The étale symmetric Künneth theorem

Let k be an algebraically closed field, l≠chark\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$l\ne {{\,\textrm{char}\,}}k$$\end{document} a prime number, and X a quasi-projective scheme over k. We show that the étale homotopy type of the dth symmetric power of X is Z/l\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{Z}/l$$\end{document}-homologically equivalent to the dth strict symmetric power of the étale homotopy type of X. We deduce that the Z/l\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{Z}/l$$\end{document}-local étale homotopy type of a motivic Eilenberg–Mac Lane space is an ordinary Eilenberg–Mac Lane space.


Introduction
In the first part of this paper we show that the étale homotopy type of the dth symmetric power of a quasi-projective scheme X over a separably closed field k is Z/l-homologically equivalent to the dth symmetric power of the étale homotopy type of X , where l = char k is any prime. Symbolically, where ét ∞ is the étale homotopy type, S d is the dth symmetric power (more precisely the strict symmetric power), and L Z/l is Z/l-localization á la Bousfield-Kan. The étale homotopy type ét ∞ X of a scheme X is a pro-space originally defined by Artin and Mazur [1] and later refined by Friedlander [11]. It is characterized by the property that the (nonabelian) étale cohomology of X with constant coefficients coincides with the cohomology of ét ∞ X . The formula ( * ) is related to a theorem of Deligne about the étale cohomology of symmetric powers [2, XVII, Théorème 5.5.21], but there are three significant differences: (1) Deligne's theorem is about cohomology with proper support, and so does not say anything about the cohomology of non-proper schemes. (2) We give an equivalence at the level of homotopy types, whereas Deligne only gives an equivalence at the level of cochains. (3) Deligne's theorem works over an arbitrary quasi-compact quasi-separated base and with arbitrary Noetherian torsion coefficients; for our result the base must be a separably closed field whose characteristic is prime to the torsion of the coefficients.
While there may be a relative version of ( * ) over a base, the localization away from the residual characteristics cannot be avoided when dealing with non-proper schemes.
In his proof, after reducing to the case where the base is a field k, Deligne employs Witt vectors to further reduce to the case where k has characteristic zero (concluding with a transcendental argument). In this step it is crucial that X be proper over k. Our arguments are thus necessarily quite different. We use the existence a schematic topology, finer than the étale topology but cohomologically equivalent to it, for which the quotient map X d → S d X is a covering; this is the qfh topology used extensively by Voevodsky in his work on triangulated categories of motives.
Combining ( * ) with the motivic Dold-Thom theorem, we show that if k is algebraically closed and A is an abelian group on which the characteristic exponent of k acts invertibly, then the Z/l-local étale homotopy type of a motivic Eilenberg-Mac Lane space K (A(q), p) is the Z/l-localization of an ordinary Eilenberg-Mac Lane space K (A, p).

Conventions
Throughout this paper, we use the language of ∞-categories developed in [24] and [23]. Although our main results can be stated in more classical language, their proofs use the flexibility of higher category theory in an essential way. We warn the reader that this is the default language in this paper, so for example the word "colimit" always means "homotopy colimit", "unique" means "unique up to a contractible space of choices", etc. We will use the following notation: • S is the ∞-category of small ∞-groupoids, which we also call spaces; • Set is the category of simplicial sets; • Top ∞ is the ∞-category of ∞-topoi and geometric morphisms [24, §6.3]; • C ≤n is the subcategory of n-truncated objects in an ∞-category C; • C ω is the subcategory of compact objects in an ∞-category C with filtered colimits; • hC is the homotopy 1-category of an ∞-category C; • X ∧ is the hypercompletion of an ∞-topos X [24, §6.5.2].

Historical note
The first draft of this paper was written in 2011 as a step towards the computation of the motivic Steenrod algebra in positive characteristic. Afterwards I realized that the technology of étale homotopy types could be avoided completely using the Bloch-Kato conjecture, which was the approach taken in [16]. Since I had no other application in mind I did not attempt to turn this draft into a publishable paper. More recently however, the main result of this paper was used by Zargar [31] to compute the weight 0 homotopy groups of the motivic sphere spectrum in positive characteristic. Given this new application, it seemed important that this paper be published after all. I want to thank Chuck Weibel for encouraging me to finally take this paper out of its draft state.

Homotopy types of schemes
Let τ be a pretopology on the category of schemes (in the sense of [2, II, Définition 1.3]). If X is a scheme, the small τ -site of X is the full subcategory of Sch X spanned by the members of the τ -coverings of X and equipped with the Grothendieck topology induced by τ (we assume that this is an essentially small category). We denote by X τ the ∞-topos of sheaves of spaces on the small τ -site of X . The assignment X → X τ is functorial: a morphism of schemes f : X → Y induces a geometric morphism of ∞-topoi f * : X τ → Y τ given by Recall that the functor S → Top ∞ associating to an ∞-groupoid its classifying ∞-topos admits a pro-left adjoint ∞ : Top ∞ → Pro(S) associating to any ∞-topos its shape (see [24, §7. 1.6] or [15]). The τ -homotopy type τ ∞ X of a scheme X is the shape of the ∞-topos X τ : τ ∞ X = ∞ (X τ ).
This construction defines a functor τ ∞ : Sch → Pro(S). Let X be an ∞-topos and let c : S → X be the constant sheaf functor. By definition of the shape, we have Map X ( * , cK ) Map Pro(S) ( ∞ X, K ) for every K ∈ S. In particular, the cohomology of X with coefficients in an abelian group can be computed as the continuous cohomology of the pro-space ∞ X. If X is locally connected (i.e., if for every X ∈ X the pro-set π 0 ∞ (X /X ) is constant), we have more generally that the category Fun( ∞ X, Set) of discrete local systems on ∞ X is equivalent to the category of locally constant sheaves of sets on X [15,Theorem 3.13], and, if A is such a sheaf of abelian groups, then H * (X, A) coincides with the continuous cohomology of the pro-space ∞ X with coefficients in the corresponding local system [15,Proposition 2.15].

Remark 1.1
In the definition of τ ∞ X , we could have used any τ -site of X -schemes containing the small one. For if X τ is the resulting ∞-topos of sheaves, the canonical geometric morphism X τ → X τ is obviously a shape equivalence. It follows that the functor τ ∞ depends only on the Grothendieck topology induced by τ .

Remark 1.2
For schemes over a fixed base scheme S, we can define in the same way a relative version of the τ -homotopy type functor taking values in the ∞-category Pro(S τ ).

Remark 1.3 Let X ∧
τ be the hypercompletion of X τ . By the generalized Verdier hypercovering theorem [9, Theorem7.6(b)] ∞ (X ∧ τ ) is corepresented by the simplicially enriched diagram 0 : HC τ (X ) → Set where HC τ (X ) is the cofiltered simplicial category of τ -hypercovers of X and 0 (U • ) is the simplicial set that has in degree n the colimit of the presheaf U n . See [15, §5] for details.
If X is a CW complex, its dth symmetric power S d X is the set of orbits of the action of the symmetric group d on X d , endowed with the quotient topology. Even though the action of d on X d is not free, it is well known that the homotopy type of S d X is an invariant of the homotopy type of X . More generally, if G is a group acting on a CW complex X , the orbit space X /G can be written as the homotopy colimit where O(G) is the orbit category of G (whose objects are the subgroups of G and whose morphisms are the G-equivariant maps between the corresponding quotients) and X H is the subspace of H -fixed points [ We will relate strict symmetric powers to the notion of strictly commutative monoid in Sect. 7. Note that S 0 X is a final object of C and that S 1 X X . For example, in an ∞-category of sheaves of spaces on a site, S d is computed by applying S d objectwise and sheafifying the result, and in a 1-category it is the usual symmetric power, namely the coequalizer of the action groupoid d × X d ⇒ X d . We note that any functor that preserves colimits and finite products commutes with S d .

Homological localizations of pro-spaces
Let Pro(S) denote the ∞-category of pro-spaces. Recall that this is the ∞-category freely generated by S under cofiltered limits and that it is equivalent to the full subcategory of Fun(S, S) op spanned by accessible left exact functors [22,Proposition 3.1.6]. Any such functor is equivalent to Y → colim i∈I Map(X i , Y ) for some small cofiltered diagram X : I → S. Moreover, combining [24, Proposition 5.3.1.16] and the proof of [2, Proposition 8.1.6], we can always find such a corepresentation where I is a cofiltered poset such that, for each i ∈ I , there are only finitely many j with i ≤ j; such a poset is called cofinite.
In [19], Isaksen constructs a proper model structure on the category Pro(Set ) of prosimplicial sets, called the strict model structure, with the following properties: • a pro-simplicial set X is fibrant if and only if it is isomorphic to a diagram (X s ) s∈I such that I is a cofinite cofiltered poset and X s → lim s<t X t is a Kan fibration for all s ∈ I ; • the inclusion Set → Pro(Set ) is a left Quillen functor; • it is a simplicial model structure with simplicial mapping sets defined by Denote by Pro (S) the ∞-category associated to this model category, and by c : S → Pro (S) the left derived functor of the inclusion Set → Pro(Set ). Since Pro (S) admits cofiltered limits, there is a unique functor ϕ : Pro(S) → Pro (S) that preserves cofiltered limits and such that ϕ • j c, where j : S → Pro(S) is the Yoneda embedding. Proof Let X ∈ Pro(Set ) be fibrant. Then X is isomorphic to a diagram (X s ) indexed by a cofinite cofiltered poset and such that X s → lim s<t X t is a Kan fibration for all s, and so, for all Z ∈ Pro(Set ), Map (Z , X s ) → lim s<t Map (Z , X t ) is a Kan fibration. It follows that the limit Map (Z , X ) lim s Map (Z , X s ) in Set is in fact a limit in S, so that X lim s c(X s ) in Pro (S). This shows that ϕ is essentially surjective.
Let X ∈ Pro(S) and choose a corepresentation X : I → S where I is a cofinite cofiltered poset. Using the model structure on Set , X can be strictified to a diagram X : I → Set such that X s → lim s<t X t is a Kan fibration for all s ∈ I . By the first part of the proof, we then have X lim s c(X s ) in Pro (S), whence ϕ X X . Given also Y ∈ Pro(S), we have where in the last step we used that filtered colimits of simplicial sets are always colimits in S. This shows that ϕ is fully faithful.
Let S <∞ ⊂ S be the ∞-category of truncated spaces. A pro-truncated space is a pro-object in S <∞ . It is clear that the full embedding Pro(S <∞ ) → Pro(S) admits a left adjoint that preserves cofiltered limits and sends a constant pro-space to its Postnikov tower; it also preserves finite products since truncations do. The τ <∞ -equivalences in Pro(S ≥1 * ) are precisely those maps that become -isomorphisms in Pro(hS ≥1 * ) in the sense of Artin and Mazur [1,Definition 4.2].

Remark 3.2
The model structure on Pro(Set ) defined in [17] is the left Bousfield localization of the strict model structure at the class of τ <∞ -equivalences. It is therefore a model for the ∞-category Pro(S <∞ ) of pro-truncated spaces.
Let R be a commutative ring. A morphism f : X → Y in Pro(S) is called an R-homological equivalence (resp. an R-cohomological equivalence) if it induces an equivalence of homology pro-groups H * (X , R) H * (Y , R) (resp. an equivalence of cohomology groups H * (Y , R) H * (X , R)). By [20,Proposition 5.5], f is an R-homological equivalence if and only if it induces isomorphisms in cohomology with coefficients in arbitrary R-modules. A pro-space X is called R-local if it is local with respect to the class of R-homological equivalences, i.e., if for every R-homological equivalence Y → Z the induced map Map(Z , X ) → Map(Y , X ) is an equivalence in S. A pro-space is called R-profinite 1 if it is local with respect to the class of R-cohomological equivalences. We denote by Pro(S) R (resp. Pro(S) R ) the ∞-category of R-local (resp. R-profinite) pro-spaces.
The characterization of R-homological equivalences in terms of cohomology shows that any τ <∞ -equivalence is an R-homological equivalence. We thus have a chain of full embeddings Pro(Set ) [20, Theorems 6.3 and 6.7]. We also give a self-contained proof in Proposition 3.7 below. For the last statement, we must show that the canonical map is an equivalence for all X , Y ∈ Pro(S). Since both sides are R-local, it suffices to show that this map an R-homological equivalence. By definition of L R , the canonical map C * (X , R) → C * (L R X , R) induces an isomorphism on homology pro-groups. Since C * (−, R) : S → D(R) ≥0 is a symmetric monoidal functor, we have a natural equivalence is a pro-homology isomorphism by definition of L R , it remains to show that the tensor product in Pro(D(R) ≥0 ) preserves pro-homology isomorphisms. A morphism in Pro(D(R) ≥0 ) is a pro-homology isomorphism if and only if it induces an equivalence on n-truncations for all n, so the claim follows from the fact that the canonical map The fact that L R preserves finite products is very useful and we will use it often. It implies in particular that L R preserves commutative monoids and commutes with the formation of strict symmetric powers. Here is another consequence:

Corollary 3.4 Finite products distribute over finite colimits in Pro(S) R .
Proof Finite colimits are universal in Pro(S), i.e., are preserved by any base change (since pushouts and pullbacks can be computed levelwise). The result follows using that L R preserves finite products.

Remark 3.5
Let X be an ∞-topos and let X ∧ be its hypercompletion. The geometric morphism X ∧ → X induces an equivalence of pro-truncated shapes (since truncated objects are hypercomplete [24, Lemma 6.5.2.9]), and a fortiori also of R-local and R-profinite shapes for any commutative ring R.

Remark 3.6
Let l be a prime number. The Bockstein long exact sequences show that any Z/l-cohomological equivalence is also a Z/l n -cohomological equivalence for all n ≥ 1. In particular, if X is an ∞-topos, its Z/l-profinite shape L Z/l ∞ X remembers the cohomology of X with l-adic coefficients.
As shown in [20,Proposition 7.3], if R is solid (e.g., R = Z/n for some integer n) and X ∈ S is connected, then L R X is the pro-truncation of the Bousfield-Kan R-tower of X [3, I, §4]. It follows that the limit of the pro-space L R X is the Bousfield-Kan R-completion R ∞ X .
The existence of the localization functors L R and L R is a special case of a more general result which we now formulate. If C is any locally small ∞-category, Pro(C) op is the full subcategory of Fun(C, S) spanned by small filtered colimits of corepresentable functors [24, Note that K-equivalences are preserved by cofiltered limits, since K ⊂ C and the objects of C are cocompact in Pro(C). We denote by the full subcategory of K-local objects.

Proposition 3.7 Let C be a presentable ∞-category and K a collection of objects of C.
Suppose that K is the essential image of an accessible functor. Then the inclusion Pro(C) K ⊂ Pro(C) admits a left adjoint. Moreover, Proof With no assumptions on K, there is an obvious inclusion Pro( K) ⊂ Pro(S) K . If K is small, then every functor K → S is a small colimit of corepresentables, so the inclusion Pro( K) ⊂ Pro(C) has a left adjoint L given by restricting a functor C → S to K. If X ∈ Pro(C) K , then the canonical map X → L X is a K-equivalence between K-local objects, hence it is an equivalence. This proves the proposition when K is small.
The proof of the general case follows [20,Proposition6.10]. For any X ∈ Pro(C), we must κ such that X is a cofiltered limit of λ 0 -compact objects of C, and let X → Y 0 be the K λ 0 -localization of X . Inductively, choose λ n λ n−1 such that Y n−1 is a cofiltered limit of λ n -compact objects, and let X → Y n be the K λ n -localization of X . Finally, let Y = lim n Y n . Then Y is K-local, and it remains to show that the induced map X → Y is a K-equivalence.
Let K ∈ K be the image of L ∈ L. For any n ≥ 0, let L n be the λ n -filtered ∞-category (L λ n ) /L . Since ϕ preserves λ n -filtered colimits, K is the colimit of ϕ|L n for any n. The conclusion now follows by evaluating the colimit since X → Y n is a K λ n -equivalence and X is a cofiltered limit of λ n -compact objects for any n. Setting m = n − 1, we have since Y n−1 is a cofiltered limit of λ n -compact objects. This concludes the proof of the existence of the left adjoint. By construction, Y is in fact K λ -local for any large enough λ κ, so we have also proved that This implies that Pro(C) K ⊂ Pro( K), since we already know it when K is small. Proposition 3.7 applies in particular whenever K is small. For example, if K is the collection of Eilenberg-Mac Lane spaces K (R, n) with n ≥ 0, then Pro(S) K = Pro(S) R . Proposition 3.7 also applies with K the collection of Eilenberg-Mac Lane spaces K (M, n) for M any R-module and n ≥ 0, this being the image of a filtered-colimit-preserving functor from a countable disjoint union of copies of the category of R-modules. In this case, Pro(S) K = Pro(S) R .

Remark 3.8
If l is a prime number, the spaces that can be obtained from the Eilenberg-Mac Lane spaces K (Z/l, n) using finite limits are precisely the truncated spaces with finite π 0 and whose homotopy groups are finite l-groups. Hence, Pro(S) Z/l coincides with the ∞-category of l-profinite spaces studied in [22, §3]. Lemma 3.9 Let C be a presentable ∞-category, let (K α ) α be a small filtered diagram of collections of objects of C satisfying the assumption of Proposition 3.7, and let K = α K α . Then the localization functors induce an equivalence Proof Note that K also satisfies the assumption of Proposition 3.7, so that Pro(C) K = Pro( K). Since K = colim α K α , we have an equivalence which implies that the functor Pro(C) K → lim α Pro(C) K α is fully faithful. It remains to show that a functor F : K → S is a small filtered colimit of corepresentables if each of its By Proposition 3.7, the inclusion Pro(S ≤n ) R ⊂ Pro(S) admits a left adjoint L R,≤n . We define Pro(S ≤n ) R and the localization functor L R ≤n similarly (using only the R-module R). By Lemma 3.9, the localization functors L R,≤n induce an equivalence and similarly for Pro(S) R .
The following proposition shows that the localizations L R and L R agree in many cases of interest, partially answering [20, Question 11.2].

Proposition 3.11
Let F be a prime field and let X be a pro-space whose F-homology progroups are pro-finite-dimensional vector spaces. Then L F X is F-profinite. In other words, the canonical map L F X → L F X is an equivalence.
Proof First we claim that any F-profinite pro-space with profinite π 0 satisfies the given condition on X . Such a pro-space is a cofiltered limit of spaces with finite π 0 that are obtained from K (F, n)'s using finite limits. By [3, Proposition 5.3], each connected component of such a space is obtained from the point by a finite sequence of principal fibrations with fibers K (F, n) with n ≥ 1. Using Eilenberg-Moore [22, Corollary 1.1.10], it thus suffices to show that H m (K (F, n), F) is finite-dimensional for every m ≥ 0 and n ≥ 1, which is a well-known computation. Thus, both X and L F X have pro-finite-dimensional F-homology pro-groups. It follows that the canonical map X → L F X induces an isomorphism on F-cohomology ind-groups, whence on F-homology pro-groups.

Remark 3.12
It is clear that the class of pro-spaces X satisfying the hypothesis of Proposition 3.11 is preserved by L F , retracts, finite products, and finite colimits (it suffices to verify the latter for pushouts).

Proposition 3.13
Let F be a prime field and let X be a pro-space whose F-homology progroups in degrees ≤ n are pro-finite-dimensional vector spaces. Then the canonical map L F,≤n X → L F ≤n X is an equivalence. Furthermore, L F,≤n preserves finite products of such pro-spaces.
Proof As in the proof of Proposition 3.11, L F ≤n X has pro-finite-dimensional F-homology pro-groups, hence the canonical map X → L F ≤n X induces an isomorphism on F-cohomology ind-groups in degrees ≤ n. Since these groups are finite-dimensional F-vector spaces, this remains true for cohomology with coefficients in any F-vector space, which implies the first statement. For the second statement, we must show that the canonical map Dualizing and taking the colimit, we find that for all m ≤ n and similarly for Y , so we are done.

The h and qfh topologies
Let X be a Noetherian scheme. An h covering of X is a finite family {U i → X } of morphisms of finite type such that the induced morphism i U i → X is universally submersive (a morphism of schemes f : Y → X is submersive if it is surjective and if the underlying topological space of X has the quotient topology). If in addition each U i → X is quasi-finite, it is a qfh covering. These notions of coverings define pretopologies on Noetherian schemes which we denote by h and qfh, respectively. The h and qfh topologies are both finer than the fppf topology, and they are not subcanonical.

Proposition 4.1 Let X be a Noetherian scheme.
(1) The canonical map qfh ∞ X → ét ∞ X induces an isomorphism in cohomology with any local system of abelian coefficients. In particular, for any commutative ring R, qfh ∞ X induces an isomorphism in cohomology with any local system of torsion abelian coefficients. In particular, for any torsion commutative ring R, Proof Recall that the cohomology of ∞ X with coefficients in a local system coincides with the cohomology of X with coefficients in the associated locally constant sheaf (see Sect. 1). The first statements are thus translations of [29,Theorem 3.4.4] and [29,Theorem 3.4.5], respectively (the excellence of X is a standing assumption in loc. cit., but it is not used in the proof of (1); see also [28, §10] for self-contained proofs). The statements about the Rlocal shapes follow immediately, since L R inverts morphisms that induce isomorphisms in cohomology with coefficients in any R-module (see Sect. 3).

Remark 4.2
Voevodsky's proof also shows that H 1 et (X , G) H 1 qfh (X , G) for any locally constant étale sheaf of groups G. It follows from [17,Proposition 18.4

] (and Remark 3.2) that
For C a small ∞-category, we denote by r : C → PSh(C) the Yoneda embedding, and if τ is a topology on C, we denote by r τ = a τ r the τ -sheafified Yoneda embedding. The advantage of the qfh topology over the étale topology is that it can often cover singular schemes by smooth schemes. Let us make this explicit in the case of quotients of smooth schemes by finite group actions. We first recall the classical existence result for such quotients.
A groupoid scheme X • is a simplicial scheme such that, for every scheme Y , Hom(Y , X • ) is a groupoid. If P is any property of morphisms of schemes that is stable under base change, we say that X • has property P if every face map in X • has property P (of course, it suffices that d 0 : X 1 → X 0 have property P).

Lemma 4.4 Let S be a scheme and let X • be a finite and locally free groupoid scheme over S. Suppose that for any x
Proof By a well-known theorem of de Jong [5, Theorem 4.1], every scheme of finite type over a perfect field k is h-locally smooth, hence is h-locally in C. It follows that there is an induced h topology on C whose covering sieves are the restrictions of h-covering sieves in Sch ft k , or equivalently those sieves that generate an h-covering sieve in Sch ft k . By the comparison lemma [2, III, Théorème 4.1], the restriction functor i * and its right adjoint i * restrict to an equivalence between the subcategories of h sheaves of sets, hence between the ∞-categories of hypercomplete h sheaves (since they are the hypercompletions of the associated 1-localic ∞-topoi).

The Étale homotopy type of symmetric powers
Proposition 5.1 Let k be a separably closed field, l = char k a prime number, and X and Y schemes of finite type over k. Let τ ∈ {h, qfh,ét}. Then Proof By Proposition 4.1, it suffices to prove the lemma for τ =ét. Since L Z/l preserves finite products and X´e t and X ∧ et have the same Z/l-local shape (see Remark 3.5), it suffices to show that the canonical map is a Z/l-homological equivalence or, equivalently, that it induces an isomorphism in cohomology with coefficients in any Z/l-module M. Both sides of (5.2) are corepresented by cofiltered diagrams of simplicial sets having finitely many simplices in each degree (by Remark 1.3 and the fact that any étale hypercovering of a Noetherian scheme is refined by one that is degreewise Noetherian). If K is any such pro-space, C * (K , Z/l) is a cofiltered diagram of degreewise finite chain complexes of vector spaces. On the one hand, this implies so we may assume that M = Z/l. On the other hand, it implies that the Küneth map The composition of this isomorphism with the map induced by (5.2) in cohomology is the canonical map , which is also an isomorphism by [6, Th. finitude, Corollaire 1.11].

Remark 5.3
Let X be a Noetherian scheme and let τ ∈ {h, qfh,ét}. We observed in the proof of Proposition 5.1 that the pro-space ∞ (X ∧ τ ) is the limit of a cofiltered diagram of spaces whose integral homology groups are finitely generated. It follows from Proposition 3.11 that L F τ ∞ X is F-profinite for any prime field F, which answers [20, Question 11.3] quite generally. Now let τ and σ be pretopologies on Sch S with τ finer than σ , and let C be a small full subcategory of Sch S closed under σ -coverings (but not necessarily under τ -coverings). Then the functor τ ∞ : C → Pro(S) takes values in a cocomplete ∞-category and is a σ -cosheaf according to Lemma 1.5, so it lifts uniquely to a left adjoint functor C Pro(S).

Remark 5.4
If C is such that C /X contains the small τ -site of X for any X ∈ C, then τ ∞ is simply the composition where for X an ∞-topos the inclusion X → Top ∞ is X → X /X . Indeed, this composition preserves colimits (by [24, Proposition 6.3.5.14]), and it restricts to τ ∞ on C (cf. Remark 1.1). However, the reader should be warned that we will use τ ∞ in situations where this hypothesis on C is not satisfied.

Remark 5.5
The extension τ ∞ involves taking infinite colimits in Pro(S), which are somewhat ill-behaved (they are not universal, for example). As we will see in Sect. 9, it is sometimes advantageous to consider a variant of τ ∞ taking values in ind-pro-spaces.

Theorem 5.6
Let k be a separably closed field, l = char k a prime number, and X a quasiprojective scheme over k. Let τ ∈ {h, qfh,ét}. Then for any d ≥ 0 there is a canonical equivalence Proof Let C be the category of quasi-projective schemes over k. By Corollary 4.6, the representable sheaf functor r qfh : C → Shv qfh (C) preserves strict symmetric powers. Using Proposition 5.1 and the fact that L Z/l qfh ∞ preserves colimits, we deduce that L Z/l qfh ∞ preserves strict symmetric powers on C. For τ ∈ {h, qfh,ét}, we get the first and last equivalences being from Proposition 4.1. The functor L Z/l itself also preserves finite products (Proposition 3.3) and hence strict symmetric powers, so we are done.

Remark 5.7 It is possible to define a natural map
in Pro(S) inducing the equivalence of Theorem 5.6. It suffices to make the square commute. Using the model for the τ -homotopy type discussed in Remark 1.3 and the commutativity of the functor of connected components with symmetric powers, the task to accomplish is the following: associate to any τ -hypercover U • → X a τ -hypercover V • → S d X refining S d U • → S d X and such that V • × S d X X d → X d refines U d • → X d , in a simplicially enriched functorial way (i.e., we must define a simplicial functor HC τ (X ) → HC τ (S d X ) and the refinements must be natural). If τ = h or τ = qfh, S d U • → S d X is itself a τ -hypercover and we are done, but things get more complicated for τ =ét as symmetric powers of étale maps are not étale anymore.
We refer to [21, §4.5] and [27, §3] for more details on the following ideas. Given a finite group G and quasi-projective G-schemes U and X , a map f : U → X is G-equivariant if and only if it admits descent data for the action groupoid of G on X . The map f is fixed-point reflecting if it admits descent data for theČech groupoid of the quotient map X → X /G (this condition can be expressed more explicitly using the fact that G×X → X × X /G X is faithfully flat: f is fixed-point reflecting if and only if it is G-equivariant and induces a fiberwise isomorphism between the stabilizer schemes). Since étale morphisms descend effectively along universally open surjective morphisms [26,Theorem 5.19], such as X → X /G, the condition that f reflects fixed points is equivalent to the induced map U /G → X /G being étale and the square in which the vertical maps are d -equivariant fixed-point reflecting étale covers (because fixed-point reflecting morphisms are stable under base change). Finally, let V n = W n / d . It is then easy to prove that V • → S d X is an étale hypercover with the desired functoriality. Using the commutativity of (5.8), one can also show that the map induced by

A 1 -localization
Let S be a quasi-compact quasi-separated scheme and C a full subcategory of Sch S such that (1) objects of C are of finite presentation over S; (2) S ∈ C and A 1 S ∈ C; (3) if X ∈ C and U → X is étale, separated, and of finite presentation, then U ∈ C; (4) C is closed under finite products and finite coproducts.
Following [30, §0], we call such a category C admissible. Note that every smooth S-scheme admits an open covering by schemes in C. Let Shv Nis (C) denote the ∞-topos of sheaves of spaces on C for the Nisnevich topology, and let Shv Nis (C) A 1 ⊂ Shv Nis (C) be the full subcategory of A 1 -invariant Nisnevich sheaves. We shall denote by L Nis,A 1 : PSh(C) → Shv Nis (C) A 1 the left adjoint to the inclusion.
From now on we fix a prime number l different from the residual characteristics of S. In Sect. 1, we defined the functor The functorÉt l is called the Z/l-local étale homotopy type functor. Note that if S is Noetherian (resp. Noetherian and excellent), we could also use qfh ∞ (resp. h ∞ ) instead of ét ∞ in the above diagram, according to Proposition 4.1.

Remark 6.1
The Z/l-profinite completion L Z/lÉ t l is the ∞-categorical incarnation of the étale realization functor defined by Isaksen in [18] as a left Quillen functor, but our results do not require this stronger completion. Note thatÉt l X is already Z/l-profinite if S is Noetherian and X ∈ (Shv Nis (C) A 1 ) ω , by Remarks 5.3 and 3.12.
We now assume that S = Spec k where k is a separably closed field.

Lemma 6.2 The restriction ofÉt l to the subcategory of compact objects (Shv Nis (C) A 1 ) ω preserves finite products.
Proof By Proposition 5.1, the functor L Z/l ét ∞ preserves finite products on C. Since the functor L Nis,A 1 r : C → Shv Nis (C) A 1 also preserves finite products, the restriction ofÉt l to the image of C preserves finite products. Finally, since (Shv Nis (C) A 1 ) ω is the closure of the image of C under finite colimits and retracts, the result follows from Corollary 3.4.
Let p ≥ q ≥ 0. We define the Z/l-local mixed spheres S p,q l ∈ Pro(S) Z/l, * by where μ is the group of roots of unity in k and T l μ = lim n μ l n is its l-adic Tate module. Here we regard Z l and T l μ as pro-groups. Of course, S p,q l L Z/l S p , but if q > 0 this equivalence depends on infinitely many noncanonical choices ( viz., an isomorphism Z l T l μ). By Proposition 3.11, S p,q l is Z/l-profinite. Note that the functorÉt l preserves pointed objects, since ét ∞ (Spec k) * .
Proof This is obvious if q = 0. By Lemma 6.2, it remains to treat the case p = q = 1. The étale μ l n -torsor l n : G m → G m is classified by a morphism ét ∞ G m → K (μ l n , 1). In the limit over n ≥ 0, we obtain a morphism of pro-spaces ϕ : ét ∞ G m → K (T l μ, 1). We claim that ϕ is a Z/l-homological equivalence, i.e., it induces an isomorphism in cohomology with coefficients in any Z/l-module M. By [14, VII, Proposition 1.3(i)(c)], we have In fact, this computation shows that the morphism ét ∞ G m → K (μ l , 1) induces an isomorphism on H i (−, M) for i ≤ 1. The same is true for the projection K (T l μ, 1) → K (μ l , 1), hence also for ϕ. Since both the source and the target of ϕ have vanishing cohomology in degrees ≥ 2, this completes the proof.

Group completion and strictly commutative monoids
Let C be an ∞-category with finite products. Recall from [23, §2.4.2] that a commutative monoid in C is a functor M : Fin * → C such that for all n ≥ 0 the canonical map M( n ) → M( 1 ) n is an equivalence. We let CMon(C) denote the full subcategory of Fun(Fin * , C) spanned by the commutative monoids.
A commutative monoid M in C has an underlying simplicial object, namely its restriction along the functor Cut : op → Fin * sending [n] to the finite set of cuts of [n] pointed at the trivial cut, which can be identified with n . The commutative monoid M is called grouplike if its underlying simplicial object is a groupoid object in the sense of [24, Definition 6.1.2.7]. This is equivalent to requiring both shearing maps M × M → M × M to be equivalences. We denote by CMon gp (C) ⊂ CMon(C) the full subcategory of grouplike objects.
If f : C → D preserves finite products (and C and D admit finite products), then it induces a functor CMon(C) → CMon(D) by postcomposition; this functor clearly preserves grouplike objects and hence restricts to a functor CMon gp (C) → CMon gp (D). We will continue to use f to denote either induced functor. Proof The functors f and g induce adjoint functors between ∞-categories of Fin * -diagrams, and it remains to observe that they both preserve the full subcategory of (grouplike) commutative monoids. Definition 7.2 An ∞-category C is distributive if it is presentable and if finite products in C distribute over colimits. A functor f : C → D between distributive ∞-categories is distributive if it preserves colimits and finite products.
For example, for any ∞-topos X, the truncation functors τ ≤n : X → X ≤n are distributive, and for any admissible category C ⊂ Sch S and any topology τ on C, the localization functor L τ,A 1 : PSh(C) → Shv τ (C) A 1 is distributive.
If C is distributive, then the ∞-category CMon(C) is presentable by [

Remark 7.4
If X is an ∞-topos, group completion of commutative monoids in X preserves 0-truncated objects. As in the proof of Lemma 2.5, it suffices to prove this for X = S, where it follows from the McDuff-Segal group completion theorem (see [25] for a modern proof of the latter).
We can define a generalized "free Z-module" functor in any distributive ∞-category as follows. Let FFree N (resp. FFree Z ) be the full subcategory of CMon(Set) spanned by (N n , +) (resp. by (Z n , +)) for n ≥ 0. If C is an ∞-category with finite products, we shall denote by Assume now that C is distributive. The forgetful functors Mod N (C) → C and Mod Z (C) → C then preserve limits and sifted colimits, hence admit left adjoints N : C → Mod N (C) and Z : C → Mod Z (C).
Since the ∞-categories Mod N (C) and Mod Z (C) are pointed, we also have reduced versions NX andZX when X is a pointed object of C. More precisely,Ñ is the unique colimitpreserving extension of N to C * , and similarly forZ.

Lemma 7.5 If C is an ∞-category with finite products, the square
admits a left adjoint such that the following square commutes: CMon gp (C) CMon(C).
forget gp forget In particular,ZX (ÑX ) gp for any X ∈ C * .
Proof Since FFree N is semiadditive and FFree Z is additive, the forgetful functors CMon gp (C) → CMon(C) → C induce equivalences The key point is that the ∞-category FFree Z is obtained from FFree N be group-completing the mapping spaces, so that FFree N → FFree Z is the universal finite-product-preserving functor to an additive ∞-category. Therefore, the forgetful functor can be identified with the functor Mod Z (CMon gp (C)) Mod N (CMon gp (C)) → Mod N (CMon(C)).
This description immediately implies the claims: the first claim follows since a finite-productpreserving functor M : FFree op N → CMon(C) lands in CMon gp (C) if and only if M(N) is grouplike, and the second claims follows since group completion preserves finite products.
We can describe free strictly commutative monoids more concretely using strict symmetric powers (see Sect. 2): Lemma 7.6 Let C be a distributive ∞-category. Then the composite functor Proof It suffices to check this for the universal X , which lives in the distributive ∞-category Fun(Fin, S). We may thus assume C = S. In that case, the forgetful functor Mod N (S) → S is modeled by the right Quillen functor Mod N (Set ) → Set [24, Proposition 5.5.9.1], whose left adjoint is given by the desired formula. Since the functor S d on S can be computed using symmetric powers of CW complexes, this completes the proof. Lemma 7.6 shows that the endofunctor X → d≥0 S d X of any distributive ∞-category has a canonical structure of monad. Its multiplication involves a canonical equivalence S d (X Y ) e+ f =d S e X × S f Y and a canonical map S d S e X → S de X .

Remark 7.7
If C is presentable and A is a small ∞-category with finite products, we have where ⊗ denotes the tensor product of presentable ∞-categories (this follows immediately from [23,Proposition 4.8.1.17]). Hence, the presentable ∞-category Mod Z (C) of grouplike strictly commutative monoids in C is a module over Mod Z (S), which is the ∞-category of connective H Z-modules. For X ∈ Mod Z (C) and A a connective H Z-module, we will write X ⊗ A for the result of the action of A on X . Note that the construction X → X ⊗ A is preserved by any colimit-preserving functor f : C → D.
In particular, if C is distributive and X ∈ C, one can form the strictly commutative monoid ZX ⊗ A for any connective H Z-module A, which can be described more concretely as follows. Any connective H Z-module A can be obtained from Z in the following steps: (1) take finite products of copies of Z to get finitely generated free Z-modules; (2) take filtered colimits of finitely generated free Z-modules to get arbitrary flat Z-modules; (3) take colimits of simplicial diagrams of free Z-modules to get arbitrary connective H Zmodules [24, Lemma 5.5.8.13].
Since the forgetful functor Mod Z (C) → C preserves finite products and sifted colimits, the object ZX ⊗ A in C can be obtained from ZX using the same steps.
In the distributive ∞-category S, ZX ⊗ A has its "usual" meaning. For instance, if A is an abelian group, thenZS p ⊗ A is an Eilenberg-Mac Lane space K (A, p).

Sheaves with transfers
Let S be a Noetherian scheme, C ⊂ Sch S an admissible category consisting of separated S-schemes, and R a commutative ring. We denote by Cor(C, R) the additive category whose objects are those of C and whose morphisms are the finite correspondences with coefficients in R [4, §9], 4 We denote by PSh * (C) the ∞-category of pointed presheaves on C, by the ∞-category of presheaves with R-transfers, and by PSh tr (C, R) : u tr the free-forgetful adjunction. The functor u tr preserves limits and sifted colimits and factors through the ∞-category CMon gp (PSh(C)); in fact, it factors through the ∞-category of grouplike strictly commutative monoids, using the finite-product-preserving functor Since finite products and finite coproducts coincide in semiadditive ∞-categories, the functor u tr : PSh tr (C, R) → Mod Z (PSh(C)) preserves all colimits. For τ a topology on C, we denote by Shv tr τ (C, R) the ∞-category of τ -sheaves with Rtransfers on C, and by Shv tr τ (C, R) A 1 the ∞-category of homotopy invariant τ -sheaves with R-transfers on C; these are defined by the cartesian squares Furthermore, by [24, Proposition 5.4.6.6], the forgetful functors u tr in the above diagram are accessible. Since they preserve limits, they admit left adjoint functors, which we will denote by R tr (it will always be clear from the context which category R tr is defined on).
We say that a topology τ on C is compatible with R-transfers if for any presheaf with R-transfers F on C, the canonical map is an equivalence in Shv * τ (C).
Proof If τ is compatible with transfers, then for any F ∈ PSh * (C), Since a τ (U + ) a τ (X + ), this proves the "only if" direction. Conversely, define so that Shv tr τ (C, R) ⊂ PSh tr (C, R) is the subcategory of E-local objects, and suppose that the functor a τ u tr sends elements of E to equivalences. LetĒ be the strong saturation of E, i.e., the smallest class of morphisms containing E, satisfying the 2-out-of-3 property, and closed under colimits in Fun( 1 , PSh tr (C, R)). By [24, Proposition 5.5.4.15(4) and Proposition 5.2.7.12], the localization functor a tr τ : PSh tr (C, R) → Shv tr τ (C, R) is the universal functor sending elements ofĒ to equivalences. We claim that a τ u tr sends morphisms inĒ to equivalences. It will suffice to show that the class of morphisms f such that a τ u tr ( f ) is an equivalence is closed under the 2-out-of-3 property (which is obvious) and colimits. The functor u tr : PSh tr (C, R) → PSh * (C, R) does not preserve colimits, but it preserves sifted colimits and transforms finite coproducts into finite products. Since a τ is left exact and any colimit can be built out of finite coproducts and sifted colimits, this proves the claim. Thus, there exists a functor f : Shv tr commute. Since the horizontal compositions are the identity, f u tr and a τ u tr u tr a tr τ . Lemma 8.2 Suppose that τ is compatible with R-transfers. Then the square It will suffice to show that a functor f exists as indicated. Define so that Shv tr τ (C, R) A 1 ⊂ PSh tr (C, R) is the full subcategory of (E τ ∪ E A 1 )-local objects. The functor L τ,A 1 u tr carries morphisms in E τ and E A 1 to equivalences: for E τ , this is because τ is compatible with R-transfers and for E A 1 it is because u tr R tr (X × A 1 ) + → u tr R tr X + is an A 1 -homotopy equivalence (see the last part of the proof of [ where S p,q ∈ Shv * Nis (C) A 1 is the motivic p-sphere of weight q. Although this construction depends on the coefficient ring R in general, it does not if either the schemes in C are regular or if the positive residual characteristics of S are invertible in R [4, Remark 9.1.3(3)]; the latter will always be the case in what follows.
Pro (S ≤n ) Z/l also preserves finite products. It follows that finite products distribute over finite colimits in each of these ∞-categories, so that their ind-completions are distributive.
Consider the colimit-preserving functor By construction,Ét l is the composition ofÉt The next theorem is our étale version of [30,Proposition3.41]. We point out that the category C in the statement below need not be closed under symmetric powers, so the theorem applies directly with C the category of smooth separated k-schemes with no need for resolutions of singularities.  (2) given X , Y , A, and B, the square Proof Any k-scheme of finite type is Zariski-locally quasi-projective, so we can assume that the schemes in C are quasi-projective without changing the categories and functors involved. As X varies, the source and target of θ X ,A are functors taking values in strictly commutative monoids in lim n Ind(Pro (S ≤n ) Z/l ), and as such they preserve colimits: for the left-hand side this is clear and for the right-hand side it follows from Corollary 8.3. In particular, these functors are left Kan extended from their restriction to C. To show the existence of θ X ,A , it will therefore suffice to define θ X ,A for X representable, i.e., X = L Nis,A 1 r (Z ) + for some Z ∈ C, and this construction should be natural in Z and A. Furthermore, since u tr is H Z-linear andÉt × l is distributive, we havé in Shv Nis (C) ≤0 . To show that θ X ,Z [1/e] is an equivalence, it suffices to show that Et × l L Nis,A 1 (ϕ) is an equivalence. By definition ofÉt × l , the functorÉt × l L Nis,A 1 factors through a ∧ h i ! : Shv Nis (C) → Shv ∧ h (Sch ft k ), so it suffices to show that a ∧ h i ! (ϕ) is an equivalence. This follows from Corollary 4.6 and Proposition 4.7 (since k is perfect and every smooth k-scheme is Zariski-locally in C).
The strategy to prove (1) and (2) is the following: we first reduce as above to the representable case, where the statements follow from properties of the motivic Dold-Thom equivalence. For (1), we may assume that X is represented by Z ∈ C. Then the adjunction map X → u tr Z[1/e] tr X corresponds, through the Dold-Thom equivalence, to the map which proves the result. For (2), we may assume that X and Y are represented by Z and W and that A = B = Z[1/e]. Moreover, we can replaceZ withÑ. It then suffices to note that the pairing u tr Z[1/e] tr X ∧ u tr Z[1/e] tr Y → u tr Z[1/e] tr (X ∧ Y ) arising from the monoidal structure of Z[1/e] tr is induced, via the Dold-Thom equivalence, by the obvious maps S a Z × S b W → S ab (Z × W ).
For A a connective H Z-module, let A l be the pro-H Z-module lim n A/l n . Note that A l A if A admits an H Z/l n -module structure for some n ≥ 1. Given q ∈ Z, we let A l (q) = A l ⊗ Z l T l μ ⊗q (which is noncanonically isomorphic to A l ). For example, Z/l n (q) μ ⊗q l n . Theorem 9.4 Let k be an algebraically closed field of characteristic exponent e, l = e a prime number, and C ⊂ Sch k an admissible subcategory consisting of semi-normal separated schemes. For any connective H Z[1/e]-module A and any integers p, q with p ≥ 1 and p ≥ q ≥ 0, there is a canonical equivalencé Et l K (A(q), p) C τ <∞ K (A l (q), p) of pointed objects in Pro(S) Z/l , natural in A, andÉt l preserves smash products between such spaces. Furthermore, under these equivalences, É t l (X ) ∧Ét l (Y ) wheneverÉt × l X andÉt × l Y belong to the essential image of L ind j ind . The remaining statements are easily deduced from properties (1) and (2) in Theorem 9.2.
In conclusion, let us emphasize the two most important special cases of Theorem 9.4: Corollary 9.6 Let k be an algebraically closed field of characteristic exponent e, l = e a prime number, and C ⊂ Sch k an admissible subcategory consisting of semi-normal separated schemes.
(1) For any Z[1/e] ⊂ ⊂ Z (l) , there is a canonical equivalencé where T l μ ⊗q is the l-adic Tate module of μ ⊗q viewed as a pro-group.
(2) For any n ≥ 1, there is a canonical equivalencé Et l K (Z/l n (q), p) C K (μ ⊗q l n , p).
In particular,Ét l K (Z/l n (q), p) C is a constant pro-space.
In both cases, the Z/l-local étale homotopy type is already Z/l-profinite.
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