory of rational spaces can be described in terms of the homotopy theory of differential
graded Lie algebras over Q. This result of Quillen was the impetus for later work of many authors, relating
differential graded Lie algebras to the study of deformation problems in algebraic geometry. In [49], wemade this relationship explicit by constructing an equivalence of ∞-categories Liek ' Modulik , where k isany field of characteristic zero. Here Liek denotes the ∞-category of differential graded Lie algebras overk, and Modulik the ∞-category of formal moduli problems over k. In §1, we will apply this result (in thespecial case k = Q) to recover Quillen’s results. Along the way, we will discuss Sullivan’s approach to rational
homotopy theory and its relationship with the theory of coaffine stacks developed in [47].
In §2 we will turn our attention to the case of p-adic homotopy theory. We will say that a space X
is p-finite if it has finitely many connected components and finitely many nonzero homotopy groups, eachof which is a finite p-group (Definition 2.4.1). If X is a p-finite space, then Mandell shows that X can berecovered as the mapping space
MapCAlgF (C ∗ (X; Fp ), Fp ) ' MapCAlgF (C ∗ (X; Fp ), Fp )ppwhere CAlgk denotes the ∞-category of E∞ -algebras over k and Fp denotes the algebraic closure of Fp . Hereit is important to work over Fp rather than Fp : the functor X 7→ C ∗ (X; Fp ) is not fully faithful (even when2

restricted to p-finite spaces). We can explain this point as follows: if k is an arbitrary field of characteristicp, then the homotopy theory of E∞ -algebras over k is most naturally related not to the homotopy theoryof (p-finite) spaces, but to the theory of (p-finite) spaces equipped with a continuous action of the absolute
Galois group Gal(k/k). More generally, we will show that if k is a commutative ring in which p is nilpotent,
p−fc
then there is a fully faithful embedding from Shvkp−fc into CAlgop
denotesk (Corollary 2.6.12); here Shvk
the ∞-category of p-constructible étale sheaves (of spaces) on Spec R (see Definition 2.4.1).
In order to apply the results of §2 to the study of an arbitrary space X, we need to study the problem ofapproximating X by p-finite spaces. For this, it is convenient to introduce the notion of a p-profinite space.
By definition, a p-profinite space is a Pro-object in the ∞-category Sp−fc of p-finite spaces. The collectionof p-profinite spaces can be organized into an ∞-category SPro(p) = Pro(Sp−fc ). In §3 we will study the∞-category SPro(p) , and show that it behaves in many respects like the usual ∞-category of spaces. Using
Corollary 2.6.12, we will construct a fully faithful embedding
SPro(p) → CAlgopk
X 7→ C ∗ (X; k),where k is any separably closed field (Proposition 3.1.16). Moreover, if k is algebraically closed, we can
explicitly describe the essential image of this functor (Theorem 3.5.8). We then recover some results of [54]by restricting our attention to p-profintie spaces of finite type (Corollary 3.5.15).

Notation and Terminology
We will use the language of ∞-categories freely throughout this paper. We refer the reader to [43] for ageneral introduction to the theory, and to [44] for a development of the theory of structured ring spectrafrom the ∞-categorical point of view. We will also assume that the reader is familiar with the formalism ofspectral algebraic geometry developed in the earlier papers in this series. For convenience, we will adopt thefollowing reference conventions:(T ) We will indicate references to [43] using the letter T.
(A) We will indicate references to [44] using the letter A.
(V ) We will indicate references to [45] using the Roman numeral V.
(V II) We will indicate references to [46] using the Roman numeral VII.
(V III) We will indicate references to [47] using the Roman numeral VIII.
(IX) We will indicate references to [48] using the Roman numeral IX.
(X) We will indicate references to [49] using the Roman numeral X.
(XI) We will indicate references to [50] using the Roman numeral XI.
(XII) We will indicate references to [51] using the Roman numeral XII.
For example, Theorem T.6.1.0.6 refers to Theorem 6.1.0.6 of [43].
If C is an ∞-category, we let C' denote the largest Kan complex contained in C: that is, the ∞category obtained from C by discarding all non-invertible morphisms. We will say that a map of simplicial
sets f : S → T is left cofinal if, for every right fibration X → T , the induced map of simplicial sets
FunT (T, X) → FunT (S, X) is a homotopy equivalence of Kan complexes (in [43], we referred to a map withthis property as cofinal). We will say that f is right cofinal if the induced map S op → T op is left cofinal:that is, if f induces a homotopy equivalence FunT (T, X) → FunT (S, X) for every left fibration X → T . If S

3

and T are ∞-categories, then f is left cofinal if and only if for every object t ∈ T , the fiber product S ×T Tt/
is weakly contractible (Theorem T.4.1.3.1).
Throughout this paper, we let CAlg denote the ∞-category of E∞ -rings. If R is an E∞ -ring, we let
CAlgR = CAlg(ModR ) denote the ∞-category of E∞ -algebras over R. We let Specét R denote the affine
spectral Deligne-Mumford stack associated to R. This can be identified with the pair (Shvét
R , O), whereétShvét⊆Fun(CAlg
,
S)
is
the
full
subcategory
spannedby
thosefunctorswhich
aresheaveswithrespect
to theRRététale topology, and O is the sheaf of E∞ -rings on Shvétdeterminedbytheforgetful
functor
CAlg
→
CAlg.
R
R

1

Rational Homotopy Theory

Definition 1.0.1. Let Y be a simply connected space. We say that Y is rational if, for each n ≥ 2, thehomotopy group πn Y is a vector space over the field Q of rational numbers (since Y is simply connected,
the homotopy groups πn (Y, y) are canonically independent of the choice of base point y ∈ Y ). We let Srat
denote the full subcategory of S spanned by the simply connected rational spaces, and Srat
∗ the ∞-categoryof pointed objects of Srat (that is, the ∞-category of pointed simply connected rational spaces).
Definition 1.0.2. Let k be a field of characteristic zero, and let Liek denote the ∞-category of differential
graded Lie algebas over k (Definition X.2.1.14). We will say that a differential graded Lie algebra g∗ isconnected if the graded Lie algebra H∗ (g∗ ) is concentrated in positive degrees: that is, if the homology groups
Hn (g∗ ) vanish for n ≤ 0 (here our notation indicates passage to the homology groups of the underlying chain
complex of g∗ , rather than the Lie algebra homology of g∗ ). We let Lie≥1
k denote the full subcategory of Liek
spanned by the connected differential graded Lie algebras.
Quillen’s work on rational homotopy theory establishes a close connection between rational spaces anddifferential graded Lie algebras. We can formulate his main result as follows:
Theorem 1.0.3 (Quillen). The ∞-category Sratof rational pointed spaces is equivalent to the ∞-category
∗
Lie≥1
of
connected
differential
gradedLiealgebrasover the field Q of rational numbers.
Q
In §X.2, we studied a different interpretation of the homotopy theory of differential graded Lie algebras:
for any field k of characteristic zero, there is a canonical equivalence of ∞-categories Ψ : Liek → Modulik ,where Modulik denotes the ∞-category of formal moduli problems over k (Theorem X.2.0.2). In this section,
we will explore the relationship between this statement and Quillen’s work. To this end, we will associate to
every field k of characteristic zero an ∞-category RType(k), which we refer to as the ∞-category of k-rational
homotopy types. By definition, RType(k) is a full subcategory of the ∞-category Fun(CAlgcn
k , S). Our main
results can be summarized as follows:(a) If k = Q is the field of rational numbers, then the evaluation map X 7→ X(Q) induces an equivalenceof ∞-categories RType(Q) → Srat (Theorem 1.3.6). Restricting to pointed objects, we obtain an
equivalence RType(Q)∗ → Srat
∗ .
(b) For any field k of characteristic zero, and k-rational homotopy type X, and any base point η ∈ X(k),we can associate a formal moduli problem X ∨ ∈ Modulik , which we call the formal completion of X at
the point k. The construction (X, η) 7→ X ∨ induces a fully faithful embedding RType(k)∗ → Moduli≥2k ,where Moduli≥2
denotesthefullsubcategoryofModulispannedby
thoseformalmoduliproblemskkhaving a 2-connective tangent complex (Theorem 1.5.3).
(c) For every field k of characteristic zero, the equivalence Ψ : Liek → Modulik restricts to an equivalence≥2
Lie≥1
k → Modulik .
Taking k to be the field of rational numbers, assertions (a), (b), and (c) give a diagram of equivalences
α

β

γ

≥2
≥1
Srat
∗ ← RType(Q)∗ → ModuliQ ← LieQ ,4thereby giving a proof of Theorem 1.0.3.
Let us now outline the contents of this section. We will begin in §1.1 by reviewing a convergencecriterion for the cohomological Eilenberg-Moore spectral sequence, which plays an important role in ourstudy of rational and p-adic homotopy theory. In §1.2, we define the ∞-category RType(k) of k-rational
homotopy types and establish some of its formal properties. In §1.3, we review the Sullivan model for rational
homotopy theory, and use it to construct the equivalence RType(Q) → Srat described in (a). The functor
β : RType(k)∗ → Moduli≥2
k of (b) will be constructed in §1.5. The proof that β is an equivalence will require
some basic facts about the homotopy theory of differential graded Lie algebras, which we review in §1.4.

1.1

Cohomological Eilenberg-Moore Spectral Sequences

Notation 1.1.1. Let C be an ∞-category. For every Kan complex X, we let CX denote the ∞-category
Fun(X, C) of functors from X to C. If f : X → Y is a map of Kan complexes, then composition with f
induces a functor CY → CX , which we will denote by f∗ .
Assume that, for each vertex y ∈ Y , the ∞-category C admits colimits indexed by the Kan complex
X ×Y Y/y . Then the functor f ∗ : CY → CX admits a left adjoint, given by left Kan extension along f ; wewill denote this functor by f! . If for each y ∈ Y the ∞-category C admits limits indexed by X ×Y Yy/ , thenf ∗ admits a right adjoint (given by right Kan extension along f ), which we will denote by f∗ .
Let k be a field and let Modk denote the ∞-category of k-module spectra. For every Kan complex X,
we let ModXk denote the ∞-category Fun(X, Modk ). The symmetric monoidal structure on Modk induces aX
Xsymmetric monoidal structure on ModX
k . We will denote the unit object of Modk by kX (that is, kX ∈ Modk
denotes the constant functor X → Modk taking the value k).
Y
Remark 1.1.2. In the situation of Notation 1.1.1, the functor f∗ : ModX
k → Modk is lax symmetricY
monoidal. In particular, A = f∗ (kX ) is a commutative algebra object of Modk , and f∗ determines a functor
Y
ModX
k → ModA (Modk ).

Notation 1.1.3. Let X be a Kan complex and let f : X → ∆0 be the projection map. For each F ∈ ModXk ,we setC∗ (X; F) = f! FC ∗ (X; F) = f∗ F .
In the special case F = kX , we will denote C∗ (X; kX ) and C ∗ (X; kX ) by C∗ (X; k) and C ∗ (X; k), respectively.
We regard C ∗ (X; k) as a commutative algebra object of Modk .
Remark 1.1.4. If we identify Modk with the underlying ∞-category of the model category Vectdg( k) of chain
complexes of k-vector spaces, then C∗ (X; k) and C ∗ (X; k) can be represented by the objects of Vectdg( k)given by the usual k-valued chain and cochain complexes associated to the simplicial set X. In particular,
we have canonical isomorphisms of k-vector spaces
πn C ∗ (X; k) ' H−n (X; k).

πn C∗ (X; k) ' Hn (X; k)

Lemma 1.1.5. Suppose we are given a homotopy pullback diagram of Kan complexes σ :
X0

f0

g0

g


X

/ Y0

f


/ Y.

Let C be an ∞-category, and assume that for each y ∈ Y the ∞-category C admits limits indexed by the Kancomplex X ×Y Yy/ . Then the diagram of ∞-categories
CY

/ CX


0
CY


0
/ CX
5

is right adjointable.
∗

Proof. Let F : X → C be a functor; we wish to show that the canonical map g 0 f∗ F → f 0 ∗ g ∗ F is an0
equivalence in CY . Unwinding the definition, we must show that for each vertex y 0 ∈ Y 0 , the map
0lim(F |X ×Y Yg0 (y)/ ) → lim(F |X 0 ×Y 0 Yy/
)
←−
←−
0
is an equivalence in C. This follows from the fact that the map X 0 ×Y 0 Yy/→ X ×Y Yg0 (y)/ is a homotopy
equivalence, since we have assumed that σ is a homotopy pullback diagram.

Remark 1.1.6. Let X be a Kan complex. The stable ∞-category ModX
k admits an accessible t-structure
X
X
X((ModXk )≥0 , (Modk )≤0 ), where (Modk )≥0 is the full subcategory of Modk spanned by those functors F :X → Modk such that F(x) ∈ (Modk )≥0 for all x ∈ X, and (Modk )≤0 is defined similarly. The heard ofModX
k can be identified with the category of functors from the fundamental groupoid of X to the categoryof vector spaces over k. If X is connected and contains a vertex x, then we can identify the heart of ModXk
with the category of representations (in k-vector spaces) of the fundamental group π1 (X, x).
If f : X → Y is a map of Kan complexes, then the functor f ∗ : ModYk → ModX
k is t-exact. It follows
Y
X
Y
→
Mod
is
left
t-exact
andthefunctor
f
:
Mod
→
Mod
that the functor f∗ : ModX!
k is right t-exact.
kkkDefinition 1.1.7. Let X be a connected Kan complex containing a vertex x. We will say that an object
F ∈ ModX
k is nilpotent if the following conditions are satisfied:(1) For each integer n, regard V = πn F as a representation of the fundamental group G = π1 (X, x). Then
V has a finite filtration by subrepresentations
0 = V0 ⊆ V1 ⊆ · · · ⊆ Vm = V
such that each quotient Vi /Vi−1 is isomorphic to k (endowed with the trivial G-action).
(2) The homotopy groups πn F vanish for n  0.
Remark 1.1.8. In the situation of Definition 1.1.7, the condition that F be nilpotent does not depend on
the choice of vertex x ∈ X.
Proposition 1.1.9. Let f : X → Y be a map of Kan complexes. Assume that Y is connected and let
F ∈ ModYk be nilpotent. Then the canonical map
θF : C ∗ (Y ; F) ⊗C ∗ (Y ;k) C ∗ (X; k) → C ∗ (X; f ∗ F)
is an equivalence in Modk .
Proof. Let C ⊆ ModYk be the full subcategory spanned by those objects F ∈ ModYk for which θF is anequivalence. It is clear that C is a stable subcategory of ModYk which contains kY .
Let F ∈ ModYk be nilpotent; we wish to show that F ∈ C. Replacing F by a shift if necessary, we canassume that F ∈ (ModYk )≤0 . Let KF be the cofiber of the map θF . We will show that KF ∈ (Modk )≤−n forevery integer n, so that KF ' 0. To prove this, we observe that the fiber sequenceτ≥−n F → F → τ≤−n−1 F
in ModYk determines a fiber sequenceKτ≥−n F → KF → Kτ≤−n−1 F
in Modk . For each n ≥ 0, the truncation τ≥−n F is a successive extension of (shifted) copies of kY , so thatτ≥−n F ∈ C and therefore Kτ≥−n F ' 0. It follows that KF ' Kτ≤−n−1 F . We may therefore replace F byτ≤−n−1 F and thereby assume that πi F ' 0 for i ≥ −n.
To prove that KF ∈ (Modk )≤−n , it will suffice to show that C ∗ (X; f ∗ F) and C ∗ (Y ; F) ⊗C ∗ (Y ;k) C ∗ (X; k)belong to (Modk )≤−n−1 . In the first case, this follows from the left t-exactness of the functor C ∗ (X; •). In thesecond case, it follows from the left t-exactness of the functor C ∗ (Y ; •) together with Corollary VIII.4.1.11(note that C ∗ (Y ; k) is a coconnective k-algebra, since we have assumed that Y is connected).
6

Corollary 1.1.10. Let k be a field, and let f : X → Y be a Kan fibration between Kan complexes. Assume
that:(a) The set of components π0 Y is finite.
(b) For each point y ∈ Y and each n ≥ 0, the vector space V = Hn (Xy ; k) admits a finite filtration
0 = V0 ⊆ V1 ⊆ · · · ⊆ Vm ⊆ 0
by π1 (Y, y)-invariant subspaces such that each quotient Vi /Vi−1 is a one-dimensional vector space overk endowed with the trivial action of π1 (Y, y).
Then, for any map g : Y 0 → Y , the canonical map
θ : C ∗ (Y 0 ; k) ⊗C ∗ (Y ;k) C ∗ (X; k) → C ∗ (Y 0 ×Y X; k)is an equivalence.
Proof. Using (a), we can immediately reduce to the case where Y is connected. Using Proposition 1.1.9, we∗
can identify θ with the canonical map C ∗ (Y 0 ; g ∗ f∗ kX ) → C ∗ (Y 0 ; f∗0 g 0 kX ), where f 0 : Y 0 ×Y X → Y 0 and
00g : Y ×Y X → X denote the projection maps. It follows from Lemma 1.1.5 that θ is an equivalence.
Remark 1.1.11. Under the ∗hypotheses of Corollary 1.1.10, Proposition A.7.2.1.19 gives a spectral sequence{Erp,q }r≥2 with E2p,∗ ' TorpH (Y ;k) (H∗ (X; k), H∗ (Y 0 ; k)) which converges to H∗ (Y 0 ×Y X; k). This spectral
sequence is called the cohomological Eilenberg-Moore spectral sequence.
Corollary 1.1.12 (KünnethFormula). Let k be a field, let X and Z be Kan complexes, and assume that thecohomology groups Hn (X; k) are finite dimensional for n ≥ 0. Then the canonical map
C ∗ (X; k) ⊗k C ∗ (Z; k) → C ∗ (X × Z; k)
is an equivalence of E∞ -algebras over k.
Proof. Apply Corollary 1.1.10 in the case Y = ∆0 , Y 0 = Z.
Remark 1.1.13. In the situation of Corollary 1.1.12, the spectral sequence of Remark 1.1.11 degenerates,
and we obtain the usual KünnethisomorphismsM
Hn (X × Z; k) 'Hi (X; k) ⊗k Hj (Z; k).
i+j=n

For n ≥ 1 and A an abelian group, we let K(A, n) denote the associated Eilenberg-MacLane space: it ischaracterized up to equivalence by the requirements that K(A, n) be connected and have homotopy groups(
A if i = nπi K(A, n) '0
otherwise.
When A is a field k, we have a tautological cohomology classηn ∈ Hn (K(A, n); k),which classifies a map k[−n] → C ∗ (K(A, n), k) in Modk and therefore a map Sym∗ (k[−n]) → C ∗ (K(A, n); k)
in CAlgk .
Proposition 1.1.14. For each n ≥ 1, the map
φ : Sym∗ (Q[−n]) → C ∗ (K(Q, n); Q)
is an equivalence of E∞ -rings.
7

Lemma 1.1.15. Let k be a field, let A be a coconnective E1 -algebra over k. Let f : M → M 0 be a morphism
in RModA and let N ∈ LModA . Assume that N is nonzero, that πi M ' πi M 0 ' πi N ' 0 for i > 0, andthat the induced map M ⊗A N → M 0 ⊗A N is an equivalence. Then f is an equivalence.
Proof. Let K be the fiber of f , and assume (for a contradiction) that K 6= 0. Then there exists some largest
integer n such that πn K 6= 0. Since N 6= 0, there exists some largest integer m such that πm N 6= 0. Corollary
VIII.4.1.11 implies that the map(πn )K ⊗k (πm N ) → πm+n (K ⊗A N ) ' 0
is injective, which is a contradiction.
Proof of Proposition 1.1.14. We proceed by induction on n. Consider first the case n = 1. The space K(Q, 1)
is given by the filtered limit of the diagram1
1
1
K(Z, 1) → K( Z, 1) → K( Z, 1) → K( Z, 1) → · · ·2
624
Each space appearing in this diagram can be identified with the circle S 1 , and each map in the diagram
induces an isomorphism on rational cohomology. It follows that(
Q if i ∈ {0, 1}ii1H (K(Q, 1); Q) ' H (S ; Q) '0
otherwise.
For m ≥ 0, we can identify Symm (Q[−1]) with V [−m], where V is the mth exterior power
L of Q i(as a vector
space over Q). It follows that Symm (Q[−1]) ' 0 for m ≥ 2, so that the map ψ :i≤1 Sym (Q[−1]) →Sym∗ (Q[−1]) is an equivalence. It is easy to see that φ ◦ ψ is an equivalence, so that φ is an equivalence as
well.
Now suppose that n ≥ 2. We have a homotopy pullback diagram of spaces
K(Q, n − 1)

/∗


∗


/ K(Q, n).

The inductive hypothesis guarantees that each cohomology group Hi (K(Q, n − 1); Q) is finite dimensional,
so that Corollary 1.1.10 gives an equivalenceC ∗ (K(Q, n − 1); Q) ' Q ⊗C ∗ (K(Q,n);Q) Q .
Let V = C ∗ (K(Q, n); Q) ⊗Sym∗ (Q[−n]) Q. It follows from Corollary VIII.4.1.11 that πi V ' 0 for i > 0. Using
the inductive hypothesis, we deduce thatC ∗ (K(Q, n − 1); Q) ' Sym∗ (Q[1 − n]) ' Q ⊗Sym∗ (Q[−n]) Q ' Q ⊗C ∗ (K(Q,n);Q) V.
Since K(Q, n) is connected, C ∗ (K(Q, n); Q) is coconnective and Lemma 1.1.14 implies φ induces an equivalence V ' Q. Using Lemma 1.1.14 again, we deduce that φ is itself an equivalence.
Corollary 1.1.16. Let V be a finite-dimensional vector space over Q and let V ∨ denote the dual space ofV . Then for n ≥ 1, the canonical map of E∞ -algebras over Q
Sym∗ (V ∨ [−n]) → C ∗ (K(V, n); Q)
is an equivalence.
8

1.2

k-Rational Homotopy Types

Let k be a field of characteristic zero, which we regard as fixed throughout this section. We let CAlgk denotethe ∞-category of E∞ -algebras over k, CAlgcn
k the full subcategory of CAlgk spanned by the connective
E∞ -algebras over k, and CAlg0k the full subcategory of CAlgcn
k spanned by the discrete E∞ -algebras over k(so that CAlg0k is equivalent to the nerve of the ordinary category of commutative k-algebras).
Definition 1.2.1. Let X : CAlgcn
k → S be a functor. We will say that X is a k-rational homotopy type if
the following conditions are satisfied:(a) The functor X is a left Kan extension of its restriction X0 = X| CAlg0k to discrete E∞ -algebras.
(b) For every R ∈ CAlg0k , the space X0 (R) is simply connected.
(c) For every integer n ≥ 2, there exists a vector space V over k such that the functor R 7→ πn X0 (R) is
given by R 7→ V ⊗k R.
We let RType(k) denote the full subcategory of Fun(CAlgcn
k , S) spanned by the k-rational homotopy types.
We will refer to RType(k) as the ∞-category of k-rational homotopy types.
Example 1.2.2. If V is a vector space over k, we define a functor K(V, n) : CAlgcn
k → S by the formulaK(V, n)(R) = Ω∞−n (V ⊗k R).
If n ≥ 2, then K(V, n) is a k-rational homotopy type. If V is finite-dimensional, then K(V, 0) is the functor
corepresented by the discrete E∞ -algebra Sym∗ V ∨ .
Our goal in this section is to study the ∞-category RType(k) and establish some of its properties.
Remark 1.2.3. Let Vectk denote the category of vector spaces over k, let Ab denote the category of abeliangroups, and define a functor T : Vectk → Fun(CAlg0k , Ab) by the formula
T (V )(R) = V ⊗k R.
If V is finite dimensional, then T carries V to the functor represented by the group scheme Spec V ∨ . In
particular, T carries finite-dimensional vector spaces to compact objects of Fun(CAlg0k , Ab). Since k has
characteristic zero, the functor T is fully faithful when restricted to finite dimensional vector spaces. Since
T commutes with filtered colimits, we conclude that T is fully faithful in general.
Now suppose that X : CAlgcn
k → S is a k-rational homotopy type and n ≥ 2 an integer. Then there
exists a vector space V over k such that the functor R 7→ X(R) is given by V ⊗k R for R ∈ CAlg0k . The
above argument shows that the vector space V is determined by X (up to unique isomorphism). We willemphasize this dependence by writing V = πn X.
Definition 1.2.4. Let X be a k-rational homotopy type. We will say that X is of finite type if the vector
spaces πn X are finite-dimensional for each n ≥ 2.
Remark 1.2.5. Recall that a functor X : CAlgcn
k → S is said to be a coaffine stack if it is corepresentable bya coconnective E∞ -algebra A ∈ CAlgk . We will say that a coaffine stack X is simply connected if π−1 A ' 0.
Using Propositions VIII.4.4.8 and VIII.4.4.6, we see that an arbitrary functor X : CAlgcn
k → S is a simply
connected coaffine stack if and only if it satisfies conditions (a) and (b) of Definition 1.2.1, together with thefollowing version of (c):(c0 ) For every integer n ≥ 2, there exists a vector space W over k such that the functor R 7→ πn X0 (R) is
given by R 7→ Homk (W, R). Here X0 denotes the restriction X| CAlg0k .
We say that a simply connected coaffine stack X is of finite type if the vector spaces W appearing in (c0 ) are
finite-dimensional. The following conditions on a functor X : CAlgcn
k → S are equivalent:9(i) The functor X is a k-rational homotopy type of finite type.
(ii) The functor X is a coaffine stack which is simply connected and of finite type.
In particular, if X is a k-rational homotopy type of finite type, then X is a coaffine stack.
We now establish some basic formal properties of k-rational homotopy types.
Proposition 1.2.6. Let k be a field of characteristic zero and let X : CAlgcn
k → S be a k-rational homotopy
type. Then X commutes with sifted colimits.
Lemma 1.2.7. Let A• be a simplicial abelian group, and suppose that the unnormalized chain complex· · · → A2 → A1 → A0
is an acyclic resolution of some abelian group A. Then for each integer n ≥ 0, the canonical map|K(A• , n)| → K(A, n)
is a homotopy equivalence.
Proof. We proceed by induction on n. If n = 0, then |K(A• , n)| can be identified with the geometricrealization |A• |, and its homotopy groups are computed by the chain complex· · · → A2 → A1 → A0 .
Now suppose that n > 0, and consider the map θn : |K(A• , n)| → K(A, n). Using Corollary A.5.1.3.7, we seethat the map Ω(θn ) : Ω|K(A• , n)| → ΩK(A, n) can be identified with θn−1 , which is a homotopy equivalenceby the inductive hypothesis. Since the spaces |K(A• , n)| and K(A, n) are both connected, we conclude that
θn is a homotopy equivalence.
Proof of Proposition 1.2.6. Let Polyk denote the full subcategory of CAlgcn
k spanned by those k-algebrasof the form k[x1 , . . . , xn ] for some n ≥ 0. Then Polyk is a subcategory of compact projective generators
cn
cn
for CAlgcn
k : that is, the inclusion Polyk ,→ CAlgk extends to an equivalence PΣ (Polyk ) ' CAlgk (seecn0Proposition A.7.1.4.20). Let X0 = X| Polyk and let X : CAlgk → S be a functor which extends X0 andcommutes with sifted colimits. Then X 0 is a left Kan extension of X0 , so that the identity transformationfrom X 0 to itself extends to a natural transformation α : X 0 → X. We wish to show that α is an equivalence,or equivalently that X is a left Kan extension of X0 . Since X is a left Kan extension of X| CAlg0k , it will
suffice to show that X| CAlg0k is a left Kan extension of X0 (Proposition T.4.3.2.8). For this, it suffices toshow that for every object R ∈ CAlg0k , the canonical map X 0 (R) → X(R) is an equivalence.
We first treat the case where R = Sym∗ V for some k-vector space V . Then R ' lim Sym∗ Vα , where the−→colimit is taken over the filtered partially ordered set of all finite-dimensional subspaces Vα ⊆ V . We thenhave a commutative diagram/ X 0 (R)lim X 0 (Sym∗ Vα )−→lim X(Sym∗ Vα )−→
/ X(R).

Since each Sym∗ Vα belongs to Polyk , the left vertical map is a homotopy equivalence. The upper horizontal
map is a homotopy equivalence because the functor X 0 commutes with filtered colimits. It will therefore
suffice to show that the lower horizontal map is a homotopy equivalence. Since the domain and codomainof this map are both simply connected, we are reduced to proving that the map
lim πn X(Sym∗ Vα ) → πn X(R)−→10

is an isomorphism for each n ≥ 2. This is clear, since the functor W 7→ (πn X) ⊗k (Sym∗ W ) commutes withfiltered colimits.
Now suppose that R ∈ CAlg0k is arbitrary. Choose a representation of R as the geometric realization of asimplicial object R• of CAlgcn
k , where each Rn a polynomial algebra (on a possibly infinite set of generators)
over k. We then have a commutative diagram|X 0 (R• )|/ X 0 (R)
|X(R• )|
/ X(R).

The upper horizontal map is a homotopy equivalence because X 0 commutes with sifted colimits, and theleft vertical map is a homotopy equivalence by the previous step in the proof. We are therefore reduced toproving that the map θ : |X(R• )| → X(R) is a homotopy equivalence.
To show that θ is a homotopy equivalence, it suffices to show that θ induces a homotopy equivalenceτ≤m |X(R• )| → τ≤m X(R) for each integer m ≥ 0. Note that τ≤m |X(R• )| is equivalent to the m-truncationof |τ≤m X(R• )|. It will therefore suffice to show that each of the maps
θm : |τ≤m X(R• )| → τ≤m X(R)is a homotopy equivalence. We proceed by induction on m, the case m ≤ 1 being trivial. We have a simplicial
diagram of fiber sequencesτ≤m X(R• ) → τ≤m−1 X(R• ) → K((πm X) ⊗k R• , m + 1).
Since each of the spaces K((πm X) ⊗k R• , m + 1) is connected, Lemma A.5.3.6.17 gives a fiber sequence ofgeometric realizations|τ≤m X(R• )| → |τ≤m−1 X(R• )| → |K((πm X) ⊗k R• , m + 1)|.
Consequently, to show that θm is an equivalence, it will suffice to show that the vertical maps in the diagram|τ≤m−1 X(R• )|/ |K((πm X) ⊗ −kR• , m + 1)|τ≤m−1 X(R)
/ K((πm X) ⊗k R• , m + 1)

are homotopy equivalences. For the left vertical map this follows from the inductive hypothesis, and for theright vertical map it follows from Lemma 1.2.7.
Corollary 1.2.8. Let X be a k-rational homotopy type. For each R ∈ CAlgcn
k , the space X(R) is simply
connected.
Proof. Write R as a geometric realization |R• |, where each Rn is discrete. Proposition 1.2.6 implies that
X(R) ' |X(R• )|. Since each of the spaces X(Rn ) is simply connected, we conclude that X(R) is simply
connected.
Corollary 1.2.9. Let X be k-rational homotopy type, and suppose we are given a pullback diagram σ :
A0

/A


B0


/B
11

in CAlgcn
k . Suppose that the right vertical map induces a surjection π0 A → π0 B. Then the diagram X(σ)
X(A0 )/ X(A)


X(B 0 )


/ X(B)is also a pullback square.
Proof. Choose a base point η ∈ X(k). Using η, we can lift X to a functor CAlgcn
k → S∗ taking values in
the ∞-category of pointed spaces. Choose a simplicial object B• of CAlgcn
whosegeometric realization iskequivalent to B, where each Bn is discrete. Set
A• = A ×B B•

B•0 = B 0 ×B B•

A0• = A0 ×B B• .

Then σ can be obtained as the geometric realization of a simplicial diagram σ• :
A0•

/ A•


B•0


/ B• .

Proposition 1.2.6 implies that X commutes with sifted colimits. It follows that X(σ) can be identified withthe geometric realization |X(σ• )|. We wish to show that |X(σ• )| is a pullback diagram. Each of the spaces
X(B• ) is connected (and equipped with a base point determined by our chosen point η ∈ X(k)). Applying
Lemma A.5.3.6.17, we are reduced to proving that each of the diagrams X(σn ) is a pullback square. We may
therefore replace B by Bn and thereby reduce to the case where B is discrete. Using a similar argument, we
may suppose that A and B 0 are also discrete (so that A0 ' A ×B B 0 is likewise discrete). Let Y denote thefiber product X(A) ×X(B) X(B 0 ). The homotopy groups of Y fit into a long exact sequence
πn+1 X(A) × πn+1 X(B 0 ) → πn+1 X(B) → πn Y → πn X(A) × πn X(B 0 ) → πn X(B).
Since X is a k-rational homotopy type, we can rewrite this sequence asβα(πn+1 X) ⊗k (A ⊕ B 0 ) → (πn+1 X) ⊗k B → πn Y → (πn X) ⊗k (A ⊕ B 0 ) → (πn X) ⊗k B.
Since the map A → B is surjective, α is a surjection for each n ≥ 1. It follows that Y is simply connected
and that for each n ≥ 2 we have a canonical isormophism πn Y ' ker(β) ' (πn X) ⊗k A0 . From this, we
immediately deduce that the map X(A0 ) → Y is a homotopy equivalence.
Corollary 1.2.10. Let X be a k-rational homotopy type and let f : A → B be a morphism in CAlgcn
k . If f
is n-connective for some n ≥ 0, then the induced map X(A) → X(B) is (n + 2)-connective.
Proof. We proceed by induction on n. If n > 0, then the diagonal map δ : A → A ×B A is (n − 1)-connective.
The inductive hypothesis then implies that X(δ) : X(A) → X(A×B A) is (n+1)-connective. Using Corollary
1.2.9 we can identify X(δ) with the diagonal map X(A) → X(A) ×X(B) X(A). Since X(f ) : X(A) → X(B)is surjective on connected components (both spaces are simply connected, by Corollary 1.2.8), the (n + 1)connectivity of X(A) → X(A) ×X(B) X(A) implies that (n + 2)-connectivity of X(f ).
It remains to treat the case n = 0. Since X(A) and X(B) are simply connected, it suffices to show thatthe map X(f ) induces a surjection π2 X(A) → π2 X(B). Choose an equivalence B ' |B• | for some simplicial
object B• of CAlgcn
k , where each Bn is a polynomial algebra over k (possibly on an infinite set of generators).
Proposition 1.2.6 implies that X(B) is equivalent to the geometric realization of the simplicial space X(B• ),where each X(Bn ) is simply connected. It follows that the map π2 X(B0 ) → π2 X(B) is surjective. Since f isconnective, the map B0 → B factors through A, so that the map π2 X(A) → π2 X(B) is also surjective.
12

Corollary 1.2.11. Let X be a k-rational homotopy type. For every A ∈ CAlgcn
k , the canonical map X(A) →lim X(τ≤n A) is a homotopy equivalence.
←−
We now turn our attention to the compactness properties of k-rational homotopy types of finite type.
Lemma 1.2.12. Let X be a k-rational homotopy type of finite type. Then, for each n ≥ 0, the functor(X, K(V, n))
V 7→ MapFun(CAlgcn
k ,S)commutes with filtered colimits,
Proof. Since X is a coaffine stack (Remark 1.2.5), Proposition VIII.4.4.4 implies that X is given by thegeometric realization of a simplicial object X• of Fun(CAlgconn, S), where each Xn is the set-valued functorkcorepresented by some discrete k-algebra Rn . Since K(V, n) is n-truncated, the mapping space
MapFun(CAlg0k ,S) (X, K(V, n))is equivalent to the finite limit
MapFun(CAlgcn(Xm , K(V, n)).
k ,S)

lim
←−

[m]∈∆≤n+1

It will therefore suffice to show that each of the functors
V 7→ MapFun(CAlgcn(Xm , K(V, n)) ' K(Rm ⊗k V, n)k ,S)commutes with filtered colimits, which is clear.
Definition 1.2.13. Let X be a k-rational homotopy type and let n ≥ 0 be an integer. We will say that X
is n-truncated if, for each R ∈ CAlg0k , the space X(R) is n-truncated. We let RType(k)≤n denote the fullsubcategory of RType(k) spanned by the the n-truncated k-algebraic homotopy types.
Remark 1.2.14. The inclusion functor RType(k)≤n ,→ RType(k) admits a left adjoint X 7→ τ≤n X, whereτ≤n X denotes a left Kan extension of the functor CAlg0k → S given by R 7→ τ≤n X(R).
Remark 1.2.15. Let X be an n-truncated k-rational homotopy type and let V = πn X. To each R ∈ CAlg0k
we can associate a fiber sequence
X(R) → τ≤n−1 X(R) → K(V ⊗k R, n + 1),
depending functorially on R. Taking left Kan extensions, we obtain a commutative diagram of k-rational
homotopy types σ/ τ≤n−1 XX∗/ K(V, n + 1).

Let C ⊆ CAlgcn
k denote the full subcategory spanned by those connective E∞ -algebras A for which thediagram/ (τ≤n−1 X)(A)
X(A)

∗


/ K(V, n + 1)(A).

is a pullback square. Using Lemma A.5.3.6.17, we see that C is closed under sifted colimits. Since C contains
CAlg0k , it follows that C = CAlgcn
k . That is, we have a fiber sequence of functors
X → τ≤n−1 X → K(V, n + 1).
13

Lemma 1.2.16. Let k be a field of characteristic zero. Then the ∞-category RType(k) is closed underfiltered colimits in Fun(CAlgcn
k , S). Moreover, for each n ≥ 2, the functor X 7→ πn X commutes with filteredcolimits. In particular, of each of the subcategories RType(k)≤n is also closed under filtered colimits.
Proof. Let {Xα } be a filtered diagram of k-rational homotopy types, and let X denote its colimit in0
Fun(CAlgcn
k , S). It is clear that X satisfies conditions (a) and (b) of Definition 1.2.1. Moreover, for R ∈ CAlgk ,we have a canonical isomorphism
πn X(R) ' lim πn (Xα (R)) ' (πn Xα ) ⊗k R ' V ⊗k R,−→
where V denotes the direct limit lim πn Xα (see Remark 1.2.3). It follows that X is a k-rational homotopy−→
type and that the canonical map
lim πn Xα = V → πn X−→
is an isomorphism of k-vector spaces.
Lemma 1.2.17. Let k be a field of characteristic zero, let n ≥ 0, and let X be an n-truncated k-algebraic
homotopy type. If X is of finite type, then X is a compact object of RType(k)≤n .
Proof. Let θ : RType(k) → S denote the functor corepresented by X. We will prove that for each integer
m, the restriction θ| RType(k)≤m commutes with filtered colimits. We proceed by induction on m, the casem ≤ 1 being trivial. To carry out the inductive step, suppose we are given a filtered diagram {Yα } in
RType(k)≤m having colimit Y , and set Vα = πn Yα . We then have a fiber sequence of functors
Yα → τ≤n−1 Yα → K(Vα , n + 1)
depending functorially on α. Set V = lim Vα so that V ' πn Y . We then have another fiber sequence−→
Y → τ≤n−1 Y → K(V, n + 1). Since θ commutes with limits, we have a map of fiber sequenceslim θ(Yα )−→/ lim θ(τ≤n−1 Yα )
−→

α


θ(Y )



/ lim θ(K(Vα , n + 1))−→γβ/ θ(τ≤n−1 Y )/ θ(K(V, n + 1)).

The map β is a homotopy equivalence by the inductive hypothesis, and γ is a homotopy equivalence byLemma 1.2.12. It follows that α is a homotopy equivalence, as desired.
ft
For each integer n ≥ 0, we let RType(k)ft≤n denote the intersection RType(k) ∩ RType(k)≤n : that is,
the ∞-category of k-rational homotopy types whcih are n-truncated and of finite type.

Proposition 1.2.18. Let k be a field of characteristic zero. For each integer n, the inclusion RType(k)ft≤n ,→RType(k)≤n induces an equivalence of ∞-categories θ : Ind(RType(k)ft)'
RType(k).
≤n≤nRemark 1.2.19. Since RType(k)ft≤n is closed under retracts in RType(k)≤n , it follows from Proposition
1.2.18 and Lemma 1.2.17 that an object X ∈ RType(k)≤n is compact if and only if it is finite type.
Remark 1.2.20. One can show that the ∞-category RType(k) itself is compactly generated, but we will
not need this.
Proof. Using Lemma 1.2.17 and Proposition T.5.1.3.1, we deduce that θ is fully faithful. Let C ⊆ RType(k)≤ndenote the essential image of θ. We wish to show that C contains every n-truncated k-algebraic homotopy
type X. The proof proceeds by induction on n, the case n ≤ 1 being trivial. Set V = πn X so that we havea fiber sequence of functors
X → τ≤n−1 X → K(V, n + 1).

14

Using the inductive hypothesis, we can write τ≤n−1 X as a filtered colimit lim Yα , where each Yα is (n − 1)−→
truncated and of finite type. For each α, let Xα denote the fiber of the composite map
Yα → τ≤n−1 X → K(V, n + 1).
Then X ' lim Xα . Since C is closed under filtered colimits, it will suffice to prove that each Xα belongs to−→
C. Using Lemma 1.2.12, we see that there exists a finite-dimensional subspace W ⊆ V such that the map
Yα → K(V, n + 1) factors through K(W, n + 1). Write V ' W ⊕ W 0 , so that we have an equivalence offunctors
Xα ' Zα × K(W 0 , n)where Zα denotes the fiber of the map Yα → K(W, n + 1). Then Zα is of finite type. Since C is closed
under products, we are reduced to proving that K(W 0 , n) belongs to C. This is clear, since K(W 0 , n) 'lim K(Wβ0 , n), where Wβ0 ranges over all finite-dimensional subspaces of W 0 .
−→Proposition 1.2.21. Let k be a field of characteristic zero. Then the canonical map from RType(k) to thehomotopy limit of the towerτ≤2· · · → RType(k)≤3 → RType(k)≤2 → RType(k)≤1 ' ∗is an equivalence.
Proof. Let RType0 (k) denote the full subcategory of Fun(CAlg0k , S) spanned by those functors which satisfy
conditions (b) and (c) of Definition 1.2.1, and for each integer n ≥ 0 define RType0 (k)≤n similarly. Using
Proposition T.4.3.2.15, we see that the restriction maps
RType(k) → RType0 (k)RType(k)≤n → RType0 (k)≤nare equivalences. It will therefore suffice to show that RType0 (k) is equivalent to the homotopy inverse limitof the tower of ∞-categoriesτ≤2· · · → RType0 (k)≤3 → RType0 (k)≤2 → RType0 (k)≤1 ' ∗.
This follows from the observation that Fun(CAlg0k , S) is given by the homotopy inverse limit of the tower{Fun(CAlg0k , τ≤n S).

1.3

Rational Homotopy Theory and E∞ -Algebras

Let Y be a simply connected rational space. We will say that X is of finite type if the homotopy group πn X
is a finite-dimensional vector space over Q for each n ≥ 2. Sullivan has shown that the homotopy theory ofrational spaces of finite type can be described algebraically, using commutative differential graded algebras
over Q. His main result can be formulated as follows:
Theorem 1.3.1 (Sullivan). Let Q be the field of rational numbers. Let Srat
ft denote the full subcategory ofS spanned by those spaces X which are simply connected and such that each homotopy group πi X is a finite
dimensional vector space over Q. Then the construction X 7→ C ∗ (X; Q) determines a fully faithful embedding Srat
ft → CAlgQ , whose essential image is the collection of those Q-algebras A satisfying the following
conditions:(a) The Q-vector spaces πi A are finite dimensional for all i.

if i > 00(b) We have πi A ' Q if i = 00
if i = −1.

15

In this section, we will review the proof of Theorem 1.3.1, and deduce from it an analogous descriptionof the entire ∞-category Srat of rational spaces (Theorem 1.3.6).
Remark 1.3.2. Let k be a field. The construction X 7→ C ∗ (X; k) determines a functor from S to CAlgopk .
This functor admits a right adjoint, given by the formula A 7→ MapCAlgk (A, k). In particular, for every
space X we have a canonical unit map X → MapCAlgk (C ∗ (X; k), k), which carries a vertex x ∈ X to theevaluation map C ∗ (X; k) → C ∗ ({x}; k) ' k.
Proposition 1.3.3. Let X be a simply connected rational space of finite type. Then the unit map
uX : X → MapCAlgQ (C ∗ (X; Q), Q)
is a homotopy equivalence.
Proof. The map X → τ≤n X induces an isomorphism on homotopy groups in degrees ≤ n, hence an isomorphism Hi (X; Q) → Hi (τ≤n X; Q) for i ≤ n. It follows that C ∗ (X; Q) ' lim C ∗ (τ≤n X; Q), so that uX can be−→
identified with the limit of the maps {uτ≤n X }n≥0 . It will therefore suffice to prove that each of the mapsuτ≤n X is a homotopy equivalence. We proceed by induction on n, the case n = 1 being obvious. Let n > 1
and set V = πn X, so that V is a finite-dimensional vector space over Q. We have a pullback diagram ofspaces/∗τ≤n X
/ K(V, n + 1)τ≤n−1 Xso that Corollary 1.1.10 yields a pushout diagramC ∗ (K(V, n + 1), Q)/QC ∗ (τ≤n−1 X; Q)/ C ∗ (τ≤n X; Q).

Consequently, to show that uτ≤n X is a homotopy equivalence, it suffices to show that uτ≤n−1 X and uK(V,n+1)
are homotopy equivalences. In the first case, this follows from the inductive hypothesis. In the second, it
follows from Corollary 1.1.16.
Corollary 1.3.4. Let X be a simply connected rational space of finite type, and let Y be an arbitrary space.
Then the functor C ∗ (•; Q) induces a homotopy equivalence
MapS (Y, X) → MapCAlgQ (C ∗ (X; Q), C ∗ (Y ; Q)).
rat
Proof of Theorem 1.3.1. Corollary 1.3.4 implies that C ∗ | Srat
ft is fully faithful. We next show that if X ∈ Sft ,then C ∗ (X; Q) satisfies conditions (a) and (b). Since X is simply connected, condition (b) is obvious. We
wish to prove that H−i (X; Q) is finite dimensional for every integer i. Replacing X by τ≤n X for n ≥ i, we
can assume that X is n-truncated, and we proceed by induction on n. If n = 1, then X ' ∗ and the result
is obvious. Otherwise, set V = πn X so that we have a pullback diagram

X

/∗


τ≤n−1 X
/ K(V, n + 1)

16

and therefore (by Corollaries 1.1.10 and 1.1.16) a pushout diagram of Q-algebras
Sym∗ (V [−n − 1])/QC ∗ (τ≤n−1 X; Q)/ C ∗ (X; Q).

The desired result now follows from the inductive hypothesis and Lemma X.4.1.16.
We now show that if A ∈ CAlgQ satisfies conditions (a) and (b), then A belongs to the essential of∗ rat
C | Sft . Let X = cSpec A be the coaffine stack determined by A. Proposition VIII.4.4.18 implies that X
has finite type, so that each homotopy group πi X is representable by a commutative unipotent algebraic
group over Q and therefore πi X(Q) is a finite dimensional vector space over Q for each i. Proposition
VIII.4.4.12 implies that X(Q) is simply connected, so that X(Q) ∈ S0 . Let A0 = C ∗ (X(Q); Q), and let
X 0 be the coaffine stack determined by A0 . The canonical map A → A0 induces a map of coaffine stacks
v : X 0 → X. Since C ∗ | S0 is fully faithful, the induced map X 0 (Q) → X(Q) is a homotopy equivalence.
For each i ≥ 1, we have a map πi (v) : πi X 0 → πi X of unipotent algebraic groups over Q which induces anisomorphism on Q-points. It follows that πi (v) is an isomorphism of algebraic groups and therefore that v
is an equivalence of coaffine stacks, so that A ' A0 = C ∗ (X(Q); Q) lies in the essential image of C ∗ | Srat
ft as
desired.
In the language of coaffine stacks, Theorem 1.3.1 has the following interpretation:
Theorem 1.3.5. Let C ⊆ Fun(CAlgcn
Q , S) denote the full subcategory spanned by the simply connected coaffine
stacks of finite type. The evaluation map X 7→ X(Q) determines an equivalence of ∞-categories C → Srat
ft .
Proof. It is clear that if X ∈ C, the X(Q) ∈ Srat
ft . Let D denote the full subcategory of CAlgQ spannedby those E∞ -algebras A ∈ CAlgQ satisfying conditions (a) and (b) of Theorem 1.3.1. Using Propositions
VIII.4.4.12 and VIII.4.4.18, we see that the construction A 7→ cSpec A induces an equivalence of ∞-categories
Dop → C. It will therefore suffice to prove that the composite functor Dop → C → C0 is an equivalence of∞-categories. Unwinding the definitions, we see that this composite functor is given by
A 7→ MapCAlgQ (A, Q).
We now observe that this construction is right adjoint to the equivalence of ∞-categories appearing in
Theorem 1.3.1.
We now use the theory of k-rational homotopy types developed in §1.2 to remove the finiteness restrictions
from Theorem 1.3.5.
Theorem 1.3.6. Let Q denote the field of rational numbers. Then the construction X 7→ X(Q) induces anequivalence of ∞-categories e : RType(Q) → Srat .
Proof. We first show that the functor X 7→ X(Q) is fully faithful. Fix Q-rational homotopy types X andY ; we wish to show that the canonical map
uX,Y : MapRType(Q) (X, Y ) → MapS (X(Q), Y (Q))
is a homotopy equivalence. Note that uX,Y is given by the homotopy limit of the tower of maps {uX,τ≤n Y }n≥0 .
It will therefore suffice to show that each of the maps uX,τ≤n Y is a homotopy equivalence. We proceed byinduction on n, the case n ≤ 1 being trivial. To carry out the inductive step, write πn Y (R) ' V ⊗Q R forsome Q-vector space V . We then have a fiber sequence of functors
τ≤n Y → τ≤n−1 Y → K(V, n + 1).

17

The inductive hypothesis implies that uX,τ≤n−1 Y is a homotopy equivalence. We are therefore reduced
to proving that uX,K(V,n+1) is a homotopy equivalence. In other words, we may assume without loss ofgenerality that Y has the form K(V, m) for some Q-vector space V and some m ≥ 3.
Replacing X by τ≤m X, we may assume without loss of generality that X is m-truncated. Applying
Proposition 1.2.18, we can write X as a filtered colimit lim Xα , where each Xα is an m-truncated Q-rational−→
homotopy type of finite type. Then uX,Y is equivalent to the limit of the maps uXα ,Y , so it will suffice toshow that each uXα ,Y is a homotopy equivalence. Replacing X by Xα , we are reduced to the case where X
has finite type.
Write V as a direct limit lim Vβ of finite-dimensional subspaces of V . We have a commutative diagram−→lim MapRType(Q) (X, K(Vβ , m))−→/ MapRType(Q) (X, K(V, m))lim MapS (X(Q), K(Vβ , m))−→/ MapS (X(Q), K(V, m).

uX,K(V,m)Using Lemma 1.2.12, we see that the upper horizontal map is a homotopy equivalence. We claim that thelower horizontal map is also a homotopy equivalence. Using Whitehead’s theorem, we are reduced to provingthat the canonical map
γ : lim Hi (X(Q); Vβ ) → Hi (X(Q); V )−→βis an isomorphism for each i ≤ m. Using the universal coefficient formula, we can identify γ with thecanonical map
lim HomQ (Hi (X(Q); Q), Vβ ) → HomQ (Hi (X(Q); Q), V ).
−→βThis map is an isomorphism since the group Hi (X(Q); Q) is a finite-dimensional vector space (by Theorem
1.3.1). Consequently, we can identify uX,K(V,m) with the filtered colimit of the maps uX,K(Vβ ,m) . Replacing
V by Vβ , we can reduce to the case where Y = K(V, m) for some finite-dimensional vector space V . In thiscase, both X and Y can be extended to coaffine stacks (Remark 1.2.5), so that Theorem 1.3.5 guarantees
that uX,Y is a homotopy equivalence.
To complete the proof, it will suffice to show that if Z is a simply connected space whose homotopy
groups are rational vector spaces, then Z has the form X(Q) for some Q-rational homotopy type X. We
will construct X as the inverse limit of a tower· · · → X4 → X3 → X2 → X1 ' ∗,where each Xn satisfies Xn (Q) ' τ≤n Z. The construction proceeds by recursion. Assume that Xn−1 has
been constructed. The existence of a homotopy equivalence Xn−1 (Q) ' τ≤n−1 Z guarantees that Xn−1γis (n − 1)-truncated. Let V = πn Z, so that we have a fiber sequence τ≤n Z → τ≤n−1 Z → K(V, n + 1).
Using the first part of the proof, we see that γ is obtained from a map of Q-rational homotopy typesγ
e : Xn−1 → K(V, n + 1). We now define Xn to be fiber of the map γe. Using the fact that Xn−1 is(n − 1)-truncated, we immediately deduce that Xn is a Q-rational homotopy type. By construction, we have
Xn (Q) ' τ≤n Z.

1.4

Differential Graded Lie Algebras

Let k be a field of characteristic zero, which we regard as fixed throughout this section. In this section, we
will study connectivity and finiteness properties of differential graded Lie algebras over k.
Notation 1.4.1. Let g∗ be a differential graded Lie algebra over k. We let H∗ (g∗ ) denote the homology of
the underlying chain complex of g∗ (so that H∗ (g∗ ) has the structure of a graded Lie algebra). We let C ∗ (g∗ )denote the cohomological Chevalley-Eilenberg complex of g∗ (see Construction X.2.2.13), which we regard18

as an augmented commutative differential graded algebra over k. Our notation is potentially confusing: takenote that the cohomology groups of C ∗ (g∗ ) are not given by the k-linear duals of the vector spaces Hn (g∗ ).
We will generally not distinguish between the commutative differential graded algebra C ∗ (g∗ ) and its
aug op∗image in the ∞-category CAlgaug
admits a rightk . Recall that the construction C : Liek → (CAlgk )aug opadjoint D : (CAlgk ) → Liek , which we refer to as Koszul duality.
Definition 1.4.2. Let g∗ be a differential graded Lie algebra and let n be an integer. We will say that g∗is:• n-connective if the underlying chain complex of g∗ is n-connective: that is, if the homology groups
Hm (g∗ ) vanish for m < n.
• connective if it is 0-connective; that is, if Hm (g∗ ) ' 0 for m < 0.
• connected if it is 1-connective: that is, if Hm (g∗ ) ' 0 for m ≤ 0.
• n-truncated if the underlying chain complex of g∗ is n-truncated: that is, if the homology groups
Hm (g∗ ) vanish for m > n.
• truncated if it is n-truncated for some integer n: that is, if Hm (g∗ ) ' 0 for m  0.
• of finite type if the homology groups Hm (g∗ ) are finite-dimensional for each integer m.
The main result of this section can be formulated as follows:
Theorem 1.4.3. Let g∗ be differential graded Lie algebra which is connected, truncated, and of finite type.
Then the unit map g∗ → DC ∗ (g∗ ) is an equivalence.
Remark 1.4.4. The conclusion of Theorem 1.4.3 remains valid if we assume only that g∗ is connected andof finite type: the condition of truncatedness is superfluous. However, we will not need this stronger result.
The proof of Theorem 1.4.3 will require a few remarks about Postnikov towers of a differential graded
Lie algebra.
Construction 1.4.5. Let g∗ be a differential graded Lie algebra, given by a chain complex
d

d

d

d−1

· · · → g2 →2 g1 →1 g0 →0 g−1 → g−2 → · · · .
For each integer n ≥ 0, we let τ≥n g∗ denote the subcomplex of g∗ , given by
dn+2

dn+1

· · · → gn+2 → gn+1 → ker(dn ) → 0 → · · · .
We note that τ≥n g∗ is stable under the Lie bracket (using the assumption that n ≥ 0), and therefore inherits
the structure of a differential graded Lie algebra. By construction we have(
Hm (g∗ ) if m ≥ n
Hm (τ≥n g∗ ) '0
if m < n.
From this we deduce the following:(a) Every quasi-isomorphism of differential graded Lie algebras h∗ → g∗ induces a quasi-isomorphismτ≥n h∗ → τ≥n g∗ . Consequently, τ≥n determines a functor from the ∞-category Liek to itself, which wewill also denote by τ≥n .
(b) If g∗ is n-connective, then the inclusion τ≥n g∗ ,→ g∗ is a quasi-isomorphism. In particular, every
n-connective differential graded Lie algebra g∗ is quasi-isomorphic to a differential graded Lie algebra
which is concentrated in homological degrees ≥ n.
19

Remark 1.4.6. Let n ≥ 0 be an integer. We let Lie≥ndenote the full subcategory of Liek consisting the nk
connective differential graded Lie algebras over k. The truncation functor τ≥n defines a map Liek → Lie≥nk ,equipped with a natural transformation α : τ≥n → id. For every differential graded Lie algebra g∗ , thecanonical mapsατ≥n g∗ , τ≥n (αg∗ ) : τ≥n τ≥n g∗ → τ≥n g∗
are equivalences (in fact, they are given by isomorphisms in the ordinary category Liedgk ). It follows that τ≥ndetermines a colocalization functor on the ∞-category Liek (see Proposition T.5.2.7.4) with having essential≥nimage Lie≥nk . That is, we can identify τ≥n with a right adjoint to the inclusion Liek ,→ Liek .
≥nFrom this we conclude that Liek is closed under small colimits in Liek . Since the functor τ≥n commutes with filtered colimits, its essential image Lie≥nis an accessible subcategory of Liek , and therefore akpresentable ∞-category.
Remark 1.4.7. Let Free : Modk → Liek denote a left adjoint to the forgetful functor θ : Liek → Modk .
dgNote that Free can be identified with the left derived functor of the left Quillen functor F : Vectdg
k → Liek ,which assigns to each differential graded vector space V∗ the free differential graded Lie algebra generatedby V∗ . Note that if V∗ is concentrated in degrees ≥ n for some n ≥ 0, then so is F (V∗ ). It follows that forn ≥ 0, the adjunction
Free /
Lie
Mod o
k

θ

k

restricts to an adjunctiono(Modk )≥n , Lie≥nk.

/

The forgetful functor θ is conservative and preserves small sifted colimits (Proposition X.2.1.16). Since
the ∞-category (Modk )≥n has compact projective generators (given by V [n], where V is a finite-dimensional
vector space over k), the ∞-category Lie≥n
has a set of compact projective generators given by Free(V [n]),kwhere V ranges over isomorphism classes of finite-dimensional vector spaces over k (Proposition A.7.1.4.12).
Remark 1.4.8. Let m ≥ n ≥ 0 be integers, and let g∗ be an n-connective differential graded Lie algebra
over k. The following conditions are equivalent:(a) When viewed as an object of the ∞-category Lie≥nk , g∗ is (m − n)-truncated. That is, for every nconnective differential graded Lie algebra h∗ , the mapping space MapLiek (h∗ , g∗ ) is (m − n)-truncated.
(b) For every finite-dimensional vector space V , the mapping space MapLie k (Free(V [n]), g∗ ) is (m − n)truncated.
(c) The differential graded Lie algebra g∗ is m-truncated in the sense of Definition 1.4.2: that is, Hp (g∗ ) ' 0
for p > m.
The equivalence of (a) and (b) follows from the fact that Lie≥nk is generated by the objects Free(V [n]) undersmall colimits, and the equivalence of (b) and (c) follows from the existence of isomorphisms
Homk (V, Hp (g∗ )) ' πp−n MapLiek (Free(V [n]), g∗ ).
≤m
Let (Lie≥ndenote the full subcategory of Lie≥nk )k spanned by those objects which satisfy the equivalent≤mconditions of Remark 1.4.8. Then the inclusion functor (Lie≥n,→ (Lie≥nk )k ) admits a left adjoint, which
we will denote by τ≤m . Using Remark T.5.5.8.26, we see that for each g∗ ∈ Lie≥nk , the truncation τ≤m g∗ ischaracterized up to equivalence by the requirement that there exist a map g∗ → τ≤m g∗ which induces anisomorphism on homology in degrees ≤ m, and where τ≤m g∗ has trivial homology in degrees ≥ m. From
this description, we see that τ≤m g∗ does not depend on the integer n. It will be convenient to have a more
explicit construction of τ≤m g∗ :20

Construction 1.4.9. Let g∗ be a differential graded Lie algebra and suppose that gm ' 0 for m < 0. Let
us denote the underlying chain complex of g∗ bydd· · · → g2 →2 g1 →1 g0 →0.
For each integer n ≥ 0, we let Tm g∗ denote the quotient chain complex given by· · · → 0 → coker(dn+1 ) → gn−1 → · · · → g0 → 0.
Then Tm g∗ inherits the structure of a differential graded Lie algebra. We have an evident map g∗ → Tm g∗
whcih identifies Tm g∗ with the truncation τ≤m g∗ defined above. This map is a quasi-isomorphism if andonly if g∗ is m-truncated.
Remark 1.4.10. Let n ≥ 0 be an integer. Since Lie≥nis projectively generated, Postnikov towers in Lie≥nkk
are convergent (see Remark T.5.5.8.26). It follows that Lie≥nis equivalent to the homotopy inverse limit ofkthe tower of ∞-categories
≤n+2
≤n+1
≤n
· · · → (Lie≥n→ (Lie≥n→ (Lie≥n
.
k )
k )
k )

Notation 1.4.11. Let V be a vector space over k. For each integer n, we let V [n] denote the chain complex
consisting of the single vector space V , concentrated in homological degree n. We will regard V [n] as adifferential graded Lie algebra over k, where the differential and Lie bracket on V [n] are zero.
Proposition 1.4.12. Let g∗ be a differential graded Lie algebra over k. Assume that g∗ is connected andn-truncated for some integer n ≥ 1, and let V = Hn (g∗ ). Then there exists a fiber sequenceg∗ → h∗ → V [n + 1]
in the ∞-category Liek , where h∗ is connected and (n − 1)-truncated.
Proof. Using Constructions 1.4.5 and 1.4.9, we may reduce to the case where gm = 0 unless 1 ≤ m ≤ n.
Then we can regard V as a subspace of gn ; let i : V ,→ gn denote the inclusion map. Let h∗ denote the chain
complexi· · · → 0 → V → gn → gn−1 → · · · .
The Lie algebra structure on g∗ extends uniquely to a Lie algebra structure on h∗ (satisfying [v, x] = 0 for allv ∈ V ). As a differential graded Lie algebra, h∗ is equipped with an evident map h∗ → V [n + 1]. This map isfibration (with respect to the model structure of Proposition X.2.1.10) with fiber g∗ . Since the ∞-category
Liedg
k is right proper, we obtain a homotopy fiber sequenceg∗ → h∗ → V [n + 1]
in Liedg
k , hence a fiber sequence in the underlying ∞-category Liek . Since Hm (h∗ ) '(Hm (g∗ )0

if m 6= n,
if m = n,

we conclude that h∗ is connected and (n − 1)-truncated.
Remark 1.4.13. In the proof of Proposition 1.4.12, suppose that the vector space V is finite-dimensional,
so that C ∗ (V [n + 1]) can be identified with the symmetric algebra Sym∗ (V ∨ [−n − 2]) (equipped with thetrivial differential). The diagram
Sym∗ (V ∨ [−n − 2])/ C ∗ (h∗ )k/ C ∗ (g∗ )21

is a pushout diagram in the ordinary category of commutative differential graded algebras over k. Since
C ∗ (h∗ ) is cofibrant when viewed as a differential graded module over Sym∗ (V ∨ )[−n − 2], this diagram is alsoa homotopy pushout square. It follows that we have a cofiber sequenceC ∗ (V [n + 1]) → C ∗ (h∗ ) → C ∗ (g∗ )in the ∞-category CAlgaug
k .
Corollary 1.4.14. Let g∗ be a differential graded Lie algebra over k. Assume that there exists an integern > 0 such that Hm (g)∗ ' 0 for m 6= n. Then g∗ is equivalent (in the ∞-category Liek ) to V [n], where
V = Hn (g∗ ).
Proof. Since n > 0, we see that g∗ is connected and n-truncated. Applying Proposition 1.4.12 we obtain afiber sequenceg∗ → 0 → V [n + 1],
from which we immediately deduce that g∗ is equivalent to V [n].
Proof of Theorem 1.4.3. Assume that g∗ is connected, n-truncated, and of finite type. We wish to provethat the unit map ug∗ : g∗ → DC ∗ (g∗ ) is an equivalence of differential graded Lie algebras over k. The
proof proceeds by induction on n, the case n ≤ 0 being trivial. To carry out the inductive step, we apply
Proposition 1.4.12 to choose a fiber sequenceg∗ → h∗ → V [n + 1]
where h∗ is (n − 1)-truncated. Since g∗ has finite type, the vector space V is finite-dimensional and h∗ isalso finite type. Applying Remark 1.4.13, we obtain a cofiber sequence C ∗ (V [n + 1]) → C ∗ (h∗ ) → C ∗ (g∗ ).
It follows that we have a commutative diagram of fiber sequences
/ h∗

g∗


ug∗

DC ∗ (g∗ )/ V [n + 1]

uh∗

/ DC ∗ (h∗ )uV [n+1]

/ DC ∗ (V [n + 1]).

Since the map uh∗ is an equivalence by the inductive hypothesis, we are reduced to proving that uV [n+1] isan equivalence.
∨
Let A = Sym∗ (V ∨ [−n−2]) ∈ CAlgaug
k . Then the cotangent complex LA/k is equivalent to V [−n−2]⊗k A.
Applying Proposition X.2.3.9, we see that the underlying k-module of D(A) is equivalent to V [n + 1].
Applying Corollary 1.4.14 we obtain an equivalence of differential graded Lie algebras D(A) ' V [n + 1].
∗
Note that1]) is a commutative differential graded algebra with trivial differential, given by the
QC (V [n +product n≥0 Sym∗ (V ∨ [−n − 2]). Since n 6= 2, the canonical map v : A → C ∗ D(A) ' C ∗ (V [n + 1]) is anequivalence in CAlgaug
k . Since the composition
V [n + 1]

uV [n+1]→

D(v)DC ∗ (V [n + 1]) → A

is homotopic to the the identity map, we conclude that uV [n+1] is also an equivalence, as desired.
We will also need the following characterization of compact objects in the setting of differential graded
Lie algebras:≤m
Proposition 1.4.15. Let m ≥ n > 0. Then the ∞-category (Lie≥nis compactly generated. Moreover,
k )
≥n ≤m
an object g∗ ∈ (Liek )is compact if and only if it is of finite type.

22

≤m
Proof. Remark 1.4.7 implies that Lie≥nis compactly generated. Since the inclusion (Lie≥n,→ Lie≥n
k
k )
k
≥n ≤m
commutes with filtered colimits, the truncation functor τ≤m carries compact objects of of (Liek ) . Since≤mevery object of Lie≥ncan be written as a filtered colimit of compact objects, we deduce that (Lie≥niskk )compactly generated.
≤m
We now show that every object of (Lie≥nwhich has finite type is compact. This can be formulatedk )as follows:≤m(∗) Let g∗ ∈ Lie≥nbe of finite type. Then τ≤m g∗ is a compact object of (Lie≥n.
kk )To prove (∗), we let C denote the full subcategory of (Modk )≥n spanned by those k-modules V whose
homotopy groups are finite-dimensional as k-vector spaces, and let D denote the full subcategory of Lie≥nkspanned by those objects which are of finite type. Note that C is closed under the formation of geometricrealizations in (Modk )≥n . Since the forgetful functor Lie≥n→ (Modk )≥n preserves geometric realizations
kof simplicial objects (Proposition X.2.1.16), we conclude that D is closed under geometric realizations in≥nLie≥ndenote the left adjoint to the forgetful functor (see Remark 1.4.7).
k . Let Free : (Modk )≥n → Liek
We claim that Free carries C into D. To prove this, let V ∈ C; we wish to show that Free(V ) has finite
type. Using Remark X.2.1.8, we are reduced to proving that the universal enveloping algebra U (Free(V ))has finiteLtype. The conclusion now follows from the observation that U (Free(V )) is equivalent to the tensor
algebra p≥0 V ⊗p . The forgetful functor and its left adjoint Free therefore restrict to give a pair of adjoint
functors C oFG/ D . Since is closed under geometric realizations in Lie≥n . We have an evident forgetfulkfunctor G : D → C, which is conservative and preserves geometric realizations. Let g∗ be as in (∗). Using
Proposition A.6.2.2.11, we can write g∗ as the geometric realization of a simplical object X• of Lie≥nk , whereeach Xq has the form F (V ) for some V ∈ C. Then τ≤m g∗ is given by the geometric realization of τ≤m X• in≤m≤mthe ∞-category (Lie≥n. Since (Lie≥nis equivalent to an (m − n + 1)-category (Remark 1.4.8), wek )k )τ≤m g∗ is equivalent to the finite colimit lim[q]∈∆opτ≤m Xq (see the proof of Proposition A.1.3.2.9). It−→≤m−n+1≤mwill therefore suffice to show that each τ≤m Xq is a compact object (Lie≥n. We may therefore replacek )g∗ by Xq and thereby reduce to the case where g∗ = F (V ) for some V ∈ C. Replacing V by τ≤m V , we canreduce to the case where V is perfect as a k-module. Then V is a compact object of (Modk )≥n . Since the≥nforgetful functor Lie≥nk → (Modk )≥n commutes with filtered colimits, F (V ) is a compact object of Liek . It≥n ≤mfollows that τ≤m F (V ) is a compact object of (Liek ) , as desired. This completes the proof of (∗).
≤m
We now complete the proof by showing that every compact object g∗ of (Lie≥n
has finite type.
k )We begin by constructing a transfinite sequence of differential graded Lie algebras g(α)∗ equipped with acompatible family of maps ρα : g(α)∗ → g∗ . We set g(0)∗ = 0, and g(λ)∗ = limβ<λ g(β)∗ when λ is a limit−→
ordinal. Suppose that g(α)∗ has been defined and that the map ρα is not a quasi-isomorphism. Let q be
the smallest integer such that the map u : Hq (g(α)∗ ) → Hq (g∗ ) induced by ρα is not an isomorphism. If u
fails to be surjective, then we choose a cycle y ∈ gq whose homology class does not lie in the image of u.
We then define g(α + 1)∗ to be the differential graded Lie algebra obtained from g(α)∗ by freely adjoining
a generator Yα in degree q satisfying dY = 0, and define ρα+1 to be the unique extension of ρα satisfying
ρα+1 (Y ) = y. If u is surjective but fails to be injective, we can choose a cycle z ∈ g(α)q and an elementy ∈ gq+1 satisfying dy = ρα (z). We then define g(α + 1)∗ to be the differential graded Lie algebra obtained
from g(α)∗ by freely adjoining an element Y in degree q + 1 satisfying dYα = z, and ρα+1 to be the uniqueextension of ρα satisfying ρα+1 (Y ) = y.
If β is a sufficiently large ordinal, then ρβ is a quasi-isomorphism. Note that, as a graded Lie algebra,g(β)∗ is freely generated by the elements {Yα }α<β . Let us say that a subset S ⊆ {α : α < β} is saturated
if, for each γ ∈ S, the element dYγ belongs to the Lie subalgebra of g(β)∗ generated by the elements{Yα : α ∈ S}. In this case, we let g(S)∗ denote the differential graded Lie subalgebra of g(β)∗ generated by
the elements {Yα : α ∈ S}. Note that g(β)∗ is given by the filtered colimit
lim g(S)∗−→S

23

where S ranges over all finite saturated subsets of {α : α < β} (where this limit is formed in the ordinarydg
category Liedgk ). Since the collection of weak equivalences in Liek is closed under filtered colimits, we havean equivalence limS g(S)∗ → g∗ in the ∞-category Lie≥nk . Then g∗ τ≤m g∗ is equivalent to the filtered colimit−→≤mof the objects τ≤m g(S)∗ . Since g∗ is a compact object of (Lie≥n, we conclude that g∗ is equivalent to ak )retract of τ≤m g(S)∗ for some finite saturated subset S ⊆ {α : α < β}. It will therefore suffice to show thatg(S)∗ is of finite type. This is clear, since each of the vector spaces g(S)q is finite dimensional over k.

1.5Comparison with Formal Moduli Problems

Let k be a field of characteristic zero and let RType(k) be the ∞-category of k-rational homotopy types. We
let RType(k)∗ denote the ∞-category of pointed objects of RType(k). In other words, RType(k)∗ is the fullsubcategory of Fun(CAlgcn , S∗ ) spanned by those functors which satisfy (a), (b), and (c) of Definition 1.2.1.
We will refer to the objects of RType(k)∗ as pointed k-rational homotopy types.
augspanned by the small E∞ -algebras
Construction 1.5.1. Let CAlgsm
k denote the full subcategory of CAlgk
cnover k (see Proposition X.1.1.11). If X : CAlg → S∗ is a pointed k-rational homotopy type, we let X ∨∨denote the functor CAlgsm
k → S given by the formula X (A) = fib(X(A) → X(k)). Here the fiber is taken∨over the chosen base point of X(k). We will refer to X as the formal completion of X.

Proposition 1.5.2. Let X be a pointed k-rational homotopy type. Then the formal completion X ∨ is aformal moduli problem over k. Moreover, the tangent complex TX ∨ is 2-connective, and we have canonicalisomorphisms πn TX ∨ ' πn X for n ≥ 2.
Proof. It is clear that the space X ∨ (k) is contractible. Suppose we are given a pullback diagram σ :
A0

/ B0


A


/B

φ

0
∨
in CAlgsm
k , where φ induces a surjection π0 B → π0 B. We wish to show that X (σ) is a pullback diagramof spaces: that is, that the upper horizontal morphism in the diagram τ :X ∨ (A0 )/ X ∨ (B 0 ) ×X ∨ (B) X ∨ (A)
X(A0 )

/ X(B 0 ) ×X(B) X(A)is a homotopy equivalence. Since τ is a pullback square, it suffices to show that the bottom horizontal
map is a homotopy equivalence, which follows from Corollary 1.2.9. This completes the proof that X ∨ is a
k-rational homotopy type.
Let us now study the tangent complex of X ∨ . For each m ≥ 0, let k ⊕ k[m] denote the trivial square-zeroextension of k by k[m]. We then have Ω∞−m TX ∨ ' X ∨ (k ⊕ k[m]) ' fib(X(k ⊕ k[m]) → X(k)). SinceX(k ⊕ k[m]) and X(k) are simply connected (Corollary 1.2.8), we conclude that Ω∞−m TX ∨ is connected.
It follows that πn TX ∨ ' 0 for n ≤ 0. Let k[]/(2 ) ' k ⊕ k[0] denote the ring of dual numbers, so that
πn TX ∨ ' πn X ∨ (k[]/(2 )) for n ≥ 0. The fiber sequenceX ∨ (k[]/(2 )) → X(k[]/(2 )) → X(k)gives a long exact sequence of homotopy groups(πn+1 X) ⊗k k[]/(2 ) → πn+1 (X) → πn X ∨ (k[]/(2 )) → (πn X) ⊗k k[]/(2 ) → πn X.
24For n ≥ 1, this sequence gives an isomorphism
πn TX ∨ ' πn X ⊗k (k[]/(2 )) ' πn X.

Let Modulik denote the ∞-category of formal moduli problems over k, and Moduli≥2
k the full subcategoryof Modulik spanned by those formal moduli problems Y for which TY is 2-connective. Proposition 1.5.2
implies that the formal completion construction X 7→ X ∨ determines a functor RType(k)∗ → Moduli≥2k .
Our goal in this section is to prove the following result:
Theorem 1.5.3. Let k be a field of characteristic zero. Then the formal completion functor X 7→ X ∨induces an equivalence of ∞-categories
θ : RType(k)∗ → Moduli≥2k .
Proof of Theorem 1.0.3. Combining Theorem X.2.0.2, Theorem 1.5.3, and Theorem 1.3.6, we obtain a diagram of categorical equivalences≥2
Srat ← RType(Q) → Moduli≥2
Q ← LieQ .

Remark 1.5.4. Let Y be a simply connected rational space, so that Y ' X(Q) for some Q-rational
homotopy type X (uniquely determined up to equivalence, by Theorem 1.3.6). Let X ∨ denote the formal
completion of X, so that X ∨ is the formal moduli problem associated to g∗ for some differential graded Liealgebra g∗ over Q. Using Theorem X.2.0.2 and Proposition 1.5.2, we see that for for each n ≥ 2 there arecanonical isomorphisms
πn Y ' πn X(Q) ' πn X ' πn TX ∨ ' Hn−1 (g∗ ).
That is, the homology groups of the differential graded Lie algebra g∗ can be identified with the homotopy
groups of Y . With more effort, one can show that this is an isomorphism of graded Lie algebras (where the
Lie bracket on π∗ Y is defined by means of the classical Whitehead product).
≥2
Proof of Theorem 1.5.3. For each m ≥ 2, let (Moduli≥2k )≤m denote the full subcategory of Modulik spannedby those formal moduli problems Y for which the tangent complex TY is m-truncated. It follows from
Proposition 1.5.2 that θ induces a functor θm : (RType(k)≤m )∗ → (Moduli≥2k )≤m . Using Theorems X.2.0.2,
Proposition 1.2.21, and Remark 1.4.10, we see that θ can be identified with the limit of the tower of functors{θm }m≥2 . It will therefore suffice to show that each θm is an equivalence of ∞-categories.
Let C denote the full subcategory of Modulik spanned by those formal moduli problems Y for which
the homotopy groups πn TY are finite-dimensional vector spaces over k, which vanish unless 2 ≤ n ≤ m.
Proposition 1.5.2 implies that θm restricts to a functor θ0 : (RType(k)ft≤m )∗ → C. Using Proposition 1.4.15 and≥2Theorem X.2.0.2, we see the inclusion C ,→ (Modulik )≤m induces an equivalence of ∞-categories Ind(C) 'ft(Moduli≥2
k )≤m . Proposition 1.2.18 implies that the inclusion (RType(k)≤m )∗ ,→ (RType≤m )∗ induces anft
equivalence of ∞-categories Ind((RType(K)≤m )∗ ) → (RType≤m )∗ . Since the functor θm commutes withfiltered colimits, we are reduced to proving that θ0 is an equivalence of ∞-categories.
Let D denote the full subcategory of Liek spanned by those differential graded Lie algebras which areconnected, m-truncated, and of finite type. For g∗ ∈ D, then C ∗ (g∗ ) is a 2-coconnective E∞ -algebra overk having finite type. We let T (g∗ ) denote the associated coaffine stack cSpec C ∗ (g∗ ): that is, T (g∗ ) denote∗∗the functor CAlgcn
k → S given by R 7→ MapCAlgk (C (g∗ ), R). Since C (g∗ ) has finite type, we can regard
T (g∗ ) as a k-rational homotopy type, which is equipped with a canonical base point determined by theaugmentation on C ∗ (g). We may therefore regard the construction g∗ → T (g∗ ) as a functor

T : D → RType(k)∗ .
25

We next claim that the diagram σ :/ LiekDTRType(k)∗Ψθ/ Modulik

commutes up to homotopy. To prove this, we note that the two maps from D to Modulik determined bythis diagram can be identified with functors
D × CAlgsm
k →S
given by(g∗ , R) 7→ MapLiek (D(R), g∗ )(g∗ , R) 7→ MapCAlgaug(C ∗ (g)∗ , R).
k

To prove that these functors are equivalent, it suffices to show that the canonical maps
MapCAlgaug(C ∗ (g∗ ), R) → MapCAlgaug(C ∗ (g∗ ), C ∗ D(R)) ' MapLiek (D(R), DC ∗ (g∗ )) ← MapLiek (D(R), g∗ )kk
are homotopy equivalences for g∗ ∈ D and R ∈ CAlgsm
k . This follows from the fact that the unit maps
R 7→ C ∗ D(R) and g∗ → DC ∗ (g∗ ) are equivalences (in the first case, this follows from Proposition X.1.3.5
and Theorem X.2.3.1; in the second case, it follows from Theorem 1.4.3).
Using the commutativity of the diagram σ together with Proposition 1.5.2, we see that for g∗ ∈ D,
we have canonical isomorphisms Hn (g∗ ) ' πn T (g∗ ). It follows that the functor T carries D into the fullsubcategory (RType(k)ft≤m )∗ ⊆ RType(k)∗ . The diagram

T

D(RType(k)ft
≤m )∗
8
0
θ

Ψ

/& C

commutes up to homtopy, where the horizontal map is an equivalence of ∞-categories (Theorem X.2.0.2).
We are therefore reduced to proving that T induces an equivalence of ∞-categories D → (RType(k)ft≤m )∗ .
We first show that the functor T is fully faithful. According to Corollary VIII.4.4.7, the functor A 7→
cSpec A is fully faithful when restricted to coconnective E∞ -algebras over k. It will therefore suffice to showthat the functor C ∗ : Liek → CAlgaug
is fully faithful when restricted to D. Equivalently, we wish to showkthat for every pair of objects g∗ , h∗ ∈ D, the canonical map
MapLiek (g∗ , h∗ ) → MapCAlgaug(C ∗ (g∗ ), C ∗ (g∗ )) ' MapLiek (g∗ , DC ∗ (h∗ ))kis an equivalence. This follows immediately from Theorem 1.4.3.
It remains to prove that T induces an essentially surjective functor D → (RType(k)ft≤m )∗ . Let X be apointed k-rational homotopy type which is m-truncated and of finite type. According to Remark 1.2.5, X isa coaffine stack, having the form cSpec A for some coconnective E∞ -algebra A over k. The base point of X
determines an augmentation on A. Let g∗ = D(A). For each R ∈ CAlgsm
k , we have a canonical homotopy
equivalence
MapCAlgaug(A, R) ' MapCAlgaug(A, C ∗ D(R)) ' MapLiek (D(R), D(A)),
kkso that the formal completion X ∨ of X is represented by the differential graded Lie algebra D(A). Since
X is m-truncated and of finite type, Proposition 1.5.2 implies that the homotopy groups πn TX ∨ are finitedimensional and vanish unless 2 ≤ n ≤ m, so that g∗ ∈ D. We will complete the proof by showing thatX ' T (g∗ ). Let Y = T (g∗ ) ' cSpec C ∗ (g∗ ). Then the unit map A → C ∗ (g∗ ) induces a map of coaffine stacks
β : Y → X; we will prove that this map is an equivalence. Since C ∗ (g∗ ) is of finite type, Y is a k-rational26

homotopy type. It will therefore suffice to show that β induces a homotopy equivalence Y (B) → X(B) for
each B ∈ CAlg0k . Since both of these spaces are simply connected, we are reduced to proving that the map
πn Y (B) → πn X(B) is an isomorphism for each n ≥ 2. Equivalently, we wish to show that β induces anisomorphism πn Y → πn X of vector spaces over k. Using Proposition 1.5.2, we see that this is equivalent
to the requirement that β induces an equivalence between the formal completions Y ∨ → X ∨ , which is animmediate consequence of our construction.

2

E∞ -Algebras in Positive Characteristicét
Let R be an E∞ -ring, and let Shvét
R ⊆ Fun(CAlgR , S) denote the ∞-category of étale sheaves on R. Thenétthe affine spectral Deligne-Mumford stack Spec R is given by (Shvét
R , O), where O is a sheaf of E∞ -rings onét opShvét.
We
will
identifyOwithafunctor
(Shv
)
→
CAlg
.
It
is
naturalto ask the following question:
R
R
R

Question 2.0.5. Given an object U ∈ Shvét
R , to what extent can U be recovered from the E∞ -algebra O(U )?
For example, suppose that R is an algebraically closed field. Then the global sections functor
Γ : Shvét
R →S∗
is an equivalence of ∞-categories. For U ∈ Shvét
R , we have a canonical equivalence O(U ) ' C (Γ(U ); R).
Question 2.0.5 then asks to what extent a space X can be recovered from the cochain algebra C ∗ (X; R).
When R is a field of characteristic zero, this question can be attacked using the ideas of §1.
In this section, we will focus instead on the case where R has characteristic p. We begin by introducing
some tools for studying the ∞-category CAlgR . Suppose that R is a commutative ring and let A be an E∞ algebra over R. Then π∗ A has the structure of a graded-commutative algebra over R. We can regard π∗ A
as a concrete invariant of A, which can be studied by means of classical (Z-graded) commutative algebra.
However, if we are given a prime number p such that p = 0 in R, then there is a richer story. In thiscase, the commutative ring π∗ R is equipped with additional “power operations,” (which, in the case where
A = C ∗ (X; R), reduce to the classical Steenrod operations on the cohomology of spaces). We will recall theconstruction of these power operations in §2.1 and §2.2.
In the situation of Question 2.0.5, it is generally not reasonable to expect that an object U ∈ Shvét
R can berecovered from O(U ) ∈ CAlgR unless U satisfies a reasonable finiteness condition. In §2.3 we will introduce
the notion of a finitely constructible object of a coherent ∞-topos X. Our main result (Theorem 2.3.24)
asserts that an object U ∈ Shvét
R is finitely constructible if and only if there exists a finite stratification of
SpecZ R such that the restriction of U to each stratum is locally constant, and the stalks of U have finitely
many finite homotopy groups.
Now let p be a prime number. We will say that an object U ∈ Shvét
R is p-constructible if it is finitely
constructible, and the homotopy groups of each stalk of U are finite p-groups (see Definition 2.4.1). Our
main result (Corollary 2.6.12) can be formulated as follows:(∗) Let R be a commutative ring, let p be a prime number which is nilpotent in R, and let O be thestructure sheaf of Specét R. Then the functor U 7→ O(U ) induces a fully faithful embedding from theop∞-category of p-constructible objects of Shvét
R to the ∞-category CAlgR of E∞ -algebras over R.
Remark 2.0.6. The formulation of Corollary 2.6.12 is actually a little more general than (∗): we willconsider more generally the case where R is an E∞ -ring which is p-thin (see Definition 2.4.2).
The proof of (∗) relies heavily on the excellent behavior of the functor U 7→ O(U ). More specifically, we
will need to know the following:(a) Let f : R → R0 be a map of commutative rings in which p is nilpotent, let O be the structure sheafof Specét R and O0 the structure sheaf of Specét R0 . If U ∈ Shvét
R is p-constructible, then the canonicalmap
R0 ⊗R O(U ) → O0 (f ∗ U )27

is an equivalence of E∞ -algebras over R0 .
(b) If p is nilpotent in R and U ∈ Shvét
R is p-constructible, then O(U ) is a compact object of CAlgR .
We will prove (a) in §2.4 (Theorem 2.4.9), and (b) in §2.5 (Theorem 2.5.1). In §2.6, we will combine (a)and (b) to give the proof of (∗).
Remark 2.0.7. The results of this section were inspired by Mandell’s work on p-adic homotopy theory (see[54]). The main ingredient in our proof of (∗) is Mandell’s “generators and relations” description of the E∞ algebra C ∗ (X; Fp ) in the case where X is an Eilenberg-MacLane space K(Z /p Z, n) (Theorem 2.2.17), which
we formulate (without proof) in §2.2. We can recover Mandell’s applications of this result by specializing to
the case where R is a separably closed field of characteristic p. We will make a detailed study of this case in
§3.

2.1

Norm Maps

Let M be an abelian group equipped with an action of a group G. We can associate to M the subgroup
M G = {x ∈ M : (∀g ∈ G)[g(x) = x]} consisting of G-invariant elements, as well as the quotient group
MG = M/K, where K is the subgroup of M generated by all elements of the form g(x) − x. When the group
G is finite, there is a canonical norm map
N m : MG → M G ,Pwhich is induced by the map from M to itself given by x 7→ g∈G g(x). In this section, we will describe ananalogous construction in the ∞-categorical setting and the closely Tate construction.
Notation 2.1.1. Let C be an ∞-category and X a Kan complex. We let CX denote the ∞-category
Fun(X, C) of all maps from X to C. If f : X → Y is a map of Kan complexes, then composition with f
induces a map f ∗ : CY → CX . If we assume that C admits small limits and colimits, then f ∗ admits both aright and left adjoint, which we denote by f∗ and f! , respectively.
Example 2.1.2. Let G be a group and BG its classifying space (which we regard as a Kan complex). Recall
that if C is an ∞-category, then a G-equivariant object of C is an object of CBG . Let f : BG → ∆0 be theprojection map. If C admits small limits and colimits, then we have functors f∗ , f! : CBG → C. We will
denote these functors by M 7→ M G and M 7→ MG , respectively.
We can now formulate our problem more precisely. Let G be a finite group, f : BG → ∆0 the projection
map, and C be a sufficiently nice ∞-category. We wish to associate to the pair (G, C) a natural transformation
N m : f! → f∗ . That is, we wish to construct a natural map MG → M G for each G-equivariant object M ∈ C.
It will be convenient to construct this natural transformation more generally for any map f : X → Y having
reasonably simple homotopy fibers. We will proceed in several steps, each time allowing slightly more generalhomotopy fibers.
Construction 2.1.3. Let C be an ∞-category which has both an initial object and a final object. It follows
that for any map of Kan complexes f : X → Y with (−1)-truncated homotopy fibers, the pullback functorf ∗ : CY → CX admits left and right adjoints f! and f∗ , given by left and right Kan extension along f .
Let X ×Y X denote the homotopy fiber product of X with itself over Y and let δ : X → X ×Y X be thediagonal map. Since f is (−1)-truncated, δ is a homotopy equivalence. It follows that the Kan extension
functors δ! , δ∗ : CX → CX×Y X are both homotopy inverse to δ ∗ , so there is a canonical equivalence δ∗ → δ! .
Let p0 , p1 : X ×Y X → X be the projection onto the first and second factor, respectively. We have a naturaltransformation of functorsp∗0 → δ∗ δ ∗ p∗0 ' δ∗ ' δ! ' δ! δ ∗ p∗1 → p∗128

which is adjoint to a natural transformation β : idCX → (p0 )∗ p∗1 . Since we have a homotopy pullback diagramp0

X ×Y X
p1


X

/X
f

f


/ Y,

Lemma 1.1.5 implies that the canonical map f ∗ f∗ → (p0 )∗ p∗1 is an equivalence, so that β determines anatural transformation idCX → f ∗ f∗ , which is in turn adjoint to a map N mf : f! → f∗ . We will refer to
N mf as the norm map determined by f .
Example 2.1.4. Let C be an ∞-category with initial and final objects and let f : X → Y be a homotopy
equivalence of Kan complexes. Then the natural transformation N mf : f! → f∗ of Construction 2.1.3 isthe equivalence determined by the observation that f! and f∗ are both homotopy inverse to f ∗ : in otherwords, it is determined by the requirement that the induced map f ∗ f! → f ∗ f∗ is homotopy inverse to thecomposition of counit and unit mapsf ∗ f∗ → idCX → f ∗ f! .
Example 2.1.5. Let Y = ∆0 and let C be an ∞-category with initial and final objects. If f : X → Y is a(−1)-truncated map of Kan complexes, then X is either empty or contractible. If X is contractible, then thenorm map N mf is the equivalence described in Example 2.1.4. If X = ∅, then CX ' ∆0 and the functors f!
and f∗ can be identified with initial and final objects of C ' CY , respectively. In this case, the norm map
N mf is determined up to a contractible space of choices, since it is a map from an initial object of C to a
final object of C.
Proposition 2.1.6. Let C be an ∞-category with an initial and final object. The following conditions areequivalent:(1) For every map of Kan complexes f : X → Y with (−1)-truncated homotopy fibers, the norm map
N mf : f! → f∗ is an equivalence.
(2) Condition (1) holds whenever Y = ∆0 .
(3) The ∞-category C is pointed.
Proof. The equivalence of (1) and (2) is easy, and the equivalence of (2) and (3) follows from Example
2.1.5.
When the hypotheses of Proposition 2.1.6 are satisfied, it is possible to perform a more elaborate version
of Construction 2.1.3.
Construction 2.1.7. Let C be an ∞-category which admits finite products and coproducts. It follows thatfor any map of Kan complexes f : X → Y whose homotopy fibers are 0-truncated and have finitely many
path components, the pullback functor f ∗ : CY → CX admits left and right adjoints f! and f∗ , given by left
and right Kan extension along f .
Let X ×Y X denote the homotopy fiber product of X with itself over Y and let δ : X → X ×Y X be
the diagonal map. Since f is 0-truncated, the map δ is (−1)-truncated, so that Construction 2.1.3 defines anorm map N mδ : δ! → δ∗ . Assume that C is pointed. Proposition 2.1.6 implies that N mδ is an equivalence,and therefore admits a homotopy inverse N m−1
δ : δ∗ → δ! .
Let p0 , p1 : X ×Y X → X be the projection onto the first and second factor, respectively. We have anatural transformation of functorsp∗0 → δ∗ δ ∗ p∗0

N m−1
δ

→ δ∗ ' δ! ' δ! δ ∗ p∗1 → p∗129which is adjoint to a natural transformation β : idCX → (p0 )∗ p∗1 . Since we have a homotopy pullback diagram/X

p0

X ×Y X
p1

f


X


/ Y,

f

Lemma 1.1.5 implies that the canonical map f ∗ f∗ → (p0 )∗ p∗1 is an equivalence, so that β determines a map
idCX → f ∗ f∗ , which is adjoint to a natural transformation N mf : f! → f∗ . We will refer to N mf as thenorm map determined by f .
Remark 2.1.8. In the situation of Construction 2.1.7, assume that f is (−1)-truncated. Then our definition of N mf is unambiguous: in other words, the natural transformations N mf : f! → f∗ described inConstructions 2.1.3 and 2.1.7 agree. This follows immediately from Example 2.1.4.
Remark 2.1.9. Suppose we are given a homotopy pullback diagram
f0

X0
p0


X

f

/ Y0

/Y

p

where the homotopy fibers of f are 0-truncated and have finitely many homotopy groups. Let C be a pointed∞-category which admits finite products and coproducts. Using Lemma 1.1.5, it is not difficult to show that∗∗the natural transformation f!0 ◦ p0 → f∗0 ◦ p0 determined by N mf 0 is homotopic to the composition
N mf

∗

∗

f!0 ◦ p0 → p∗ ◦ f! → p∗ ◦ f∗ → f∗0 ◦ p0 .
Example 2.1.10. Let C be a pointed ∞-category which admits finite products and coproducts, let S be a
finite set (regarded as a discrete simplicial set), and let f : S → ∆0 be the canonical projection map. We
can identify objects of CS with tuples C = (Cs ∈ C)s∈S . The norm map f! (C) → f∗ (C) can be identified
with the map
a
Y
Cs →
Ct ,
s∈S

t∈S

which classifies a collection of maps φs,t : Cs → Ct in C where φs,t = id if s = t and the zero map otherwise.
Proposition 2.1.11. Let C be a pointed ∞-category which admits finite products and coproducts. The
following conditions are equivalent:(1) For every map of Kan complexes f : X → Y whose homotopy fibers are discrete and have finitely many
connected components, the norm map N mf : f! → f∗ is an equivalence.
(2) Condition (1) holds whenever Y = ∆0 .
(3) For every finite collection of objects {Cs ∈ C}s∈S , the map
Y
a
Cs →
Ct
s∈S

t∈S

described in Example 2.1.10 is an equivalence.
Proof. The implication (1) ⇒ (2) is obvious and the converse follows from Remark 2.1.9. The equivalenceof (2) and (3) follows from Example 2.1.10.
30

Definition 2.1.12. We will say that an ∞-category C is semiadditive if it satisfies the equivalent conditionsof Proposition 2.1.11.
Remark 2.1.13. Let C be a semiadditive ∞-category. Suppose we are given a pair of objects C, D ∈ C anda finite collection of maps {φs : C → D}s∈S . Then we can define a new map φ : C → D by the compositionC→Y(φs )s∈S Y

C −→

s∈S

s∈S

D'

a

D → D,

s∈S

where the first map is the diagonal of C and the last the codiagonal of D. This construction determines amap
YMapC (C, D) → MapC (C, D),s∈S

which endows MapC (C, D) with the structure of aP
commutative monoid up to homotopy. We will denotethe image of a collection of morphisms (φs )s∈S by s∈S φs .
It is possible to make a much stronger assertion: the addition on MapC (C, D) is not only commutative
and associative up to homotopy, but up to coherent homotopy. That is, each mapping space in C can be
regarded as a commutative algebra object of S, and the composition of morphisms in C is multilinear. Since
we do not need this for the time being, we omit the proof.
Remark 2.1.14. Let C be an ∞-category which admits finite products and coproducts. Since products andcoproducts in C are also products and coproducts in the homotopy category hC, we see that C is semiadditive
if and only if (the nerve of) the category hC is semiadditive.
Example 2.1.15. Let A be an additive category (see Definition A.1.1.2.1). Then the ∞-category N(A) issemiadditive.
Example 2.1.16. Let C be a stable ∞-category. Then the homotopy category hC is additive (Lemma
A.1.1.2.8). Combining this with Example 2.1.15 and Remark 2.1.14, we deduce that C is semiadditive.
Definition 2.1.17. Let X be a Kan complex. We will say that X is a finite groupoid if the following
conditions are satisfied:(1) The set of connected components π0 X is finite.
(2) For every point x ∈ X, the fundamental group π1 (X, x) is finite.
(3) The homotopy groups πn (X, x) vanish for n ≥ 2.
More generally, we say that a map of Kan complexes f : X → Y is a relative finite groupoid if the homotopy
fibers of f are finite groupoids.
Construction 2.1.18. Let C be a semiadditive ∞-category which admits limits and colimits indexed byfinite groupoids. It follows that for any map of Kan complexes f : X → Y which is a relative finite groupoid,the pullback functor f ∗ : CY → CX admits left and right adjoints f! and f∗ , given by left and right Kan
extension along f .
Let X ×Y X denote the homotopy fiber product of X with itself over Y and let δ : X → X ×Y X be
the diagonal map. Since f is a relative finite groupoid, the homotopy fibers of δ are homotopy equivalent tofinite discrete spaces. Construction 2.1.7 defines a norm map N mδ : δ! → δ∗ . Since C is semiadditive, thenatural transformation N mδ is an equivalence and therefore admits a homotopy inverse N m−1
δ : δ∗ → δ! .
Let p0 , p1 : X ×Y X → X be the projection onto the first and second factor, respectively. We have anatural transformation of functorsp∗0 → δ∗ δ ∗ p∗0

N m−1
δ

→ δ∗ ' δ! ' δ! δ ∗ p∗1 → p∗1
31which is adjoint to a natural transformation β : idCX → (p0 )∗ p∗1 . Since we have a homotopy pullback diagramp0

X ×Y X
p1

/X
f


X

f


/ Y,

Lemma 1.1.5 implies that the canonical map f ∗ f∗ → (p0 )∗ p∗1 is an equivalence, so that β determines a map
idCX → f ∗ f∗ , which is adjoint to a natural transformation N mf : f! → f∗ . We will refer to N mf as thenorm map determined by f .
Remark 2.1.19. In the situation of Construction 2.1.18, assume that f is 0-truncated. Then the definition
of N mf given in Construction 2.1.18 agrees with that given in Construction 2.1.7 (and, if f is (−1)-truncated,with that given in Construction 2.1.3): this follows easily from Remark 2.1.8.
Remark 2.1.20. In the situation of Construction 2.1.18, suppose we are given a homotopy pullback diagram
f0

X0
p0


X

/ Y0
p

f


/ Y.

∗

∗

Using Lemma 1.1.5, we deduce that the natural transformation f!0 ◦ p0 → f∗0 ◦ p0 determined by N mf 0 is
homotopic to the composition
N mf

∗

∗

f!0 ◦ p0 → p∗ ◦ f! → p∗ ◦ f∗ → f∗0 ◦ p0 .
Example 2.1.21. Let G be a finite group. Then the classifying space BG is a finite groupoid. Letf : BG → ∆0 be the projection map. If C is semiadditive ∞-category which admits limits and colimits
indexed by finite groupoids, then Construction 2.1.18 determines a natural transformation N mf : f! → f∗ .
In particular, for every G-equivariant object M ∈ C, we obtain a canonical map N m : MG → M G .
Remark 2.1.22. Let C be a semiadditive ∞-category which admits limits and colimits indexed by finitegroupoids, and let G be a finite group. Let M be a G-equivariant object of C, and abuse notation byidentifying M with its image in C. We have canonical maps e : M → MG and e0 : M G → M . Unwinding
the definitions, we see that the composition
e

Nm

e0

M → MG → M G → M
P
is given by g∈G φg , where for each g ∈ G we let φg : M → M be the map given by evaluation on the
1-simplex of BG corresponding to g; here the sum is formed with respect to the addition described in Remark
2.1.13.
If C is equivalent to the nerve of an ordinary category, then the map M → MG is a categorical epimorphism
and the map M G → M is a categorical monomorphism. P
It follows that the map N m : MG → M G is0determined (up to homotopy) by the
P formula e ◦ N m ◦ e ' g φg . Moreover, the map N m exists by virtueof the observation that the map g φg : M → M is invariant under left and right composition with themaps φg .
Definition 2.1.23. Let C be a stable ∞-category which admits limits and colimits indexed by finitegroupoids. Let G be a finite group, and let M be a G-equivariant object of C. We will denote the cofiber ofthe norm map N m : MG → M G by M tG . We refer to the formation M 7→ M tG as the Tate construction.

32

Remark 2.1.24. Let C be a semiadditive ∞-category which admits limits and colimits indexed by finitegroupoids. Assume that for every finite group G and every G-equivariant object M of C, the norm map
N m : MG → M G is an equivalence (if C is stable, this is equivalent to the requirement that the Tate
construction M tG vanish). It follows that for every relative finite groupoid f : X → Y , the norm map
N mf : f! → f∗ is an equivalence of functors from CX to CY . We can then repeat Construction 2.1.18 todefine a norm map N mf : f! → f∗ for maps f : X → Y whose homotopy fibers are finite 2-groupoids. If
C also admits limits and colimits indexed by finite 2-groupoids, then we can repeat Construction 2.1.18 todefine a norm map N mf : f! → f∗ whenever f is a relative finite 2-groupoid. This condition is satisfied, forexample, if C is a Q-linear ∞-category (here Q denotes the field of rational numbers), but is generally notsatisfied for stable ∞-categories defined in positive or mixed characteristics. However, it is always satisfies
in the setting of K(n)-local stable homotopy theory. We will study this construction in more detail in [30].
Example 2.1.25. Let C be a semiadditive ∞-category which admits limits and colimits indexed by finitegroupoids. Let G be a finite group, let i : ∆0 → BG be the inclusion of the base point and let f : BG → ∆0`∆0beand let N = i! M ∈ CBG , so that N ' i! M '' g∈G M 'Q the projection map. Let X ∈ C ' C
g∈G M ' i∗ N . Unwinding the definitions, we see that the norm map f! (N ) → f∗ (N ) is given by thecomposition
f! (N ) = f! i! M ' (id! M ) ' (id∗ M ) ' f∗ i∗ (M ) ' f∗ (N )and is therefore an equivalence. If C is stable, we conclude that the Tate construction N tG is a zero object
of C.

2.2

Power Operations on E∞ -Algebras

Let k be a field and let A be an E∞ -algebra over k. Then the homotopy groups π∗ A have the structure ofa graded commutative ring. However, we can say much more in the case where k has characteristic p > 0.
In this case, the homotopy groups π∗ A are equipped with power operations (which reduce to the classical
Steenrod operations in the case where A = C ∗ (X; k) is the E∞ -algebra of cochains on a space X). In thissection, we will review the construction of these power operations and use them to formulate (without proof)a theorem of Mandell, giving a generators-and-relations presentation of C ∗ (X; k) in the case where X is an
Eilenberg-MacLane space K(Z /p Z, n).
Construction 2.2.1. Given a group G, we let EG denote the nerve of the category whose objects are theelements of G, where there is a unique isomorphism between every pair of objects. Then EG is a contractible
Kan complex with a free action of the group G.
Let n ≥ 0 be an integer and let Σn be the symmetric group on n letters. For every simplicial set K, we
let K (n) denote the quotient (K n × EΣn )/Σn . We refer to K (n) as the nth extended power of K. It is amodel for the homotopy coinvariants for the action of the symmetric group Σn on K n . It follows that if K
is an ∞-category, we can identify K (n) with the nth symmetric power Symn (K) in the ∞-category Cat∞ .
Let C be a symmetric monoidal ∞-category. Then we can identify C with a commutative algebra objectof Cat∞ . It follows that C is equipped with a canonical map θ : C(n) ' Symn (C) → C. In particular, forevery diagram K → C, composition with θ yields a map K (n) → C.
In particular, if C ∈ C is an object classified by a map ∆0 → C, we obtain a map f : BΣn ' (∆0 )(n) → C
which classifies a Σn -equivariant object of C, which we will denote by C ⊗n . We let Symn (C) ∈ C denote acolimit of f (provided that such a colimit exists).
Construction 2.2.2. Let C be a symmetric monoidal ∞-category. Assume that C is stable and admits limits
and colimits indexed by finite groupoids. Suppose we are given a homomorphism of finite groups η : G → Σn ,so that η induces a diagonal map δ : C ×BG → C(n) . Composing δ with the extended power map C(n) → C ofConstruction 2.2.1, we obtain a map C → Fun(BG, C). We let Tbη : C → C be the composition of this functor
with the Tate cohomology construction Fun(BG, C) → C of Definition 2.1.23. In other words, Tbη : C → C isthe functor given by C 7→ (C ⊗n )tG , where G acts on C ⊗n via the representation η.

33

Let p be a prime number, and let η : Z /p Z → Σp correspond to the action of Z /p Z on itself bytranslation. In this case, we will denote the functor Tbη : C → C by Tbp .
Proposition 2.2.3. Let C be a symmetric monoidal ∞-category. Assume that C is stable, admits limits andcolimits indexed by finite groupoids, and that the tensor product on C is exact in each variable. For every
prime number p, the functor Tbp : C → C of Construction 2.2.2 is exact.
Proof. It is easy to see that Tbp preserves zero objects. It will therefore suffice to show that if σ :
X

f

/Y

/Z


0

is an exact triangle in C, the Tbp (σ) is an exact triangle in C. We can identify f with a map ∆1 → C, so that f
determines an extended power f (p) : (∆1 )(p) → C. We can identify the objects of the ∞-category (∆1 )p withp-tuples (i1 , . . . , ip ), where each ij belongs to the linearly ordered set [1] = {0, 1}. For 0 ≤ j ≤ p, let K ≤jdenote the full subcategory of (∆1 )p spanned by those tuples (i1 , . . . , ip ) such that i1 + · · · + ip ≤ j and K j
the full subcategory spanned by those tuples where i1 + · · · + ip = j. These subsets are invariant under theaction of the group G = Z /p Z by cyclic permutations. We let K≤j denote the quotient (K ≤j × EG)/G and
define Kj similarly. Then f (p) determines a map F≤p : K≤p → C, which restricts to diagrams F≤j : K≤j → C
0for each j ≥ 0. Let F≤j: BG → C be a left Kan extension of F≤j along the projection map K≤j → BG, and000for j > 0 let Fj be the cofiber of the map F≤j−1→ F≤j. Unwinding the definitions, we obtain identifications
0
Tbp (X) ' (F≤0)tG

0Tbp (Y ) = (F≤p)tG

Tbp (Z) = (Fp0 )tG .

0
It follows that the fiber of the map Tbp (Y ) → Tbp (Z) can be identified with (F≤p−1)tG . We wish to show that00
the evident transformation F≤0 → F≤p−1 induces an equivalence after applying the Tate construction. For
00this, it suffices to show more generally that F≤j−1→ F≤jinduces an equivalence of Tate constructions for
0 tG
0 < j < p: that is, we wish to show that Fj ' 0. Let U : K≤j → C be a left Kan extension of F≤j−1 andlet Fj denote the cofiber of the canonical map U → F≤j . It follows that Fj0 can be identified with the left
Kan extension of Fj along the map K≤j → BG. We observe that Fj is a left Kan extension of the restriction
Fj |Kj . It follows that Fj0 is a left Kan extension of Fj |Kj along the map Kj → BG. Since p is prime and
0 < j < p, the simplicial set Kj is a union of finitely many contractible Kan complexes, indexed by thecollection of G-orbits on the set of subsets of {1, . . . , p} which have cardinality j. It follows from Example
tG
2.1.25 that Kj0 ' 0.

Remark 2.2.4. Let C be a symmetric monoidal ∞-category. Assume that C is stable and admits limits
and colimits indexed by finite 1-groupoids. For every homomorphism of finite groups η : G → Σn and every
commutative algebra object A ∈ CAlg(C), we obtain a canonical mapnTbη (A)[−1] ' (A⊗n )tG [−1] → A⊗n
G → Sym (A) → A.

This construction is functorial in A, and determines a natural transformation Tbη [−1]◦θ → θ, where θ denotes
the forgetful functor CAlg(C) → C.
Example 2.2.5. Let κ be an E∞ -ring and let p be a prime number. Let Modk = LModk be the ∞-categoryof k-module spectra, which we will view as left modules over k. The construction M 7→ Tbp (M ) is an exact
functor from LModk to itself, which does not commute with filtered colimits. Let Tp : LModk → LModk
be a left Kan extension of Tbp | LModperf, where LModperf
is the full subcategory of LModk spanned by thekkperfect k-modules. Then Tp commutes with colimits, and we have a natural transformation Tp → Tbp which34

induces an equivalence Tp (M ) → Tbp (M ) whenever M ∈ LModk is perfect. Using Proposition A.7.1.2.4, wededuce that Tp is given by the formula M 7→ B ⊗k M , where B ∈ k BModk (Sp) is a k-bimodule spectrum.
Unwinding the definitions, we obtain an equivalence of left k-modules B ' Tp (k) ' Tbp (k) ' k tG , where
G = Z /p Z acts trivially on k ' k ⊗p ∈ LModk .
Construction 2.2.6. Suppose that k is a discrete commutative ring and let G = Z /p Z. Then we haveb −i (G; k), where H
b i (G; •) denotes the usual Tate cohomology functor. In
canonical isomorphisms πi k tG ' H
b −1 (G; k) with the kernel of the norm map
particular, we have a canonical isomorphism of π1 k tG ' H
k ' H0 (G; k) → H0 (G; k) ' k, which is given by multiplication by p. If we assume that k is a ring of characteristic p (that is, that p = 0
in k), then the element 1 ∈ k gives a canonical element of π1 k tG , which induces a map of right k-module
spectra k → B[−1]. It follows that for each A ∈ CAlgk we obtain a canonical map of spectrapA ' k ⊗k A → B[−1] ⊗k A ' Tp (A)[−1] → Tbp (A)[−1] = (A⊗p )tG [−1] → A⊗p
G → Sym (A) → A.

We will denote this map by P 0 : A → A. This construction is functorial in A, so that we can regard P 0 as
an endomorphism of the forgetful functor CAlgk → Sp.
Remark 2.2.7. Let k be a commutative ring of characteristic p and let A be an E∞ -algebra over k. Thenthe map of spectra P 0 : A → A induces a homomorphism of abelian groups πn A → πn A for every integer n,which we will also denote by P 0 . Note that a class η ∈ πn A is represented by a map of k-modules k[n] → A,
and that P 0 (η) corresponds to the compositionpk[n] → Tp (k[n])[−1] ' Tbp (k[n]) ' (k[pn]tG )[−1] → k[pn]G → A⊗p
G → Sym (A) → A.

If n > 0, then the map k[n] → k[pn]G is nullhomotopic (since πn k[pn]G ' Hn−pn (G; k) ' 0), so thatP 0 (η) = 0. If n = 0, then the map P 0 : π0 A → π0 A coincides with the Frobenius map x 7→ xp .
Warning 2.2.8. In the situation of Construction 2.2.6, the map P 0 : A → A is a morphism of spectra, but
not of k-module spectra. For example. if A is discrete, then Remark 2.2.7 allows us to identify P 0 with
the Frobenius map x 7→ xp . This map is generally not k-linear: for λ ∈ k, we have (λx)p = λp xp , which is
generally not equal to λxp .
b n−1 (Z /p Z; k), we obtain an
Remark 2.2.9. Construction 2.2.6 can be generalized: given any class x ∈ H
associated map P (x) : A → A[n], which induces group homomorphisms πm A → πm−n A. These operations
depend functorially on A and generate an algebra (the extended Steenrod algebra) of “power operations”
which act on the homotopy groups of every E∞ -algebra over k.
Warning 2.2.10. If k is a discrete commutative ring of characteristic p and A is a discrete commutative
k-algebra, then the Frobenius map P 0 : A → A is a ring homomorphism. If A is not discrete, then the
map P 0 : A → A usually cannot be refined to a map of E∞ -rings. In fact, P 0 need not even induce a ringhomomorphism at the level of homotopy groups. We have instead a Cartan formulaX
P 0 (xy) =P i (x)P j (y),i+j=0where P n denotes the power operation obtained from a suitable generator of the Tate cohomology groupb (2p−2)n−1 (G; k) (see Remark 2.2.9).
H
Remark 2.2.11. Let k be a commutative ring of characteristic p and let A be an E∞ -algebra over k. We canidentify the ∞-category ModA of A-modules with the stabilization of the ∞-category CAlgaug
A of augmented
A-algebras. It follows that P 0 induces a natural transformation from the forgetful functor ModA → Sp to35

itself. Proposition A.7.1.2.4 implies that the ∞-category of colimit-preserving functors ModA → Sp can be
identified with the ∞-category ModA . Under this equivalence, the forgetful functor ModA → Sp correponds
to A, regarded as a module over itself. It follows that P 0 determines an endomorphism of A as an A-module,
which is given by multiplication by an element η ∈ π0 A. This element is characterized by the requirementthat for every A-module M and every element (a, x) ∈ πn (A ⊕ M ), we have P 0 (a, x) = (P 0 (a), ηx). Taking
M = A[1], n = 1, and x to be a generator of π1 M ' π0 A, we conclude that η = 0 (since P 0 acts trivially on
π1 (A ⊕ M ), by Remark 2.2.7).
Notation 2.2.12. Let k be a commutative ring. For every Kan complex X, we let C ∗ (X; k) denote theE∞ -algebra over k of cochains on X. If X is equipped with a base point, then there is a natural augmentation∗on the algebra C ∗ (X; k); we will denote the fiber of this augmentation by Cred
(X; k). We have canonicalisomorphisms∗
πi C ∗ (X; k) ' H−i (X; k)πi Cred(X; k) ' H−ired (X; k),where H−i (X; k) and H−ired (X; k) denote the cohomology and reduced cohomology groups of X, respectively.
Suppose now that X is an Eilenberg-MacLane space K(Z /p Z; n) (when n = 0, this means that X is
homotopy equivalent to Z /p Z, as a discrete space). The identity map idX determines a cohomology classηn ∈ Hnred (X; Z /p Z). Multiplication by ηn determines a k-module homomorphism φn : k → Hnred (X; k).
Elementary obstruction theory shows that this map is an isomorphism for n > 0, and that the cohomology
groups Hi (X; k) vanish for 0 < i < n.
We will need the following well-known fact concerning the behavior of the power operation P 0 :Proposition 2.2.13. Let k be a commutative ring of characteristic p. Let n ≥ 0 and let X be an EilenbergMacLane space K(Z /p Z, n). Then the diagram
k

F

/k

P0


/ Hnred (X; k)

φn

φn


Hnred (X; k)is commutative, where F : k → k denotes the Frobenius map x 7→ xp and φn is defined as in Notation 2.2.12.
Proof. We proceed by induction on n. When n = 0, the desired result follows from Remark 2.2.7. Assume
that n > 0, and let Y be the Eilenberg-MacLane space K(Z /p Z, n − 1). We have a pullback diagram ofpointed spaces
/∗
Y

/X


∗
which induces a diagram of spectra∗Cred(Y ) o
O

0O

0o

∗
Cred
(X)

∗
∗
given by a map Cred
(X) → Cred(Y )[−1]. From this we obtain a transgression map t : Hnred (X; k) →n−1
Hred (Y ; k). Note that φn−1 ' t ◦ φn . It follows that t is injective (and an isomorphism if n > 1). It follows
from naturality that the diagram

Hnred (X; k)

P0

t


n−1
Hred
(Y ; k)/ Hnred (X; k)
t

P

0

36


/ Hn−1 (Y ; k)redcommutes. We are therefore reduced to verifying the commutativity of the diagram
k

F

/k

0


/ Hn−1 (Y ; k),
red

φn−1


Hn−1
red (Y ; k)

φn−1

P

which follows from the inductive hypothesis.
Corollary 2.2.14. Let k = Fp and let X ∈ S be arbitrary. Then P 0 : Hn (X; k) → Hn (X; k) is the identity
map.
Proof. Every cohomology class η ∈ Hn (X; k) classifies a map X → K(Z /p Z, n). We may therefore replace
X by K(Z /p Z, n) and thereby reduce to the situation of Proposition 2.2.13.
Construction 2.2.15. Let k be a commutative ring of characteristic p > 0, letθ : CAlgk = CAlg(Modk (Sp)) → Sp
be the forgetful functor, and let Free : Sp → CAlgk be a left adjoint to θ. Let P 0 be the endomorphismof θ defined in Construction 2.2.6. Since Fun(CAlgk , Sp) is a stable ∞-category, we can also regard thedifference id −P 0 as an endomorphism of θ. Passing to left adjoints, we obtain a natural transformation℘ : Free → Free, which we will refer to as the Artin-Schreier transformation.
Example 2.2.16. Let k be a commutative ring of characteristic p > 0 and let S denote the sphere spectrum.
Then Free(S) ' k{x} is the free E∞ -algebra over k on one generator; in particular, k{x} is a connective
E∞ -ring with π0 k{x} ' k[x]. Using Remark 2.2.7, we deduce that the Artin-Schreier transformation ℘ :Free → Free induces a map k{x} → k{x} which is determined, up to homotopy, by the requirement thatx 7→ x − xp ∈ k[x] ' π0 k{x}.
Note that the free algebra functor Free : Sp → CAlgk carries the zero spectrum to the object k ' Free(0).
It follows that Free induces a functor
Sp ' Sp/0 → (CAlgk )/k = CAlgaug
k ,which we will also denote by Free.
If A ∈ CAlgaug
and I is the augmentation ideal of A, then we have a canonical homotopy equivalencek
MapCAlgaug(Free(M ), A) ' MapSp (M, I)kfor every spectrum M . In particular, if the augmentation ideal I satisfies πm I ' 0 for m > −n, then the
mapping space MapCAlgaug(Free(S −n ), A) homotopy equivalent to the discrete space π−n I. It follows that ifk
X is an Eilenberg-MacLane space K(Z /p Z, n), then the mapping space MapCAlgaug(Free(S −n ), C ∗ (X; k))knn
is homotopy equivalent to the discrete set Hred (X; k). The morphism φn : k → Hred (X; k) of Notation 2.2.12
determines a map k → MapCAlgaug(Free(S −n ), C ∗ (X; k)), which is a homotopy equivalence if n > 0 (and thekinclusion of a summand when n = 0).
The element 1 ∈ k determines a map of augmented k-algebras Free(S −n ) → C ∗ (X; k). Using Proposition
2.2.13, we see that the composition℘

Free(S −n ) → Free(S −n ) → C ∗ (X; k)is given by 0 ∈ Hnred (X; k): that is, it factors through the augmentation map Free(S −n ) → k. Since themapping space MapCAlgaug(Free(S −n ), C ∗ (X; k)) is homotopy equivalent to a discrete space, this choice ofk37

factorization is essentially unique. We obtain a commutative diagram℘

Free(S −n )/ Free(S −n )/ C ∗ (X; k)kof E∞ -algebras over k. We will need the following fundamental result:
Theorem 2.2.17 (Mandell). Let k be a commutative ring of characteristic p and let X = K(Z /p Z, n) bean Eilenberg-MacLane space. Then the diagram℘

Free(S −n )/ Free(S −n )/ C ∗ (X; k)kis a pushout square in CAlgk .

For a proof, we refer the reader to Theorem 6.2 of [54].
Remark 2.2.18. Theorem 2.2.17 asserts that for an Eilenberg-MacLane space X = K(Z /p Z, n), thecochain algebra C ∗ (X; k) can be described as the E∞ -algebra Free(S −n ) ⊗Free(S −n ) k, which is generated bya single element η (in degree −n) which is fixed by the operation P 0 . The proof given in [54] proceeds by
explicitly computing the homotopy groups of the algebra Free(S −n ) ⊗Free(S −n ) k and comparing them withthe cohomology groups of X. It would be desirable to have a less computationally intensive proof, but we
do not know of one.

2.3

Finitely Constructible Sheaves

Let X = (X, OX ) be a spectral Deligne-Mumford stack. Then we can think of X as the ∞-category of S-valued
sheaves on X (with respect to the étale topology). In this section, we will introduce a full subcategory Xfc ,which we call the ∞-category of finitely constructible sheaves on X. Roughly speaking, a sheaf F ∈ X is
finitely constructible if it is truncated, its stalks have finite homotopy groups, and it is locally constant alongeach stratum of a suitable stratification of X (actually, we demand the existence of this stratification only
locally on X). However, it will be convenient to begin with a much more general definition.
Definition 2.3.1. Let X be a coherent ∞-topos. We will say that an object X ∈ X is finitely constructible
if it is coherent and n-truncated for some integer n. We let Xfc denote the full subcategory of X spanned by
the finitely constructible objects.
Example 2.3.2. A space X is a finitely constructible object of the ∞-topos S if and only if it satisfies thefollowing conditions:(1) The space X is n-truncated for some integer n.
(2) The set π0 X is finite.
(3) For each vertex x ∈ X and each integer m ≥ 1, the group πm (X, x) is finite.
We will say that a space X is π-finite if it satisfies these conditions.
Remark 2.3.3. Let X be a coherent ∞-topos. If X ∈ X is finitely constructible and U ∈ X is a coherent
object, then X × U is finitely constructible as an object of X/U .
38

Definition 2.3.4. Let X be a locally coherent ∞-topos. We will say that an object X ∈ X is finitely
constructible if, for every coherent object U ∈ X, the product X × U is a finitely constructible object of X/U .
(When X is coherent, this definition is equivalent to Definition 2.3.1, by virtue of Remark 2.3.3.)
Finite constructibility is a local condition:
Proposition 2.3.5. Let X be a locally coherent ∞-topos and let X ∈ X be an object. Suppose that there
exists a covering {Uα ∈ X}α∈A of X such that each Uα ∈ X is coherent and each product Uα × X is a finitely
constructible object of X/Uα . Then X ∈ X is finitely constructible.
Proof. We may assume without loss of generality that X is coherent. Since X is quasi-compact, we may
assume without loss of generality that the set A is finite. It follows that there exists an integer n suchthat each Uα × X is an n-truncated object of X/Uα . It follows from Proposition T.6.2.3.17 that X is ann-truncated object of X. The coherence of X follows from Corollary VII.3.11.
Proposition 2.3.6. Let f : (X, OX ) → (Y, OY ) be a map of spectral Deligne-Mumford stacks. Then the
pullback functor f ∗ carries finitely constructible objects of Y to finitely constructible objects of X.
Proof. The assertion is local on X and Y, by Proposition 2.3.5. We may therefore assume that (X, OX ) and(Y, OY ) are affine. Since it is clear that f ∗ preserves n-truncatedness, it suffices to show that f ∗ preserves
coherence. This follows from Example VIII.1.4.6.
étNotation 2.3.7. If R is an E∞ -ring, we let Shvét
R denote the full subcategory of Fun(CAlgR , S) spannedby those functors which are sheaves with respect to the étale topology. This is a coherent ∞-topos; we will
denote its subcategory of finitely constructible objects by Shvfc
R . It follows from Proposition 2.3.6 that weasa
covariantfunctorfrom CAlg to the ∞-category Cat∞ of small
can regard the construction R 7→ Shvfc
R∞-categories. We will denote this functor by Shvfc .

We can now formulate the first main result of this section.
Theorem 2.3.8. The functor Shvfc : CAlg → Cat∞ preserves small filtered colimits.
The proof of Theorem 2.3.8 will require a number of preliminaries. First, we establish some compactness
properties of finitely constructible sheaves.
Proposition 2.3.9. Let X be an n-coherent ∞-topos for some n ≥ 0, and let Γ : X → S be the global
sections functor (that is, Γ is the functor corepresented by the final object 1 ∈ X). Then the restriction of Γto τ≤n−1 X commutes with filtered colimits.
Corollary 2.3.10. Let X be a locally n-coherent ∞-topos. Then:(1) If U ∈ X is an n-coherent object, then τ≤n−1 U is a compact object of τ≤n−1 X.
(2) The ∞-category τ≤n−1 X is generated, under small colimits, by objects of the form τ≤n−1 U , where
U ∈ X is n-coherent.
(3) The ∞-category τ≤n−1 X is compactly generated.
Proof. Assertion (1) follows immediately from Proposition 2.3.9. Consider an arbitrary object X ∈ X. Since
X is locally coherent, we can choose a hypercovering X• of X such that each Xm is a coproduct of n-coherent
objects of X. If we assume that X ∈ τ≤n−1 X, then the mapτ≤n−1 |X• | → τ≤n−1 X ' X
is an equivalence. Consequently, X is the geometric realization of a simplicial object of τ≤n−1 X, each term ofwhich is a coproduct of objects having the form τ≤n−1 U , where U is n-coherent. This proves (2). Assertion(3) follows immediately from (1) and (2).
39Remark 2.3.11. Let X be a locally (n + 1)-coherent ∞-topos. Let C be the smallest full subcategory ofτ≤n X which is closed under finite colimits and contains τ≤n U for every (n + 1)-coherent object U ∈ X. Then
C is the full subcategory of τ≤n X spanned by the compact objects. To see this, we first note that every objectof C is compact in τ≤n X. It follows from Proposition T.5.3.5.11 that the inclusion C ,→ τ≤n X extends to a
fully faithful embedding φ : Ind(C) → τ≤n X. Proposition T.5.5.1.9 implies that φ preserves small colimits,so that φ is an equivalence of ∞-categories by Corollary 2.3.10. It follows that the ∞-category of compact
objects of τ≤n X can be identified with an idempotent completion of C. In particular, every object X ∈ τ≤n X
is the colimit (in τ≤n X) of a diagram p : Idem → C, where Idem is the simplicial set of Definition T.4.4.5.2.
Since τ≤n X is equivalent to an (n + 1)-category, X is the colimit of the restriction of p to the (n + 1)-skeleton
of Idem, which is a finite simplicial set. Since C is closed under finite colimits, we conclude that X ∈ C as
desired.
Corollary 2.3.12. Let f : (X, OX ) → (Y, OY ) be a morphism of spectral Deligne-Mumford stacks (or spectral
schemes) and let n ≥ 0 be an integer. Suppose that f is n-quasi-compact. Then:(1) The pullback functor f ∗ carries n-coherent objects to n-coherent objects.
(2) The pullback functor f ∗ carries compact objects of τ≤n−1 Y to compact objects of τ≤n−1 X.
(3) The pushforward functor f∗ : τ≤n−1 X → τ≤n−1 X commutes with filtered colimits.
Proof. Assertion (1) follows from the definition of n-quasi-compactness (see Proposition VIII.1.4.4). Assertion (2) follows from (1) and Remark 2.3.11, and assertion (3) from (2) and Proposition T.5.5.7.2.
Remark 2.3.13. Proposition 2.3.9 implies that if F is a finitely constructible object of a coherent ∞-topos
X, then F is a compact object of τ≤n X for some n ≥ 0. It follows that the ∞-category Xfc is essentiallysmall.
We now turn to the proof of Proposition 2.3.9.
Lemma 2.3.14. Let n ≥ 0 be an integer, and let X be an ∞-topos which we assume to be locally (n − 1)coherent if n > 0. Let f : U → X be a morphism in X. If f is (n − 2)-truncated, X is n-coherent, and U is(n − 1)-coherent, then U is n-coherent.
Proof. We proceed by induction on n. In the case n = 0, the map f is an equivalence and the n-coherenceof U follows from the n-coherence of X. Assume therefore that n > 0, so that U is quasi-compact. We
wish to show that if we are given maps V1 → U , V2 → U , where V1 and V2 are (n − 1)-coherent objects ofX, then the fiber product V1 ×U V2 is also (n − 1)-coherent. Since U is (n − 1)-coherent, the fiber product
V1 ×U V2 is automatically (n − 2)-coherent. The map V1 ×U V2 → V1 ×X V2 is a pullback of the diagonal
map U → U ×X U and therefore (n − 3)-truncated. Since X is n-coherent, the fiber product V1 ×X V2 is(n − 1)-coherent, and the desired result follows from the inductive hypothesis.
Remark 2.3.15. Let X be a coherent ∞-topos and let U ∈ X be an n-truncated object. Then U is coherent
if and only if it is (n + 1)-coherent. This follows by applying Lemma 2.3.14 in the case where X is a final
object of X.
Proof of Proposition 2.3.9. We proceed by induction on n. In the case n = 0, our assumption guarantees
that X is quasi-compact and the desired result follows immediately from the definition. Let us therefore
assume that n > 0. Let J be a small filtered ∞-category, let U, U 0 : Fun(J, τ≤n−1 X) → S be given by
U 0 (F ) = Γ(lim F (J)).
−→

U (F ) = lim Γ(F (J))
−→
J∈J

J∈J

There is an evident natural transformation βF : U (F ) → U 0 (F ); we wish to show that βF is a homotopy
equivalence. Assume for the moment that βF is surjective on connected components. It then suffices toshow that for every pair of points η, η 0 ∈ U (F ), the map βF induces a homotopy equivalence of path spaces40

φ : {η} ×U (F ) {η 0 } → {η} ×U 0 (F ) {η 0 }. Since J is filtered, we may assume without loss of generality that ηand η 0 are the images of points η0 , η00 ∈ Γ(F (J)). Since J is filtered, the map JJ/ → J is left cofinal; wemay therefore replace J by JJ/ and thereby assume that J is a final object of J. In this case, η0 and η00
determine natural transformations ∗ → F , where ∗ denotes the constant functor J → X taking the value
1. Let F 0 = ∗ ×F ∗. Unwinding the definitions, we see that φ can be identified with the map βF 0 . The
desired result then follows from the inductive hypothesis, since X is (n − 1)-truncated and F 0 takes valuesin τ≤n−2 X.
It remains to prove that βF is surjective on connected components. Choose a point η ∈ U 0 (F ), corresponding to a map α : 1 → limJ∈J F (J). We wish to show that α factors (up to homotopy) through F (J) for−→ `some J ∈ J. Note that the map J∈J F (J) → limJ∈J F (J) is an effective epimorphism. Since X is locally−→`(n − 1)-coherent, there exists a collection of (n − 1)-coherent objects {Ui ∈ X}i∈I such that i∈I Ui → 1
is an effective epimorphism and each of the composite maps Ui → 1 → limJ∈J F (J) factors through F (Ji ),−→`for some Ji ∈ J. Since X is quasi-compact, we can assume that the set I is finite. Let U = i∈I Ui .
Since J is filtered, there exists an object J0 ∈ J and maps Ji → J0 for i ∈ I, so that the composite map
α
U → 1 → limJ∈J F (J) factors through F (J0 ). Since J is filtered, the map JJ0 / → J is left cofinal; we may−→
therefore replace J by JJ0 / and thereby reduce to the case where J0 is an initial object of J. Let F0 : J → X
be the constant functor taking the value F (J0 ), and let F• be the simplicial object of Fun(J, X) given by theČech nerve of the map F0 → F . Let U• be the Čech nerve of the map U → 1, so that we obtain a map ofsimplicial objects of X γ : U• → limJ∈J F• (J). We will prove the following assertion by induction on m ≥ 0:−→(∗) Let ∆s,≤m denote the subcategory of ∆ whose objects are linearly ordered sets [j] for j ≤ m, and whosemorphisms are given by injective maps [j] → [j 0 ]. Let U•≤m be the restriction of U• to N(∆s,≤m )op ,define F•≤m similarly, and let γ ≤m : U•≤m → limJ∈J F•≤m (J) be the map induced by γ. Then there−→
exists an object Jm ∈ J such that γ ≤m factors through F•≤m (Jm ).
Assertion (∗) is obvious when m = 0, since the functor F0 is constant. Assume that γ ≤m−1 factors through≤m−1
F•(Jm−1 ) for some Jm−1 ∈ J. Replacing J by JJm−1 / , we may assume that Jm−1 is an initial object ofJ, so that we have a canonical map δJ : U•≤m−1 → F•≤m−1 (J) for all J ∈ J. Let M (J) ∈ X denote the mth
matching object of F• (J), for each j ∈ J, so that δJ determines a map θ : Um → M (J). Using Proposition
T.A.2.9.14, we see that promoting δJ to a natural transformation U•≤m → F•≤m (J) is equivalent to choosing
a point of the mapping space MapX/M (J) (Um , Fm (J)) ' MapX/Um (Um , Fm (J) ×M (J) Um ). Consequently, toprove (∗), it suffices to show that the map
lim MapX/Um (Um , Fm (J) ×M (J) Um ) → MapX/Um (Um , lim Fm (J) ×M (J) Um )−→−→is a homotopy equivalence. Since X is n-coherent and U ∈ X is (n − 1)-coherent, the ∞-topos X/Um is(n − 1)-coherent. By the inductive hypothesis, it suffices to show that the the objects Fm (J) ×M (J) Um are(n − 2)-truncated objects of X/Um . For this, it suffices to show that the map Fm (J) → M (J) is (n − 2)mtruncated. This map is a pullback of the diagonal F (J) → F (J)∂ ∆ , and therefore (n − m − 1)-truncated(since F (J) is assumed to be (n − 1)-truncated). This completes the proof of (∗).
Applying (∗) in the case m = n and composing with the natural map lim F•≤n → F , we deduce the−→existence of an object Jn ∈ J and a commutative diagram σ :/ F (Jn )lim U•≤n
−→

1

η


/ lim
F (J).
−→J∈J

Since the map U → 1 is an effective epimorphism, we deduce that τ≤n−1 lim U•≤n ' |U• | ' 1. Applying the−→truncation functor τ≤n−1 to the diagram σ, we conclude that η factors through F (Jn ).
41

Lemma 2.3.16. Let U : CAlg → Cat∞ be the functor given by U (R) = CAlgét
R . Then U commutes withfiltered colimits.
Proof. Suppose we are given a filtered diagram F : J → CAlg having colimit R. We wish to prove that the
canonical mapét
φ : lim CAlgét
F (J) → CAlgR−→J

is an equivalence of ∞-categories. Since every étale R-algebra is a compact object of CAlgR , Lemma XII.2.3.4
implies that φ is fully faithful. It follows from Proposition VII.8.10 that φ is essentially surjective.
d∞ whose objects are ∞-topoi and
For the next statement, we let L Top denote the subcategory of Cat∗
whose morphisms are functors f : X → Y which preserve small colimits and finite limits.
étLemma 2.3.17. Let Shvét : CAlg → L Top be the functor given by R 7→ Shvétcommutes with
R . Then Shv
filtered colimits.

Proof. Let X be an ∞-topos and let F : J → CAlg be a filtered diagram having colimit R. We wish to showthat the canonical map
MapL Top (Shvétθ : MapL Top (Shvét
F (J) , X)
R , X) → lim←−
Jis a homotopy equivalence. We have a commutative diagram
MapL Top (Shvét
R , X)

ψφ
ét op
MapCatd ∞ ((CAlgR ) , X)

/ lim MapL Top (Shvét
F (J) , X)
←−J

θ

θ

0


op
/ lim Map d ((CAlgét
F (J) ) , X)
Cat∞
←−J

where θ0 is a homotopy equivalence by Lemma 2.3.16 and the vertical maps are fully faithful by Propositionop→ X such
T.6.2.3.20. To complete the proof, it suffices to show that if we are given a functor f : (CAlgétR)0that θ (f ) lies in the essential image of ψ, then f belongs to the essential image of φ. In view of Proposition
T.6.2.3.20, we must verify two conditions:op(1) The functor f is left exact. That is, for every finite simplicial set K and every map p : K → (CAlgétR) ,we must show that the canonical map f (lim(p)) → lim(f ◦ p) is an equivalence in X. Since K is finite,←−←−
Lemma 2.3.16 implies that p is homotopic to a composition
p0

p00

op
op
K → (CAlgét→ (CAlgét
F (J) )
R)

for some J ∈ J. Since p00 is left exact, it suffices to show that f ◦ p00 is left exact. This follows from ourassumption that θ0 (f ) belongs to the essential image of ψ (using Proposition T.6.2.3.20).
00(2) For every collection of morphisms {gα : R0 → Rα}α∈A`in CAlgét
R which generate a covering sieve on R00(with respect to the étale topology), the induced map α∈A f (Rα ) → f (R ) is an effective epimorphism
in X. Without loss of generality, we may assume that the set A is finite. Using Lemma 2.3.16, we
00
may assume that there is an object J ∈ J, an object R00 ∈ CAlgét
F (J) , and morphisms gα,0 : R0 → Rα,0étZin CAlgF (J) such that each gα is the image of gα0 . Let X denote the Zariski spectrum Spec (π0 R00 ).
00
Since the maps S
π0 R00 → π0 Rα,0are étale , the image of each SpecZ (π0 Rα,0
) is an open subset Uα ⊆ X.
00
Let X = X − α∈A Uα , so that X is a closed subset of X defined by an ideal I ⊆ π0 R00 . Since themorphisms gα are a covering of R0 , the image of I generates the unit ideal in π0 R0 . It follows that
there is a map J → J 0 in J such that the image of I in π0 (R00 ⊗F (J) F (J 0 )) generates the unit ideal.

42

Replacing J by J 0 , we can assume that X 0 = ∅, so that the maps gα,0 generate a covering sieve in0
CAlgét
F (J) . Since θ (f ) belongs to the essential image of ψ, the induced functorfop
op
fJ : (CAlgét→ (CAlgét→X
F (J) )
R)

satisfies the hypotheses of Proposition T.6.2.3.20, which implies that the map
a
a
0
0
f (Rα
)'
fJ (Rα,0) → fJ (R00 ) ' f (R0 )
α∈A

α∈A

is an effective epimorphism as desired.

Lemma 2.3.18. Let J be a filtered ∞-category and let F : J → CAlg be a diagram having a colimit R.
(1) The canonical map
ShvétShvétF (J)R → lim←−J∈J

is an equivalence of ∞-categories; here the limit is taken with respect to the pushforward functorsét0π∗ : Shvét
F (J) → ShvF (J 0 ) associated to maps J → J in J.
(2) For each n ≥ −2, the canonical mapτ Shvétτ≤n Shvét
F (J)R → lim←− ≤nJ∈J

is an equivalence of ∞-categories.
Proof. Assertion (1) is a reformulation of Lemma 2.3.17 (see Theorem T.6.3.3.1), and assertion (2) follows
immediately from (1).
Lemma 2.3.19. Let J be a filtered ∞-category and let F : J → CAlg be a diagram having a colimit R.
For every E∞ -ring R, let Shvc,≤ndenote the full subcategory of Shvét
R spanned by the compact objects ofRétτ≤n ShvR . Then the canonical map
c,≤n
c,≤n
lim ShvF(J) → ShvR−→J∈J

is an equivalence of ∞-categories.
Proof. Since each τ≤n Shvét
R is compactly generated (Corollary 2.3.10), the desired result follows from Lemma
2.3.18 (see Lemma A.6.3.7.9 and Remark A.6.3.7.10).
fc,≤n
Proof of Theorem 2.3.8. Let R be an E∞ -ring. For n ≥ −2, we define a full subcategory ShvR
⊆ ShvétR
as follows:fc,≤n(a) If n ≥ 0, an object F ∈ Shvét
if and only if F is coherent and n-truncated.
R belongs to ShvRfc,≤−1(b) An object F ∈ Shvét
if and only if it is coherent and there exists a (−1)-truncatedR belongs to ShvR00map F → F , where F is corepresentable by an étale R-algebra.
fc,≤−2(c) IIf n = −2, then an object F ∈ Shvét
if and only if it is corepresentable by an
R belongs to ShvRétale R-algebra.

43

ShvfcTo prove that Shvfc
F (J) , it will suffice to show that for each n ≥ −2, the map
R ' lim
−→J
fc,≤n
fc,≤n
θ : lim ShvF(J) → ShvR−→J

is an equivalence of ∞-categories. It follows from Lemma 2.3.19 that θ is fully faithful. We prove theessential surjectivity by induction on n. The case n = −2 follows from Lemma 2.3.16. Assume therefore
that n ≥ −1, and let F ∈ Shvfc,≤n. Since F is quasi-compact, we can choose an effective epimorphismRu : F0 → F, where F0 is representable by an étale R-algebra. Let F• be the Čech nerve of u. Note that themap v : F1 → F0 × F0 is (n − 1)-truncated if n ≥ 0. If n = −1, then F1 is representable and v is (−1)fc,≤n−1
truncated. It follows that Fn belongs to ShvR
. Choose m ≥ max{n + 2, 2} and let F0• be the m-skeletal
op
op
category object F• | N(∆≤m ) . Since N(∆≤m ) is a finite simplicial set, the inductive hypothesis implies
that F0• is equivalent to a compositionG0

φ∗fc,≤n−1fc,≤n−1N(∆≤m )op →• ShvF
→ ShvR(J)

for some functor G0• . For each j ≤ m, the morphism
βj : G0j → G01 ×G00 · · · ×G00 G01
is such that φ∗ (βj ) is an equivalence. Altering our choice of J if necessary, we may assume that each βj is. Let G• : N(∆)op → Shvfc,≤n−1bean equivalence: that is, G0• is an m-skeletal category object of Shvfc,≤n−1
F (J)
F (J)
0
0
∗
∗
a right Kan extension of G• . Since φ is left exact, the image φ G• is a right Kan extension of F• so thatφ∗ G• ' F• by Proposition XII.2.1.4. Since φ∗ preserves colimits, we obtain an equivalenceφ∗ | G• | ' |φ∗ G• | ' | F• | ' F .
fc,≤n
It will therefore suffice to show that G = | G• | belongs to ShvF
(J) .
2
2
For 0 ≤ i ≤ 2, the inclusion Λi ,→ ∆ induces a map γi : G2 → G1 ×G0 G1 . Since F• is a groupoid objectof C, the image φ∗ (γi ) is an equivalence. Altering our choice of J if necessary, we may assume that each γiétis an equivalence: that is, G• is a groupoid object of Shvét
F (J) . Since ShvF (J) is an ∞-topos, this groupoid
object is effective; it follows that the natural map G1 → G0 ×G G0 is an equivalence.
We next prove that G is coherent. Since the map ψ : G0 → G is an effective epimorphism and G0 iscoherent, it will suffice to show that ψ is relatively coherent (Proposition VII.3.9). Because ψ is an effective
epimorphism, we are reduced to proving that the projection map G0 ×G G0 → G0 is relatively coherent(Corollary VII.3.11). This follows from the coherence of G0 and G1 .
Suppose next that n = −1, so that there exists a (−1)-truncated map q : F → F, where F is representableby an étale R-algebra. Altering J if necessary, we can use Lemma 2.3.16 to guarantee that F = φ∗ G, where
G is representable by an étale F (J)-algebra. The map q determines a finite diagram

F1

// F

0

/ F.

Using the inductive hypothesis and altering our choice of J, we can assume that this diagram is the image
under φ∗ of a diagram/ G.
// G 0G1
Since G is discrete, this diagram determines a map q 0 : G → G. Let δ : G → G ×G G be the diagonal map.
Since q is (−1)-truncated, φ∗ (δ) is an equivalence. Altering our choice of J, we may assume that δ is anétopequivalence. It follows that q 0 is (−1)-truncated so that G ∈ Shvfc≤−1 ((CAlgF (J) ) ) as desired.
Now suppose that n = 0. We wish to show that G is discrete. By assumption, each Gi is discrete; it will
therefore suffice to show that G1 is an equivalence relation on G0 : that is, the map r : G1 → G0 × G0 is (−1)truncated. Let δ : G1 → G1 ×G0 × G0 G1 be the diagonal map. Since F is discrete, φ∗ (r) is (−1)-truncated44and therefore φ∗ (δ) is an equivalence. Altering our choice of J if necessary, we may assume that δ is anequivalence, which implies that r is (−1)-truncated.
Suppose now that n > 0. We wish to show that G is n-truncated. We have an effective epimorphism
G0 → G, where G0 is n-truncated (in fact, (n − 1)-truncated). It will therefore suffice to show that the map
G0 → G is (n − 1)-truncated. This is equivalent to the statement that the projection map G0 ×G G0 → G0 is(n − 1)-truncated, which follows immediately from the fact that both G0 and G1 are n-truncated.
Our final goal in this section is to obtain a structure theorem (Theorem 2.3.24) for finitely constructible
objects of Shvét
R , where R is a commutative ring. We begin by studying some prototypical examples.
Definition 2.3.20. Let X be an ∞-topos, and let q ∗ : S → X be a left adjoint to the global sections functor
Γ : X → S. We will say that an object X ∈ X is finite constant if there exists an equivalence X ' q ∗ Y ,where Y ∈ S is π-finite (see Example 2.3.2). We say that X ∈ X is finite locally constant if there exists acollection of objects {Uα ∈ X}α∈A such that X × Uα is a finite`constant object of the ∞-topos X/Uα for eachα ∈ A, and the objects {Uα }α∈A cover X (that is, the map α Uα → 1 is an effective epimorphism, where
1 denotes the final object of X).
Lemma 2.3.21. Let X be a locally coherent ∞-topos, and let X ∈ X be a finite locally constant object. Then
X is finitely constructible.
Proof. Using Lemma 2.3.5, we can reduce to the case where X is finite constant and X is coherent. Letq ∗ : S → X be a geometric morphism and assume that X = π ∗ Y for some π-finite space Y . Then X isn-truncated; we must show that X is coherent. We prove that X is m-coherent using induction on m. Whenm = 0, we must show that X is quasi-compact. Choose a finite set S and an effective epimorphism S → Y
in S. Then we have an effective epimorphism f ∗ S → X in X, so it will suffice to show that f ∗ S is finitely
constructible. Since f ∗ S is a coproduct of finitely many copies of the final object of X, the desired result
follows from the coherence of X.
Now suppose that m > 0. According to Proposition VII.3.9, it will suffice to show that the map f ∗ S → X
is relatively m-coherent. Using Corollary VII.3.11, we are reduced to showing that the projection map
f ∗ S ×X f ∗ S → f ∗ S is relatively (m − 1)-coherent. In other words, we must show that for every elements ∈ S, the fiber product f ∗ S ×X f ∗ {s} is an (m − 1)-coherent object of X. This follows from the inductive
hypothesis, since f ∗ S ×X f ∗ {s} ' f ∗ (S ×Y {s}).
Proposition 2.3.22. Let X be an (n + 1)-coherent ∞-topos, let U ∈ X be a quasi-compact (−1)-truncatedobject, and leti∗ : X → X /Uj ∗ : X → X/U
be the associated geometric morphisms. Then an object X ∈ X is n-coherent if and only if i∗ X and j ∗ X aren-coherent objects of X /U and X/U , respectively. In particular, the ∞-topoi X/U and X /U are n-coherent.
Proof. We first prove that U is (n + 1)-coherent. For this, we show that U is m-coherent for m ≤ n + 1 using
induction on m. In the case m = 0, this follows from our hypothesis that U is quasi-compact. If m ≥ 1,
we must show that for every pair of (m − 1)-coherent objects V, V 0 ∈ X/U , the fiber product V ×U V 0 is
(m − 1)-coherent. Since U is (−1)-truncated, we have an equivalence V ×U V 0 ' V × V 0 , and the desired
result follows from our assumption that X is (n + 1)-coherent.
Since X is (n + 1)-coherent and U is n-coherent, the product X × U ∈ X is n-coherent whenever X is
n-coherent. In other words, j ∗ X is n-coherent whenever X is n-coherent. We prove the remaining assertions
by induction on n.
Suppose first that n = 0 and that X is quasi-compact; we must show that i∗ X is quasi-compact.
Suppose`we are given a collection of maps {Vα → i∗ X}α∈A which induce an effective epimorphism
V→i∗ X inαα∈A`∗
X /U . We wish to prove that there exists a finite subset A0 ⊆ A such that the map α∈A0 Vα → i X is also
an effective epimorphism. Let us identify X /U with a full subcategory of X, so that we have a canonical map
X → i∗ X.`For each α ∈ A, let Vα0 = X ×i∗ X Vα ∈ X. If A is empty, there is nothing to prove. Otherwise,
the map α∈A Vα0 → X is an effective epimorphism (because it becomes an effective epimorphism after45

pullback along i or j). Since X is quasi-compact, there exists a finite subset A0 ⊆ A`such that the map`0∗∗
α∈A0 Vα → X is an effective epimorphism. Applying the functor i , we deduce thatα∈A0 Vα → i X isan effective epimorphism.
Now assume that n = 0 and that i∗ X and j ∗ X are quasi-compact objects of X /U and X/U`, respectively.
We wish to show that X is quasi-compact. Suppose we are given an effective epimorphism α∈A Vα → X
in X. Then the induced mapsaai∗ Vα → i∗ X
j ∗ Vα → j ∗ X
α∈A

α∈A

are effective epimorphisms in X /U and X/U , respectively. Using the quasi-compactness of i∗ X and j ∗ X, weconclude that there is a finite subset A0 ⊆ A such that the mapsaaj ∗ Vα → j ∗ X
i∗ Vα → i∗ X
α∈A0

α∈A0

`
are effective epimorphisms, from which it follows that the map α∈A0 Vα → X is also an effective epimorphism.
Now suppose that n > 0 and that X ∈ X is such that the objects i∗ X ∈ X /U and j ∗ X ∈ X/U are
n-coherent. We wish to show that X is n-coherent. Choose (n − 1)-coherent objects Y, Y 0 ∈ X equippedwith maps Y → X ← Y 0 ; we need to show that the fiber product Y ×X Y 0 is (n − 1)-coherent. The
inductive hypothesis implies that i∗ Y , i∗ Y 0 , j ∗ Y and j ∗ Y 0 are (n − 1)-coherent, so that the fiber productsi∗ (Y ×X Y 0 ) ' i∗ Y ×i∗ X i∗ Y 0 and j ∗ (Y ×X Y 0 ) ' j ∗ Y ×j ∗ X j ∗ Y 0 are (n − 1)-coherent. Applying the inductive
hypothesis again, we conclude that Y ×X Y 0 is (n − 1)-coherent.
We now complete the proof by showing that if X is n-coherent, then i∗ X is an n-coherent object of X /U .
Let C ⊆ (X /U )/i∗ X be the full subcategory spanned by those morphisms i∗ V → i∗ X obtained by applyingi∗ to a morphism V → X in X, where V is (n − 1)-coherent. Since X is n-coherent, the subcategory C
is stable under products, and the inductive hypothesis implies that C consists of (n − 1)-coherent objectsof (X/U )/i∗ X . Consequently, to prove that i∗ X is n-coherent,
it will suffice to show that for every object`
X 0 ∈ (X /U )/i∗ X , there exists an effective epimorphism Wα`→ X 0 , where each Wα belongs to C (Corollary
VII.3.10). To prove this, choose an effective epimorphism
Wα → X 0 ×i∗ X X in X, where each Wα is∗(n − 1)-coherent. We now complete the proof by taking Vα = i Wα .
Corollary 2.3.23. Let X be a coherent ∞-topos containing a quasi-compact (−1)-truncated object U , and leti∗ : X → X /U and j ∗ : X → X/U be as in Proposition 2.3.22. Then an object X ∈ X is finitely constructible
if and only if both i∗ X and j ∗ X are finitely constructible.
We can now state our main result:
Theorem 2.3.24. Let R be a commutative ring and let F ∈ Shvét
R . The following conditions are equivalent:(1) There exists a finite sequence of elements x1 , x2 , . . . , xn ∈ R which generate the unit ideal and acollection of Galois extensions (R/(x1 , . . . , xi−1 ))[x−1i ] → Ri (see Definition XI.4.16), such that theétpullback functors φ∗i : Shvét
→
Shv
carry
F
to
finiteconstant objects φ∗i F ∈ Shvét
R
Ri
Ri .
(2) There exists a finite sequence of elements x1 , x2 , . . . , xn ∈ R which generate the unit ideal such that,
ét
ét
∗
if Ri = R/(x1 , . . . , xi−1 )[x−1i ], then the pullback maps φi : ShvR → ShvRi carry F to finite locallyconstant objects φ∗i F ∈ ShvétRi .
(3) There exists a finite sequence of elements x1 , x2 , . . . , xn ∈ R which generate the unit ideal such that, if
ét
ét
∗
Ri = R/(x1 , . . . , xi−1 )[x−1i ], then the pullback maps φi : ShvR → ShvRi carry F to finitely constructibleét∗objects φi F ∈ ShvRi .
(4) The sheaf F is finitely constructible.

46

Proof. The implication (1) ⇒ (2) is obvious, the implication (2) ⇒ (3) follows from Lemma 2.3.21, and theimplication (3) ⇒ (4) follows from Corollary 2.3.23. We will prove that (4) ⇒ (1).
Assume that F is finitely constructible. Write R as the union of its finitely generated subrings Rα . Using
Theorem 2.3.8, we may suppose that F is the image of a finitely constructible object Fα ∈ Shvfc
Rα for someindex α. Replacing R by Rα and F by Fα we can reduce to the case where R is a finitely generated algebra
over the ring Z of natural numbers; in particular, R is Noetherian.
Choose a sequence of elements x1 , x2 , . . . ∈ R of maximal length having the following properties:(a) For each i > 0, the element xi does not belong to the ideal generated by {xj }j<i .
(b) For each i > 0, there exists a Galois covering (R/(x1 , . . . , xi−1 ))[x−1i ] → Ri such that the image of F
in Shvét
isfiniteconstant.
Ri
Since R is Noetherian, condition (a) guarantees that the sequence has some finite length n. We willcomplete the proof by showing that the elements x1 , x2 , . . . , xn ∈ R generate the unit ideal. Replacing R by
R/(x1 , . . . , xn ), we may assume that n = 0; we wish to show that R = 0. Let I be the nilradical of R. Sinceét
ShvétR ' ShvR/I , we may replace R by R/I and thereby assume that R is reduced.
Assume that R 6= 0. Since R is Noetherian, we can choose a minimal prime ideal p ⊆ R. Let k denotethe localization Rp . Since R is reduced, k is a field. Let k 0 be a separable closure of k. Then Shvét
k0 ' S andcorresponds
toanobject
Y
∈
S.
Since
F
is
finitely
constructible,
the
space
Y is
the image of F in Shvét
0
k
0
ét
π-finite. Let F0 be the image of F in Shvét
and
let
F
∈
Shv
be
the
finiteconstant
sheaf
associated
to
Y.
k
0
k
.
Using
Theorem
Then F0 and F00 are both finitely constructible and they have equivalent images in Shvét0k00
2.3.8, we conclude that F0 and F00 become equivalent in Shvét
k00 for some finite separable extension k of k.
00
Enlarging k is necessary, we may assume that it is Galois over k.
Choose an element x ∈ R − p and a Galois extension R[x−1 ] → R0 such that k 00 ' R0 ⊗R[x−1 ] k. Let F1ét0
be the image of F in Shvét
R0 , and let F 1 be the finite constant object of ShvR0 corresponding to π-finite space
Y . Note that k 00 is isomorphic to a filtered colimit of localizations R0 [y −1 ], where y ranges over elementsof R − p. Since the images of F1 and F01 in Shvét
k00 are equivalent, Theorem 2.3.8 implies that there exists
0 −1] is aan element y ∈ R − p such that the images of F1 and F01 in Shvét
R0 [y −1 ] are equivalent. Then R [y−1
Galois covering of R[(xy) ] over which the sheaf F becomes finite constant, contradicting our assumptionthat n = 0.
Corollary 2.3.25. Let X = (X, OX ) be a spectral algebraic space which is quasi-compact and quasi-separated,
and let F ∈ X be an object. The following conditions are equivalent:(1) There exists a sequence of quasi-compact open substacks∅ = U0 ⊆ U1 ⊆ · · · ⊆ Un = X
such that, if Yi denotes the reduced closed substack of Ui complementary to Ui−1 , then there is asurjective finite étale map Zi → Yi such that the restriction of F to Zi is finite constant.
(2) There exists a sequence of quasi-compact open substacks∅ = U0 ⊆ U1 ⊆ · · · ⊆ Un = X
such that, if Yi denotes the reduced closed substack of Ui complementary to Ui−1 , then the restrictionof F to Yi is finite locally constant.
(3) There exists a sequence of quasi-compact open substacks∅ = U0 ⊆ U1 ⊆ · · · ⊆ Un = X
such that, if Yi denotes the reduced closed substack of Ui complementary to Ui−1 , then the restrictionof F to Yi is finitely constructible.
47

(4) The sheaf F is finitely constructible.
Proof. The implication (1) ⇒ (2) is obvious, the implication (2) ⇒ (3) follows from Lemma 2.3.21, and theimplication (3) ⇒ (4) follows from Corollary 2.3.23. To prove that (4) ⇒ (1), we first use Theorem XII.1.3.8
to reduce to the case where X = Specét R is affine. Replacing R by π0 R, we may suppose that R is discrete,
in which case the desired result follows immediately from Theorem 2.3.24.

2.4

A Universal Coefficient Theorem

To every space X and every E∞ -ring k, we can associate an E∞ -ring C ∗ (X; k) of k-valued cochains on X.
The functor X 7→ C ∗ (X; k) is canonically determined by the following pair of properties:(a) If X = ∗ consists of a single point, then C ∗ (X; k) ' k.
(b) The construction X 7→ C ∗ (X; k) carries colimit in S to limits in CAlg.
ét∗
Write Specét k = (Shvétk , O), and let q S → Shvk be a left adjoint to the global sections functor (so∗that q carries a space X ∈ S to the sheafification of the constant presheaf CAlgétk → S taking the value
X). The functor X 7→ O(q ∗ X) satisfies conditions (a) and (b), so that we have a functorial equivalenceC ∗ (X; k) ' O(q ∗ X). We may therefore think of the structure sheaf O of Specét k as a kind of generalizedversion of cohomology with coefficients in k: rather than being defined on spaces, it is defined on étale sheavesof spaces over k. Our goal in this section is to initiate the study of this cohomology theory.
Our main result (Theorem 2.4.9) asserts that, under some reasonable hypotheses, it satisfies a universal
coefficient formula. Before we can formulate this result, we need some definitions.

Definition 2.4.1. Let p be a prime number. We will say that a space X is p-finite if it satisfies the following
conditions:(1) The space X is n-truncated for some integer n.
(2) The set π0 X is finite.
(3) For each vertex x ∈ X and each integer m ≥ 1, the group πm (X, x) is a finite p-group.
Let R be a connective E∞ -ring. We say that an object X ∈ Shvét
R is p-constructible if it is finitely
constructible and, for every separably closed field k and every map φ : R → k, the stalk φ∗ X ∈ Shvétk ' S isa p-finite space.
Definition 2.4.2. Let p be a prime number and let R be an E∞ -ring. We will say that R is p-thin if thefollowing conditions are satisfied:(1) The E∞ -ring R is connective and truncated: that is, πn R vanishes for n < 0 and n  0.
(2) The prime number p is nilpotent in the commutative ring π0 R.
(3) The E∞ -ring R lies in the essential image of the forgetful functor SCR → CAlg; here SCR denotes the∞-category of simplicial commutative rings.
Remark 2.4.3. In this paper, we will prove many results about p-constructible objects of Shvét
R under theassumption that R is p-thin. We conjecture that these results are valid more generally under the assumptionthat R satisfies conditions (1) and (2) of Definition 2.4.2.
Remark 2.4.4. Let R be a p-thin E∞ -ring, and let x ∈ π0 R. Then R admits the structure of a simplicial
commutative ring, and x determines a map Z[X] → R in the ∞-category of simplicial commutative rings.
Let us regard Z as a Z[X]-module, with X acting by 0. Then
R0 = R ⊗Z[X] Z

R1 = R ⊗Z[X] Z[X ±1 ]are p-thin E∞ -rings.
48

Remark 2.4.5. Let R be a p-thin E∞ -ring. Then every étale R-algebra R0 is also p-thin. The only
nontrivial point is to show that R0 admits the structure of a simplicial commutative ring. This follows fromthe classification of étale morphisms in both SCR and CAlg (see Theorem A.7.5.0.6 and Corollary V.4.3.12).
Definition 2.4.6. Let f : R → R0 be a map of E∞ -rings. We say that f is formally étale if the relative
cotangent complex LR0 /R is trivial. In this case, we will also say that R0 is formally étale over R.
Remark 2.4.7. Let R be an E∞ -ring and let A, A0 ∈ CAlgR . Then A × A0 is formally étale over R if andonly if A and A0 are formally étale over R.
Remark 2.4.8. Suppose we are given a pushout diagram of E∞ -rings
R

/ R0


A


/ A0 .

Then LA0 /A ' LR0 /R ⊗k0 A0 . Consequently, if R0 is formally étale over R, then A0 is formally étale over A.
We can now formulate our main result:
Theorem 2.4.9. Let R be an E∞ -ring, p a prime number, and X ∈ Shvét
R . Assume that R is p-thin andthat X is p-constructible. Write Specét R = (Shvét
R , O). Then:(a) The E∞ -ring O(X) is formally étale over R.
(b) If M ∈ (ModR )≤0 and M denotes the associated quasi-coherent sheaf on Specét R, then the canonicalmap O(X) ⊗R M → M(X) is an equivalence.
Corollary 2.4.10. Let φ : R → R0 be a map of E∞ -rings, and let p be a prime number. Assume that
ét
0
ét 0
R is p-thin and that R0 is truncated. Write Specét R = (Shvét
R , O) and Spec R = (ShvR0 , O ). For every0étp-constructible X ∈ ShvR , the canonical map R0 ⊗R O(X) → O (φ∗ X) is an equivalence, and O0 (φ∗ X) is
formally étale over R0 .
The proof of Theorem 2.4.9 will occupy our attention throughout this section. We begin by establishingan analogue of Theorem 2.3.24 for p-constructible sheaves.
Proposition 2.4.11. Let R be a commutative ring and let X ∈ Shvét
R . Then X is p-constructible if andonly if the following condition is satisfied:(∗) There exists a finite sequence of elements x1 , x2 , . . . , xn ∈ R which generate the unit ideal and acollection of maps (R/(x1 , . . . , xi−1 ))[x−1i ] → Ri which are finite étale and faithfully flat, such that
ét
ét
∗
each pullback map φi : ShvR → ShvRi carries X to the constant sheaf associated to a p-finite space.
Proof. Suppose first that X is p-constructible. Using Theorem 2.3.24, we can choose a finite sequence ofelements x1 , x2 , . . . , xn ∈ R which generate the unit ideal and a collection of maps (R/(x1 , . . . , xi−1 ))[x−1i ]→ét
Ri which are finite étale and faithfully flat, such that the pullback maps φ∗i : Shvét
→
Shv
carry
X
to theR
Riconstant sheaves associated to certain π-finite spaces Yi ∈ S. We may assume without loss of generality thatxi does not belong to the radical of the ideal generated by {xj }j<i , so that each of the rings Ri is nonzero.
It follows that there exist ring homomorphisms Ri → ki , where each ki is a separably closed field. Then
Yi ' ηi∗ X where ηi denotes the composite map R → Ri → ki . Since X is p-constructible, we conclude thateach Yi is p-finite.
Conversely, assume that (∗) is satisfied. Theorem 2.3.24 implies that F is finitely constructible. If
η : R → k is a ring homomorphism where k is a separably closed field, then η factors through some Ri(where Ri is as in the statement of (2)), so that η ∗ X ∈ S is p-finite.
49

The proof of Theorem 2.4.9 will proceed by dévissage, using Proposition 2.4.11 to reduce to the case of
the constant sheaf associated to a p-finite space. We therefore begin by studying the constant case. Our
first step is to prove the following:
Proposition 2.4.12. Let k be a commutative ring, let p be a prime number which vanishes in k, and let X
be a p-finite space. Then the cochain algebra C ∗ (X; k) formally étale over k.
Lemma 2.4.13. Let k be a commutative ring, p a prime number which vanishes in k, and let X =K(Z /p Z, n) be an Eilenberg-MacLane space. Then the cochain algebra C ∗ (X; k) is formally étale overk.
Proof. Let Free : Sp → CAlgk denote the free algebra functor, so that Theorem 2.2.17 provides a pushout
diagram of E∞ -algebras over k℘/ Free(S −n )Free(S −n )/ C ∗ (X; k).

k

Let R = R0 = Free(S −n ), and regard ℘ as a map from R to R0 . Let η ∈ π−n R and η 0 ∈ π−n R0 be thegenerators, so that ℘(η) = η 0 − P 0 (η 0 ). We have canonical maps α : R → R ⊕ R[−n], β : R0 → R0 ⊕ R0 [−n],characterized up to homotopy byβ(η 0 ) = (η 0 , 1) ∈ π−n (R0 ⊕ R0 [−n]).

α(η) = (η, 1) ∈ π−n (R ⊕ R[−n])These maps induce equivalences LR/k ' R[−n], LR0 /k ' R0 [−n].
Let φ : R ⊕ R[−n] → R0 ⊕ R0 [−n] be the map induced by ℘. Consider the diagram
/ R0

℘

R
α


R ⊕ R[−n]βφ/ R0 ⊕ R0 [−n].

It follows from Remark 2.2.11 that this diagram commutes up to homotopy, so that ℘ induces the identitymap
R0 [−n] ' R[−n] ⊗R R0 ' LR/k ⊗R R0 → LR0 /k ' R0 [−n].
It follows that LR0 /R ' 0, so that R0 is formally étale over R. Using Remark 2.4.8, we deduce that C ∗ (X; k)is formally étale over k.
Lemma 2.4.14. Suppose we are given maps of E∞ -algebras k → k 0 → k 00 , where k 0 is formally étale overk. Then k 00 is formally étale over k if and only if it is formally étale over k 0 .
Proof. This follows by inspecting the fiber sequence
Lk0 /k ⊗k0 k 00 → Lk00 /k → Lk00 /k0 .

Lemma 2.4.15. Suppose we are given a pushout diagram of E∞ -algebras
k

/ k0


A


/ A0 .

If k 0 and A are formally étale over k, then A0 is formally étale over k.
50

Proof. Combine Lemma 2.4.14 and Remark 2.4.8.
Lemma 2.4.16. Let X be a connected p-finite space. Then there exists a finite sequence of maps
X = X0 → X1 → · · · → Xn = ∗and pullback diagrams
Xi/ Xi+1∗/ K(Z /p Z, mi )for some integers mi ≥ 2.
Proof. Choose a base point x ∈ X. Since X is p-finite, it is m-truncated for some m. We proceed byinduction on m and on the order pa of the finite p-group πm (X, x). If m = 0 then X is contractible and there
is nothing to prove. Assume therefore that m > 0. If a = 0 then X is (m − 1)-truncated and the desired
result follows from the inductive hypothesis. Assume that a > 0, so that the order of the group πm (X, x) is
divisible by p. Let G = π1 (X, x), and let πm (X, x)G denote the group of G-fixed points of πm (X, x). Since
G is a finite p-group, the order of πm (X, x)G is congruent to the order of πm (X, x) modulo p. Since a > 0,
we conclude that the order of the group πm (X, x)G is divisible by p so we can choose a cyclic subgroupZ ⊆ πm (X, x)G of order p. The theory of principal fibrations implies that there exists a fiber sequence of
spacesf
X → X1 → K(Z /p Z, m + 1)
where f induces isomorphisms πi (X, x) ' πi (X1 , f (x)) for i 6= m and a surjection πm (X, x) → πm (X1 , f (x))
having kernel Z. The desired result now follows by applying the inductive hypothesis to X1 .
Proof of Proposition 2.4.12. Using Remark 2.4.8, we can reduce to the case where k = Fp (in particular, k
is a field). Remark 2.4.7 allows us to assume that X is connected. Choose a finite sequence of maps
X = X0 → X1 → · · · → Xn = ∗as in Lemma 2.4.16. By Lemma 2.4.14, it suffices to show that each of the maps C ∗ (Xi+1 ; k) → C ∗ (Xi ; k)is formally étale . Corollary 1.1.10 implies that we have a pushout diagram of E∞ -ringsC ∗ (K(Z /p Z; m)/kC ∗ (Xi ; k)
/ C ∗ (Xi+1 ; k).

Using Lemma 2.4.15, we reduce to proving that k is formally étale over C ∗ (Z /p Z; k). This follows from
Lemmas 2.4.14 and 2.4.13.
We now turn to the verification of condition (b) in Theorem 2.4.9. Recall that if k is a connective Ek -ring,then a k-module M is said to have Tor-amplitude ≤ m if the tensor product functor N 7→ M ⊗k M carries(Modk )≤0 into (Modk )≤m .
Proposition 2.4.17. Let R be a connective E∞ -ring. Let M and N be R-modules. Assume that M ∈(ModR )≤n and that N has Tor-amplitude ≤ m, so that M ⊗R N ∈ (ModR )≤n+m . Write Specét R =ét(Shvét
R , O), and let M and N be the quasi-coherent sheaves on Spec R associated to M and N , respectively.
Then:(1) The homotopy groups πi M(X) vanish for i > n and every X ∈ ShvétR.
51

(2) If X ∈ Shvét
R is coherent, then the canonical map
M(X) ⊗R N → (M ⊗O N)(X)
is an equivalence of R-modules.
Proof. We first prove (1). Let X denote the full subcategory of Shvét
R spanned by those objects X such thatét
M(X) is n-truncated. Then X is closed under small colimits in Shvét
R . To prove that X = ShvR , it sufficesét
to show that X contains every representable sheaf X ∈ ShvR . This is clear, since if X is represented by anétale R-algebra R0 , then M(X) ' R0 ⊗R M is n-truncated by virtue of our assumption on M and the factthat R0 is flat over R.
We now prove (2). Since N has Tor-amplitude ≤ m, the module M(X) ⊗R N is (m + n)-truncatedby (1). Note that M ⊗O N is the quasi-coherent sheaf associated to the tensor product M ⊗R N which is(n + m)-truncated, so that (M ⊗O N)(X) is (n + m)-truncated for all X ∈ Shvét
R by (1). Let KX denote thefiber of the map M(X) ⊗R N → (M ⊗O N)(X), so that KX is (n + m)-truncated for all X.
We will prove by induction on r that if X is r-coherent, then πi KX ' 0 for i ≥ n + m − r. The
assertion is obvious if r < 0. Assume that r ≥ 0, so that X is quasi-compact. We may therefore choose
an effective epimorphism u : X0 → X, where X0 is representable. Let X• denote the Čech nerve of u.
Let M 0 = lim[j]∈∆op M(Xj ). Since each M(Xi ) is n-truncated, the map M(X) → M 0 has (n − r − 1)←−≤r
truncated fibers, so that the induced map M(X) ⊗k N → M 0 ⊗R N has (n + m − r − 1)-truncated fibers. Let
P = lim[j]∈∆op (M ⊗O N)(Xj ); the same reasoning shows that the map (M ⊗O N)(X) → P has (n+m−r−1)←−≤r
truncated fibers. It follows that the map KX → K 0 has (n + m − r − 1)-truncated fibers, where K 0 =lim[j]∈∆op KXj . It will therefore suffice to show that K 0 is (n + m − r − 1)-truncated.
−→≤r
Since X is r-coherent, each Xj is (r − 1)-coherent and therefore KXj is (n + m − r)-truncated by the
inductive hypothesis. It follows that K 0 is (n + m − r)-truncated, and that πn+m−r K 0 is the inverse limitof the system of abelian groups {πn+m−r KXj }[j]∈∆op. In particular, the map πn+m−r K 0 → πn+m−r KX0 is≤rinjective. Since X0 is representable, KX0 ' 0 so that πn+m−r K 0 ' 0 and K 0 is (n + m − r − 1)-truncated,as desired.
Corollary 2.4.18 (Universal Coefficient Formula). Let X be a π-finite space and let φ : R → R0 be a mapof E∞ -rings. Assume that k and k 0 are connective and truncated, and that there exists a map of E∞ -rings
Z → R (this is automatic, for example, if R is discrete). Then the induced map C ∗ (X; R)⊗R R0 → C ∗ (X; R0 )is an equivalence.
Proof. Using the commutative diagramC ∗ (X; R) ⊗R R0
5(/ C ∗ (X; R0 )C ∗ (X; Z) ⊗Z R0

we can reduce to the case where R is the ring Z of integers. In this case, R0 has finite Tor-amplitude overR, so the desired result follows from Proposition 2.4.17.
In what follows, it will be convenient to give a name to those sheaves which satisfy the conclusions ofTheorem 2.4.9.
Definition 2.4.19. Let R be a connective E∞ -ring and write Specét R = (Shvét
R , O). We will say that an
object X ∈ Shvét
R is pseudo-affine if it satisfies the following conditions:(a) The E∞ -ring O(X) is formally étale over R.

52

(b) If M ∈ (ModR )≤0 and M is the associated quasi-coherent sheaf on Specét R, then the canonical map
O(X) ⊗R M → M(X) is an equivalence.
Example 2.4.20. Let R be a connective E∞ -ring. Then every affine object X ∈ Shvét
R is pseudo-affine.
If X is corepresentable by an étale R-algebra R0 , then O(X) ' R0 is obviously formally étale over R, andcondition (b) of Definition 2.4.19 is satisfied for all R-modules M .
Remark 2.4.21. Let R be a connective E∞ -ring and suppose that X ∈ Shvét
R satisfies condition (a) of
Proposition 2.4.19. Then X is pseudo-affine if and only if it satisfies the following weaker version of (b):(b0 ) If M is a discrete R-module and M denotes the associated quasi-coherent sheaf on X, then the canonicalmap O(X) ⊗R M → M(X) is an equivalence.
To prove this, we first note that (b0 ) and Proposition 2.4.17 imply that O(X) has Tor-amplitude ≤ 0over R (see Proposition A.7.2.5.23). Let us now suppose that M is an arbitrary object of (ModR )≤0 and
let KM denote the fiber of the map O(X) ⊗R M → M(X). Since O(X) has Tor-amplitude ≤ 0, thetensor product O ⊗R M is 0-truncated. Since M(X) is 0-truncated (by Proposition 2.4.17), we conclude
that KM is 0-truncated for all M ∈ (ModR )≤0 . We now prove that KM is (−r)-truncated using induction
on r. We have a cofiber sequence of R-modules π0 M → M → τ≤−1 M , hence another cofiber sequence
Kπ0 M → KM → Kτ≤−1 M . Since Kπ0 M ' 0 by virtue of assumption (b), we conclude that KM ' Kτ≤−1 M .
Since τ≤−1 M ∈ (ModR )≤−1 , the inductive hypothesis implies that KM ' Kτ≤−1 M is (−r)-truncated as
desired.
Theorem 2.4.9 asserts that if R is a p-thin E∞ -ring, then every p-constructible object of Shvét
R is pseudoaffine. In order to verify this, we will need to establish some stability properties for the collection ofpseudo-affine sheaves.
Lemma 2.4.22. Let k be a field of characteristic p, let Y be a p-finite space, and let X ∈ Shvétk be theconstant sheaf associated to Y . Then X is pseudo-affine.
Proof. Let φ∗ : S → Shvétk be a geometric morphism (that is, a left adjoint to the functor given by evaluation∗∗at k), and write Specét k = (Shvétk , O), so that we have an equivalence C (Z; k) ' O(φ Z) depending∗
functorially on Z. Then O(X) ' C (Y ; k) is formally étale over k by Proposition 2.4.12. Since k is afield, every object M ∈ (Modk )≤0 has Tor-amplitude ≤ 0. Condition (b) of Definition 2.4.19 now follows
immediately from Proposition 2.4.17.
Suppose we are given a map f : R → R0 of connective E∞ -rings, and write
0Specét R0 = (Shvét
R0 , O ).

Specét R = (Shvét
R , O)

Suppose that M 0 is a R0 -module, and let M denote the same spectrum regarded as a R-module. Let M and
M0 be the associated quasi-coherent sheaves on Specét R and Specét R0 . Then M is the pushforward of M0 :0 ∗in other words, for each X ∈ Shvét
R we have a canonical equivalence M(X) ' M (f X). If X is pseudo-affineand M is truncated, then M(X) ' O(X) ⊗R M and we obtain an equivalence α : O(X) ⊗R M → M0 (f ∗ X).
Assume that R0 is n-truncated for some n. Applying this argument to the module M 0 = R0 , we obtain an
equivalence O(X) ⊗R R0 ' O0 (X), so that O0 (X) has finite Tor-amplitude over R0 and is formally étale overR0 by Remark 2.4.8. In this case, for any truncated R0 -module M 0 , we can identify α with the map
O(X) ⊗R M ' (O(X) ⊗R R0 ) ⊗R0 M 0 ' O0 (f ∗ X) ⊗R0 M 0 → M0 (f ∗ (X)).
This proves the following:
Lemma 2.4.23. Let f : R → R0 be a map of connective E∞ -rings. Assume that R0 is n-truncated for someététn. Then the pullback functor f ∗ : Shvét
R → ShvR0 carries pseudo-affine objects of ShvR to pseudo-affineétobjects of ShvR0 .
53

If R is a commutative ring and p is a prime number which vanishes in R, then there is a map ofcommutative rings Fp → R, where Fp denotes the field with p elements. Combining Lemmas 2.4.22 and2.4.23, we obtain:
Lemma 2.4.24. Let R be a commutative ring, p a prime number which vanishes in R, Y a p-finite space,and X ∈ Shvét
R the constant sheaf associated to Y . Then X is pseudo-affine.
We next observe that the property of being pseudo-affine is local:
Lemma 2.4.25. Let R be a connective
QE∞ -ring, let {φα : R → Rα }α∈A be a finite collection of étale mapsof E∞ -rings such that the map R → α Rα is faithfully flat, and let X ∈ Shvét
R be coherent. Then X is
pseudo-affine if and only if each pullback φ∗α X ∈ Shvétispseudo-affine.
Rα
ét
ét
Proof. Write Specét R = (Shvét
R , O) and Spec Rα = (ShvRα , Oα ). Using Proposition 2.4.17, we deduce that∗Oα (φα X) ' O(X) ⊗R Rα so we have equivalences

LOα (φ∗α X)/Rα ' LOX (X)/R ⊗R Rα .
Q
Since the map R → α Rkα is faithfully flat, we deduce that O(X) is formally étale over R if and only ifeach Oα (φ∗α X) is formally étale over Rα .
Assume now that X is pseudo-affine and let α ∈ A. We wish to prove that φ∗α X is pseudo-affine. We
have already verified condition (a) of Definition 2.4.19. To prove (b), let M 0 ∈ (ModRα )≤0 , let M be its
image in ModR , and let M and M0 denote the associated quasi-coherent sheaves on Specét R and Specét Rα ,respectively. We wish to show that the canonical map β : Oα (φ∗α (X))⊗Rα M 0 → M0 (φ∗α (X)) is an equivalence.
Using Proposition 2.4.17, we can identify β with the map O(X) ⊗R M → M(X), which is an equivalence byvirtue of our assumption that X is pseudo-affine.
Conversely, assume that each φ∗α is pseudo-affine. We must show that X satisfies condition (b) of Definition 2.4.19. Let M ∈ (ModR )≤0 and let M be the associated quasi-coherent
sheaf on X; we wish to show
Q
that the map O(X) ⊗R M → M(X) is an equivalence. Since R → α Rα is faithfully flat, it suffices to showthat for each α ∈ A, the induced map
γ : (O(X) ⊗R Rα ) ⊗Rα (Rα ⊗R M ) → Rα ⊗R M(X).
Let M 0 = Rα ⊗R M and let M0 be the associated quasi-coherent sheaf on Specét Rα .
Since Rα is flat over R, Proposition 2.4.17 allows us to identify γ with the map
Oα (φ∗α X) ⊗Rα M 0 → M0 (φ∗α X),
which is an equivalence by virtue of our assumption that φ∗α X is pseudo-affine.
Lemma 2.4.26. Let p be a prime number, R a p-thin E∞ -ring, and X ∈ Shvét
R a sheaf which is locallyconstant and p-constructible. Then X is pseudo-affine.
Proof. Using Lemma 2.4.25, we can reduce to the case where X is the constant sheaf associated to a pfinite space Y . Write Specét R = (Shvét
R , O). Let M be an R-module and let M denote the corresponding
quasi-coherent sheaf on Specét R. We will say that M is good if it satisfies the following conditions:(i) The canonical map O(X) ⊗R M → M(X) is an equivalence.
(ii) The tensor product LO(X)/R ⊗R M vanishes.
To prove that X is pseudo-coherent, it will suffice to show that (i) is satisfied whenever M is discrete,and condition (ii) is satisfied when M = R (see Remark 2.4.21). Since R is connective and truncated, it
can be obtained as a successive extension of finitely many discrete R-modules. Because the collection of54

good R-modules is closed under extensions, it will suffice to show that every discrete R-module M is good.
Choose m ≥ 0 such that pm = 0 in π0 R. The module M then has a finite filtration
0 = pm M ⊆ pm−1 M ⊆ · · · ⊆ M.
Consequently, to show that M is good, it will suffice to show that each quotient pa−1 M/pa M is good. We may
therefore assume without loss of generality that M is a module over the commutative ring k = (π0 R)/(p).
Using Corollary 2.4.18, we can replace R by k and thereby reduce to the situation of Lemma 2.4.24.
Proof of Theorem 2.4.9. Using Theorem 2.3.24, we may assume that there is a finite sequence of elements
x1 , . . . , xm ∈ π0 R which generate the unit ideal in R such that, if we let Ri = (π0 R)/(x1 , . . . , xi−1 )[x−1
i ],then the map of E∞ -rings φi : R → Ri is such that φ∗i X ∈ Shvétislocally
constantandp-constructible.
Ri
Let us assume that m is chosen as small as possible; we proceed by induction on m. If m = 0, then R ' 0
and there is nothing to prove. Let us therefore assume that m > 0. Let Z[y] denote the polynomial ring in
one variable over Z. Since R admits the structure of a simplicial commutative ring, there exists a map of
E∞ -rings Z[y] → R such that y 7→ x1 ∈ π0 R. Consider also the map of commutative rings Z[y] → Z given
by y 7→ 0, and let R0 denote the E∞ -ring given by R ⊗Z[y] Z. Then R0 is also p-thin, and π0 R0 ' (π0 R)/(x1 ).
ét0∗Let ψ 0 : R → R0 and ψ 00 : R → R[x−11 ] be the canonical maps. Then ψ X ∈ ShvR0 is pseudo-affine by theét00∗
inductive hypothesis, and ψ X ∈ ShvR[x−1 ] is pseudo-affine by Lemma 2.4.26. Let O0 and O00 denote the1structures sheaves of Specét R0 and Specét R[x−11 ], respectively.
We now prove that X is pseudo-affine. Let A = O(X). We first show that A is formally étale over R.
Consider the relative cotangent complex LA/R . Note that R0 has Tor-amplitude ≤ 1 over R (since Z has∗
Tor-amplitude ≤ 1 over Z[y]). It follows from Proposition 2.4.17 that the canonical map A⊗R R0 → O0 (ψ 0 X)
00is an equivalence, so that the inductive hypothesis implies that A ⊗R R is formally étale over R . It follows
that the tensor product LA/R ⊗R R0 vanishes. Note that as an R-module, R0 is given by the cofiber of themap x1 : R → R. We therefore have a fiber sequencex

LA/R →1 LA/R → LA/R ⊗R R0 ,so that multiplication by x1 is an equivalence from LA/R to itself. It follows that the canonical map−1
LA/R → LA/R ⊗R R[x−11 ] ' LA[x−1 ]/R[x−1 ] is an equivalence. Since R[x1 ] is flat over R, Proposition 2.4.17
1

∗

1

00
−1
00
00∗
gives an equivalence A[x−11 ] ' O (ψ X). Since ψ X is pseudo-affine, we conclude that A[x1 ] is formally−1étale over R[x1 ], so that LA/R ' 0.
Now suppose that M ∈ (ModR )≤0 , and let M be the associated quasi-coherent sheaf on Specét R. We wish
to prove that the canonical map u : A ⊗R M → M(X) is an equivalence. Let K denote the fiber of u; we wish
to show that K ' 0. Let M 0 = R0 ⊗R M and let M0 be the associated quasi-coherent sheaf on Specét R0 . Using
Proposition 2.4.17, we can identify K ⊗R R0 with the fiber of the map u0 : (A ⊗R R0 ) ⊗R0 M 0 → M0 (ψ 0∗ X).
Since ψ 0∗ X is pseudo-affine, the map u0 is an equivalence. Using the fiber sequencex

K →1 K → K ⊗R R0 ,we deduce that multiplication by x1 induces an equivalence from K to itself, so that the map K → K ⊗R
−1R[x−1
1 ] is an equivalence. It will therefore suffice to show that K ⊗R R[x1 ] ' 0: that is, that the map−1u000 : A[x−11 ] ⊗R M → M(X) ⊗R R[x1 ]is an equivalence. This follows from Proposition 2.4.17, together with our assumption that ψ 00∗ X is mild.

2.5

Compactness of Relative Cochain Algebras

ét
Let R be a connective E∞ -ring, write Specét R = (Shvét
R , O), and let X ∈ ShvR . In §2.4, we proved that
if R is p-thin and X is p-constructible, then O(X) is well-behaved as an E∞ -algebra over R: it is formally55

étale (Theorem 2.4.9) and its definition is compatible with base change in R (Corollary 2.4.10). In thissection, we will continue our study of R-algebras having the form O(X). Our main result can be stated asfollows:
Theorem 2.5.1. Let R be an E∞ -ring, let p be a prime number such that R is p-thin, and let X ∈ Shvét
R bep-constructible. If we let O denote the structure sheaf of Specét R, then O(X) is a compact object of CAlgR .
The proof of Theorem 2.5.1 will require several preliminary results. We begin by studying the behaviorof O(X) as a functor of X.
Proposition 2.5.2. Let R be an E∞ -ring which is p-thin and let O denote the structure sheaf of Specét R.
Let τ :
/X
X0

Y0


/Y

be a pullback square of p-constructible objects of Shvét
R . Then the diagram σ :O(X 0 ) o
O

O(X)
O

O(Y 0 ) oO(Y )is a pushout square in CAlgR .
Proof. Using Theorem 2.3.24, we conclude that there exists a sequence of elements x1 , . . . , xm ∈ π0 R whichétgenerate the unit ideal such that, if we set ki = (π0 R)/(x1 , . . . , xi−1 )[x−1i ], the image of X in Shvki is locallyconstant. Choose m as small as possible. We proceed by induction on m.
If m = 0, then R ' 0 and there is nothing to prove. Assume therefore that m > 0. Since R is p-thin,it admits the structure of a simplicial commutative ring. Choose a map of simplicial commutative rings
Z[t] → R which carries t to x1 . Let R0 denote the tensor product R ⊗Z[t] Z, where Z[t] → Z is given by
t 7→ 0. Then R0 is p-thin and π0 R0 ' R/(x1 ). Using the inductive hypothesis and Corollary 2.4.10, we
deduce that the diagram
O(X 0 ) ⊗R R0 o
O(X) ⊗R R0
O
O
O(Y 0 ) ⊗R R0 o

O(Y ) ⊗R R0

is a pushout square of E∞ -algebras over R0 . If we let K denote the total homotopy fiber of σ (regarded as
an R-module), then K ⊗R R0 ' 0. It follows that multiplication by x1 induces an equivalence from K to
itself, so that K ' K ⊗R R[x−11 ]. It will therefore suffice to show that σ is a pullback square after tensoring−1with R[x−11 ]. Using Corollary 2.4.10, we can replace R by R[x1 ] and thereby reduce to the case where X
is locally constant.
Using the same argument, we may reduce to the case where Y and Y 0 are locally constant. Let φ∗ :S → Shvét
R be a geometric morphism. The assertion that σ is a pushout square is local with respect to theétale topology. We may therefore assume without loss of generality that X, Y , and Y 0 are constant: that is,
X = φ∗ X0 , Y = φ∗ Y0 , and Y 0 = φ∗ Y00 for some p-finite spaces X0 , Y0 , Y00 ∈ S.
Choose an integer r such that X0 and Y0 are r-truncated. Since X0 is p-finite, Proposition 2.3.9 implies
that X0 is a compact object of τ≤r S, so there exists a compact object K ∈ S such that X0 = τ≤r K. The map
α = X → Y induces a morphism φ∗ K → φ∗ X0 → φ∗ Y0 . Since K is finite, we may assume (after passing toan étale covering of R if necessary) that this morphism is induced by a map of spaces β : K → Y0 . Since Y0
is r-truncated, β we deduce that β factors (in an essentially unique way) through some map β 0 : X0 → Y0 ,56

and that α can be identified with φ∗ (β 0 ). Similarly, we may assume that the map Y 0 → Y arises from a mapof p-finite spaces Y00 → Y0 . Then the diagram τ is obtained from a pullback diagram of p-finite spaces τ0 :
X00

/ X0


Y00


/ Y0 .

We may therefore identify σ with the diagramC ∗ (X00 ; R) oOC ∗ (X0 ; R)O

C ∗ (Y00 ; R) oC ∗ (Y0 ; R).

Let K denote the fiber of the map C ∗ (X0 ; R) ⊗C ∗ (Y0 ;R) C ∗ (Y00 ; R) → C ∗ (X00 ; R), and let us say that an
R-module M is good if M ⊗R K ' 0. We wish to prove that R is good. Since R is truncated and thecollection of good R-modules is closed under extensions, it will suffice to show that every discrete R-module
is good. Since p is nilpotent in R, every discrete R-module M admits a finite filtration
0 = pm M ⊆ pm−1 M ⊆ · · · ⊆ pM ⊆ M.
It therefore suffices to show that every module over the commutative ring (π0 R)/(p) is good. Using Corollary
2.4.18, we can replace R by (π0 R)/(p), and thereby reduce to the case where R is a commutative ring inwhich p = 0. In that case, there exists a ring homomorphism Fp → R (where Fp denotes the finite fieldwith p elements). Using Corollary 2.4.18 again, we can reduce to the case where R = Fp . In this case, σ isa pushout diagram by virtue of Corollary 1.1.10.
Corollary 2.5.3. Let R be an E∞ -ring which is p-thin, let O be the structure sheaf of Specét R, and let
op
X ∈ Shvét
R be p-constructible. If X is m-truncated, then O(X) is an m-truncated object of CAlgR . In otherwords, for every E∞ -algebra A over R, the mapping space MapCAlgR (O(X), A) is m-truncated.
Proof. Combine Proposition 2.5.2 with Proposition T.5.5.6.16.
Lemma 2.5.4. Let R → R be a square-zero extension of E∞ -rings, and let A be a formally étale R-algebra.
Then R is a compact object of CAlgR if and only if A = A ⊗R R is a compact object of CAlgR .
Proof. The “only if” direction is obvious (and does not require the assumption that A is formally étale over
R). For the converse, assume that A is a compact object of CAlgR . Suppose we are given a filtered diagram{Bα } in CAlgR having colimit B. Let B α = Bα ⊗R R, and let B = B ⊗R R. Then each Bα is a square-zeroextension of B α , so the canonical maps
MapCAlgR (A, Bα ) → MapCAlgR (A, B α ) ' MapCAlgR (A, B α )are homotopy equivalences. Similarly, the natural map MapCAlgR (A, B) → MapCAlgR (A, B) is a homotopy
equivalence. We have a commutative diagramlim MapCAlgR (A, Bα )−→/ lim MapCAlg (A, B α )R−→MapCAlgR (A, B)/ MapCAlg (A, B)
R

in which the horizontal maps are homotopy equivalences, and the right vertical map is a homotopy equivalenceby virtue of our assumption that A is a compact object of CAlgR . It follows that the left vertical map is a
homotopy equivalence, as desired.
57

Notation 2.5.5. Let R be an E∞ -ring and let x ∈ π0 R. If M is an R-module, we let M [x−1 ] denote the∧tensor product M ⊗R R[x−1 ] and M(x)the completion of M with respect to the ideal (x) generated by x (see§XII.4.1). Note that if A is an E∞ -algebra over R, then we can also regard A[x−1 ] and A∧(x) as E∞ -algebras
over R.
Lemma 2.5.6. Let R be an E∞ -ring, let x ∈ π0 R, and let M be an R-module. Then the diagram
M

/ M∧


M [x−1 ]/ M ∧ [x−1 ]

(x)

(x)

is a pullback square of R-modules.
∧−1∧ [x−1 ] M [xProof. We freely use the terminology of §XII.4.1. Let N = M(x)×M(x)]; we wish to show thatthe canonical map φ : M → N is an equivalence. Let K denote the fiber of φ. Then we have a fiber sequence

K → K 0 → K 00 ,∧∧where K 0 is the fiber of the map M → M [x−1 ] and K 00 is the fiber of the map M(x)→ M(x)[x−1 ]. Since K 000
and K are both (x)-nilpotent, we conclude that K is (x)-nilpotent. We also have a fiber sequence

K → L → L0 ,∧∧where L is the fiber of the completion map M → M(x)and L0 is the fiber of the map M [x−1 ] → M(x)[x−1 ].
0
The definition of completion guarantees that L is (x)-local. Since L is the fiber of a map between (x)-local
R-modules, it is (x)-local. It follows that K is (x)-local. Since K is also (x)-nilpotent, the identity map
id : K → K is nullhomotopic, so that K ' 0 and M ' N .

Lemma 2.5.7. Let R be an E∞ -ring, let x ∈ π0 R, and suppose we are given a filtered diagram of R-modules{Mα } having a colimit M . Then the diagram σ :lim(Mα )∧
(x)
−→

/ M∧


−1
lim(Mα )∧](x) [x−→/ M ∧ [x−1 ]

(x)

(x)

is a pullback square.
Proof. The diagram σ is a filtered colimit of diagrams σn :lim(Mα )∧
(x)
−→

/ M∧

(x)

xn


lim(Mα )∧
(x)
−→



xn

/ M∧

(x)

in which the vertical maps are induced by multiplication by xn . Let τn denote the diagram/M

lim Mα−→xnxn

lim Mα
−→


/ M.
58

The diagrams τn are evidently pullback squares (since the horizontal maps are equivalences). There aretransformations τn → σn which induce equivalences after passing to the fibers of the vertical maps. It
follows that each σn is a pullback square. Passing to the limit, we deduce that σ is a pullback square.
Lemma 2.5.8. Let f : R → R0 be a finite étale map between commutative rings and let O and O denote thestructure sheaves of Specét R and Specét R0 , respectively. Let X 0 ∈ Shvét
R0 and let A ∈ CAlgR . Suppose thatX 0 is p-constructible for some prime number p which vanishes in R. Then the canonical map
MapCAlgR (O(f∗ X 0 ), A) → MapCAlgR0 (R0 ⊗R O(f∗ X 0 ), R0 ⊗R A) → MapCAlgR0 (O0 (X 0 ), R0 ⊗R A)
is a homotopy equivalence.
Proof. Since f is finite étale , there exists an étale covering {R → Rα } such that each tensor productnαR0 ⊗R Rα is equivalent to Rα
for some integer nα ≥ 0. Since the property of being p-constructible is local
and stable under finite products, we conclude that f∗ X 0 is p-constructible. Using Corollary 2.4.10, we seethat O(f∗ X 0 ) and O0 (X 0 ) are compatible with base change in R. It follows that the desired conclusion is
local with respect to the étale topology on R. We may therefore replace R by Rα and thereby reduce to the0case where R0 ' Rn for someQ integer n. In this case, X corresponds to an n-tuple of p-constructible objects0ét0Xi ∈ ShvR , and O (X ) ' 1≤i≤n O(Xi ). It follows thatY
Y
MapCAlgR0 (O0 (X 0 ), R0 ⊗R A) ' MapCAlgR0 (O(Xi ), An ) 'MapCAlgR (O(Xi ), A).
1≤i≤n

1≤i≤n

Consequently,
we are reduced to showing that O(f∗ X ) is a coproduct of the objects O(Xi ) in CAlgR . Since
Qf∗ X 0 ' 1≤i≤n Xi , this follows from Proposition 2.5.2.
0Lemma 2.5.9. Let C be a full subcategory of the ∞-category S of spaces. Assume that C is closed underretracts and finite limits, and that C contains the Eilenberg-MacLane space K(Z /p Z, n) for every integern ≥ 0. Then C contains every p-finite space.
Proof. Let X be a p-finite space. Let m be the smallest integer such that X is m-truncated; we proceed
by induction on m. The` case` m = 0 is trivial and left to the reader. Assume that m > 0 and write X
as a disjoint union X1 · · · Xn where each XQ
i is connected, and choose a base point xi in each Xi . Weproceed by induction on the order of the group 1≤i≤n πm (Xi , xi ). Since X is not (m − 1)-truncated, thegroup πm (Xj , xj ) does not vanish for some j, and therefore has order divisible by p. Since π1 (Xj , xj ) is a
finite p-group, the number of π1 (Xj , xj )-fixed elements of πm (Xj , xj ) is congruent to the order of πm (Xj , xj )modulo p, and therefore divisible by p. It follows that there exists a subgroup N ⊆ πm (Xj , xj ) of order p on
which π1 (Xj , xj ) acts trivially. It follows that there is a fiber sequence of spacesφηjXj → Xj0 → K(Z /p Z, m + 1),
with(
πi (Xj0 , φ(xj )) 'πi (Xj0 , xj )πi (Xj0 , xj )/N

if i 6= m
if i = m.

`
Let Y be the disjoint union of Xj0 with i6=j Xi . The inductive hypothesis implies that Y ∈ C. Let
η : Y → K(Z /p Z, m + 1) be a map which is given by ηj on the component Xj0 , and is nullhomotopic onevery other component, and let Y 0 be the homotopy fiber of η. Since C contains K(Z /p Z, m + 1) and isstable under`finite limits, we conclude that Y 0 ∈ C. Note that Y 0 homotopy equivalent to the disjoint unionof Xj with i6=j (Xi × K(Z /p Z, m + 1)). It follows that X is a retract of Y 0 , so that X ∈ C as desired.
Lemma 2.5.10. Let φ : R → A be a map of E∞ -rings which exhibits A as an m-truncated object of Q
CAlgopR.
Suppose we are given a finite collection of morphisms {R → Rα } such that the induced map R → Rα isfaithfully flat, and each tensor product Rα ⊗R A is a compact object of CAlgRα . Then A is a compact objectof CAlgR .
59

Q
Proof. Let R0 = α Rα , so that R → R0 is faithfully flat and R0 ⊗R A is a compact object of CAlgR0 . LetR• be the Čech nerve (in CAlgop ) of the map R → R0 , and set A• = A ⊗R R• . Let F : CAlgR → S be thefunctor corepresented by A. Then F is the totalization of a complicial object F • of Fun(CAlgR , S), whereF n is given by the formula F n (B) = MapCAlgR (A, Rn ⊗R B) ' MapCAlgRn (Rn ⊗R A, Rn ⊗R B). Since eachRn ⊗R A is a compact object of CAlgRn , we conclude that each F n commutes with filtered colimits. Since A
is an m-truncated object of (CAlgR )op , each of the functors F i takes values in τ≤m S, so that F is equivalent
to the finite limit lim[n]∈∆F n . It follows that F commutes with filtered colimits.
←−≤m+1
Lemma 2.5.11. Let R be a commutative ring, let O denote the structure sheaf of Specét R, and let X ∈ Shvét
R.
Assume that X is locally constant and p-constructible for some prime number p which vanishes in R. Then
O(X) is a compact object of CAlgR .
Proof. Using Lemma 2.5.10, Corollary 2.5.3, and Corollary 2.4.10, we can reduce to the case where X isthe constant sheaf associated to a p-finite space X0 . In this case, we can identify O(X) with C ∗ (X0 ; R) 'C ∗ (X0 ; Fp ) ⊗Fp R. It will therefore suffice to show that C ∗ (X0 ; Fp ) is a compact object of CAlgFp . Using
Corollary 1.1.10, we see that the collection of p-finite spaces X0 for which C ∗ (X0 ; Fp ) is compact in CAlgFp
is closed under retracts and finite limits. Using Lemma 2.5.9, we are reduced to proving that each of
the cochain algebras C ∗ (K(Z /p Z, n); Fp ) is a compact object of CAlgFp , which follows immediately from
Theorem 2.2.17.
Proposition 2.5.12. Let R be a Noetherian commutative ring, let p be a prime number which vanishes in
R. Choose an element x ∈ R and let O denote the structure sheaf of Specét R[x−1 ]. Let X ∈ ShvétR[x−1 ] be−10p-constructible, and suppose there is a Galois covering φ : R[x ] → R0 such that the image of X in Shvét
R00
is constant. Then the construction−1
A 7→ MapCAlgR[x−1 ] (O(X), A∧])(x) [xdetermines a functor CAlgR → S which commutes with filtered colimits.
Proof. Let ShvétR[x−1 ] oφ∗φ∗

/ Shvét0 be the geometric morphism determined by φ, and let T : Shvét −1 →
R0
R[x ]

ét
∗
ShvétR[x−1 ] be the monad given by φ∗ ◦ φ . For every object Y ∈ ShvR[x−1 ] , we can associate an augmented•
n+1
cosimplicial object Y , given informally by [n] 7→ T
(Y ). This augmented cosimplicial object is φ∗ -split,∗∗ •so that the map φ Y → lim φ Y is an equivalence. If Y is n-truncated for some integer n, then we have←−φ∗ Y ' lim φ∗ Y • in the ∞-category τ≤n Shvét
R00 which is equivalent to an (n + 1)-category. It follows that←−φ∗ Y is equivalent to the finite limit lim[m]∈∆φ∗ Y m . Since φ∗ is left exact and conservative, we conclude←−≤n+1
that Y ' lim[m]∈∆
Y m.
←−
≤n+1
Let us say that a p-constructible sheaf Y ∈ Shvét
R is good if the construction−1])
A 7→ MapCAlgR[x−1 ] (O(Y ), A∧(x) [xcommutes with filtered colimits. Proposition 2.5.2 implies that the collection of good objects of X is closed
under finite limits. It follows that to prove that Y is good, it suffices to show that each Y m ' T m (Y ) isgood.
Let G denote a Galois group for the covering R[x−1 ] → R00 . It follows that G acts on the ∞-topos Shvét
R00 .
Q
ét
∗
g
Unwinding the definitions, we see that for each Z ∈ ShvR00 , we have φ φ∗ Z ' g∈G Z . In particular, if Z
is the constant sheaf associated to a p-finite space Z0 , then φ∗ φ∗ Z is the constant sheaf associated to theQ|G|p-finite space Z0 ' g∈G Z0 . Since φ∗ X is constant and p-constructible, we deduce that each φ∗ (T n X)
is constant and p-constructible. Consequently, to prove that X is good, it will suffice to show that φ∗ Z is60

good whenever Z ∈ X0 is the constant sheaf associated to a p-finite space Z0 ∈ S. In other words, we mustshow that the functor−1])
A 7→ MapCAlgR[x−1 ] (O(φ∗ Z), A∧(x) [xcommutes with filtered colimits. Let O0 denote the structure sheaf of Specét R00 . Using Lemma 2.5.8, we
obtain canonical homotopy equivalences−1])
MapCAlgR[x−1 ] (O(φ∗ Z), A∧(x) [x' MapCAlgR0 (O0 (Z), R00 ⊗R A∧(x) )0' MapCAlgR0 (C ∗ (Z0 ; R00 ), R00 ⊗R A∧(x) ).
0−1

By assumption, the map R[x ] → R00 is finite étale. It follows that the map R → R00 is quasi-finite,
and therefore admits a factorization R → R0 → R00 where R0 is a finite R-module and the map of schemes
SpecZ R00 → SpecZ R0 is an open immersion (Theorem VII.7.11). It follows that j : SpecZ R00 → SpecZ R0 [x−1 ]is also an open immersion. But R00 is finitely generated as a module over R[x−1 ] and therefore over R0 [x−1 ],so the image of j is closed. It follows that R00 ' R0 [x−1 , e−1 ] for some idempotent element e ∈ R0 [x−1 ]. Then
m
0
0
0
m

x e is the image of some element e ∈ R . Replacing R by R /(x − e), we can reduce to the case wheree = 1, so that R00 ' R0 [x−1 ].
The ring R0 is finitely presented as an R-algebra. Write R0 = R[y1 , . . . , ym ]/(f1 , . . . , fk ). Let ψ :Z[z1 , . . . , zk ] → R[y1 , . . . , ym ] be the map which sends zi to the polynomial fi and let R00 denote the tensor
product R[y1 , . . . , ym ] ⊗Z[z1 ,...,zk ] Z, where Z[z1 , . . . , zk ] → Z is the map which sends each zi to 0 ∈ Z.
Z[z ,...,z ]kThen R00 is an E∞ -algebra over R with πi R00 ' Tori 1(R[y1 , . . . , ym ], Z). In particular, we deduce00
000that π0 R ' R . Each πi R is a finite module over R[y1 , . . . , ym ] which is annihilated by each fi , hence
a finite module over R0 and therefore also a finite module over R. It follows that R00 is an almost perfect
R-module (Proposition A.7.2.5.17). Since Z has Tor-amplitude ≤ k over Z[z1 , . . . , zk ], the algebra R00 has
Tor-amplitude ≤ k over R[y1 , . . . , ym ] and is therefore of finite Tor-amplitude over R. Using Proposition
A.7.2.5.23, we conclude that R00 is perfect as an R-module.
Corollary 2.4.10 gives an equivalence C ∗ (Z0 , R00 ) ' C ∗ (Z0 , R00 ) ⊗R00 R00 . Consequently, we wish to provethat the functor
A 7→ MapCAlgR00 (C ∗ (Z0 ; R00 ), R00 ⊗R A∧(x) )commutes with filtered colimits. Note that we have an equivalence R00 ' π0 R00 [x−1 ]. Since R00 is k-truncated,
it follows that the map R00 [x−1 ] → R00 is a composition of square-zero extensions (Corollary A.7.4.1.28). Since
C ∗ (Z0 ; R00 ) is formally étale over R00 , we have canonical homotopy equivalences∗000∧
MapCAlgR00 (C ∗ (Z0 ; R00 ), R00 [x−1 ] ⊗R A∧(x) → MapCAlgR00 (C (Z0 ; R ), R0 ⊗R A(x) ).
00∧
Because R00 is perfect as an R-module, the canonical map R00 ⊗R A∧(x) → (R ⊗R A)(x) is an equivalence foreach A ∈ CAlgR . It will therefore suffice to show that the construction∧
B 7→ MapCAlgR00 (C ∗ (Z0 ; R00 ), B(x)[x−1 ])

determines a functor θZ0 : CAlgR00 → S which commutes with filtered colimits.
Using Proposition 2.5.2, we see that the functor Z0 7→ θZ0 commutes with finite limits. It follows thatthe collection of p-finite spaces Z0 for which θZ0 commutes with filtered colimits is closed under retracts andfinite limits. Using Lemma 2.5.9, we are reduce to proving that θZ0 commutes with filtered colimits in thecase where Z0 is an Eilenberg-MacLane space K(Z /p Z, n).
The sequence of spaces {K(Z /p Z, n)}n≥0 is a spectrum object in the ∞-category of p-finite spaces. It
follows from Proposition 2.5.2 that for every E∞ -algebra B over R00 , we obtain a spectrum U (B), given byΩ∞−n U (B) = MapCAlgR00 (C ∗ (K(Z /p Z, n), R00 ), B). Theorem 2.2.17 implies that each C ∗ (K(Z /p Z, n); R00 )is a compact object of CAlgR00 , so that U commutes with filtered colimits. Suppose we are given a filtereddiagram {B(α)} of objects of CAlgR00 having colimit B. We wish to show that the induced map−1∧
lim U (B(α)∧] → U (B(x))[x−1 ](x) [x−→61

is an equivalence. Since U commutes with filtered colimits, this is equivalent to the assertion that U carries−1∧
the map lim B(α)∧] → B(x)[x−1 ] to an equivalence of spectra. According to Lemma 2.5.6, we have a(x) [x−→pullback diagram/ lim B(α)∧ [x−1 ]lim B(α)∧
(x)
(x)
−→
−→

/ B ∧ [x−1 ]


∧
B(x)

(x)

in CAlgR00 , hence a pullback diagram of spectra
U (lim B(α)∧
(x) )
−→

/ U (lim B(α)∧ [x−1 ])
(x)
−→

ψ


/ U (B ∧ [x−1 ]).

∧U (B(x))(x)

It will therefore suffice to show that ψ is an equivalence. Using the fact that U commutes with filteredcolimits, we can identify ψ with the canonical map∧
lim U (B(α)∧(x) ) → U (B(x) ).
−→Consider the map of commutative rings Z[t] → R given by t 7→ x. For n ≥ 0, set Rn = R ⊗Z[t] Z[t]/(tn ).
Z[t]/(tn+1 ) → Z[t]/(tn ) is a square-zero extension. It follows that for B ∈ CAlgR00 , we have a tower of squarezero extensions· · · → B ⊗R R 3 → B ⊗R R 2 → B ⊗R R 1∧
with limit B(x). Since each C ∗ (K(Z /p Z, n), R00 ) is formally étale over R00 , it follows that the induced map∧
U (B(x) ) → U (B ⊗R R1 ) is an equivalence of spectra. Consequently, we can identify ψ with the map

lim U (B(α) ⊗R R1 ) → U (B ⊗R R1 ).
−→This map is an equivalence by virtue of the fact that U commutes with filtered colimits.
Proof of Theorem 2.5.1. Let R be p-thin. Then the map R → (π0 R)/(p) is a composition of finitely manysquare-zero extensions. Using Lemma 2.5.4, we are reduced to proving that O(X) ⊗R ((π0 R)/(p)) is acompact (π0 R)/(p)-algebra. Using Corollary 2.4.10, we can replace R by (π0 R)/(p) and thereby reduce tothe case where R is a commutative ring in which p = 0. Then R is the union finitely generated subrings
Ri . Using Theorem 2.3.8, we can assume that X lies in the essential image of one of the pullback functorsétShvétRi → ShvR . Using Corollary 2.4.10, we can replace R by Ri and thereby reduce to the case where R is
Noetherian.
Let S be the collection of all ideals I ⊆ R such that O(X) ⊗R R/I is not a compact object of CAlgR/I .
We wish to show that S is empty. Assume otherwise; then, since R is Noetherian, there is a maximal element
I ∈ S. Replacing R by R/I, we may assume S does not contain any nonzero ideals of R.
If R = 0, there is nothing to prove. Otherwise, since X is p-constructible, Theorem 2.3.24 implies that
there exists a nonzero element x ∈ R and a Galois extension R[x−1 ] → R00 such that the image of X in
ShvétR00 is a constant sheaf.
Choose a map of commutative rings Z[t] → R carrying t to x. For every E∞ -algebra A over R, let
An denote the relative tensor product B ⊗Z[t] Z[t]/(tn ). Using Proposition XII.4.2.7, we deduce that that(x)-adic completion of A can be identified with the limit of the tower· · · → A3 → A2 → A1 .
Note that each map in this tower is a square-zero extension.
62

Let F : CAlgR → S be the functor corepresented by O(X). Let {Aα } be a filtered diagram in CAlgR
having colimit A. It follows from Lemma 2.5.6 that the diagram
A

/ A∧


A[x−1 ]


/ A∧ [x−1 ]

(x)

(x)

is a pullback square in CAlgR . Since F is a corepresentable functor, the canonical map
F (A) → F (A[x−1 ]) ×F (A∧(x) [x−1 ]) F (A∧(x) )is a homotopy equivalence. We wish to prove that the canonical map
β : lim F (Aα ) → F (A[x−1 ]) ×F (A∧(x) [x−1 ]) F (A∧(x) )−→is a homotopy equivalence. Since each Aα fits into a pullback diagram
Aα/ (Aα )∧

Aα [x−1 ]/ (Aα )∧ [x−1 ],

(x)

(x)

we can identify β with the map−1lim(F (Aα [x−1 ]) ×F ((Aα )∧(x) [x−1 ]) F ((Aα )∧]) ×F (A∧(x) [x−1 ] F (A∧(x) ) → F (A[x(x) ).
−→

It will therefore suffice to show that the maps
β0 : lim F (Aα [x−1 ]) → F (A[x−1 ])−→−1−1
β1 : lim F ((Aα )∧]) → F (A∧])(x) [x(x) [x−→∧β2 : lim F ((Aα )∧(x) ) → F (A(x) )−→are homotopy equivalences.
By construction, the image of X in ShvétR[x−1 ] ) is locally constant. Lemma 2.5.11 and Corollary 2.4.10
imply that β0 is a homotopy equivalence. Similarly, Proposition 2.5.12 implies that β1 is a homotopy
equivalence. We complete the proof by showing that β2 is a homotopy equivalence. Since the functor F iscorepresentable, we have F (A∧F (An ). Since each of the maps An+1 → An is a square-zero extension(x) ) ' lim←−
1and O(X) is formally étale over R, the tower {F (An )} is constant, so that the evident map F (A∧(x) ) → F (A )∧1
is a homotopy equivalence. Similarly, each of the maps F ((Aα )(x) ) → F (Aα ) is a homotopy equivalence.
We may therefore identify β2 with the canonical map lim F (A1α ) → F (A1 ). To prove that this map is a−→
homotopy equivalence, it suffices to show that O(X) ⊗R R1 is a compact object of CAlgR1 . Note that R1
is a square-zero extension of π0 R1 ' R/(x). According to Lemma 2.5.4, we are reduced to proving that
O(X) ⊗R R/(x) is a compact object of CAlgR/(x) . This follows from the observation that (x) is a nonzero
ideal in R.

63

2.6

Affine Behavior of p-Constructible Morphisms

In §2.4, we introduced the definition of a p-constructible object of an ∞-topos of the form Shvét
R . We canrelativize this notion as follows:
Definition 2.6.1. Let X = (X, OX ) be a spectral Deligne-Mumford stack and let p be a prime number. We
will say that an object U ∈ X is p-constructible if, for every étale morphism f : Specét R → X, the pullbackf ∗ U is a p-constructible object of the ∞-topos Shvét
R (see Definition 2.4.1).
We say that a morphism of spectral Deligne-Mumford stacks g : Y → X is p-constructible if Y is equivalent
to a spectral Deligne-Mumford stack of the form (X/U , OX |U ), where U ∈ X is p-constructible.
Remark 2.6.2. If X = (X, OX ) is a spectral Deligne-Mumford stack, then an object U ∈ X is p-constructible
if and only if U is finitely constructible and, for every separably closed field k and every map f : Specét k → X,
the stalk f ∗ U ∈ Shvétk ' S is a p-finite space.
Remark 2.6.3. Every p-constructible morphism of spectral Deligne-Mumford stacks in étale.
Remark 2.6.4. If (X, OX ) ' Specét R is affine, then an object U ∈ X is p-constructible in the sense of
Definition 2.6.1 if and only if it is p-constructible in the sense of Definition 2.4.1.
Remark 2.6.5. Let (X, OX ) be a spectral Deligne-Mumford stack and let U ∈ X be an object. Using
Proposition 2.3.5, we see that the condition that U be p-constructible is local for the étale topology.
Definition 2.6.6. Let X be a spectral Deligne-Mumford
stack and p a prime number. We will say that X`
is p-thin if there exists a surjective étale map Specét Rα → X, where each Rα is a p-thin E∞ -ring.
Warning 2.6.7. If R is a p-thin E∞ -ring, then the affine spectral Deligne-Mumford stack Specét R is p-thin.
It seems likely that the converse is false, because condition (3) of Definition 2.4.2 is somewhat unnatural.
Remark 2.6.8. Let f : Y → X be an étale map between spectral Deligne-Mumford stacks. If X is p-thin,then Y is also p-thin (see Remark 2.4.5). If f is surjective, then the converse holds.
Example 2.6.9. Let (X, OX ) be a spectral Deligne-Mumford stack. Suppose that OX is discrete (that is, we
can think of OX ) as a sheaf of commutative rings on X) and that the map pn : OX → OX is zero for n  0.
Then (X, OX ) is p-thin.
We can now formulate the first main result of this section.
Theorem 2.6.10. Let p be a prime number, let q : Y = (Y, OY ) → X be a p-constructible morphism of p-thin
spectral Deligne-Mumford stacks, and let f : Z = (Z, OZ ) → X be an ∞-quasi-compact morphism between
spectral Deligne-Mumford stacks. If Z is p-thin, then the canonical map
MapStk/ X (Z, Y) → MapCAlg(QCoh(X)) (q∗ OY , f∗ OZ )is a homotopy equivalence.
Remark 2.6.11. In the situation of Theorem 2.6.10, the quasi-coherence of the pushforwards q∗ OY andf∗ OZ follows from Corollary VIII.2.5.22.
Corollary 2.6.12. Let R be a p-thin E∞ -ring, and let O denote the structure sheaf of Specét R. Then thep−fc
construction U 7→ O(U ) induces a fully faithful embedding from the full subcategory ShvR
⊆ Shvét
R spanned
opby the p-constructible sheaves to the ∞-category CAlgR .
To prove Theorem 2.6.10, we are free to work locally on X and Z, and thereby reduce to the case where
each is the spectrum of a p-thin E∞ -ring. Using Theorem 2.4.9, we can replace Y by Z ×X Z, we can reduceto the case where X = Z = Specét R. In this case, we are reduced to proving the following:

64

Theorem 2.6.13. Let R be a p-thin E∞ -ring, let O denote the structure sheaf of Specét R, and let X ∈ ShvétR
be a p-constructible sheaf. Then the canonical map X(R) → MapCAlgR (O(X), R) is a homotopy equivalence.
op
Proof. The functor O : Shvét
R → CAlgR preserves small colimits. It follows from Remark T.5.5.2.10 that O
opadmits a right adjoint Sol : CAlgR → Shvét
R . Unwinding the definitions, we see that Sol carries an object
0
A ∈ CAlgop
to
the
sheaf
Sol(A)givenbySol(A)(R) = MapCAlgR (A, R0 ). Theorem 2.6.13 is an immediateR
consequence of the following assertion:ét(∗) If X ∈ Shvét
R is p-constructible, then the unit map uX : X → Sol(O(X)) is an equivalence in ShvR .

We first treat the case where R is a separably closed field of characteristic p. In this case, we have an equivét
alence of ∞-topoi Shvét
R ' S. We can identify p-constructible sheaves X ∈ ShvR with p-finite spaces, so that∗
O(X) ' C (X; R). Note that Sol(O(X)) can be identified with the mapping space MapCAlgR (C ∗ (X; R), R).
Using Proposition 2.5.2, we see that the construction X 7→ Sol(O(X)) commutes with finite limits. Consequently, to prove that uX is an equivalence for every p-finite space X, it suffices to treat the case where X is an
Eilenberg-MacLane space K(Z /p Z, m) (see Lemma 2.5.9). Using Theorem 2.2.17, we can identify Sol(O(X))
A
with the homotopy fiber of the map K(R, m) → K(R, m), where A is induced by the Artin-Schreier map
x 7→ x − xp from R to itself. Since R is a separably closed field, the Artin-Scheier map is surjective withuX
kernel Fp ⊆ R, so that Sol(O(X)) ' K(Fp , m). Since composite map K(Z /p Z, m) →K(Fp , m) → K(R, m)
is given by the canonical embedding of Z /p Z into R, we conclude that uX is a homotopy equivalence as
desired.
We now treat the general case. Write Specét π0 R = (X, OX ) and let φ∗ : Shvét
R → X be the equivalence of∞-topoi induced by the truncation map R 7→ π0 R. Since R is truncated, the map R → π0 R is a compositionof square-zero extensions. Since O(X) is formally étale over R (Theorem 2.4.9), Corollary 2.4.10 implies thatφ∗ Sol(O(X)) ' Sol(OX (π∗ X)). We may therefore replace R by π0 R and thereby reduce to the case where
R is discrete.
Define full subcategories Shvfc,≤m⊆ Shvét
R as in the proof of Theorem 2.3.8. We will prove that theRfc,≤musing induction on m. Whenunit map uX is an equivalence for all p-constructible sheaves X in ShvR0m = −2, the sheaf X is representable by an étale R-algebra R , and the desired result follows from Theorem
VII.5.14. Suppose that m ≥ −1. Since X is quasi-compact, we can choose a representable sheaf X0 and an
effective epimorphism f : X0 → X. Let X• denote the Čech nerve of f . Using Proposition 2.5.2, we deducethat Sol(O(X• )) is the Čech nerve of the map Sol(O(X0 )) → Sol(O(X)). Each Xi belongs to Shvfc,≤m−1,
R
so that uXi is an equivalence by the inductive hypothesis. It follows that X• can be identified with theuXČech nerve of the composite map X0 → X →Sol(O(X)). Consequently, to show that uX is an equivalence,
it will suffice to show that uX is an effective epimorphism. Unwinding the definitions, this amounts to thefollowing assertion:(∗0 ) Let R0 be an étale R-algebra suppose we are given a map of R-algebras φ : O(X) → R0 . Then, forevery maximal ideal m ⊆ R0 , there exists an étale map ψ : R0 → A such that A/mA 6= 0, and thecomposition ψ ◦ φ is homotopic to the map given by a point η ∈ X(A).
Let φ : O(X) → R0 be as in (∗0 ). Using Theorem 2.3.8, we deduce that there exists a subring R0 ⊆ R
which is finitely generated over Z such that X is the restriction of a p-constructible sheaf Y ∈ Shvét
R0 .
Enlarging R0 if necessary, we can assume that R0 is given by R ⊗R0 R00 for some étale R0 -algebra R00 . Let
O00 denote the structure sheaf of Specét R0 . Using Theorem 2.5.1, we can assume that φ arises from a map
φ0 : O00 (Y ) → R00 . To prove (∗), we may replace R by R0 , X by Y , and R0 by R00 . We may therefore
assume without loss of generality that the commutative ring R is finitely generated over Z; in particular, R
is Noetherian.
For every map of commutative rings ψ : R0 → A, let XA denote the image of X in the ∞-topos ∞-topos
ShvétA , and let F (A) denote the homotopy fiber of the map of spaces XA (A) → MapCAlgR (O(X), A) over the
point given by ψ ◦ φ. Theorem 2.3.8 implies that the functor A 7→ XA (A) commutes with filtered colimits,
and Theorem 2.5.1 implies that the functor A 7→ MapCAlgR (O(X), A) commutes with filtered colimits. It
65

follows that F commutes with filtered colimits. We wish to prove that for every maximal ideal m ⊆ R0 , there
exists an étale R0 -algebra A such that A/mA 6= 0 and F (A) is nonempty.
Let k denote the residue field R0 /m and let k denote a separable closure of k. Using the first part ofthe proof together with Corollary 2.4.10, we deduce that F (k) is nonempty. Since F commutes with filteredcolimits, there exists a finite separable extension ek of k such that F (ek) 6= ∅. Using the structure theory ofe such that R/me Re'eétale morphism (Proposition VII.8.10), we see that there exists an étale R0 -algebra R
k.
∧
∨
e
e
e
e
e
e
In particular, mR is a maximal ideal of R. Let R denote the completion of R at mR. Then R is theinverse limit of a tower of square-zero extensions
e 3R
e → R/me 2R
e → R/me R.
e· · · → R/me∨ ) →
Since OX (X) is formally étale over R, we conclude that the reduction map MapCAlgR (OX (X), Re∧ is a complete local Noetherian ring, it is Henselian
MapCAlg (OX (X), ek) is a homotopy equivalence. Since RRe∧ ) → Xe (e(Proposition VII.7.16) so that the map XRe∧ (Rk k) is a homotopy equivalence (Proposition XI.3.22).
∧eee∧ ) is nonempty.
It follows that the map F (R ) → F (k) is a homotopy equivalence. In particular, F (ReSince R is an étale R-algebra, it is a finitely generated Z-algebra, and therefore an excellent ring (seee→Re∨ is geometrically regular. Using Popescu’s theorem ([67]), we see that[56]). It follows that the map Re∨ can be obtained as filtered colimit of smooth R-algebras.
e
R
Since F commutes with filtered colimits, weee∨ such that F (B) is nonempty.
conclude that there exists a smooth R-algebra
B equipped with a map B → Re∨ /mRe∨ ' eSince Rk 6= 0, we conclude that B/mB 6= 0. We can therefore choose a map of rings B → A
e (and therefore also over R0 ). Since F (B) 6= ∅, we conclude thatsuch that A/mA 6= 0 and A is étale over R
F (A) 6= ∅, as required by (∗).
Suppose we are given a p-constructible morphism q : (Y, OY ) → X. Assume that the structure sheaf of
X is truncated, so that OY is also truncated and therefore q∗ OY is a quasi-coherent sheaf on X (Corollary
VIII.2.5.22). It follows from Theorem 2.6.13 that, under some mild hypotheses, we can recover Y as a sortof relative spectrum of the quasi-coherent sheaf of E∞ -rings q∗ OY on X. In other words, p-constructible
morphisms behave, in some respects, as if they are affine. We formulate another result to this effect.
Notation 2.6.14. Let q : Y = (Y, OY ) → X be a map of spectral Deligne-Mumford stacks. Assume thatq is ∞-quasi-compact and that the structure sheaf OY is truncated. Then q∗ OY is a commutative algebra
object of QCoh(X) (Corollary VIII.2.5.22). We let Modq∗ OY (QCoh(X)≤0 ) denote the full subcategory ofModq∗ OY (QCoh(X)) spanned by those q∗ OY -modules whose image in QCoh(X) belongs to QCoh(X)≤0 , sothat q∗ induces a functor
QCoh(Y)≤0 → Modq∗ OY (QCoh(X)≤0 ).
Theorem 2.6.15. Let p be a prime number and let q : Y = (Y, OY ) → X be a p-constructible morphism
between p-thin spectral Deligne-Mumford stacks. Then the pushforward functor q∗ induces an equivalence of∞-categories
QCoh(Y)≤0 → Modq∗ OY (QCoh(X)≤0 ).
The proof of Theorem 2.6.15 will require some preliminaries.
Lemma 2.6.16. Let R be a p-thin E∞ -ring and let X be a spectral Deligne-Mumford stack over R. Letx ∈ π0 R be an element, define R0 , R1 ∈ CAlgR as in Remark 2.4.4, and set
X0 = Specét R0 ×Specét R X

X1 = Specét R1 ×Specét R X,

so that we have a closed immersion i : X0 → X and an open immersion j : X1 → X. Let F be a quasi-coherent
sheaf on X. Then:(1) The sheaf F belongs to QCoh(X)≤0 if and only if j ∗ F ∈ QCoh(X1 )≤0 and i∗ F ∈ QCoh(X0 )≤1 .
66

(2) The sheaf F is zero if and only if j ∗ F and i∗ F are zero.
(3) A map α : F → F0 in QCoh(X) is an equivalence if and only if i∗ F and j ∗ F are equivalences.
Proof. The implication (2) ⇒ (3) is obvious, and the implication (1) ⇒ (2) follows from the fact thatthe t-structure on QCoh(X) is right complete. Assertion (1) is local on X; we may therefore suppose that
X = Specét A is affine. Then X0 ' Specét A0 and X1 ' Specét A1 are also affine. We will identify x with its
image in π0 A, so that A1 ' A[x−1 ] and we have a fiber sequence of A-modulesx

A → A → A0 .
Let us identify F with an A-module M , so that j ∗ F can be identified with M [ x1 ] = A[ x1 ] ⊗A M and i∗ Fxwith the cofiber cofib(M → M ). It is now clear that F ∈ QCoh(X)≤0 implies that j ∗ F ∈ QCoh(X1 )≤0 andxi∗ F ∈ QCoh(X0 )≤1 . To prove the converse, let us assume that M [ x1 ] ∈ (ModA )≤0 and that cofib(M →M ) ∈ (ModA )≤1 . We wish to show that M ∈ (ModA )≤0 . Form a fiber sequence
1
K → M → M [ ];x
we will show that K ∈ (ModA )≤0 . Note that K ' Q[−1] ⊗A M , where Q denotes the cofiber of the map
A → A[ x1 ]. Then Q ' lim Qm , where Qm is the cofiber of the map xm : A → A. It will therefore suffice−→
to prove that Qm ⊗A M ∈ (ModA )≤1 for each m. When m = 1, this follows from our assumption onx
cofib(M → M ). The general case follows by induction on m, using the existence of fiber sequences
Qm → Qm+1 → Q1 .

Lemma 2.6.17. Let G be a finite p-group and let V be an Fp -vector space equipped with an action of G. If
V 6= 0, then V contains a nonzero element which is fixed by the action of G.
Proof. Replacing V by the G-submodule generated by any nonzero element v ∈ V , we may suppose that
V is a finite set. Every nontrivial orbit for the action of G on V has cardinality divisible by p. Since thecardinality of V is divisible by p, we conclude that the cardinality of the set of G-fixed points is divisible byp. In particular, 0 ∈ V cannot be the only fixed point.
Proof of Theorem 2.6.15. The assertion is local on X. We may therefore assume without loss of generalitythat X = Specét R, where R is a p-thin E∞ -ring.
Consider the pushforward functor G : ModOY → Modq∗ OY (in the setting of sheaves which are notnecessarily quasi-coherent), and let
G0 : QCoh(Y)≤0 → Modq∗ OY (QCoh(X)≤0 )denote the restriction of G. We wish to show that G0 is an equivalence of ∞-categories. The functor G
admits a left adjoint F , given by
F (M) = OY ⊗q∗ q∗ OY q ∗ M .
From this description it is clear that F carries quasi-coherent sheaves of q∗ OY -modules to quasi-coherent
sheaves on Y. The main step of the proof is to establish the following:(∗) Let M ∈ Modq∗ OY (QCoh(X)≤0 ). Then F (M) ∈ QCoh(X)≤0 . Moreover, the unit map M → (G◦F )(M)
is an equivalence.
Write X = (X, OX ) and Y = (X/U , OX |U ) for some p-constructible sheaf U ∈ X ' Shvét
R . By repeateduse of Lemma 2.6.16, we are reduced to proving (∗) in the special case where U is locally constant and p = 0
in π0 R. Since the assertion is local on X, we may reduce further to the case where U is the constant sheaf
associated to a p-finite space K. In this case, we can identify q∗ OY with the E∞ -ring C ∗ (K; R). We can
now rephrase (∗) as follows:67(∗0 ) Let M be a C ∗ (K; R)-module with πm M ' 0 for m > 0. Then, for each x ∈ K, the moduleC ∗ ({x}; R) ⊗C ∗ (K;R) M belongs to (ModR )≤0 . Moreover, the canonical map
M → lim (C ∗ (K; {x}) ⊗C ∗ (K;R) M )←−x∈Kis an equivalence.
Since R is strongly commutative, it admits the structure of a simplicial commutative ring. In particular,
we have a map of simplicial commutative rings Z → R. Let e0 , ep : Z[X] → Z be the maps carrying thevariable X to 0 and p, respectively. Since p = 0 in π0 R, the diagram
Z[X]

e0

ep


Z

/Z

/R

commutes up to homotopy (in the ∞-category of simplicial commutative rings), so there exists a map
Fp ' (Z ⊗Z[X] Z) → R. Corollary 2.4.18 supplies an equivalence of E∞ -ringsC ∗ (K; R) ' R ⊗Fp C ∗ (K; Fp ).
It will therefore suffice to prove (∗0 ) in the special case where R = Fp .
By breaking K up into its connected components, we may reduce to the case where K is connected.
In this case, C ∗ (K; Fp ) is a coconnective E∞ -algebra over Fp , in the sense of Definition VIII.4.1.1. Using
Proposition VIII.4.1.9 (and Remark VIII.4.1.10), we may assume that M is given by the colimit of a sequenceof C ∗ (K; Fp )-modules
0 = M (0) → M (1) → M (2) → · · · ,where each M (n) fits into a fiber sequence
M (n) → M (n + 1) → C ∗ (K; Fp ) ⊗Fp V (n)for some V (n) ∈ (ModFp )≤1 . Fix a point x ∈ K, and consider the corresponding map C ∗ (K; Fp ) →C ∗ ({x}; Fp ) ' Fp . ThenFp ⊗C ∗ (K;Fp ) M ' lim Fp ⊗C ∗ (K;Fp ) M (n).
−→Consequently, to prove that πm (Fp ⊗C ∗ (K;Fp ) M ) vanishes for m > 0, it will suffice to prove that
πm (Fp ⊗C ∗ (K;Fp ) M (n)) ' 0
for m > 0 and each n ≥ 0. We proceed by induction on n, the case n = 0 being trivial. The inductive step
follows from the observation that
πm (Fp ⊗C ∗ (K;Fp ) C ∗ (K; Fp ) ⊗Fp V (n)) ' πm V (n) ' 0
for m > 0.
Let N denote the local system of Fp -module spectra on K given by x 7→ C ∗ ({x}; Fp ) ⊗C ∗ (K;Fp ) M .
To complete the proof of (∗0 ), we wish to show that the canonical map θM : M → C ∗ (K; N ) is anequivalence. For each n ≥ 0, let N (n) denote the local system of Fp -module spectra on K given by
x 7→ C ∗ ({x}; Fp ) ⊗C ∗ (K;Fp ) M (n). Then Then N ' lim N (n), and the above argument shows that each−→
N (n) takes values in (ModFp )≤0 . Since K is p-finite, the functor C ∗ (K; •) commutes with filtered colimits
when restricted to (ModK
Fp )≤0 . It will therefore suffice to show that each of the maps θM (n) is an equivalence.
We proceed by induction on n, the case n = 0 being trivial. To carry out the inductive step, it suffices toshow that each of the maps θC ∗ (K;Fp )⊗Fp V (n) is an equivalence. Since Fp is a field, we can write V (n) as
68

a direct sum of Fp -modules of the form Fp [m], where m ≤ 0. Using the fact that the functor C ∗ (K; •)commutes with direct sums when restricted to (ModK
Fp )≤0 , we are reduced to proving that θC ∗ (K;Fp )[m] is anequivalence for m ≤ 0, which is clear. This completes the proof of (∗0 ) (and therefore also the proof of (∗)).
It follows from (∗) that the functor G0 admits a left adjoint F0 : Modq∗ OY (QCoh(X)≤0 ) → QCoh(Y)≤0 ,and that the counit map id → G0 ◦ F0 is an equivalence. In particular, F0 is fully faithful. To complete
the proof that G0 is an equivalence of ∞-categories, it will suffice to show that G0 is conservative. Supposethat M ∈ QCoh(Y)≤0 and that q∗ M ' 0; we wish to prove that M ' 0. We may again use Lemma 2.6.16
to reduce to the case where U is locally constant and p = 0 in π0 R. The assertion is local on X, so we may
pass to an étale cover and thereby reduce to the case where U is the constant sheaf determined by a p-finite
space K. In this case, we can identify M with a local system of R-modules on K. We wish to prove that
if M 6= 0, then C ∗ (K; M) 6= 0. Since M is truncated, there exists a largest integer n such that πn M 6= 0.
Choose a point x ∈ K such that πn M(x) 6= 0. Let V = πn M(x). We will view V as an Fp -vector spaceequipped with an action of the fundamental group G = π1 (K, x). Since πm M ' 0 for all m > n, the vector
space H0 (G; V ) appears as a direct summand in πn C ∗ (K; M). Since G is a finite p-group, Lemma 2.6.17
implies that C ∗ (K; M) 6= 0, as desired.

3

p-Profinite Homotopy Theory

Let p be a prime number and let Sp−fc denote the ∞-category of p-finite spaces (Definition 2.4.1). We letSPro(p) denote the ∞-category Pro(Sp−fc ). We will refer to the objects of SPro(p) as p-profinite spaces.
Our goal in this section is to develop the homotopy theory of p-profinite spaces. Our main result can bestated as follows:(∗) Let k be an algebraically closed field of characteristic p. Then the functor X 7→ C ∗ (X; k) on p-finite
spaces extends to a fully faithful embedding SPro(p) → CAlgop
k . Moreover, the essential image of thisembedding has a simple algebraic description.
Let us now outline the organization of this section. We begin in §3.1 by reviewing the definition of
Pro(C), where C is an ∞-category. With an eye toward some later applications, we treat the case where C
is an accessible ∞-category which is not essentially small. We then extend Theorem 2.6.10 to the setting of
proconstructible sheaves. Specializing to the case of an algebraically closed (or, more generally, a separablyclosed) field k, we obtain the fully faithful embedding SPro(p) → CAlgop
k described in (∗) (Proposition 3.1.16).
In §3.2, we begin our study of p-profinite spaces in proper. To every p-profinite space X we can associate
an ordinary space Mat(X) = MapSPro(p) (∗, X), which we call the materialization of X. Given a pointη ∈ Mat(X), we define
π0 X = π0 Mat(X)
πn (X, η) = πn (Mat(X), η)
for n ≥ 1. We will see that π0 X is equipped with the structure of a totally disconnected compact Hausdorff
space, and that each πn (X, η) has the structure of a p-profinite group. Moreover, these homotopy groups
control the structure of X: if f : X → Y is a map of p-profinite spaces which induces an isomorphism on all
homotopy groups, then f is an equivalence (Theorem 3.2.2).
Let X be a p-profinite space. We will say that X is of finite type if it is simply connected (that is, thematerialization Mat(X) is simply connected) and the homotopy groups πn X are finitely generated modules
over the ring Zp of p-adic integers, for all n ≥ 2. In §3.3, we will show that the materialization functor
Mat : SPro(p) → S is fully faithful when restricted to p-profinite spaces of finite type (Theorem 3.3.3).
The homotopy theory of p-profinite spaces behaves in many respects like the usual homotopy theory of
spaces. For example, to every p-profinite space X on can associate a Postnikov tower· · · → τ≤2 X → τ≤1 X → τ≤0 X
having limit X. In §3.4, we will study the relative version of this Postnikov tower. That is, we will introduce
the notions of n-connective and n-truncated morphisms in SPro(p) , and show that they behave much like the69usual theory of n-connective and n-truncated morphisms of spaces. In particular, every map f : X → Y of
p-profinite spaces admits an essentially unique factorization
f0

f 00

X → Z → Y,
where f 0 is n-truncated and f 00 is (n − 1)-connective (Theorem 3.4.2).
In §3.5 we will complete the proof of (∗) by characterizing the essential image of the embedding SPro(p) →CAlgopk (at least in the case where k is algebraically closed; see Theorem 3.5.8). We will also characterize
the essential image of the p-profinite spaces of finite type. Combining this the results of §3.3, we recover thefollowing result of Mandell (see Corollary 3.5.15):(∗0 ) Let k be an algebraically closed field of characteristic p, and let C denote the full subcategory of S
spanned by those spaces X which are simply connected and for which each homotopy group πn X is a
finitely generated module over the ring Zp of p-adic integers. Then the functor X 7→ C ∗ (X; k) extends
to a fully faithful embedding C → CAlgop
k . Moreover, the essential image of this embedding admits a
simple algebraic description.
p−fc
For any p-thin E∞ -ring R, Corollary 2.6.12 supplies a fully faithful embedding ShvR
→ CAlgop
R . Theproofs of (∗) and (∗0 ) rest ultimately on a simple special case of this: namely, the case where R is analgebraically closed field of characteristic p. In §3.6, we will describe a different specialization of Corollary
2.6.12: we allow R to be arbitrary, but restrict our attention to sheaves which are constant. This leads to an
algebraic description of the p-adic étale homotopy type of Specét R (or, more generally, any p-thin spectral
Deligne-Mumford stack: see Theorem 3.6.3).

Remark 3.0.18. For another approach to the homotopy theory of profinite spaces, we refer the reader to
[80].

3.1

Pro-Constructible Sheaves and p-Profinite Spaces

Let R be an E∞ -ring and p a prime number. We let Shvp−fc
denote the full subcategory of Shvét
R spannedRby the p-constructible sheaves. If R is p-thin, then Corollary 2.6.12 gives a fully faithful embedding from
Shvp−fc
into the ∞-category CAlgop
R
R . In this section, we will consider an extension of this embedding toPro-objects of Shvp−fc
.
We
begin
with
a general review of the theory of Pro-objects.
R
Definition 3.1.1. Let C be an accessible ∞-category which admits finite limits. We let Pro(C) denote thefull subcategory of Fun(C, S)op spanned by those functors F : C → S which are accessible and preserve finite
limits. We will refer to Pro(C) as the ∞-category of Pro-objects of C.
Example 3.1.2. Let C be a small ∞-category which admits finite limits. Then C is accessible if and only
if it is idempotent complete (Proposition T.5.4.2.17). Moreover, any functor F : C → S is automaticallyaccessible. It follows that Pro(C) is the full subcategory of Fun(C, S)op spanned by those functors which
preserve finite limits. We therefore have Pro(C) ' Ind(Cop )op , where Ind(Cop ) is defined as in §T.5.3.5.
Example 3.1.3. Let C be the nerve of an accessible category C0 . Then Pro(C) is itself equivalent to thenerve of a category, which we will denote by Pro(C0 ). We can identify Pro(C0 ) with the usual category of
Pro-objects of C0 : that is, it is a category whose objects are small filtered diagrams {Xα } taking values in
C0 , with morphisms given by
HomPro(C0 ) ({Xα }, {Yβ }) = lim lim HomC0 (Xα , Yβ ).
←− −→
α
β

Remark 3.1.4. If C is an accessible ∞-category which admits finite limits, then every corepresentablefunctor C → S is accessible and preserves finite limits. It follows that the Yoneda embedding determines afunctor j : C → Pro(C), which we will also refer to as the Yoneda embedding.
70

Remark 3.1.5. Let C be an accessible ∞-category which admits finite limits. Then the collection ofleft-exact accessible functors from C to S is closed under filtered colimits. It follows that the ∞-category
Pro(C) admits small filtered limits. Moreover, filtered limits are computed “pointwise”: that is, the inclusion
Pro(C) ,→ Fun(C, S)op preserves small filtered limits.
The ∞-category Pro(C) can be characterized by the following universal property:
Proposition 3.1.6. Let C be an accessible ∞-category which admits finite limits, let D be an ∞-category
which admits small filtered limits, and let Fun0 (Pro(C), D) denote the full subcategory of Fun(Pro(C), D)
spanned by those functors which preserve small filtered limits. Then composition with the Yoneda embedding
restricts to an equivalence of ∞-categories
Fun0 (Pro(C), D) → Fun(C, D).
We can state Proposition 3.1.6 more informally as follows: the ∞-category Pro(C) is obtained from C by
freely adjoining filtered limits.
Proof. Let b
S denote the ∞-category of spaces which are not necessarily small, let E denote the smallest
full subcategory of Fun(C, bS) which contains the essential image of the Yoneda embedding and is closed
under small filtered colimits, and let Fun0 (Eop , D) denote the full subcategory of Fun(Eop , D) spanned bythose functors which preserve small filtered limits. Using Remark T.5.3.5.9, we see that composition with
the Yoneda embedding induces an equivalence of ∞-categories Fun0 (Eop , D) → Fun(C, D). It will therefore
suffice to show that E is equivalent to Pro(C)op (as subcategories of Fun(C, b
S). Using Remarks 3.1.4 and3.1.5, we see that E is contained in the essential image of Pro(C)op . We will complete the proof by verifying
the following:(∗) Let F : C → S be an accessible functor which preserves finite limits. Then F can be written as a small
filtered colimit lim Fα , where each of the functors Fα : C → S is corepresentable by an object of C.
−→
To prove (∗), choose a regular cardinal κ such that C is κ-accessible and the functor F : C → S preserves
κ-filtered colimits. Let Cκ denote the full subcategory of C spanned by the κ-compact objects. Enlarging
κ if necessary, we may assume Cκ is closed under finite limits. Let F κ denote the restriction of F to Cκ .
Since Cκ is essentially small and F κ is left exact, we can write F κ as a filtered colimit of functors Fακ ,where each Fακ is corepresentable by an object of Cκ (Corollary T.5.3.5.4). Since the Yoneda embeddinghκ : (Cκ )op → Fun(Cκ , S) is fully faithful, we can write Fακ = hκ (Cα ) for some filtered diagram {Cα } in(Cκ )op . Let h : Cop → Fun(C, S) denote the Yoneda embedding for C, and let F 0 = limα h(Cα ). We will−→complete the proof of (∗) by showing that F 0 ' F . By construction, F and F 0 have the same restriction
to Cκ . Since C is κ-accessible, it will suffice to show that the functors F and F 0 preserve small κ-filteredcolimits (Proposition T.5.3.5.10). For the functor F , this follows by assumption. To show that F 0 commuteswith κ-filtered colimits, it will suffice to show that each h(Cα ) commutes with κ-filtered colimits; this follows
from the fact that each Cα is κ-compact.
Remark 3.1.7. Let f : C → D be a functor between accessible ∞-categories which admit small limits.
fThen the composite functor C → D → Pro(D) induces a map Pro(f ) : Pro(C) → Pro(D), which commuteswith small filtered limits. If f is fully faithful, then Pro(f ) is fully faithful. If f is accessible and preserves
finite limits, then composition with f induces a functor F : Pro(D) → Pro(C). It is not difficult to see that
F (when defined) is a right adjoint to Pro(f ).
p−fc
Notation 3.1.8. Let R be an E∞ -ring, let p be a prime number, and let ShvR
denote the full subcategory
Pro(p)
étof ShvR spanned by the p-constructible sheaves. We let ShvR
denote the ∞-category Pro(Shvp−fc). We
R
Pro(p)
étwill refer to ShvR
as the p-profinite sheaves on Spec R.
ét
Let O denote the structure sheaf of Specét R, and regard O as a functor from Shvét
R to CAlgR . Using
Proposition 3.1.6, we see that O admits an essentially unique extensionb : ShvPro(p) → CAlgop
O
R
R
71

which commutes with small filtered limits.
We can now prove an extension of Corollary 2.6.12:
Pro(p)b : Shv
Theorem 3.1.9. Let p be a prime number, let R be an E∞ -ring which is p-thin, and let O
R
CAlgopbethefunctordescribed
in
Notation
3.1.8.
Then:
R→

b commutes with small limits.
(1) The functor O
Pro(p)(2) For every pro-p sheaf X ∈ ShvR
Pro(p)b : Shv(3) The functor ORb, the E∞ -ring O(X)
is formally étale over R.

→ CAlgop
R is fully faithful.

Proof. Assertion (1) follows from Propositions T.5.5.1.9 and 2.5.2, and assertion (2) follows from Theorem
2.4.9 (note that the collection of formally étale R-algebras is closed under small colimits in CAlgR ). We
p−fc
op
ét
, then O(X) is aprove (3). Let O : Shvét
R → CAlgR be the structure sheaf of Spec R. If X ∈ ShvR
compact object of CAlgR by Theorem 2.5.1. In view of Proposition T.5.3.5.11, it will suffice to show thatis fully faithful. We conclude by invoking Corollary 2.6.12.
the restriction O | Shvp−fc
R
p−fc
ét
into
If f : R → R0 is a map of E∞ -rings, then the pullback functor f ∗ : Shvét
R → ShvR0 carries ShvR
Pro(p)
Pro(p)
p−fc∗∗bb
ShvR0 , and therefore induces a functor f : ShvR
→ ShvR0. Note that f is an equivalence wheneverPro(p)
depends only on the underlying commutative ringf ∗ is an equivalence. In particular, we see that ShvR
π0 R.

Corollary 3.1.10. Let R be an E∞ -ring, and suppose we are given a flat hypercovering R → R• of R. If p
is nilpotent in π0 R, then the canonical map
Pro(p)ShvR

Pro(p)→ lim ShvR•←−

is fully faithful.
Proof. We may assume without loss of generality that R is discrete (so that R• is also discrete). Then R is
p-thin, and each Rn is p-thin. We have a commutative diagram of ∞-categories
ShvR

Pro(p)/ lim ShvPro(p)R←−
CAlgopR/ lim CAlgop• .
R←−

The vertical functors are fully faithful by Theorem 3.1.9, and the bottom horizontal map is an equivalencePro(p)
Pro(p)
of ∞-categories by Theorem VII.5.14. It follows that the functor ShvR
→ lim ShvR•
is fully faithful,←−
as desired.
Pro(p)

Warning 3.1.11. In the situation of Corollary 3.1.10, the functor ShvR
essentially surjective.

Pro(p)→ lim ShvR←−is generally not

We now specialize Theorem 3.1.9 to the case where R is a separably closed field.
Definition 3.1.12. Let p be a prime number and let Sp−fc denote the full subcategory of S spanned by thep-finite spaces (Definition 2.4.1). A p-profinite space is a pro-object of the ∞-category Sp−fc . We let SPro(p)denote the ∞-category Pro(Sp−fc ) of p-profinite spaces.
Remark 3.1.13. The Yoneda embedding is a fully faithful functor j : Sp−fc → SPro(p) . We will generally
abuse notation by identifying a p-finite space X with its image j(X) ∈ SPro(p) .
72

Notation 3.1.14. Let k be a field of characteristic p. Corollary 1.1.10 implies that the construction X 7→
C ∗ (X; k) defines a functor F0 : (Sp−fc ) → CAlgop
k which commutes with finite limits. Using Propositions
T.5.3.5.10 and T.5.5.1.9, we see that F0 admits an essentially unique extension to a functor F : SPro(p) →∗CAlgopk which commutes with small limits. We will denote this functor also by X 7→ C (X; k).
Remark 3.1.15. Let k be a separably closed field, so that the global sections functor induces an equivalencep−fc
of ∞-categories Shvét' Sp−fc , which gives rise to an equivalence
k ' S. This restricts to an equivalence Shvk
Pro(p)
Pro(p)of Pro-objects Shvk
'S
. Under this equivalence, the functor X 7→ C ∗ (X; k) of Notation 3.1.14
Pro(p)b : Shv
coincides with the functor O→ CAlgopkk appearing in Theorem 3.1.9.
Combining Theorem 3.1.9 with Remark 3.1.15, we obtain the following result:
Proposition 3.1.16. Let k be a separably closed field of characteristic p > 0. Then the construction
X 7→ C ∗ (X; k) determines a fully faithful embedding from the ∞-category SPro(p) of p-profinite spaces to the∞-category CAlgopk .
Remark 3.1.17. Proposition 3.1.16 does not really require the full strength of the hypothesis that k isseparably closed. It suffices to assume that the Galois cohomology of k is trivial modulo p: equivalently,that that Artin-Schreier map x 7→ x − xp is a surjection from k to itself.
We will study the essential image of the embedding SPro(p) → CAlgop
k in §3.5.

3.2

Whitehead’s Theorem

Let p be a prime number. In §3.1, we introduced the ∞-category SPro(p) of p-profinite spaces. Our goal inthis section is to show that the ∞-category SPro(p) behaves, in many respects, like the usual ∞-category S ofspaces. In particular, we will see that there is a good theory of connective and truncated objects in SPro(p) ,and that every object of SPro(p) has a convergent Postnikov tower.
We begin by establishing a direct connection of SPro(p) with the usual ∞-category of spaces.
Definition 3.2.1. Let p be a prime number and let SPro(p) = Pro(Sp−fc ) denote the ∞-category of p-profinite
spaces. We let Mat : SPro(p) → S denote the functor corepresented by the final object ∗ ∈ SPro(p) , so that
Mat(X) ' MapSPro(p) (∗, X). We will refer to Mat as the materialization functor.
Let X be a p-profinite space. A point of X is a point of the materialization Mat(X). We let π0 X denotethe set π0 Mat(X) of all path components of the materialization of X. Given a point η of X and an integern ≥ 1, we let πn (X, η) denote the homotopy group πn (Mat(X), η).
We can formulate our main result as follows:
Theorem 3.2.2 (Whitehead’s Theorem for p-Profinite Spaces). Let f : X → Y be a map of p-profinite
spaces. Then f is an equivalence if and only if the following conditions are satisfied:(1) The map f induces a bijection π0 X → π0 Y .
(2) For every point η of X and every n ≥ 1, the map f induces an isomorphism of profinite p-groups
πn (X, η) → πn (Y, f (η)).
Remark 3.2.3. Theorem 3.2.2 can be reformulated as follows: for every prime number p, the materialization
functor Mat : SPro(p) → S is conservative.
Before giving the proof of Theorem 3.2.2, let us describe an application. Since the ∞-category S is an
d∞ which carries each space X to the ∞-category S/X of spaces over X∞-topos, the functor Sop → Cat
preserves small limits (see Theorem T.6.1.3.9). It follows that for every Kan complex X, we have a canonicalequivalence S/X ' Fun(X, S), depending functorially on X. We can describe this equivalence informallyas follows: it assigns to a map of spaces Y → X the functor X → S which carries a point x ∈ X to thehomotopy fiber Yx = Y ×X {x}. We now prove that an analogous statement holds in the setting of p-profinite
spaces:73

Proposition 3.2.4. Let p be a prime number and let X be a p-finite space. Then:(1) Every diagram e : X → SPro(p) admits a colimit lim(e) ∈ SPro(p) . Consequently, the formation of−→colimits determines a functor lim : Fun(X, SPro(p) ) → SPro(p) .
−→(2) Let e0 denote a final object of Fun(X, SPro(p) ) (that is, the constant functor from X to SPro(p) taking
the value ∗ ∈ SPro(p) ). Then there is a canonical equivalence lim(e0 ) ' X in SPro(p) .
−→(3) The functor lim induces an equivalence of ∞-categories−→
Pro(p)

F : Fun(X, SPro(p) ) ' Fun(X, SPro(p) )/e0 → S/X.

Proof. Since the ∞-category Sp−fc admits finite limits, the ∞-category (SPro(p) )op ' Ind((Sp−fc )op ) is presentable (Theorem T.5.5.1.1). It follows that SPro(p) admits all small limits and colimits, which proves (1).
If we regard e0 as a diagram X → Sp−fc , then it has colimit X. Assertion (2) now follows from the observation that the Yoneda embedding j : Sp−fc → Fun(Sp−fc , S)op preserves all colimits which exist in Sp−fc(Proposition T.5.1.3.2).
Pro(p)
We now prove (3). We first note that the functor F admits a right adjoint G : S/X→ Fun(X, S).
Unwinding the definitions, we see that G carries a map of p-profinite spaces Y → X to the functor G(Y ) ∈Fun(X, SPro(p) ) given on objects by the formula G(Y )(x) = Y ×X {x}. We first show that the unit map
u : id → G◦F is an equivalence of functors from Fun(X, SPro(p) ) to itself. To prove this, it will suffice to showthat for every functor e : X → SPro(p) and every point x ∈ X, the canonical map ue,x : e(x) → lim(e) ×X {x}−→is an equivalence of p-profinite spaces.
Let k be a separably closed field of characteristic p, so that the functor Y 7→ C ∗ (Y ; k) is a fully faithfulembedding SPro(p) → CAlgop
k which preserves small colimits (Proposition 3.1.16). Let e : X → CAlgk be
the functor given by e(y) = C ∗ (e(y); k) (using the identification X ' X op ). We are then reduced to provingthat ue,x induces an equivalence(lim e) ⊗C ∗ (X;k) C ∗ ({x}, k) → e(x)←−of E∞ -algebras over k. Using Proposition 1.1.9, we are reduced to proving that for every point y ∈ X andevery integer n ≥ 0, the action of the fundamental group G = π1 (X, y) on the vector space V = Hn (e(y); k)is nilpotent. Let k[G] denote the group algebra of G over k, and let I ⊆ k[G] be its augmentation ideal.
Then V has a G-invariant filtration
V ⊇ IV ⊇ I 2 V ⊇ · · · ,and the action of G is trivial on each quotient I m V /I m+1 V . To complete the proof, it suffices to show that
this filtration is finite. This follows from the fact that I is a nilpotent ideal (since G is a finite p-group andk is a field of characteristic p).
Since the natural transformation u is an equivalence, the functor F is fully faithful. To complete theproof that F is an equivalence of ∞-categories, it will suffice to show that the right adjoint G is conservative.
Pro(p)
Suppose we are given a morphism Y → Y 0 in S/X
which induces an equivalence Y ×X {x} → Y 0 ×X {x}for each x ∈ X. For each x ∈ X, we have homotopy equivalences
Mat(Y ) ×X {x} ' Mat(Y ×X {x}) ' Mat(Y 0 ×X {x}) ' Mat(Y 0 ) ×X {x}.
It follows that the map Mat(Y ) → Mat(Y 0 ) is a homotopy equivalence of spaces. Invoking Theorem 3.2.2,
we conclude that the map Y → Y 0 is an equivalence.
We now turn to the proof of Theorem 3.2.2. We begin with some general remarks about filtered limitsof spaces.

74Proposition 3.2.5. Let J be a filtered ∞-category and let X : Jop → S be a diagram of spaces indexed by
Jop . Assume that:(a) For every object J ∈ J, the set π0 X(J) is nonempty and finite.
(b) For every object J ∈ J, every point η ∈ X(J), and every integer n ≥ 1, the group πn (X(J), η) is finite.
Then the limit limJ∈Jop X(J) is nonempty.
←−
Proof. According to Proposition T.5.3.1.16, there exists a filtered partially ordered set A and a left cofinal
map N(A) → J. We may therefore replace J by N(A) and thereby assume that J is the nerve of a filteredpartially ordered set.
If n ≥ 0, we will say that X is n-truncated if the space X(α) is n-truncated for each α ∈ A. We firstprove that limα∈Aop X(α) is nonempty under the additional assumption that X is n-truncated. Our proof←−
proceeds by induction on n; the case n = 0 follows from Lemma XII.A.1.8.
Suppose that X is n-truncated for n > 0. Let X 0 : N(A)op → S denote the composition of X withthe truncation functor τ≤n−1 : S → S. Our inductive hypothesis implies that the limit limα∈Aop X 0 (α) is←−
nonempty. We will prove that X is nonempty by showing that the map θ : limα∈Aop X(α) → limα∈Aop X 0 (α)←−←−
is surjective on connected components. To this end, suppose we are given a point η ∈ limα∈Aop X 0 (α), so←−that η determines a natural transformation X00 → X 0 , where X00 denotes the constant functor N(A)op → S
taking the value ∆0 . Let X0 = X ×X 0 X00 . To prove that the homotopy fiber of θ over η is nonempty, wemust show that limα∈Aop X0 (α) is nonempty. Note that for each α ∈ A, the space X0 (α) is an n-gerbe: that←−is, it is both n-truncated and n-connective. In particular, since n > 0, each of the spaces X0 (α) is connected.
Let B denote the collection of all finite subsets B ⊆ A which contain a largest element. Let K denotethe simplicial subset of N(A) given by the union of all the vertices. For each B ∈ B, let KB ⊆ N(A) denotethe union K ∪ N(B). Regard B as a partially ordered set with respect to inclusions, and define a functorop
Y : N(B)op → S by the formula Y (B) = lim(X0 |KB) (see §T.4.2.3). Using Proposition T.4.2.3.8, we obtain a←−homotopy equivalence limB∈Bop Y (B) ' limα∈A X0 (α). It will therefore suffice to show that limB∈Bop Y (B)
←−
←−
Q ←−
is nonempty. Let M = lim(X0 |K) = α∈A X0 (α) and let Z : N(Bop ) → S be the constant functor taking the←−value M . Note that B is filtered, so that limB∈Bop Z(B) ' M . We have an evident natural transformation←−of functors Y → Z which induces a map θ0 : limB∈Bop Y (B) → limB∈Bop Z(B) ' M . Since M is nonempty,←−←−
it will suffice to show that the homotopy fibers of θ0 are nonempty.
Choose a point ζ ∈ M , corresponding to a collection of points {ζα ∈ X0 (α)}α∈A . The point ζ determinesa natural transformation of functors Z0 → Z, where Z0 : N(B)op → S is the constant functor taking thevalue ∆0 . Let Y0 = Y ×Z Z0 , so that the homotopy fiber of θ0 over the point ζ is given by limB∈Bop Y0 (B).
←−Fix an element B ∈ B, so thatQ
B is a subset of A which contains
a largest element β. We have homotopy
Q0equivalences Y (B) ' X0 (β) × α∈B/ X0 (α) and Z(B) 'α∈A X0 (α). For each α ∈ B, let ζα denote theimage of ζβ under the map X0 (β) → X0 (α). Unwinding the definitions, we see that Y0 (B) can be identified
with the product over all α ∈ B − {α} of the space of paths joining ζα with ζα0 in X0 (α). Since each X0 (α) isa connected n-truncated space with finite homotopy groups, we conclude that Y0 (B) is a nonempty (n − 1)truncated space with finite homotopy groups. Since B is filtered, it follows from the inductive hypothesisthat limB∈Bop Y0 (B) is nonempty.
←−
We now treat the case of a general functor X : N(A)op → S. For each integer n, let τ≤n X denote thecomposition of X with the truncation functor τ≤n : S → S. Then X is the limit of the tower· · · → τ≤2 X → τ≤1 X → τ≤0 X,
so that limα∈Aop X(α) is given by the limit of the tower of spaces {limα∈Aop τ≤n X(α)}n≥0 . The above argu←−←−
ments show that limα∈Aop τ≤0 X(α) is nonempty and that each of the transition maps limα∈Aop τ≤n X(α) →←−←−limα∈Aop τ≤n−1 X(α) has nonempty homotopy fibers, from which it immediately follows that the tower←−{limα∈Aop τ≤n X(α)}n≥0 has nonempty limit.
←−
75

Corollary 3.2.6. Let J and X be as in Proposition 3.2.5, and let n ≥ 0 be an integer such that each X(J)is n-connective. Then X = limJ∈J X(J) is n-connective.
←−
Proof. We proceed by induction on n. If n = 0, the desired result follows from Proposition 3.2.5. Assume
therefore that n > 0. The inductive hypothesis implies that X is nonempty. It will therefore suffice toshow that, for every pair of points η, η 0 ∈ X, the path space {η} ×X {η 0 } is (n − 1)-connective. Let
X0 , X1 : Jop → S denote the constant functor taking the value ∆0 , so that η and η 0 determine naturaltransformations X0 → X ← X1 and we have a homotopy equivalence {η}×X {η 0 } ' limJ∈Jop (X0 ×X X1 )(J).
←−
Since X0 ×X X1 takes (n − 1)-connective values, the inductive hypothesis implies that {η} ×X {η 0 } is (n − 1)connective.
Corollary 3.2.7. Let p be a prime number, and let F : SPro(p) → N(Set) denote the functor given by
X 7→ π0 MapSPro(p) (∗, X). Then F commutes with filtered limits.
Proof. Let us abuse notation by identifying the ∞-category Sp−fc of p-finite spaces with a full subcategoryof the ∞-category SPro(p) = Pro(Sp−fc ) of p-profinite spaces, so that F0 = F | Sp−fc is the functor given
by X 7→ π0 X. Let F 0 : SPro(p) → N(Set) be a right Kan extension of F0 , so that the identification
F 0 | Sp−fc = F | Sp−fc extends to a natural transformation u : F → F 0 . Since F 0 commutes with filtered limits,
it will suffice to show that u is an equivalence. To this end, consider a p-profinite space K, which we canassume is given by the limit of a diagram X : Jop → Sp−fc for some filtered ∞-category J. We wish to showthat the canonical map
θ : π0 ( lim X(J)) → lim π0 X(J)
←−op
←−op
J∈J

J∈J

is a bijection. Choose a point η ∈ limJ∈Jop π0 X(J). Then η determines, for each J ∈ J, a connected←−component Xη (J) of X(J). We can regard Xη itself as a functor Jop → Sp−fc . Note that θ−1 {η} can be
identified with the set of path components of the limit Vη = limJ∈Jop Xη (J). To prove that θ is a bijection,←−
it will suffice to show that each of the spaces Vη is connected. This follows from Lemma 3.2.6, since eachXη (J) is connected by construction.
Remark 3.2.8. Let X be a p-profinite space. It follows from Corollary 3.2.7 that if X is given as afiltered limit of p-finite spaces (or even p-profinite spaces) limα Xα , then we have π0 X ' limα π0 Xα . Every←−←−point η ∈ X determines a compatible system of points ηα ∈ Xα , and Corollary 3.2.7 furnishes a canonicalisomorphism πn (X, η) ' limα πn (Xα , ηα ).
←−Remark 3.2.9. Let X = lim Xα be a p-profinite space. Using Corollary 3.2.7, we deduce that π0 X can be←−
identified with the Stone space associated to the profinite set given by the diagram {π0 Xα }. In particular,
the inverse limit topology endows π0 X with the structure of a compact, totally disconnected Hausdorff space.
To discuss the higher homotopy groups of a p-profinite space, it will be convenient to employ the languageof profinite groups. For the reader’s convenience, we give a brief review of the theory here.
Definition 3.2.10. A profinite group is a topological group G whose underlying topological space is compact,
Hausdorff, and totally disconnected. We let PFinGp denote the category whose objects are profinite groups
and whose morphisms are continuous group homomorphisms.
Example 3.2.11. Every finite group G can be regarded as a profinite group (when endowed with the discrete
topology). That is, we can regard the category FinGp of finite groups as a full subcategory of PFinGp.
Proposition 3.2.12. The inclusion FinGp ,→ PFinGp extends to an equivalence of categories Pro(FinGp) 'PFinGp.
Proof. Since the category of compact, totally disconnected Hausdorff spaces is closed under projective limits(in the larger category of all topological spaces; see Lemma XII.A.1.7), the category PFinGp is closed under
projective limits in the category of all topological groups. It follows that PFinGp admits small projective76

limits, so that the inclusion i : FinGp ,→ PFinGp extends to a functor F : Pro(FinGp) → PFinGp which
commutes with filtered limits (Proposition T.5.3.5.10). We first claim that F is fully faithful. According to
Proposition T.5.3.5.11, it will suffice to show that every finite group G is compact when viewed as an objectof PFinGpop . In other words, we claim that if we are given a filtered system of profinite groups Hα having
limit H, then the canonical map
lim HomPFinGp (Hα , G) → HomPFinGp (H, G)−→
is bijective. Proposition XII.A.1.6 implies that G is compact when viewed as an object of the oppositeof the category of compact, totally disconnected Hausdorff spaces. Consequently, every continuous grouphomomorphism f : H → G factors as a compositionf0

H → Hα → G,
for some index α. We must show that it is possible to choose α so that f 0 is a group homomorphism. To
prove this, we consider the pair of maps
u, v : Hα × Hα → G
given by u(x, y) = f 0 (xy), v(x, y) = f 0 (x)f 0 (y). Then u and v induce the same map from H × H into G.
Using Proposition XII.A.1.6 we conclude that there exists a map of indices β → α such that u and v agree
on Hβ × Hβ . Replacing α by β, we may assume that u = v so that f 0 is a group homomorphism, as desired.
It remains to prove that the functor F is essentially surjective. Fix a profinite group G, and let S
be the partially ordered set of open normal subgroups G0 ⊆ G. Then S op is filtered (since the collectionof open normal subgroups is closed under finite intersections). We may therefore view the inverse system{G/G0 }G0 ∈A as an object G ∈ Pro(FinGp). We will complete the proof by showing that the natural map
φ : G → F (G) is an isomorphism of profinite groups.
Note that every nonempty open subset of F (G) contains the inverse image of some element of G/G0 ,where G0 is an open normal subgroup of G. From this we immediately deduce that φ has dense image. Since
G is compact and F (G) is Hausdorff, it follows that φ is a quotient map: that is, φ induces an isomorphismof profinite groups G/ ker(φ) → F (G). We will complete the proof by showing that ker(φ) is trivial.
Choose a non-identity element x ∈ G; we wish to show that there exists an open normal subgroup of G
which does not contain x. Since G is totally disconnected, there exists a closed and open subset Y ⊆ G whichcontains the identity element but does not contain x. Let Y + = {gyg −1 : g ∈ G, y ∈ Y }. Then Y + image ofa continuous map G × Y → G. Since G is compact, we conclude that Y + is compact and therefore a closed
subset of G. As a union of conjugates of Y , Y + is also an open subset of G. Let G0 = {g ∈ G : gY + = Y + }.
Then G0 is a subgroup of G which does not contain x. Since Y + is conjugation-invariant, the subgroup G0
is normal. Moreover, the complement of G0 is given by the image of (G − Y + ) × Y + under the continuous
map (g, h) 7→ (gh−1 ). Since the product is a compact set, we conclude that G − G0 is compact, so that G0
is an open subgroup of G.
Definition 3.2.13. Let G be a profinite group and p a prime number. We say that G is p-profinite if, for everynopen neighborhood U ⊆ G containing the identity, there exists an integer n ≥ 0 such that {g p : g ∈ G} ⊆ U .
Proposition 3.2.14. Let p be a prime number, let FinGpp ⊆ FinGp denote the category of finite p-groups,ab
let FinGpab ⊆ FinGp be the category of finite abelian groups, and FinGpab
the categoryp = FinGpp ∩ FinGpof finite abelian p-groups. Then:(1) The equivalence Pro(FinGp) ' PFinGp restricts to an equivalence of Pro(FinGpab ) with the full subcategory of PFinGp spanned by the profinite abelian groups.
(2) The equivalence Pro(FinGp) ' PFinGp restricts to an equivalence of Pro(FinGpp ) with the full subcategory of PFinGp spanned by the p-profinite groups.
77

(3) The equivalence Pro(FinGp) ' PFinGp restricts to an equivalence of Pro(FinGpabp ) with the full subcategory of PFinGp spanned by the p-profinite abelian groups.
Proof. We will prove (2); the proofs of (1) and (3) are similar. We first show that if G is the inverse limit ofa filtered system Gα of finite p-groups, then G is p-profinite. To prove this, let U be an open neighborhoodof the identity in G. Then U contains the kernel of the map φα : G → Gα for some index α. Let pn be thennorder of Gα , so that g p = e for all g ∈ Gα . It follows that g p ∈ ker(φα ) ⊆ U for all g ∈ G.
Conversely, suppose that G is a p-profinite group. The proof of Proposition 3.2.12 shows that we canwrite G as the filtered inverse limit lim G/G0 , where G0 ranges over all open normal subgroups of G. It will←−
therefore suffice to show that G/G0 is a finite p-group. Assume otherwise: then G/G0 contains an elementng which is not annihilated by any power of p. Let g be an element of G representing g. Then g p ∈/ G0 forall n ≥ 0, contradicting our assumption that G is p-profinite.
Example 3.2.15. Let X be a p-profinite space, and choose a base point η ∈ Mat(X). Then we can identify
πn (X, η) with π0 Ωn (X), where Ωn (X) denotes the n-fold loop space of X in the ∞-category SPro(p) . It follows
that πn (X, η) is itself endowed with a totally disconnected compact Hausdorff topology. Write X = lim Xα←−and let ηα denote the image of η in Xα . Then πn (X, η) = lim πn (Xα , ηα ). It follows that πn (X, η) has the←−
structure of a p-profinite group (which is abelian if n ≥ 2).
For each integer n ≥ 0, the truncation functor τ≤n carries the ∞-category Sp−fc of p-finite spaces to itself.
This extends uniquely to a functor SPro(p) → SPro(p) which commutes with filtered limits. We will denote
this extension also by τ≤n . This is a localization functor from SPro(p) to itself; we will denote its essential
image by τ≤n SPro(p) . We will say that a p-profinite space X is n-truncated if it belongs to τ≤n SPro(p) : thatis, if the canonical map X → τ≤n X is an equivalence.
Example 3.2.16. When n = 0, the truncation τ≤0 X is equivalent to π0 X, regarded as a profinite set via
the topology described in Remark 3.2.8.
Remark 3.2.17. For each n ≥ 0, we will generally abuse notation by identifying the truncation functor τ≤ndefined above with a functor from the ∞-category SPro(p) of p-profinite spaces to itself. For every p-finite
space X, the tower· · · → τ≤2 X → τ≤1 X → τ≤0 X
is eventually constant, and its limit is equivalent to X. It follows that for X ∈ SPro(p) , the tower· · · → τ≤2 X → τ≤1 X → τ≤0 X
also has limit X (though it is generally not eventually constant).
The Postnikov tower of a p-profinite space X is particularly useful in the case where X is simply connected;
in this case, one can show that each of the maps τ≤n+1 X → τ≤n X is a principal fibration. When working
with p-profinite spaces which are not simply connected, it will be convenient to consider a refinement of the
Postnikov tower.
Definition 3.2.18. Let G and H be groups, and suppose we are given a group homomorphism ρ : G →Aut(H). We define a sequence of normal subgroups· · · ⊆ H2ρ ⊆ H1ρ ⊆ H0ρ = H
ρ
by induction as follows: for each integer k, we let Hk+1
be the smallest normal subgroup of H which containsρ−1h ρ(g)(h) for each h ∈ Hk and each g ∈ G.
If X is a space equipped with a base point η and n ≥ 1 is an integer, then we let ρη denote the associatedmap π1 (X, η) → Aut(πn (X, η)). Let X ∈ S, and suppose we are given integers n ≥ 1, c ≥ 0. We will sayρthat X is (n, c)-truncated if it is n-truncated and, for every point η ∈ X, the subgroup πn (X, η)c η ⊆ πn (X, η)is trivial.

78

Example 3.2.19. Let G be a group, and let ρ : G → Aut(G) classify the action of G on itself by conjugation.
Then the sequence of subgroups· · · ⊆ Gρ2 ⊆ Gρ1 ⊆ Gρ0 = G
is the lower central series of the group G.
Example 3.2.20. A space X ∈ S is (n, 0)-truncated if and only if it is (n − 1)-truncated. If X is simply
connected and c > 0, then X is (n, c)-truncated if and only if it is n-truncated.
Lemma 3.2.21. Let X be a p-finite space. If X is n-truncated, then it is (n, c) truncated for some integerc  0.
Proof. Since X has finitely many connected components, we may assume without loss of generality that X
is connected. Choose a point η ∈ X, and let ρ : π1 (X, η) → Aut(πn (X, η)) be the canonical map. We wish
to show that πn (X, η)ρc is trivial for c  0. Since πn (X, η) is a finite group, it will suffice to show thatthe containments πn (X, η)ρc+1 ⊆ πn (X, η)ρc are proper whenever πn (X, η)ρc 6= 0. To prove this, it suffices toconstruct a nonzero π1 (X, η)-equivariant map from πn (X, η) into the the abelian group Z /p Z (which weregard as endowed with a trivial action of π1 (X, η)). Since πn (X, η)ρc is a nonzero finite p-group, it has anonzero abelianization A. We now apply Lemma 2.6.17 to choose a nonzero fixed point for the action of
π1 (X, η) on the finite abelian group Hom(A, Z /p Z).
Notation 3.2.22. Let p be a prime number, and let n ≥ 1 and c ≥ 0 be integers. We let Sp−fc(n,c) denote thefull subcategory of S spanned by those spaces which are p-finite and (n, c)-truncated.
p−fcc
The inclusion Sp−fc
admits a left adjoint, which we will denote by τ≤n: Sp−fc → Sp−fc(n,c) ⊆ S(n,c) . Thisfunctor extends to a functor SPro(p) → Pro(Sp−fc(n,c) which commutes with filtered limits. We will denote thisp−fccextension also by τ≤n; it is left adjoint to the fully faithful embedding Pro(S(n,c)) → SPro(p) .

Example 3.2.23. When c = 0, the truncation functor τ≤n c coincides with the functor τ≤n−1 constructedabove.
Remark 3.2.24. Let n ≥ 1 be an integer. For any p-finite space X, Lemma 3.2.21 implies that the towerof truncations
2
1
0· · · → τ≤n
X → τ≤n
X → τ≤n
X = τ≤n−1 X
is eventually constant, and its limit is an n-truncation of X. It follows that when X is p-profinite, the tower
2
1
0· · · → τ≤n
X → τ≤n
X → τ≤n
X = τ≤n−1 X

has limit τ≤n X, though it is not eventually constant in general.
Let X be an arbitrary p-profinite space. Our next goal is to show that, for every pair of integers n ≥ 1
c+1c
and c ≥ 0, the canonical map τ≤n
X → τ≤n
X behaves like a principal fibration. To formulate this result
more precisely, we need to introduce a bit of notation.
Definition 3.2.25. A finite bundle of abelian groups consists of the following data:(a) A map of finite sets φ : A → S.
(b) An abelian group structure on each fiber φ−1 {s}.
We will generally abuse notation and indicate a finite bundle of abelian groups simply as a map φ : A → S.
If φ : A → S and φ0 : A0 → S 0 are finite bundles of abelian groups, then a morphism of finite bundles ofabelian groups from φ to φ0 is a commutative diagram of finite sets
A

φ

f

F


A0

/S

φ

0

79


/ S0

−1

such that, for every element s ∈ S, the induced map Fs : φ−1 {s} → φ0 {f (s)} is a group homomorphism.
We let Bun denote the category whose objects are finite bundles of abelian groups, and Pro(Bun) thecategory of Pro-objects of Bun (see Example 3.1.3). We will refer to the objects of Pro(Bun) as profinite
bundles of abelian groups.
If p is a prime number, we let Bunp denote the full subcategory of Bun spanned by those objects φ : A → S
such that each fiber φ−1 {s} is a p-group. We will refer to the objects of Bunp as finite bundles of abelianp-groups, and objects of Pro(Bunp ) as profinite bundles of abelian p-groups.
Remark 3.2.26. We can think of a finite bundle of abelian groups as a pair (φ : A → S, m : A ×S A → A),
where φ is a morphism in the category Setfin of finite sets and m exhibits A as an abelian group object
finof Setfin/S . Stone duality supplies an equivalence of the category Pro(Set ) with the category of totally
disconnected Hausdorff spaces (Proposition XII.A.1.6). Consequently, we obtain a fully faithful embedding
from Pro(Setfin ) to the category whose objects pairs (φ : A → S, m : A ×S A → A), where φ is a continuous
map between totally disconnected compact Hausdorff spaces, and m : A ×S A → A is a continuous map
which endows each fiber of φ with the structure of a topological abelian group.
Construction 3.2.27. If φ : A → S is a finite bundle of abelian groups, then we can identify A with anabelian group object of the topos Set/S . For each n ≥ 0, we let K(φ, n)`denote the associated EilenbergMacLane object of the ∞-topos S/S . More concretely, we have K(φ, n) = s∈S K(As , n), where As denotes
the abelian group φ−1 {s}. Note that if φ is a finite bundle of abelian p-groups, then the space K(φ, n) is
p-finite. The construction φ 7→ K(φ, n) determines a functor N(Bunp ) → Sp−fc , which extends to a functor
N(Pro(Bunp )) → SPro(p) which commutes with filtered limits. We will denote this extended functor byb
φ 7→ K(φ,n).
Example 3.2.28. Let A be a profinite abelian p-group: that is, A is a topological abelian group gives asa filtered limit lim Aα , where each Aα is a finite abelian p-group. Then we can regard the map φ : A → ∗←−bb
as a profinite bundle abelian p-groups. We will denote the p-profinite space K(φ,n) simply by K(A,n). Webnote that there is a canonical homotopy equivalence of spaces Mat(K(A, n)) ' K(A, n).
Construction 3.2.29. Let p be a prime number, and fix integers n, c ≥ 1. We define a full subcategory
Cpn,c ⊆ Fun(∆1 × ∆1 , S) as follows. A commutative diagram σ :
X

Y

g

/ X0
g0

f


/ Y0

in S belongs to Cpn,c if and only if the following conditions are satisfied:(i) The space X is p-finite and (n, c)-truncated.
(ii) The map g exhibits X0 as a 0-truncation of X.
(iii) The map f exhibits Y as an (n, c − 1)-truncation of X.
(iv) The map g 0 induces a bijection π0 X0 → π0 Y 0 .
(v) The diagram σ is a pullback square.
It follows from these conditions that Y 0 ' K(φ, n+1), where φ : A → π0 X is an object of Bunp such that, forevery point η ∈ X having homotopy class [η] ∈ π0 X, the fiber φ−1 {[η]} can be identified with the subgroupρηπn (X, η)c−1⊆ πn (X, η) (note that this group is abelian and canonically independent of the choice of point
η in its homotopy class).

80

Using obstruction theory, one can show that evaluation at (0, 0) ∈ ∆1 × ∆1 induces an equivalence of∞-categories Cpn,c → Sp−finn,c . In other words, we can functorially associate to every (n, c)-truncated p-finite
space X a pullback square/ π0 X
X

c−1
τ≤n
X


/ K(φ, n + 1),

where φ ∈ Bunp is defined as above. This construction admits an essentially unique extension to a functor
Pro(p)
S(n,c) → Fun(∆1 × ∆1 , SPro(p) ) which commutes with filtered limits, and carries an (n, c)-truncated pprofinite space X to a pullback square of p-profinite spacesX/ τ≤0 X


c−1
τ≤n
X


b
/ K(φ,
n + 1).

Proof of Theorem 3.2.2. The “only if” direction is obvious. To prove the converse, let us suppose thatf : X → Y is a map of p-profinite spaces satisfying conditions (1) and (2); we wish to show that f is a
homotopy equivalence. We note that f is a filtered limit of maps f≤n : τ≤n X → τ≤n Y ; it will therefore
suffice to show that each f≤n is an equivalence. For this, we proceed by induction on n.
We begin by treating the case n = 0. We are given that f induces a bijection (π0 f ) : π0 X → π0 Y . As
explained in Example 3.2.9, π0 X and π0 Y can be identified with the Stone spaces of the profinite sets τ≤0 X
and τ≤0 Y . Since π0 f is a continuous bijection between compact Hausdorff spaces, it is a homeomorphism.
Using Proposition XII.A.1.6, we deduce that f≤0 induces an isomorphism of profinite sets.
Now suppose that n ≥ 1 and that f≤n−1 is an equivalence; we wish to show that f≤n is an equivalence.
Note that f≤n is the limit of a tower of maps
c
c
c
f≤n
: τ≤n
X → τ≤nY.
c
It will therefore suffice to show that each f≤n
is an equivalence. We proceed by induction on c. In the case0
c = 0, we observe that f≤n ' f≤n−1 . For the inductive step, suppose that c > 0. Using Construction 3.2.29,
we obtain a pullback square of morphisms in SPro(p) :
c
f≤n

/ f≤0


c−1
f≤n


b Y , n + 1))/ (ψ : K(φX , n + 1) → K(φwhere φX : AX → π0 X and φY : AY → π0 Y are objects of Pro(Bunp ) and ψ is induced by f . It will therefore
suffice to show that ψ is an equivalence in SPro(p) . As explained in Example 3.2.26, we can identify φX andφY with maps of totally disconnected compact Hausdorff spaces; it will therefore suffice to show that thevertical maps in the diagram/ π0 XAX
AY
/ π0 Yare homeomorphisms. The right vertical map is a homeomorphism by the argument given above. To provethat the left vertical map is a homeomorphism, it will suffice to show that it is bijective. That is, we must
81

−1
0
show that for each [η] ∈ π0 X having image [η 0 ] ∈ π0 Y , the induced map θ : φ−1
X {[η]} → φY {[η ]} is an
isomorphism of profinite groups. Unwinding the definitions, we can identify θ with the map
ρ

ρ 0

ρ

ρ 0

η
η
πn (X, η)c−1/πn (X, η)c η → πn (Y, η)c−1/πn (Y, η 0 )c η

(here H denote the closure of H, if H is a subgroup of πn (X, η) or πn (Y, η 0 )). Assumption (2) guarantees
that f induces isomorphisms of profinite groups π1 (X, η) → π1 (Y, η 0 ) and πn (X, η) → πn (Y, η 0 ), from which
it follows immediately that θ is an isomorphism.

3.3

p-Profinite Spaces of Finite TypeLet X be a p-profinite space. In §3.2, we defined the materialization Mat(X) = MapSPro(p) (∗, X) ∈ S, andproved that the functor X 7→ Mat(X) is conservative (Theorem 3.2.2). Our goal in this section is to showthat if X satisfies some reasonable finiteness conditions, then the passage from X to Mat(X) does not lose
any information: that is, we can recover X as the p-profinite completion of Mat(X).
We begin by precisely formulating the relevant finiteness conditions. Assume that X is a simply connectedp-profinite space. Theorem 3.2.2 implies that the structure of X is “controlled” by its homotopy groups{πn X}n≥0 , each of which is a p-profinite abelian group. We therefore begin by reviewing some finitenessconditions in the setting of p-profinite algebra.
Proposition 3.3.1. Let p be a prime number and let A be a p-profinite abelian group. The following
conditions are equivalent:(1) The set Hom(A, Fp ) of continuous group homomorphisms from A to Fp is finite.
(2) The quotient A/pA is a finite abelian group.
(3) There exists a surjection of p-profinite abelian groups Znp → A for some integer n.
(4) The pro-p group A is isomorphic to a direct sum of finitely many p-profinite abelian groups of the form
Zp or Z /pk Z.
Proof. The implication (3) ⇒ (4) follows from the structure theory of finitely generated modules over thediscrete valuation ring Zp , and the implication (4) ⇒ (1) is obvious. We next show that (1) ⇒ (2). Let
pA denote the image of the multiplication-by-p map p : A → A. Since A is compact, its image pA ⊆ A
is a closed subgroup of A, so the quotient A/pA inherits the structure of a p-profinite group. Let Vectfin
Fpdenote the category of finite-dimensional vector spaces over the finite field Fp . If we write A = lim Aα where←−
each α is a finite abelian p-group, then A/pA ' lim Aα /pAα is given as a filtered inverse limit of objects of←−fin
VectfinFp , so we can regard A/pA as a pro-object in the category VectFp . Vector space duality determines anequivalence of categories
op(Vectfin' Vectfin
Fp )Fp ,which extends to an equivalence of categories
op
Pro(Vectfin' Ind(Vectfin
Fp )Fp ) ' VectFp .

The image of A/pA under this equivalence is given by Hom(A, Fp ). Condition (1) implies that Hom(A, Fp ) ∈VectfinFp ⊆ VectFp : that is, the Ind-object obtained by applying vector space duality to A/pA is constant.
It follows that A/pA is equivalent to a constant Pro-object of Vectfin
Fp : in other words, A/pA is a finitedimensional Fp -vector space.
We complete the proof by showing that (2) ⇒ (3). Choose a basis {xi }1≤i≤n for the Fp -vector space
A/pA. Each xi can be lifted to an element xi ∈ A, which determines a group homomorphism φi : Z → A.
Since A is a p-profinite group, the map φi factors through the p-profinite completion Zp of Z. We therefore
obtain a sequence of continuous maps φi : Zp → A. Since A is abelian, we can add these homomorphisms to82

obtain a continuous group homomorphism φ : Znp → A. We claim that φ is surjective. Since Znp is compact,
it will suffice to show that the image of φ is dense. In other words, we must show that if ψ : A → A0 is acontinuous surjection for some finite p-group A0 , then the composite map ψ ◦ φ : Znp → A0 is surjective. Since
the action of p is nilpotent on A0 , it suffices to show that the composite map θ : Znp → A0 /pA0 is surjective(by Nakayama’s lemma). This is clear, since θ is a composition of surjections
Znp → Fnp ' A/pA → A0 /pA0 .

Definition 3.3.2. Let p be a prime number. We will say that a p-profinite abelian group A is topologicallyfinitely generated if it satisfies the equivalent conditions of Proposition 3.3.1. We will say that a p-profinite
space X is simply connected if its materialization Mat(X) is simply conneceted. We say that X has finite
type if X is simply connected and each homotopy group πn X is topologically finitely generated.
We can now formulate the main result of this section.
Theorem 3.3.3. Let X and Y be p-profinite spaces. If X is of finite type, then the canonical map
MapSPro(p) (X, Y ) → MapS (Mat(X), Mat(Y )) is a homotopy equivalence. In particular, the materialization
functor Mat : SPro(p) → S is fully faithful when restricted to p-profinite spaces of finite type.
To prove Theorem 3.3.3, we need to study cohomology groups in the p-profinite setting.
Definition 3.3.4. Let X be a p-profinite space. For every field k of characteristic p, we let H∗ (X; k) denotethe graded abelian group given by Hn (X; k) = π−n C ∗ (X; k), where C ∗ (X; k) is the E∞ -algebra over k
introduced in Proposition 3.1.14.
Remark 3.3.5. If X is p-finite, then the groups Hn (X; k) introduced in Definition 3.3.4 agree with theusual cohomology groups of X with coefficients in K. In particular, we have a canonical isomorphism
Hn (X; Fp ) ' π0 MapSPro(p) (X, K(Z /p Z, n)).
Since both sides of this equivalence are compatible with the formation of filtered limits, we deduce that there
is a canonical isomorphism Hn (X; Fp ) ' π0 HomSPro(p) (X, K(Z /p Z, n)) for every p-profinite space X.
Remark 3.3.6. If X is a p-finite space, then H0 (X; Fp ) can be identified with the Fp -vector space of allmaps π0 X → Fp . It follows that if X is a p-profinite space, then H0 (X; Fp ) can be identified with Fp -vector
space of all continuous maps π0 X → Fp . This is a p-Boolean algebra (Definition XII.A.1.9) from which we
can functorially recover the topological space π0 X (it is given by the Zariski spectrum of H0 (X; Fp ); see
Proposition XII.A.1.12). In particular, Mat(X) is connected if and only if the unit map Fp → H0 (X; Fp ) is
an isomorphism.
Remark 3.3.7. Let X be a p-profinite space. Every map from X to K(Z /p Z, n) in the ∞-category SPro(p)induces a map from Mat(X) to Mat(K(Z /p Z, n)) ' K(Z /p Z, n) in the ∞-category of spaces. Letting nvary, we obtain a map of graded ringsH∗ (X; Fp ) → H∗ (Mat(X); Fp ).
We will deduce Theorem 3.3.3 from the following pair of results:
Proposition 3.3.8. Let X be a p-profinite space of finite type. Then the map of cohomology rings θ :H∗ (X; Fp ) → H∗ (Mat(X); Fp ) is an isomorphism. /Proposition 3.3.9. Let X and Y be p-profinite spaces, and suppose that the map
H∗ (X; Fp ) → H∗ (Mat(X); Fp )is an isomorphism. Then the canonical map θ : MapSPro(p) (X, Y ) → MapS (Mat(X), Mat(Y )) is a homotopy
equivalence.
83

We begin with the proof of Proposition 3.3.8, which will require some preliminary results.
Proposition 3.3.10. Let n ≥ 1, and consider the tower of spaces {K(Z /pa Z, n)}a≥0 having limit K(Zp , n).
For every integer m ≥ 0, the pro-system of abelian groups Hm (K(Z /pa Z, n); Fp ) is equivalent to the constantpro-system with value Hm (K(Zp , n); Fp ).
Proof. We first treat the case n = 1. We have a fiber sequence of simply connected spaces
K(Z, 1) → K(Zp , 1) → K(Zp / Z, 1).
Since multiplication by p is invertible on Zp / Z, we have Hm (K(Zp / Z, 1); Fp ) ' 0 for m > 0. Using
the Serre spectral sequence, we deduce that the map K(Z, 1) → K(Zp , 1) induces an isomorphism on
homology groups with coefficients in k. It will therefore suffice to show that, for each m ≥ 0, the pro-system{Hm (K(Z /pa Z, 1); Fp )}a≥0 is equivalent to the constant pro-system taking the value(
Fp if 0 ≤ m ≤ 1
Hm (K(Z, 1); Fp ) '0otherwise.
For each a ≥ 1, let a ∈ H1 (K(Z /pa Z, 1); Fp ) classify the unit map Z /pa Z → Z /p Z ' Fp , and let
ηa ∈ H2 (K(Z /pa Z, 1); Fp ) classify the central extension
0 → Fp → Z /pa+1 Z → Z /pa Z → 0.
Then H∗ (K(Z /pa Z, 1); Fp ) has a basis given by the products η i and a ηai for i ≥ 0. Note that the image of
ηa in H2 (K(Z /pa+1 Z, 1); Fp ) is zero (since the central extension classified by ηa splits over Z /pa+1 Z), andthe image of a in H1 (K(Z /pa+1 Z, 1)) is a+1 . It follows that the maps
Hm (K(Z /p Z, 1); Fp ) → Hm (K(Z /p2 Z, 1); Fp ) → Hm (K(Z /p3 Z, 1); Fp ) → · · ·are isomorphisms for m ≤ 1 and zero for m > 1. Passing to dual spaces, we conclude that the tower· · · → Hm (K(Z /p3 Z, 1); Fp ) → Hm (K(Z /p2 Z, 1); Fp ) → Hm (K(Z /p Z, 1); Fp )consists of isomorphisms for m ≤ 1 and zero maps for m > 1. We now complete the proof (in the case n = 1)
by observing that the maps Hm (K(Z, 1); Fp ) → Hm (K(Z /p Z, 1); Fp ) are isomorphisms for m ≤ 1.
We now treat the general case. Since ModFp is a presentable symmetric monoidal ∞-category, there
is a unique colimit-preserving symmetric monoidal functor S → ModFp . Let us denote this functor by
X 7→ C∗ (X). Note that the homology groups of a space X are given by the formula Hm (X; Fp ) = πm C∗ (X).
It will therefore suffice to prove the following:(∗) For every integer m, the Pro-object {τ≤m C∗ (K(Z /pa Z, n))}a≥0 of ModFp is equivalent to the constantFp -module spectrum τ≤m C∗ (K(Zp , n)).
Since the functor X 7→ τ≤m C∗ (X) is a successive extension of the functors X 7→ Ha (X; Fp ) for 0 ≤ a ≤ m,
the proof given above shows that (∗) holds when n = 1. To prove (∗) in general, we proceed by induction
on n. Suppose that n > 1. Let X• be a Čech nerve of the map ∗ → K(Zp , n), so that X• is a groupobject of S. Since C∗ is a symmetric monoidal functor, we deduce that C∗ (X1 ) is an associative algebra
object of ModFp ; here X1 = ∗ ×K(Zp ,n) ∗ ' K(Zp , n − 1). Since K(Zp , n) is connected, the canonicalmap |X• | → K(Zp , n) is an equivalence. Because C∗ preserves colimits, we deduce that C∗ (K(Zp , n))
is given by |C∗ (X• )|. Unwinding the definitions, we see that this geometric realization corresponds tothe bar construction Fp ⊗C∗ (K(Zp ,n−1)) Fp . Similar reasoning yields an equivalence C∗ (K(Z /pa Z, n)) 'Fp ⊗C∗ (K(Z /pa Z,n−1)) Fp for each a ≥ 0. We have a commutative diagram of Pro-objects
Fp ⊗C∗ (K(Zp ,n−1)) Fp/ {Fp ⊗C (K(Z /pa Z,n−1);F Fp }C∗ (K(Zp , n))/ {C∗ (K(Z /pa Z, n))}

∗

84

p

where the vertical maps are equivalences. Using the inductive hypothesis, we deduce that upper horizontal
map becomes an equivalence after applying the truncation functor τ≤m . It follows that the lower horizontal
map becomes an equivalence after applying the truncation functor τ≤m as well, which proves (∗).
Remark 3.3.11. Using the proof of Proposition 3.3.10, we see that the homology groups Hm (K(Zp , n); Fp )are finite dimensional vector spaces over Fp for all m ≥ 0, n ≥ 1. In the case n = 1, this follows by inspection.
In the general case, we have a pushout diagram of E∞ -algebras over Fp
C∗ (K(Zp , n − 1))

/ Fp


Fp


/ C∗ (K(Zp , n)).

The inductive hypothesis and Proposition A.7.2.5.31 imply that Fp is almost of finite presentation overC∗ (K(Zp , n − 1)), so that C∗ (K(Zp , n)) is almost of finite presentation over Fp and therefore the homology
groups of K(Zp , n) are finite dimensional over Fp by Proposition A.7.2.5.31.
Similar reasoning using the pushout diagramC∗ (K(Zp , n))/ C∗ (K(pa Zp , n))Fp/ C∗ (K(Zp /pa Zp , n); Fp ),shows that the homology groups Hm (K(Z /pa Z, n); Fp ) are finite-dimensional for n ≥ 1, m, a ≥ 0.
Proof of Proposition 3.3.8. Let us say that a p-profinite space X is good if the map θ : H∗ (X; Fp ) →b p , n) is good for each n ≥ 1,
H∗ (Mat(X); Fp ) is an isomorphism. Proposition 3.3.10 implies that K(Zand it is obvious that any p-finite space is good. Suppose we are given a pullback diagram of p-profinite
spaces/YX/ Y 0.


X0

Using Corollary 1.1.10 and Proposition 2.5.2, we deduce that if X 0 , Y , and Y 0 are good, Y 0 is simply
connected, and the homotopy fiber of the map Mat(Y ) → Mat(Y 0 ) has finite-dimensional cohomology inb p , n) for n ≥ 1 and Y 0 = ∗ (andeach degree (with coefficients in Fp ), then X is also good. Taking Y = K(Z00b p , n) is good. Similarly, if X 0 isusing Remark 3.3.11), we conclude that if X is good then X = X × K(Z0agood then X = X × K(Z /p Z, n) is good for any integer a. Combining these observations with Propositionb
3.3.1, we deduce that K(A,n) is good whenever A is topologically finitely generated.
Let X be an arbitrary p-profinite space, and write X = lim{τ≤n X}. The map Mat(X) → Mat(τ≤n X)←−
has (n + 1)-connective homotopy fibers, and therefore induces an isomorphism Hm (Mat(τ≤n X); Fp ) →Hm (Mat(X); Fp ) for m ≤ n. We therefore have a commutative diagramlim H∗ (τ≤n X; Fp )−→/ lim H∗ (Mat(τ≤n X); Fp )−→

H∗ (X; Fp )/ H∗ (Mat(X); Fp ).

where the vertical maps are isomorphisms. Consequently, to prove that X is good, it suffices to show thateach truncation τ≤n X is good.
85

Assume now that X has finite type. We prove by induction on n that the truncation τ≤n X is good, thecase n = 1 being trivial. If n > 1, then the simple-connectivity of X and Construction 3.2.29 give a pullbacksquare of p-profinite spaces
/∗
τ≤n X

b
/ K(A,
n + 1)

τ≤n−1 X

where A = πn X is topologically finitely generated. Since τ≤n−1 X is good by the inductive hypothesis andbH∗ (K(A, n); Fp ) ' H∗ (K(A,n); Fp ) is finite-dimensional in each degree (this follows from Remark 3.3.11 in
the case where A ' Zp or A ' Z /pa Z, and the general case follows from Proposition 2.5.2), we conclude
that τ≤n X is good.
Remark 3.3.12. Let A be a profinite abelian p-group which is topologically finitely generated, and letbn ≥ 1. Then the cohomology groups Hm (K(A,m); Fp ) are finite dimensional over Fp . To see this, we canuse Proposition 2.5.2 and Proposition 3.3.1 to reduce to the case where A is a cyclic module over Zp . Using
Proposition 3.3.8, we are reduced to proving that Hm (K(A, n); Fp ) is finite dimensional, which follows from
Remark 3.3.11.
Remark 3.3.13. The proof of Proposition 3.3.8 does not require the full strength of our assumption that
X is simply connected: the same result holds for a large class of p-profinite spaces which are nilpotent in asuitable sense.
We now turn to the proof of Proposition 3.3.9.
Notation 3.3.14. Let X be a space. Consider the functor Sp−fc → S given by K 7→ MapS (X, K). Thisfunctor is left exact, and can therefore be identified with an object Xp∨ ∈ SPro(p) ⊆ Fun(Sp−fc , S). We
will refer to Xp∨ as the p-profinite completion of X. The construction X 7→ Xp∧ determines a functor fromS → SPro(p) , which is left adjoint to the materialization functor Mat : SPro(p) → S. Note that we have acanonical isomorphism of cohomology groups H∗ (X; Fp ) ' H∗ (Xp∧ ; Fp ).
Lemma 3.3.15. Let f : X → Y be a map of p-profinite spaces. The following conditions are equivalent:(1) The map f is an equivalence.
(2) The map f induces an isomorphism of cohomology rings H∗ (Y ; Fp ) → H∗ (X; Fp ).
Proof. The implication (1) ⇒ (2) is trivial. Conversely, if (2) is satisfied then f induces an equivalenceC ∗ (Y ; k) → C ∗ (X; k) whenever k is a separably closed field of characteristic p, from which it follows that f
is an equivalence (Theorem 3.5.8).
Proof of Proposition 3.3.9. We wish to show that the map
MapSPro(p) (X, Y ) → MapS (Mat(X), Mat(Y )) ' MapSPro(p) (Mat(X)∧p,Y )is a homotopy equivalence. For this, it suffices to show that the counit map v : Mat(X)∧p (X) → X is anequivalence of p-profinite spaces. Using Lemma 3.3.15, we are reduced to proving checking that the map ofcohomology groups∗H∗ (X; Fp ) → H∗ (Mat(X)∧p ; Fp ) → H (Mat(X); Fp )is an isomorphism, which follows from our hypothesis.

86

3.4

Connectivity in p-Profinite Homotopy Theory

Let f : X → Y be a map of spaces, and let n ≥ −2 be an integer. Recall that f is said to be n-truncatedif, for each y ∈ Y , the homotopy fiber Xy = X ×Y {y} is an n-truncated space: that is, if the truncation
map Xy → τ≤n Xy is an equivalence. We say that f is (n + 1)-connective if each homotopy fiber Xy of f is(n + 1)-connective: that is, if the truncation τ≤n Xy is contractible. According to Example T.5.2.8.16, every
map of spaces f : X → Y admits an essentially unique factorization
f0

f 00

X → Z → Y,
where the map f 0 is (n + 1)-connective and f 00 is n-truncated. Our goal in this section is to construct ananalogous factorization in the case where f is a map of p-profinite spaces.
Definition 3.4.1. Let p be a prime number and let n ≥ −2 be an integer. We will say that a map f : X → Yof p-profinite spaces is n-truncated if it is given as a filtered limit of morphisms fα : Xα → Yα , where each fα
is an n-truncated map of p-finite spaces. If n ≥ −1, we say that f is n-connective if it is given as a filteredlimit of morphisms fβ : Xβ → Yβ , where each fβ is an n-connective map of p-finite spaces.
We say that a p-profinite space X is n-connective if the constant map X → ∗ is n-connective, and ntruncated if the constant map X → ∗ is n-truncated. We say that X is connected if it is 1-connective, andsimply connected if it is 2-connective.
Recall that a factorization system on an ∞-category C is a pair (SL , SR ) with the following properties(Definition T.5.2.8.8):(1) Both SL and SR are collections of morphisms of C which are closed under the formation of retracts.
(2) Every morphism in SL is left orthogonal to every morphism in SR (see Definition T.5.2.8.1).
f0

f 00

(3) Every morphism f : X → Y in C can be obtained as a composition X → Z → Y , where f 0 ∈ SL andf 00 ∈ SR .
We refer the reader to §T.5.2.8 for a general discussion of factorization systems in ∞-categories. The starting
point for our discussion in this section is the following observation:
Theorem 3.4.2. Let p be a prime number and let n ≥ −2 be an integer. Let ŜL denote the collectionof (n + 1)-connective morphisms in SPro(p) , and let ŜR denote the collection of n-truncated morphisms inSPro(p) . Then the pair (ŜL , ŜR ) is a factorization system on the ∞-category SPro(p) .
Theorem 3.4.2 is an immediate consequence of the following pair of results:
Proposition 3.4.3. Let p be a prime number and n ≥ −2 an integer. Let SL be the collection of (n + 1)connective morphisms between p-finite spaces and let SR be the collection of n-truncated morphisms between
p-finite spaces. Then the pair (SL , SR ) is a factorization system on the ∞-category Sp−fc of p-finite spaces.
Proposition 3.4.4. Let C be an essentially small ∞-category equipped with a factorization system (SL , SR ).
Let CL denote the full subcategory of Fun(∆1 , C) spanned by those morphisms belonging to SL , and CR thefull subcategory of Fun(∆1 , C) spanned by those morphisms belonging to SR . Then the inclusions CL , CR ,→Fun(∆1 , C) determine fully faithful embeddings
Pro(CL ), Pro(CR ) → Pro(Fun(∆1 , C)) ' Fun(∆1 , Pro(C))(see Proposition T.5.3.5.15). Let ŜL and ŜR denote the collections of morphisms in Pro(C) which belong to
the essential images of these embeddings. Then (ŜL , ŜR ) is a factorization system on Pro(C).

87Proof of Proposition 3.4.3. According to Example T.5.2.8.16, the collections of (n + 1)-connective and ntruncated morphisms in S determine a factorization system on S. The only nontrivial point is to show that
f0

f 00

if f : X → Y is a map of p-finite spaces and we factor f as a composition X → Z → Y where f 0 is
(n + 1)-connective and f 00 is n-truncated, then the space Z is also p-finite. Since Y is p-finite, it will suffice
to show that for each y ∈ Y , the homotopy fiber Zy = Z ×Y {y} is p-finite. For this, we observe that Zy is
given by the truncation τ≤n Xy , where Xy denotes the homotopy fiber X ×Y {y}.
Proof of Proposition 3.4.4. Since ∞-categories Pro(CL ) and Pro(CR ) are idempotent complete, the sets ŜLand ŜR are clearly stable under retracts. Let D denote the full subcategory of Fun(∆2 , C) spanned by thosediagrams
>Z
f0

X

f 00

f



/Y

where f 0 ∈ SL and f 00 ∈ SR . According to Proposition T.5.2.8.17, the inclusion ∆1 ' ∆{0,2} ,→ ∆2
induces an equivalence of ∞-categories D → Fun(∆1 , C). It follows that the induced map Pro(D) →Pro(Fun(∆1 , C)) ' Fun(∆1 , Pro(C)) is an equivalence. From this, we conclude that every morphism f : X →
f0

f 00

Y in Pro(C) factors as a composition X → Z → Y , where f 0 ∈ ŜL and f 00 ∈ ŜR .
It remains to prove that every morphism in ŜL is left orthogonal to every morphism in ŜR . To provethis, suppose we are given a filtered diagram {fα : Aα → Bα } in CL and a filtered diagram {gβ : Xβ → Yβ }in CR , having limits given by morphisms f : A → B and g : X → Y in Pro(C). We wish to show that the
diagram/ MapPro(C) (A, X)
MapPro(C) (B, X)
MapPro(C) (B, Y )/ MapPro(C) (B, X)

is a pullback square of spaces. Since the collection of pullback diagrams in S is closed under filtered colimits
and small limits, it suffices to prove that for every pair of indices α and β, the diagram of spaces
MapPro(C) (Bα , Xβ )/ MapPro(C) (Aα , Xβ )
MapPro(C) (Bα , Yβ )/ MapPro(C) (Bα , Xβ )is a pullback square. This follows from the fact that fα is left orthogonal to gβ in the ∞-category C.
Example 3.4.5. Let X be a p-profinite space and let n ≥ −2 be an integer. Theorem 3.4.2 implies that thef
map X → ∗ admits an (essentially unique) factorization X → Y → ∗, where f is (n + 1)-connective and Y isn-truncated. Unwinding the proof of Proposition 3.4.4, we see that Y can be identified with the truncation
τ≤n X studied in §3.2.
For applications of Theorem 3.4.2, it is useful to have some alternate descriptions of the classes of nconnective and n-truncated morphisms in SPro(p) .
Proposition 3.4.6. Let p be a prime number, let f : X → Y be a morphism of p-profinite spaces, and letn ≥ −2 be an integer. The following conditions are equivalent:(1) The morphism f is n-truncated, in the sense of Definition 3.4.1.

88

Pro(p)(2) The morphism f exhibits X as an n-truncated object of the ∞-category S/Y : that is, for every
p-profinite space Z, the map of spaces MapSPro(p) (Z, X) → MapSPro(p) (Z, Y ) is n-truncated.
(3) The induced map of materializations Mat(X) → Mat(Y ) is n-truncated.
Lemma 3.4.7. Let f : X → Y be an n-connective map of p-profinite spaces. Then the induced map
Mat(X) → Mat(Y ) is also n-connective.
Proof. Write f as a filtered limit of n-connective maps fα : Xα → Yα between p-finite spaces. Let η ∈Mat(Y ), so that η determines a compatible family of points ηα ∈ Yα . We wish to prove that the homotopy
fiber Mat(X)η = Mat(X) ×Mat(Y ) {η} is n-connective. This homotopy fiber is given as the limit of a filteredsystem n-connective, p-finite spaces Xα ×Yα {ηα }, and is therefore n-connective by Corollary 3.2.6.
Proof of Proposition 3.4.6. Suppose first that condition (1) is satisfied: that is, f is given as the limit of afiltered diagram of n-truncated morphisms fα : Xα → Yα between p-finite spaces. We will prove that (2) issatisfied. Let Z be an arbitrary p-profinite space. Then the map θ : MapSPro(p) (Z, X) → MapSPro(p) (Z, Y ) isa filtered limit of maps
θα : MapSPro(p) (Z, Xα ) → MapSPro(p) (Z, Yα ).
Consequently, to prove that θ is n-truncated, it will suffice to show that each θα is n-truncated. Write Z asa filtered limit of p-finite spaces Zβ , so that θα is a filtered colimit of maps
θα,β : MapS (Zβ , Xα ) → MapS (Zβ , Yα ).
It will therefore suffice to show that each θα,β is n-truncated, which follows immediately from our assumptionthat fα is n-truncated.
The implication (2) ⇒ (3) is obvious. We will complete the proof by showing that (3) ⇒ (1). Using
f0

f 00

Theorem 3.4.2, we can factor f as a composition X → Z → Y where f 0 is (n + 1)-connective and f 00
is n-truncated. The first part of the proof shows that the map of materializations Mat(Z) → Mat(Y )is n-truncated, so that f 0 induces an n-truncated map Mat(X) → Mat(Z). Lemma 3.4.7 implies that
Mat(X) → Mat(Z) is (n + 1)-connective. It follows that the map from Mat(X) to Mat(Z) is a homotopy
equivalence. Using Theorem 3.2.2, we deduce that f 0 is an equivalence of p-profinite spaces, so that f isn-truncated as desired.
We now study some consequences of Theorem 3.2.2.
Corollary 3.4.8. Let p be a prime number, let n ≥ −2 be an integer, and let X ∈ SPro(p) be a p-profinite
space. The following conditions are equivalent:(1) The p-profinite space X belongs to the essential image of the localization functor τ≤n : SPro(p) → SPro(p) .
(2) The p-profinite space X is n-truncated, in the sense of Definition 3.4.1.
(3) For every p-profinite space Y , the mapping space MapSPro(p) (Y, X) is n-truncated.
(4) The space Mat(X) is n-truncated. That is, for every point η ∈ X and every m > n, the homotopy
group πm (X, η) is trivial.
Proof. The equivalence (1) ⇔ (2) follows from Example 3.4.5, and the equivalences (2) ⇔ (3) ⇔ (4) follow
from Proposition 3.4.6.
We now prove an analogue of Proposition 3.4.6 for n-connective morphisms in SPro(p) .
Proposition 3.4.9. Let p be a prime number, let f : X → Y be a morphism of p-profinite spaces, and letn ≥ −1 be an integer. The following conditions are equivalent:(1) The morphism f is n-connective, in the sense of Definition 3.4.1.
89

(2) The induced map of materializations Mat(X) → Mat(Y ) is n-connective.
(3) The induced map of cohomology groups Hi (Y ; Fp ) → Hi (X; Fp ) is injective when i = n and an isomorphism for i < n.
f0

f 00

Proof. Using Theorem 3.4.2, we can write f as a composition X → Z → Y where f 0 is n-connective and f 00
is (n − 1)-truncated. Using Lemma 3.4.7 and Proposition 3.4.6, we see that the map Mat(f 0 ) : Mat(X) →Mat(Z) is n-connective and the map Mat(f 00 ) : Mat(Z) → Mat(Y ) is (n − 1)-truncated. It follows that
Mat(f ) is n-connective if and only if Mat(f 00 ) is an equivalence. Applying Theorem 3.2.2, we see that
Mat(f ) is n-connective if and only if f 00 is an equivalence, from which we deduce that (1) ⇔ (2).
Assume now that (1) is satisfied; we will prove (3). If m < n, then the map g : K(Z /p Z, m) → ∗ is(n − 1)-truncated. It follows that f is left orthogonal to g, so that composition with f induces a homotopy
equivalence
MapSPro(p) (Y, K(Z /p Z, m)) → MapSPro(p) (X, K(Z /p Z, m)).
Passing to connected components, we obtain an isomorphism Hm (Y ; Fp ) → Hm (X; Fp ). To prove thatthe map Hn (Y ; Fp ) → Hn (X; Fp ) is injective, it suffices to show that every lifting problem depicted in the
diagram
/∗
X

Y

f


/ K(Z /p Z, n),admits a solution. This is clear, since f is n-connective and the right vertical map is (n − 1)-truncated.
We now prove that (3) ⇒ (1). Note that assertion (1) is vacuous when n = −1. When n = 0, it isequivalent to the assertion that the map π0 X → π0 Y is surjective (by the first part of the proof). Using
Remark 3.3.6, we see that this is equivalent to the injectivity of the map H0 (Y ; Fp ) → H0 (X; Fp ). Let us
therefore assume that n > 0, and that f satisfies (3). To prove that f is n-connective, it will suffice to showthat every (n − 1)-truncated morphism g : U → V in SPro(p) is right orthogonal to f (Theorem 3.4.2 andProposition T.5.2.8.11). Since the collection of those morphisms g which are right orthogonal to f is closed
under limits (Proposition T.5.2.8.6), we may assume without loss of generality that U and V are p-finite
spaces. Then the map H0 (Y ; Fp ) → H0 (X; Fp ) is an isomorphism, so Remark 3.3.6 implies that f induces
a homeomorphism π0 X → π0 Y . We may therefore reduce to the case where U and V are connected. To
complete the proof, it will suffice to verify the following assertion for all m < n:(∗) If g : U → V is an m-truncated morphism between connected p-finite spaces, then g is right orthogonal
to f .
We prove (∗) using induction on m. If m = 0, then U is a connected covering space of V . Take G = π1 V ,so that there exists a subgroup H ⊆ G and a pullback diagram of spaces
U

/ BH


V


/ BG.

It will therefore suffice to show that the map BH → BG is right orthogonal to f . We proceed by induction
on the order of G. If G is trivial, there is nothing to prove. Otherwise, we can choose a central subgroup
Z ⊆ G of order p. We then have a pullback diagram
BZH

BZH/Z

ι

/ BG

ι0


/ BG/Z.

90

Since ι0 is right orthogonal to f by the inductive hypothesis, we conclude that ι is right orthogonal to f . It
will therefore suffice to show that the map BH → BZH is right orthogonal to f . Using the pullback square
BH

/ BZH


∗


/ BZH/H,

we are reduced to proving that the map ∗ → BG0 is right orthogonal to f , where G0 is either a trivial group(if Z ⊆ H) or the cyclic group Z /p Z (if Z * H). In the first case, the result is obvious. In the second case,
we must show that the map
MapSPro(p) (Y, K(Z /p Z, 1)) → MapSPro(p) (X, K(Z /p Z, 1))
is the inclusion of a summand, which follows from our assumption that the map Hi (Y ; Fp ) → Hi (X; Fp ) is
injective for m = 1 and bijective for m = 0.
We now prove (∗) in the case where m > 0. The map g factors as a composition
g0

g 00

U →V0 →V
where g 00 is (m − 1)-truncated and g 0 is m-connective. Using the inductive hypothesis, we see that g 00 is rightorthogonal to f . We may therefore replace g by g 0 and thereby reduce to the case where g is m-connective.
In this case, we have a homotopy fiber sequenceg

K(G, m) → U → V,
for some finite p-group G equipped with an action of the fundamental group π1 U . We proceed by induction
on the order of G. If G is trivial, then g is a homotopy equivalence and there is nothing to prove. Otherwise,
we can apply Lemma 2.6.17 to the subgroup of central elements of order p in G to obtain a cyclic subgroup
Z ⊆ G isomorphic to Z /p Z, on which the group π1 U acts trivially. We then have a map of fiber sequences
K(G, m)/U


K(G/Z, m)


/ U0

g

/V
id


/ V.

The map U 0 → V is right orthogonal to f by the inductive hypothesis. We are therefore reduced to provingthat the map U → U 0 is right orthogonal to f . This map is a principal fibration, fitting into a pullbacksquare/∗U
U0/ K(Z /p Z, m + 1).

We are therefore reduced to proving that the map ∗ → K(Z /p Z, m + 1) is right orthogonal to f . Unwinding
the definitions, this amounts to the assertion that the map Hi (Y ; Fp ) → Hi (X; Fp ) is an isomorphism fori ≤ m and an injection for i = m + 1. This follows from (3), since m < n.
Corollary 3.4.10. Let X be a p-profinite space and let n ≥ −1 be an integer. The following conditions areequivalent:(1) The p-profinite space X is n-connective.
91

(2) The space Mat(X) is n-connective.
(3) The map Fp → H0 (X; Fp ) is an isomorphism, and the cohomology groups Hm (X; Fp ) are trivial for
0 < m < n.
Remark 3.4.11. If X is a simply connected p-profinite space, then the pro-p-groups πn (X, η) are canonicallyindependent of the choice of base point η ∈ X. In this case, we will simply denote them by πn (X).
We conclude by applying some of the ideas above to obtain a cohomological characterization of the
p-profinite spaces having finite type, which will play an important role in §3.5.
Theorem 3.4.12. Let k be a field of characteristic p > 0 and let X be a p-profinite space. Then X is offinite type if and only if the following conditions are satisfied:(1) The canonical map Fp → H0 (X; Fp ) is an isomorphism.
(2) The cohomology group H1 (X; Fp ) is trivial.
(3) For each n ≥ 2, the cohomology group Hn (X; Fp ) is a finite-dimensional vector space over Fp .
Our proof of Theorem 3.4.12 rests on the following basic calculation:
Proposition 3.4.13. Let X be a p-profinite space which is n-connective for some n ≥ 1. There are canonicalisomorphisms
if m = 0Fp
Hm (X; Fp ) ' 0
if 0 < m < nHom(πn X, Fp ) if m = n.
Here Hom(πn X, Fp ) denotes the collection of continuous group homomorphisms from πn (X, η) to Fp (where
we regard Fp as endowed with the discrete topology), where η is an arbitrarily chosen point of X.
Proof. Choose a point η ∈ X. Using Corollary 3.4.10, we can write X as the limit of a filtered diagramof p-finite spaces Xα , each of which is n-connective. Let ηα denote the image of η in Xα , and let πn Xαdenote the finite p-group πn (Xα , ηα ). Since each Xα is n-connective, the Hurewicz and universal coefficient
theorems of classical homotopy theory give isomorphisms
if m = 0Fpm
H (Xα ; Fp ) ' 0
if 0 < m < nHom(πn Xα , Fp ) if m = n.
Since H∗ (X; Fp ) ' lim H∗ (Xα ; Fp ), we obtain canonical isomorphisms−→
if m = 0 Fpm
H (X; Fp ) ' 0
if 0 < m < nlim Hom(πn Xα , Fp ) if m = n.
−→
It now suffices to observe that the profinite group πn X is given as the limit of the filtered system of finitegroups πn X, so that Hom(πn X, Fp ) ' lim Hom(πn Xα , Fp ).
−→Corollary 3.4.14. Let X be a p-profinite space which is n-connective for some n ≥ 2. The following
conditions are equivalent:(1) The p-profinite group πn X is topologically finitely generated.
(2) The cohomology group Hn (X; Fp ) is finite dimensional over Fp .
92

Proof. Combine Propositions 3.3.1 and 3.4.13.
Proof of Theorem 3.4.12. Suppose first that X is of finite type. We will show that conditions (1), (2), and(3) are satisfied. Conditions (1) and (2) follow from the simple-connectivity of X (Corollary 3.4.10). We will
prove (3). We have a commutative diagram
Hn (τ≤n X; Fp )/ Hn (Mat(τ≤n X); Fp )

Hn (X; Fp )/ Hn (Mat(X); Fp ).

Here the vertical maps are isomorphisms by Proposition 3.3.8, and the left vertical map is an equivalence
because the map Mat(X) → Mat(τ≤n X) is (n + 1)-connective. It follows that the right vertical map is anequivalence. We may therefore replace X by τ≤n X and thereby reduce to the case where X is m-truncatedfor some integer m. We proceed by induction on m. If m = 1, then X ' ∗ and there is nothing to prove.
Assume therefore that m ≥ 2, so we have a pullback diagramX/∗τ≤m−1 Xb
/ K(A,
m + 1)bwhere A = πm X. It follows from Remark 3.3.12 that the cohomology groups Hi (K(A,m + 1); Fp ) are
finite dimensional, and from the inductive hypothesis that the cohomology groups Hi (τ≤m−1 X; Fp ) are
finite-dimensional for all i ≥ 0. Proposition 2.5.2 implies that the canonical map
C ∗ (τ≤m−1 X; Fp ) ⊗C ∗ (K(A,m+1);FFp → C ∗ (X; Fp )b
p)is an equivalence of E∞ -algebras over Fp . Using Lemma X.4.1.16, we deduce that the cohomology groups
Hi (X; Fp ) are finite-dimensional for all i ≥ 0.
Now suppose that conditions (1), (2), and (3) are satisfied. We wish to show that X is of finite type. It
follows from (1) and (2) that X is simply connected (Corollary 3.4.10). We must show that each homotopy
group πn (X) is topologically finitely generated. We proceed by induction on n, the case n ≤ 1 being vacuous.
Assume that the homotopy groups πm X are topologically generated for m < n, so that τ≤n−1 X is of finite
type. Choose a point η ∈ τ≤n−1 X and consider the associated fiber sequence
F → X → τ≤n−1 X.
The first part of the proof shows that the cohomology groups H∗ (τ≤n−1 X; Fp ) satisfy conditions (1), (2),and (3). Proposition 2.5.2 implies that the canonical map
C ∗ (X; Fp ) ⊗C ∗ (τ≤n−1 X;Fp ) Fp → C ∗ (F ; Fp )is an equivalence of E∞ -algebras over Fp . Using Lemma X.4.1.16, we deduce that the cohomology groups
Hi (F ; Fp ) are finite-dimensional for i ≥ 0. In particular, the group Hn (F ; Fp ) is finite-dimensional. Since F
is n-connective, Corollary 3.4.14 implies that πn F ' πn X is topologically finitely generated.
Remark 3.4.15. Let X ∈ S be a simply connected space, and suppose that the cohomology groups
Hn (X; Fp ) are finite-dimensional over Fp for each n ≥ 0. It follows from Theorem 3.4.12 that the p-profinite
completion Xp∧ is of finite type. Combining this with Proposition 3.3.8, we conclude that the map of spaces
u : X → Mat(Xp∧ ) induces an isomorphism H∗ (Mat(Xp∧ ); Fp ) → H∗ (X; Fp ). Choose a point η ∈ X and let
F denote the homotopy fiber of u over the image of η. Using the Serre spectral sequence, we deduce that93

the cohomology groups Hi (F ; Fp ) vanish for i > 0. It follows that multiplication by p is invertible on thehomotopy groups of F . If we assume that the homotopy groups of X are finitely generated abelian groups,
then the long exact sequence
πn (F, η) → πn X → πn Xp∧ → πn−1 (F, η)
implies that each homotopy group πn Xp∧ is given by the p-adic completion of πn X.

3.5p-adic Homotopy Theory

Let k be a separably closed field of characteristic p > 0. According to Proposition 3.1.16, the construction
X 7→ C ∗ (X; k) determines a fully faithful embedding from the ∞-category SPro(p) of p-profinite spaces tothe ∞-category CAlgop
k of E∞ -algebras over k. Our first goal in this section is to describe the essential imageof this embedding. First, we need to introduce some terminology.
Definition 3.5.1. Let k be a field of characteristic p and let V be a vector space over k. A map σ : V → V
is Frobenius-semilinear if it satisfies the conditions
σ(x + y) = σ(x) + σ(y)σ(λx) = λp σ(x)for x, y ∈ V , λ ∈ k. In this case, we let V σ denote the subset {x ∈ V : σ(x) = x}.
Example 3.5.2. Let k be a field of characteristic p and let A ∈ CAlgk be an E∞ -algebra over k. For every
integer n, the operation P 0 : πn A → πn A of Construction 2.2.6 is Frobenius-semilinear.
Lemma 3.5.3. Let k be a field of characteristic p, let V be a vector space over k, and let σ : V → V bea Frobenius-semilinear map. Then V σ is a vector space over the field Fp . Moreover, the canonical k-linear
map k ⊗Fp V σ → V is injective.
Proof. It suffices to show that if x1 , . . . , xn ∈ V σ are linearly independent over Fp , then they P
are linearly
independent over k. Suppose otherwise. Then there exists a nontrivial dependence relation
λi xi = 0
where λi ∈ k. Let us assume that n has been chosen as small as possible, so that each coefficient λi isnonzero. Without loss of generality, we may assume that λP
1 = 1. Applying the map σ, we deduce thatP
λpi xi = 0. Subtracting, we obtain a dependence relation (λi − λpi )xi = 0. Note that λ1 − λp1 = 0. By
minimality, we conclude that each difference λi − λpi = 0. This implies that each coefficient λi belongs to
Fp , contradicting our assumption that the elements xi are linearly independent over Fp .
Definition 3.5.4. Let k be a field of characteristic p, V a vector space over k, and σ : V → V a Frobeniussemilinear map. We will say that σ is solvable if the map k ⊗Fp V σ → V is an isomorphism. We will say thatan E∞ -algebra A over k is solvable if, for every integer n, the Frobenius-semilinear map P 0 : πn A → πn A issolvable.
Remark 3.5.5. Let k be a field of characteristic p and let A be a solvable E∞ -algebra over k. For n > 0,
the operation P 0 : πn A → πn A is trivial (Remark 2.2.7). Since πn A is generated (as a k-vector space) by
elements which are fixed by P 0 , we conclude that πn A ' 0.
Remark 3.5.6. Let k be a field of characteristic p. Suppose we are given a filtered diagram of k-vector
spaces {Vα } equipped with a compatible collection of Frobenius-semilinear maps σα : Vα → Vα , so that thefiltered colimit V = lim Vα inherits a Frobenius-semilinear map σ : V → V . If each σα is solvable, then σ is−→solvable.
Example 3.5.7. Let X be a space such that each cohomology group Hn (X; Fp ) is finite-dimensional. For
any field k of characteristic p, the universal coefficient theorem gives an isomorphism k ⊗Fp Hn (X; Fp ) →Hn (X; k), and the image of the map Hn (X; Fp ) → Hn (X; k) consists of elements which are fixed by theoperation P 0 : Hn (X; k) → Hn (X; k) (Corollary 2.2.14). It follows that C ∗ (X; k) ∈ CAlgk is solvable.

94

Theorem 3.5.8. Let k be a separably closed field of characteristic p > 0. Then the construction X 7→
C ∗ (X; k) determines a fully faithful embedding F : (SPro(p) )op → CAlgk , whose essential image is the collection of solvable E∞ -algebras over k.
Proof. We have already seen that F is fully faithful (Proposition 3.1.16). Let C ⊆ CAlgk be the essential
image of F and let C0 ⊆ CAlgk be the full subcategory spanned by the solvable E∞ -algebras over k. Notethat C0 is closed under filtered colimits by Remark 3.5.6 and contains C ∗ (X; k) for every p-finite space X by
Example 3.5.7. It follows that C ⊆ C0 . We will complete the proof by showing that C0 ⊆ C.
Let A ∈ CAlgk be solvable. For every integer n let VAn be the kernel of the map (1 − P 0 ) : π−n A → π−n A.
Then we have an isomorphism of vector spaces π−n A ' k ⊗Fp VAn , under which the operation P 0 corresponds
to the map k ⊗Fp VAn → k ⊗Fp VAn given by the Frobenius map on k. Since k is separably closed, the ArtinSchreier map x 7→ x − xp is a surjection from k to itself, so that (1 − P 0 ) : π−n A → π−n A is surjective forevery integer n. If n ≥ 0, then Theorem 2.2.17 gives an exact sequence of abelian groups
1−P 0

1−P 0

π1−n A −→ π1−n A → π0 MapCAlgk (C ∗ (K(Z /p Z, n); k), A) → π−n A −→ π−n A
and therefore a canonical isomorphism π0 MapCAlgk (C ∗ (K(Z /p Z, n); k), A) ' VAn .
Since F preserves small colimits, it admits a right adjoint G (Proposition T.5.5.2.9). Let φ : A → B
be a map of solvable E∞ -algebras over k such that G(φ) is an equivalence. Then for every integer n ≥ 0,composition with φ gives a bijection
π0 MapCAlgk (C ∗ (K(Z /p Z, n); k), A)'

π0 MapSPro(p) (G(A), K(Z /p Z, n))→ π0 MapSPro(p) (G(B), K(Z /p Z, n))'π0 MapCAlgk (C ∗ (K(Z /p Z, n); k), B),

It follows that φ induces an isomorphism of Fp -vector spaces VAn → VBn and therefore an isomorphism of
k-vector spaces π−n A → π−n B. For n > 0, we have πn A ' 0 ' πn B by Remark 3.5.5. It follows that themap φ : A → B is an equivalence.
Now let A ∈ CAlgk be an arbitrary solvable E∞ -algebra over k, and consider the counit map v :(F ◦ G)(A) → A. Note that (F ◦ G)(A) ∈ C ⊆ C0 and that G(v) is an equivalence (since F is fully faithful).
It follows from the above argument that v is an equivalence, so that A ' F (G(A)) belongs to the essential
image of F .
We now wish to specialize Theorem 3.5.8 to the setting of p-profinite spaces of finite type. In this case
we have a particularly convenient description of solvability:
Proposition 3.5.9. Let k be an algebraically closed field of characteristic p, let V be a finite-dimensional
vector space over k, and let F : V → V be a Frobenius-semilinear endomorphism. The following conditions
are equivalent:(1) The map F is injective.
(2) The map F is bijective.
(3) The Frobenius-semilinear automorphism F is solvable.
Lemma 3.5.10. Let k be a separably closed field of characteristic p, let V be a finite-dimensional vector
space over k, and let F : V → V be a Frobenius-semilinear endomorphism. If F is injective and V is nonzero,then V F is nonzero.
Proof. Choose a nonzero element v ∈ V . Since V is finite-dimensional, the elements {v, F (v), F 2 (v), . . .}cannot all be linearly independent. Thus there exists a nonzero dependence relation
X
λi F i (v) = 0.
0≤i≤n

95

Replacing v by F i (v) if necessary, we may assume that the coefficient λ0 is nonzero. Dividing by −λ0 , we
may assume that λ0 = 1: that is, we haveXv=
−λi F i (v).
1≤i≤n

We may assume that n is chosen as small as possible: it follows that the set {v, F (v), . . . , F n−1 (v)} is linearly
independent, and therefore λn 6= 0. Since v 6= 0, we must have n > 0.
Note that
n−1
n−2
n
n−1
n−2
f (x) = xp + λ1p xp+ λ2p xp
+ · · · + λn x
is a separable polynomial of degree pn > 1, and therefore has pn distinct roots in the field k. Consequently,
there exists a nonzero element a ∈ k such that f (a) = 0. Let2

w = av + (ap + aλ1 )F (v) + (ap + ap λp1 + aλ2 )F 2 (v) + · · · + (ap

n−1

+ ap

n−2

n−2

λp1

+ · · · + aλn−1 )F n−1 (v).

Since the elements {F i (v)}0≤i<n are linearly independent and a 6= 0, w is a nonzero element of V . An
explicit calculation givesX
w − F (w) = av +aλi F i (v) + (aλn − f (a))F n (v) = a(v + λ1 F (v) + · · · + λn F n (v)) = 0,0<i<nso that w is a nonzero element of V F .
Proof of Proposition 3.5.9. The implication (3) ⇒ (2) follows from the fact that k is perfect, and the implication (2) ⇒ (1) is obvious. Assume that (1) is satisfied; we will prove (3). Lemma 3.5.3 implies thatthe map of vector spaces u : V F ⊗Fp k → V is injective, so that V F ⊗Fp k is finite-dimensional over k andtherefore V F is finite-dimensional over Fp . Choose a basis {v1 , v2 , . . . , vm } for V F over Fp .
Let W be the cokernel of u, and note that F induces a Frobenius-semilinear endormorphism F 0 of W .
We first claim that F 0 is injective. To prove this, suppose that F 0 (w) = 0 for some w ∈ W ; we will provethat w = P
0. Let w ∈ V be a representative for w, so that F (w) belongs to the image of u and therefore has
the form 1≤i≤m ci vi for some coefficients ci ∈ k. Since k is algebraically closed, we can choose elementsbi ∈ k satisfying ci = bpi . ThenX
X pbi vi ) = F (w) −bi vi = 0.
F (w −
1≤i≤m

1≤i≤m

P

Since F is injective, we deduce that w = 1≤i≤m bi vi belongs to the image of u, so that w = 0.
0
We next claim that W F = 0. To prove this, suppose that w ∈ W satisfies F 0 (w) = w. Let w ∈ V be a
representative of w. Arguing as above, we deduce thatX
c i viF (w) = w +1≤i≤m

for some coefficients ci ∈ k. Since k is algebraically closed, we can choose elements ai ∈ k such that
api = ai − ci . Then Then
X
X p
X
X
F (w +
ai vi ) = F (w) +ai vi = w +(ci + api )vi = w +ai vi ,
1≤i≤m

1≤i≤m

1≤i≤m

1≤i≤m

P
so that w + 1≤i≤m ai vi ∈ V F . It follows that w belongs to the image of u and therefore w = 0.
Invoking Lemma 3.5.10, we conclude that W = 0. It follows that the map u is surjective and therefore
an isomorphism.

96Pro(p)

Theorem 3.5.11. Let k be an algebraically closed field of characteristic p > 0, and let Sft
be the fullPro(p)
subcategory of S
spanned by the p-profinite spaces of finite type. Then the construction X 7→ C ∗ (X; k)Pro(p) op
determines a fully faithful embedding (Sft
) → CAlgk , whose essential image is the collection of E∞ algebras A over k satisfying the following conditions:(1) The unit map k → π0 A is an isomorphism.
(2) The homotopy group π−1 A vanishes.
(3) For every integer n, the homotopy group πn A is a finite-dimensional vector space over k.
(4) The map P 0 : πn A → πn A is injective for every integer n.
Moreover, if these conditions are satisfied, then the map P 0 : πn A → πn A is bijective for every integer n.
Proof. Combine Proposition 3.5.9, Theorem 3.4.12, and Theorem 3.5.8.
We would now like to apply Theorem 3.5.11 to obtain information about the usual homotopy theory of
spaces, rather than the homotopy theory of p-profinite spaces. For this, we need to characterize the essential
image of the fully faithful embedding described in Theorem 3.3.3.
Proposition 3.5.12. Let p be a prime number and let X ∈ S. The following conditions are equivalent:(1) The space X is simply connected, and each homotopy group πn X is a finitely generated module overZp .
(2) There exists a p-profinite space Y of finite type such that X ' Mat(Y ).
Remark 3.5.13. In the situation of Proposition 3.5.12, the p-profinite space Y is canonically determinedby X: in fact, it is given by the p-profinite completion Xp∧ of X.
Proof. Let C ⊆ S be the full subcategory spanned by those spaces which satisfy condition (1), and let C0 ⊆ S
be the full subcategory spanned by those objects which satisfy condition (2). We wish to prove that C = C0 .
The containment C0 ⊆ C is obvious. To prove the converse, we first make the following observation:(∗) If we are given a small diagram p : K → C0 whose limit lim(p) belongs to C, then lim(p) belongs to C0 .
←−←−Pro(p)

To prove (∗), we let Sftdenote the full subcategory of SPro(p) spanned by the p-profinite spaces offinite type. Theorem 3.3.3 implies that the materialization functor induces an equivalence of ∞-categories
Pro(p)Matft : Sft→ C0 . Consequently, any diagram p : K → C0 is equivalent to a compositionp

Pro(p)

K → Sft

Matftlong → C0 .

Let X denote the limit lim(p) (formed in the ∞-category SPro(p) ). Since the materialization functor Mat←−
commutes with limits, we deduce that Mat(X) ' lim(p) ∈ C, so that X is of finite type and therefore←−lim(p) = Mat(X) ∈ C0 .
←−Now suppose that X is an arbitrary object of C; we wish to show that X ∈ C0 . We can realize X as thelimit of its Postnikov tower· · · → τ≤2 X → τ≤1 X ' ∗.
Using (∗), we are reduced to proving that each truncation τ≤n X belongs to C0 . We proceed by induction onn. When n = 1, the space τ≤n X is contractible and the result is obvious. Assume therefore that n > 1, sothat we have a pullback diagram
/∗
τ≤n X

τ≤n−1 X
/ K(A, n + 1)
97

where A = πn X. Using (∗), we are reduced to proving that K(A, n + 1) and τ≤n−1 X belong to C0 . In thesecond case, this follows from the inductive hypothesis. In the first case, we observe that K(A, n + 1) is thebmaterialization of K(A,n + 1).
We can summarize Proposition 3.5.12 and Theorem 3.3.3 as follows:
Pro(p)

Corollary 3.5.14. Let Sftdenote the full subcategory of SPro(p) spanned by the p-profinite spaces of finite
Pro(p)
type. Then the materialization functor Mat : SPro(p) → S induces a fully faithful embedding Sft→ S,
whose essential image is spanned by those spaces X which are simply connected and such that each homotopy
group πn X is a finitely generated module over Zp .
Corollary 3.5.15 (Mandell). Let k be an algebraically closed field of characteristic p > 0. Let C denote thefull subcategory of S spanned by those spaces X which are simply connected and for which each homotopy
group πn (X) has the structure of a finitely generated module over Zp . Then the construction X 7→ C ∗ (X; k)induces a fully faithful embedding C → CAlgop
k , whose essential image is the collection of E∞ -algebras A overk satisfying conditions (1) through (4) of Theorem 3.5.11.
Proof. Note that if the cohomology groups Hn (X; Fp ) are finite dimensional for each n ≥ 0, the canonicalmap C ∗ (X; Fp ) ⊗Fp k → C ∗ (X; k) is an equivalence. Using Notation 3.3.14, we conclude that the map
C ∗ (Xp∧ ; k) → C ∗ (X; k) is also an equivalence. The desired result now follows by combining Theorem 3.5.11
with Corollary 3.5.14.

3.6

Étale Homotopy Theory

Let X be a scheme. In [1], Artin and Mazur introduce a Pro-object of the homotopy category hS, which theycall the étale homotopy type of X. By a slight variation on this construction, one can obtain a p-profinite
space Shp (X), which we call the p-profinite étale homotopy type of X. In this section, we will use Theorem
2.6.13 to obtain a very simple description of Shp (X) in the case when p is nilpotent in the structure sheaf ofX.
We begin by reviewing the definition of the étale homotopy type.
Notation 3.6.1. Let S<∞ denote the full subcategory of S spanned by the truncated spaces, let Sfc ⊆ S<∞
denote the full subcategory spanned by the π-finite spaces (Example 2.3.2), and for each prime number p let
Sp−fc ⊆ Sfc denote the full subcategory spanned by the p-finite spaces (Definition 2.4.1).
The ∞-categories Sfc is essentially small and idempotent complete, and the ∞-category S<∞ is accessible.
Consequently, we can define ∞-categories of Pro-objects Pro(Sfc ), and Pro(S<∞ ). We will refer to Pro(Sfc )as the ∞-category of profinite spaces and Pro(S<∞ ) as the ∞-category of protruncated spaces. Using Remark
3.1.7, we obtain fully faithful embeddings
Pro(Sp−fc ) ,→ Pro(Sπ−fc ) ,→ Pro(S<∞ ) ,→ Pro(S).
Moreover, each of these functors admits a left adjoint which is given by restriction.
Let X be an ∞-topos, let f∗ : X → S be a geometric morphism (which is unique up to a contractible spaceof choices), and f ∗ its left adjoint. Then f∗ f ∗ : S → S is a left-exact accessible functor from the ∞-category
S to itself, which we can identify with an object of Pro(S). We will denote this object by Sh(X) and refer toit as the shape of X (see §T.7.1.6). Restricting the functor Sh(X) to the subcategories
Sp−fc ⊂ Sfc ⊂ S<∞ ,we obtain pro-objects
Shp (X) ∈ Pro(Sp−fc )Shfc (X) ∈ Pro(Sfc )Sh<∞ (X) ∈ Pro(S<∞ ).

We will refer to Shp (X) as the p-profinite shape of X, to Shfc (X) as the profinite shape of X, and to Sh<∞ (X)as the protruncated shape of X.
98

Let X = (X, OX ) be a spectral Deligne-Mumford stack. We let Sh(X) = Sh(X) denote the shape of theunderlying ∞-topos of X. We will refer to Sh(X) as the étale homotopy type of X. Similarly, we set
Shp (X) = Shp (X)Shfc (X) = Shfc (X)Sh<∞ (X) = Sh<∞ (X).

We refer to Shp (X), Shfc (X), and Sh<∞ (X) as the p-profinite, profinite, and protruncated étale homotopy
types of X.
Construction 3.6.2. Let R be a connective E∞ -ring, and let X = (X, OX ) be a spectral Deligne-Mumford
stack over R. Let f∗ : X → S be the global sections functor, and f ∗ : S → X its left adjoint. Then theconstruction
K 7→ OX (f ∗ K)
determines a functor F : Sop → CAlg. There is an evident map
α0 : R → Γ(X; OX ) = F (∗).
Since the functor F is a right Kan extension of its restriction to {∗}, α0 extends to a natural transformation
α : C ∗ (•; R) → F . Let 1 denote the final object of X, and let Sh(X) = f∗ f ∗ denote the étale homotopy typeof X. For every space K, we have a canonical map
αSh(X)(K) = f∗ f ∗ K ' MapX (1, f ∗ K) → MapCAlgR (OX (f ∗ K), OX (1)) → MapCAlgR (C∗ (K; R), Γ(X; OX ).
Theorem 3.6.3. Let R be an E∞ -ring which is p-thin, and let X = (X, OX ) be a p-thin spectral DeligneMumford stack over R. Then, for every p-finite space K, the canonical map
Sh(X)(K) → MapCAlgR (C ∗ (K; R), Γ(X; OX ))of Construction 3.6.2 is a homotopy equivalence. That is, the p-profinite étale homotopy type of X is givenby the formulaShp−fc (X)(K) ' MapCAlgR (C ∗ (K; R), Γ(X; OX ).
Proof. The assertion is local on X. We may therefore assume without loss of generality that X = Specét A isaffine, and Corollary 2.4.18 supplies an equivalence A ⊗R C ∗ (K; R) ' C ∗ (K; A). We may therefore replace
R by A and thereby reduce to the case where X = Specét R. The desired result now follows immediately fro
Theorem 2.6.13.
If X is a normal Noetherian scheme, then theorem of Artin and Mazur asserts that the (protruncated)étale homotopy type of X is profinite. For later reference, we record a proof of their result here. Our
treatment follows [1], with a few minor modifications.
Definition 3.6.4. Let X be a spectral Deligne-Mumford stack which is locally Noetherian. We will say that
X is normal if, for every étale map f : Specét R → X, the E∞ -ring R is a discrete and normal (that is, it isequivalent to a finite product of integrally closed integral domains).
Theorem 3.6.5 (Artin-Mazur). Let X be a quasi-compact quasi-separated spectral algebraic space. If X is
locally Noetherian and normal, then the protruncated étale homotopy type Sh<∞ (X) is profinite: that is, itbelongs to the essential image of the fully faithful embedding Pro(Sfc ) ,→ Pro(S<∞ ).
The proof of Theorem 3.6.5 reduces to the following:
Proposition 3.6.6. Let X = (X, OX ) be a quasi-compact quasi-separated spectral algebraic space which is
Noetherian and normal, and let f∗ : X → S be the global sections functor. Let Y be a truncated space. For
every point η ∈ f∗ f ∗ Y , there exists a π-finite space K and a map K → Y such that η lies in the essential
image of the induced map f∗ f ∗ K → f∗ f ∗ Y .
99

Let us show that Proposition 3.6.6 implies Theorem 3.6.5.
Proof of Theorem 3.6.5. Let us abuse notation by identifying Pro(Sfc ) and S<∞ with full subcategories ofPro(S<∞ ). We have an evident map Sh<∞ (X) → Shfc (X), and we wish to prove that it is an equivalence inPro(S<∞ ). For this, it will suffice to show that for every truncated space Y , the induced map
Shfc (X)(Y ) → Sh<∞ (X)(X)
is a homotopy equivalence. Equivalently, we must show that for each n ≥ 0, the induced mapnθ : Shfc (X)(Y ) → Shfc (X)(Y )∂ ∆ ×Sh<∞ (X)(Y )∂ ∆n Sh<∞ (X)(Y )nn

is surjective on connected components. Note that every point of the space Shfc (X)(Y )∂ ∆ ' Shfc (X)(Y ∂ ∆ )nnlies in the essential image of the map Shfc (X)(K) → Shfc (X)(Y ∂ ∆ ) for some map K → Y ∂ ∆ where K is
∂ ∆nπ-finite. It will therefore suffice to show that for every π-finite space K with a map K → Y, the induced
map
Shfc (X)(K) ×Shfc (X)(Y ∂ ∆n ) Shfc (X)(Y ) → Shfc (X)(K) ×Sh<∞ (X)(Y ∂ ∆n ) Sh<∞ (X)(Y )is surjective on connected components. Taking Z = K ×Y ∂ ∆n Y , we are reduced to proving that the map
Shfc (X)(Z) → Sh<∞ (X)(Z)is surjective on connected components, which follows immediately from Proposition 3.6.6.
Lemma 3.6.7. Let X• be a simplicial set and n ≥ 2 an integer. Assume that X• satisfies the following
conditions:(a) The simplicial set X• satisfies the Kan extension condition in dimension 2. That is, for 0 ≤ i ≤ 2,
every map Λni → X• extends to a 2-simplex of X.
(b) For each integer k, the set Xk is finite.
(c) For every x ∈ X0 , the homotopy groups πm (|X• |, x) are finite for 0 < m ≤ n.
Let G be a finite group acting on X• , and let Y• be the quotient X• /G. Then Y• also satisfies condition (a),(b), and (c).
Proof. It is obvious that Y• satisfies (b). We next prove (a) in the case i = 1 (the other cases follow by anessentially identical argument). Suppose we are given a pair of edges e : y → y 0 and e0 : y 0 → y 00 in thesimplicial set Y• . Lift e and e0 to edges e : y → y 0 and ee0 : ye0 → ye00 in the simplicial set X. Then y 0 = g(e
y0 )
0
0
0
2
e ). Since X• satisfies (a), the pair (e, e ) determines a map Λ1 → X• which
for some g ∈ G. Let e = g(eextends to a 2-simplex of X• . The image of this 2-simplex in Y• is the desired extension of (e, e0 ).
We now prove that Y• satisfies (c). Let π≤1 Y• denote the fundamental groupoid of Y• . The objects ofπ≤1 Y• are given by the elements of Y0 . Using (a), we see that every morphism in π≤1 Y• is given by an edgeof Y• . Since Y0 and Y1 are finite, we deduce that the space |Y• | has finitely many path components, eachof which has a finite fundamental group. It will therefore suffice to show that if Y•0 → Y• exhibits Y•0 as auniversal cover of some path component of Y• , then the homotopy groups πm |Y•0 | are finite for 2 ≤ m < n.
Let X•0 = X• ×Y• Y•0 . Since Y•0 is a finite-sheeted cover of Y• , X•0 is a finite-sheeted cover of X• . It follows
that X•0 also satisfies condition (c). We may therefore replace X• by X•0 , and thereby reduce to the casewhere where |Y• | is simply connected.
In the simply connected case, the finiteness of the homotopy groups πm |Y• | for m ≤ n is equivalent to
the finiteness of the homology groups Hm (Y• ; Z) for 0 < m < n. Since each of the sets Yk is finite, it iseasy to see that each Hm (Y• ; Z) is a finitely generated abelian group for every integer m. It will therefore
suffice to show that the rational homology groups Hm (Y• ; Q) vanish for 0 < m ≤ n. We now observe that
Hm (Y• ; Q) = Hm (X• /G; Q) is the space of coinvariants for the action of G on Hm (X• ; Q), and therefore
trivial (for 0 < m ≤ n) since X• satisfies condition (c).
100

Proof of Proposition 3.6.6. The object Y ∈ S is represented by a Kan complex Y• , which we will view asa hypercovering of Y . Let us view f ∗ Y• as a hypercovering in the ∞-topos X/f ∗ Y . Let 1 denote the final
object of X, so that the point η ∈ f∗ f ∗ Y can be identified with a morphism 1 → f ∗ Y in X. Let V• denotethe hypercovering of X given by Vn = 1 ×f ∗ Y Yn .
We now construct a new hypercovering U• of X equipped with a natural transformation U• → V• , where
each Un is affine. We construct U• as the union of a family of maps U•≤n : N(∆≤n )op → X. Assume thatU•≤n−1 has already been constructed, so that we can define latching and matching objects Ln (U ) and Mn (U ).
Since X is a coherent ∞-topos and Um is affine for m < n, we deduce that Mn (U ) is quasi-compact. Since
V• is a hypercovering of X, the map Vn → Mn (V ) is an effective epimorphism. It follows that the map
Vn ×Mn (V ) Mn (U ) → Mn (U ) is an effective epimorphism. We may therefore choose an object W ∈ X which
is a coproduct of affine objects equipped with a map W → Vn ×Mn (V ) Mn (U ) for which the composite map
W → Vn ×Mn (V ) Mn (U ) → Mn (U )is an effective epimorphism. Since Mn (U ) is quasi-compact, we can assume
that W is a finite coproduct`of affine objects of X, and is therefore itself affine. Set Un = Ln (U ) W , so that we have an evident
commutative diagram/ Un/ Mn (U )Ln (U )
Ln (V )


/ Mn (V ).

/ Vn

Using Proposition T.A.2.9.14, we see that this diagram determines a functor U•≤n : N(∆≤n )op → X extendingU•≤n−1 , together with a natural transformation U•≤n → V• | N(∆≤n )op . Taking the union of the maps U•≤n ,we obtain a hypercovering U• with the desired properties.
Since X is locally Noetherian, the topological space | X | is Noetherian (Remark XII.1.4.14). Let T denotethe finite set of generic points of | X |. For each affine object W ∈ X, we can identify OX (W ) with a normal
Noetherian commutative ring. We let W c denote the (finite) set of connected components of the Zariski
spectrum SpecZ OX (W ). Each of these connected components C has a generic point whose image is a pointqW (C) ∈ T . We regard qW as a map from W c to T . Note that a map of affine objects W → W 0 induces a
map of finite sets W c → W 0c fitting into a commutative diagram/ W 0c

Wc
qW

!

T.

}

qW 0

We now define a simplicial set K• by the formula
Kn = Unc .
By construction, we have a compatible family of maps Un → f ∗ Yn , which determines a map of simplicial
sets K• → Y• . Let K ∈ S denote the geometric realization of K• (that is, K is the space represented by thesimplicial set K• ). It is clear that the map U• → f ∗ Y• factors through f ∗ K• , so that the point η ∈ f∗ f ∗ Y•
lies in the essential image of the map f∗ f ∗ K → f∗ f ∗ Y . Since Y is m-truncated for some m, the map K → Y
factors through τ≤m K. We will complete the proof by showing that τ≤m K is π-finite. Without loss ofgenerality, we may assume that m ≥ 1.
−1
For each t ∈ T , we let K•t denote
subset of K• given by Knt = qU{t} ⊆ Unc = Kn . Thenn` the simplicialtK• is given by the disjoint union t∈T K• , so that K is a disjoint union of spaces {K t = |K•t |}t∈T . It will
therefore suffice to show that τ≤m K t is π-finite. Let κ denote the residue field of X at the point t. For eachn ≥ 0, let Zn denote the fiber productSpecét κ ×X (X/Un , OX |Un ),101

so that we have a canonical bijection Knt ' | Zn |. Then each Zn is finite and étale over Specét κ. We may
therefore choose a finite Galois extension κ0 of κ such that, for n ≤ m + 1, the space Zn ×Specét κ Specét κ0 is
a finite disjoint union of copies of Specét κ0 . For each n ≥ 0, let Zn denote the set of κ0 -points of Zn . ThenZ• is a simplicial set which is acted on by the Galois group G = Gal(κ0 /κ), and we have a canonical map
Z• /G → K•t which induces a bijection Zn /G ' Knt for n ≤ m + 1. Since U• is a hypercovering of X, thesimplicial set Z• is (m + 1)-connective and satisfies the Kan condition in dimension 2. Using Lemma 3.6.7,
we see that the homotopy groups πk (|Z• /G|, z) are finite for k ≤ m and any choice of base point z ∈ |Z• /G|.
From this we immediately deduce that the homotopy groups πk (|K•t |, z) are finite for k ≤ m, so that τ≤m |K•t |is π-finite.

102

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