THE ÉTALE SYMMETRIC KÜNNETH THEOREM

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1 THE ÉTALE SYMMETRIC KÜNNETH THEOREM MARC HOYOIS Abstract. Let k be a separaby cosed fied, char k a prime number, and X a quasiprojective scheme over k. We show that the étae homotopy type of the dth symmetric power of X is Z/-homoogicay equivaent to the dth symmetric power of the étae homotopy type of X. We deduce that the étae reaizations of motivic Eienberg Mac Lane spaces are ordinary Eienberg Mac Lane spaces. Contents Introduction 1 1. Homotopy types of schemes 2 2. Symmetric powers in -categories 3 3. Homoogica ocaizations of pro-spaces 4 4. The qfh topoogy 6 5. Étae homotopy type of symmetric powers 7 6. A 1 -ocaization 9 7. Group competions Étae homotopy type of motivic Eienberg Mac Lane spaces Appication to motivic cohomoogy operations 16 References 20 Introduction In the first part of this paper we show that the étae homotopy type of the dth symmetric power of a quasi-projective scheme X over a separaby cosed fied k is Z/-homoogicay equivaent to the dth symmetric power of the étae homotopy type of X, where char k is any prime. Symboicay, ( ) L Z/ Πét (S d X) L Z/ S d Πét (X), where Πét is the étae homotopy type, S d is the dth symmetric power, and L Z/ is Z/-ocaization à a Bousfied Kan. The étae homotopy type Πét X of a scheme X is a pro-space originay defined by Artin and Mazur and ater refined by Friedander to the correct notion. It is characterized by the property that the (nonabeian) étae cohomoogy of X with constant coefficients coincides with the cohomoogy of Πét X. In fact, we wi find it convenient to define Πét X by this property. In our situation, it is appropriate to think of the pro-space Πét X as a pro-sheaf of spaces on the étae site of Spec k. Such an interpretation is necessary to prove a version of ( ) over an arbitrary fied k, but we wi not consider this case here. The formua is reated to a theorem of Deigne about the étae cohomoogy of symmetric powers ([SGA4, XVII, Théorème ]), but there are three differences: (1) Deigne s theorem is about cohomoogy with proper support, and so does not say anything about the unsupported cohomoogy of non-proper schemes over k. (2) We give a cohomoogy equivaence at the eve of homotopy types, whereas Deigne ony gives an equivaence at the eve of cochains. Date: September 24,

2 2 MARC HOYOIS (3) Deigne s theorem is reative and works over an arbitrary coherent base and with arbitrary Noetherian torsion coefficients; for our resut the base must be a fied whose characteristic is prime to the torsion of the coefficients. Of course, (1) is a very significant difference, and (3) is the price we have to pay: even the usua Künneth formua in étae cohomoogy is ony known for schemes of finite type over a fied (the vaidity of the Künneth formua is in fact the ony factor restricting the base and the coefficients in (3)). Deigne s proof, after reducing to the case where the base is a fied k, uses Witt vectors to further reduce to the case where k has characteristic zero. In this step it is crucia that X be proper over k, and Deigne s argument ony generaizes to the non-proper case when the base is a fied of characteristic zero. Our argument is thus competey different. We use the existence a schematic topoogy, finer than the étae topoogy but cohomoogicay equivaent to it, for which the quotient map X d S d X is a covering; this is the qfh topoogy used extensivey by Voevodsky in his work on trianguated categories of motives. Combining ( ) with the motivic Dod Thom theorem, we show that, if A is an abeian group in which the characteristic exponent of k is invertibe, the Z/-profinite étae homotopy type of a motivic Eienberg Mac Lane space K(A(q), p) is an ordinary Eienberg Mac Lane space K(A, p). It foows that, for any n 0, there is an isomorphism H (K(A(q), p), Z/ n )[τ 1 ] H (K(A, p), µ n(k) ), where τ H 0,1 (Spec k, Z/) µ (k) is a primitive th root of unity. Moreover, if k is any perfect fied admitting resoutions of singuarities, work of Voevodsky aows us to competey identify the motivic Steenrod agebra over k. This gets rid of the expicit assumption of characteristic zero in [Voe10, Theorem 3.49] and is a first step towards the compete computation of motivic cohomoogy operations in positive characteristic. In a subsequent paper we wi show how to remove the hypothesis of resoutions of singuarities atogether. In investigating infinite symmetric powers and their preservation by certain functors, the notion of commutative monoid up to coherent homotopy arises naturay. For this reason and others that wi be apparent we wi use the anguage of -categories as deveoped in [HTT] and [HA]. We warn the reader that this is the defaut anguage in this paper, so for exampe the word coimit aways means homotopy coimit, unique means unique up to a contractibe space of choices, etc. We denote by S the -category of sma -groupoids, a.k.a. spaces. 1. Homotopy types of schemes Let τ be a pretopoogy on the category of schemes. If X is a scheme, the sma τ-site of X is the fu subcategory of Sch/X spanned by the members of the τ-coverings of X and equipped with the Grothendieck topoogy induced by τ. We denote by X τ the -topos of sheaves of spaces on the sma τ-site of X. The assignment X X τ is functoria: a morphism of schemes f : X Y induces a geometric morphism of -topoi f : X τ Y τ given by f (F)(U) = F(U Y X). Reca that the functor S Top R associating to an -groupoid its cassifying -topos admits a pro-eft adjoint Π : Top R Pro(S) associating to any -topos its shape. The τ-homotopy type Π τ X of a scheme X is the shape of the -topos X τ : Π τ X = Π X τ. This construction defines a functor Π τ : Sch Pro(S). Let X be an -topos and et c: S X be the constant sheaf functor. By definition of the shape, we have Map(, ck) Map(Π X, K) for every K S. In particuar, the cohomoogy of X with coefficients in an abeian group can be computed as the continuous cohomoogy of the pro-space Π X. If X is ocay connected (i.e., if for every X X the pro-set π 0 Π (X/X) is constant), we have more generay that the category Fun(Π X, Set) of discrete oca systems on Π X is equivaent to the category of ocay constant sheaves of sets on X, and, if A is such a sheaf of abeian groups, then H (X, A) coincides with the continuous cohomoogy of the pro-space Π X with coefficients in the corresponding oca system. Remark 1.1. In the definition of Π τ X, we coud have used any τ-site of X-schemes containing the sma one. For if X τ is the resuting -topos of sheaves, the canonica geometric morphism

3 THE ÉTALE SYMMETRIC KÜNNETH THEOREM 3 X τ X τ is obviousy a shape equivaence. It foows that the functor Π τ depends ony on the Grothendieck topoogy induced by τ. Remark 1.2. For schemes over a fixed base scheme S, we can define in the same way a reative version of the τ-homotopy type functor taking vaues in the -category Pro(S τ ). Remark 1.3. Let Xτ be the hypercompetion of X τ. By the generaized Verdier hypercovering theorem ([DHI04, Theorem 7.6 (b)]), Π Xτ is corepresented by the simpiciay enriched diagram Π 0 : HC τ (X) Set where HC τ (X) is the cofitered simpicia category of τ-hypercovers of X and Π 0 (U ) is the simpicia set which has in degree n the coimit of the presheaf U n. Remark 1.4. When τ = ét is the étae pretopoogy and X is ocay Noetherian, Π X ét is corepresented by the étae topoogica type defined by Friedander in [Fri82, 4]. This foows easiy from Remark 1.3. Lemma 1.5. Let U be a τ-covering diagram in the sma τ-site of a scheme X. Then Π τ X is the coimit of the diagram of pro-spaces Π τ U. Proof. By topos descent, the -topos X τ is the coimit in Top R of the diagram of -topoi U τ. Since Π : Top R Pro(S) is eft adjoint, it preserves this coimit. Remark 1.6. Simiary, if U X is a representabe τ-hypercover of X, then Π Xτ is the coimit of the simpicia pro-space Π (U ) τ. The trivia proof of this fact can be compared with the extremey technica proof of [Isa04, Theorem 3.4], which is the specia case τ = ét. This is a good exampe of the usefuness of the anguage of -topoi. 2. Symmetric powers in -categories If X is a CW compex, its dth symmetric power S d X is the set of orbits of the action of the symmetric group Σ d on X d, endowed with the quotient topoogy. Even though the action of Σ d on X d is not free, it is we known that the homotopy type of S d X is an invariant of the homotopy type of X. More generay, if G is a group acting on a CW compex X with at east one fixed point, the orbit space X/G can be written as the homotopy coimit (1) X/G hocoim H O(G) XH, where O(G) is the orbit category of G and X H is the subspace of H-fixed points ([Far96, Lemma A.3]). In the case of the symmetric group Σ d acting on X d, if H Σ d is a subgroup, then (X d ) H X o(h) where o(h) is the set of orbits of the action of H on {1,..., d} and where the factor of X o(h) indexed by an orbit {i 1,..., i r } is sent diagonay into the corresponding r factors of X d. Since X d has fixed points (any point of the form (x,..., x)), the formua (1) gives (2) S d X hocoim H O(Σ d ) Xo(H). This shows that S d preserves homotopy equivaences between CW compexes. In particuar, it induces a functor S d from the -category of spaces to itsef. In a genera -category C with finite products and coimits, we define the functor S d : C C by the formua (2) (note that S 0 X is a fina object of C and that S 1 X X). For exampe, in an -category of sheaves of spaces on a site, S d is computed by appying S d objectwise and sheafifying the resut, and in a 1-category it has its usua meaning, namey the coequaizer of the obvious pair Σ d X d X d. We note that any functor which preserves coimits and finite products commutes with S d. Remark 2.1. To define S d X, it suffices that X be a cocommutative comonoid in a symmetric monoida -category. Presheaves with transfers are exampes of such objects, in which case S d X agrees with the symmetric power defined in [Voe10, 2.2]. Lemma 2.2. Let X be an -topos. The incusion X 0 X preserves symmetric powers. Proof. This is obviousy true if X = S, whence if X is presheaf -topos. It remains to observe that if a: X Y is a eft exact ocaization and the resut is true in X, then it is true in Y: this foows from the fact that a preserves 0-truncated objects ([HTT, Proposition ]).

4 4 MARC HOYOIS 3. Homoogica ocaizations of pro-spaces Let Pro(S) denote the -category of pro-spaces. Reca that this is the -category freey generated by S under cofitered imits and that it is equivaent to the fu subcategory of Fun(S, S) op spanned by accessibe eft exact functors. Any such functor is equivaent to Y coim i I Map(X i, Y ) for a sma cofitered diagram X : I S. Moreover, combining [HTT, Proposition ] and the proof of [SGA4, Proposition 8.1.6], we can aways find such a corepresentation where I is a cofitered poset such that for a i I, there exist ony finitey many j with i j; such a poset is caed cofinite. In [Isa07], Isaksen constructs a proper mode structure on the category Pro(Set ) of prosimpicia sets, caed the strict mode structure, with the foowing properties: a pro-simpicia set X is fibrant iff it is isomorphic to a diagram (X s ) s I such that I is a cofinite cofitered poset and X s im s<t X t is a Kan fibration for a s I; the incusion Set Pro(Set ) is a eft Quien functor; it is a simpicia mode structure with simpicia mapping sets defined by Map (X, Y ) = Hom(X, Y ). Denote by Pro (S) the -category associated to this mode category, and by c: S Pro (S) the eft derived functor of the incusion. Since Pro (S) admits cofitered imits, there is a unique functor ϕ: Pro(S) Pro (S) which preserves cofitered imits and such that ϕ j c, where j : S Pro(S) is the Yoneda embedding. Lemma 3.1. ϕ: Pro(S) Pro (S) is an equivaence of -categories. Proof. Let X Pro(Set ) be fibrant. Then X is isomorphic to a diagram (X s ) indexed by a cofinite cofitered poset and such that X s im s<t X t is a Kan fibration for a s, and so, for a Z Pro(Set ), Map (Z, X s ) im s<t Map (Z, X t ) is a Kan fibration. It foows that the imit Map (Z, X) im s Map (Z, X s ) in Set is in fact a imit in S, so that X im s c(x s ) in Pro (S). This shows that ϕ is essentiay surjective. Let X Pro(S) and choose a corepresentation X : I S where I is a cofinite cofitered poset. Using the mode structure on Set, it is easy to show that X can be strictified to a diagram X : I Set such that X s im s<t X t is a Kan fibration for a s I. By the first part of the proof, we then have X im s c(x s) in Pro (S), whence ϕx X. Given aso Y Pro(S), we have Map(X, Y ) im t Map(X, cy t ) hoim t Map (X, Y t ) im t coim s Map(X s, Y t ), where in the ast step we used that fitered coimits of simpicia sets are aways coimits in S. This shows that ϕ is fuy faithfu. Let S < S be the -category of truncated spaces. A pro-truncated space is a pro-object in S <. It is cear that the fu embedding Pro(S < ) Pro(S) admits a eft adjoint τ < : Pro(S) Pro(S < ) which preserves cofitered imits and sends a constant pro-space to its Postnikov tower; it aso preserves finite products since truncations do. Remark 3.2. The mode structure on Pro(Set ) defined in [Isa01] is the eft Bousfied ocaization of the strict mode structure at the cass of τ < -equivaences. It is therefore a mode for the -category Pro(S < ) of pro-truncated spaces. Let R be a commutative ring. A morphism f : X Y in Pro(S) is caed an R-homoogica equivaence (resp. an R-cohomoogica equivaence) if it induces an equivaence of homoogy progroups H (X, R) H (Y, R) (resp. an equivaence of cohomoogy groups H (Y, R) H (X, R)). By [Isa05, Proposition 5.5], f is an R-homoogica equivaence iff it induces isomorphisms in cohomoogy with coefficients in arbitrary R-modues. A pro-space X is caed R-oca if it is oca with respect to the cass of R-homoogica equivaences, i.e., if for every R-homoogica equivaence Y Z the induced map Map(Z, X) Map(Y, X) is an equivaence in S. A prospace is caed R-profinite if it is oca with respect to the cass of R-cohomoogica equivaences. We denote by Pro(S) R (resp. Pro(S) R ) the -category of R-oca (resp. R-profinite) pro-spaces.

5 THE ÉTALE SYMMETRIC KÜNNETH THEOREM 5 The characterization of R-homoogica equivaences in terms of cohomoogy shows that any τ < -equivaence is an R-homoogica equivaence. We thus have a chain of fu embeddings Pro(S) R Pro(S) R Pro(S < ) Pro(S). Proposition 3.3. The incusions Pro(S) R Pro(S) and Pro(S) R Pro(S) admit eft adjoints L R and L R with the foowing properties: (1) L R preserves finite products. (2) If R is a fied and X Pro(S) is a cofitered imit of spaces whose R-cohomoogy groups are finite-dimensiona R-vector spaces, then the canonica map L R X L R X is an equivaence. Proof. The existence of the ocaization functors L R and L R foows from the existence of the corresponding eft Bousfied ocaizations of the strict mode structure on Pro(Set ) ([Isa05, Theorems 6.3 and 6.7]). (1) Foows from the Künneth formua at the chain eve. (2) The canonica map X L R X induces an isomorphism on cohomoogy. If the pro-space X is a cofitered imit of spaces with finite-dimensiona cohomoogy, it therefore induces an isomorphism on cohomoogy ind-groups, whence on homoogy pro-groups. Remark 3.4. It is cear that the cass of pro-spaces X satisfying the hypothesis of Proposition 3.3 (2), for given R, is cosed under finite products and finite coimits (it suffices to verify it for pushouts). The fact that L R preserves finite products is very usefu and we wi use it often. It impies in particuar that L R preserves commutative monoids and commutes with the formation of symmetric powers. Here is another consequence: Coroary 3.5. Finite products distribute over finite coimits in Pro(S) R. Proof. Finite coimits are universa in Pro(S), i.e., are preserved by any base change (since pushouts and pubacks can be computed evewise). The resut foows using that L R preserves finite products. Remark 3.6. If X is an -topos, it is cear that the geometric morphism X X induces an equivaence of pro-truncated shapes, and a fortiori aso of R-oca and R-profinite shapes for any commutative ring R. Remark 3.7. Let be a prime number. The Bockstein ong exact sequences show that any Z/cohomoogica equivaence is aso a Z/ n -cohomoogica equivaence for a n 1. In particuar, if X is an -topos, its Z/-profinite shape L Z/ Π X remembers the cohomoogy of X with coefficients in Z or Q. Remark 3.8. Let be a prime number and et S fc S < be the fu subcategory of truncated spaces with finite π 0 and whose homotopy groups are finite -groups. Then the -category Pro(S) Z/ can be identified with the -category Pro(S fc ) studied in [DAG13, 3]. Indeed, it is obvious that Pro(S fc ) Pro(S) Z/. Conversey, et X be Z/-profinite and et ( ) denote the eft adjoint to the incusion Pro(S fc ) Pro(S). Since K(Z/, n) S fc for a n, the canonica map X X is a Z/-cohomoogica equivaence, whence an equivaence. Proposition 3.9. Let be a prime number. Then Postnikov towers in Pro(S) Z/ are convergent. That is, the -category Pro(S) Z/ is the imit of the tower of -categories Pro(S) Z/ n Pro(S)Z/ n 1. Proof. By [DAG13, Coroary 3.4.8], the subcategory of n-truncated objects in Pro(S fc ) is Pro(S fc S n ). The resut is now obvious since S fc S <.

6 6 MARC HOYOIS 4. The qfh topoogy Let X be a Noetherian scheme. An h covering of X is a finite famiy {U i X} of morphisms of finite type such that the induced morphism i U i X is universay submersive (a morphism of schemes f : Y X is submersive if it is surjective and if the underying topoogica space of X has the quotient topoogy). If in addition each U i X is quasi-finite, it is a qfh covering. These notions of coverings define pretopoogies which we denote by h and qfh, respectivey. The h and qfh topoogies are both finer than the fpqc topoogy, and they are not subcanonica. Proposition 4.1. Let X be a Noetherian scheme. (1) The canonica map Π qfh X Πét X induces an isomorphism on cohomoogy with any oca system of abeian coefficients. In particuar, for any commutative ring R, L R Π qfh X L R Πét X. (2) The canonica map Π h X Π qfh X induces an isomorphism on cohomoogy with any oca system of torsion abeian coefficients. In particuar, for any torsion commutative ring R, L R Π h X L R Π qfh X. Proof. [Voe96, Theorem 3.4.4] and [Voe96, Theorem 3.4.5], respectivey. Remark 4.2. Combining Proposition 4.1 with a resut of Jardine ([Jar03, Theorem 12]), one can aso show that the canonica map Π qfh X Πét X is an equivaence when restricted to spaces with finite π 1. Lemma 4.3. Let X be a Noetherian scheme, et C be the category of X-schemes of finite type, and et τ {h, qfh, ét}. Then, for any n 0, the image of the canonica functor C Shv τ (C) n consists of compact objects. Proof. The category C has finite imits and the topoogy τ is finitary, and so the -topos Shv τ (C) is ocay coherent and coherent. The resut now foows from [DAG13, Proposition 2.3.9]. The advantage of the qfh topoogy over the étae topoogy is that it can often cover singuar schemes by smooth schemes. Let us make this expicit in the case of quotients of smooth schemes by finite group actions. We first reca the cassica existence resut for such quotients. A groupoid scheme X is a simpicia scheme such that, for every scheme Y, Hom(Y, X ) is a groupoid. If P is any property of morphisms of schemes which is stabe 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 et X be a finite and ocay free groupoid scheme over S. Suppose that for any x X 0, d 1 (d 1 0 (x)) is contained in an affine open subset of X 0 (for exampe, X 0 is quasi-projective over S). Then X has a coimiting cone p: X Y in the category of S-schemes. Moreover, (1) p is integra and surjective, and in particuar universay submersive; (2) the canonica morphism X cosk 0 (p) is degreewise surjective; (3) if S is ocay Noetherian and X 0 is of finite type over S, then Y is of finite type over S. Proof. The caim in parentheses foows from [EGA2, Coroaire 4.5.4] and the definition of quasiprojective morphism [EGA2, Définition 5.3.1]. An integra and surjective morphism is universay submersive because integra morphisms are cosed ([EGA2, Proposition ]). The existence of p which is integra and surjective and (2) are proved in [DG70, III, 2, 3.2] or [SGA3, Exposé 5, Théorème 4.1]. Part (3) is proved in [SGA3, Exposé 5, Lemme 6.1 (ii)]. Now et S be a Noetherian base scheme and C Sch/S a fu subcategory of S-schemes of finite type. For any X Sch/S, we denote by rx the presheaf on C defined by (rx)(u) = Map Sch/S (U, X), and if τ is a topoogy on C we denote by r τ X = a τ rx the τ-sheaf associated with rx. Proposition 4.5. Let X be a groupoid scheme in C as in Lemma 4.4. Then r qfh X r qfh coim X is a coimiting cone in the category of hypercompete qfh sheaves on C.

7 THE ÉTALE SYMMETRIC KÜNNETH THEOREM 7 Proof. By Lemma 4.4, the coimiting cone p: X coim X is a (bounded) qfh hypercover. Let C be the extension of C containing cosk 0 (p) n for every n 1. Any object in C is then qfh-covered by objects of C so that the categories of hypercompete qfh sheaves on C and C are equivaent via the restriction. We may therefore assume that C = C. But then if F is a hypercompete qfh sheaf, we have F(coim X ) im F(X ) by definition. Coroary 4.6. Suppose that C is cosed under finite products and finite coproducts and et X C be quasi-projective. Then the functor r qfh : Sch/S Shv qfh (C) preserves symmetric powers of X, i.e., it sends the schematic symmetric power S d X to the sheafy symmetric power S d r qfh X. Proof. We have r qfh S d X S d r qfh X in Shv qfh (C) 0 by Proposition 4.5, whence in Shv qfh (C) by Lemma Étae homotopy type of symmetric powers Lemma 5.1. Let k be a separaby cosed fied, char k a prime number, and X and Y schemes of finite type over k. Let τ {h, qfh, ét}. Then L Z/ Π τ (X k Y ) L Z/ Π τ X L Z/ Π τ Y. Proof. By Proposition 4.1, it suffices to prove the emma for τ = ét. Since L Z/ preserves finite products and Xét and Xét have the same Z/-oca shape (cf. Remark 3.6), it suffices to show that the canonica map (3) Π (X k Y ) ét Π X ét Π Y ét is a Z/-homoogica equivaence or, equivaenty, that it induces an isomorphism in cohomoogy with coefficients in any Z/-modue M. Both sides of (3) are corepresented by cofitered diagrams of simpicia sets having finitey many simpices in each degree (by Remark 1.3 and the fact that any étae hypercovering of a Noetherian scheme is refined by one which is degreewise Noetherian). If K is any such pro-space, C (K, Z/) is a cofitered diagram of degreewise finite chain compexes of vector spaces. On the one hand, this impies C (K, M) C (K, Z/) M, so we may assume that M = Z/. On the other hand, it impies that the Künneth map H (Π X ét, Z/) Z/ H (Π Y ét, Z/) H (Π X ét Π Y ét, Z/) is an isomorphism. The composition of this isomorphism with the map induced by (3) in cohomoogy is the canonica map H ét(x, Z/) Z/ H ét(y, Z/) H ét(x k Y, Z/) which is aso an isomorphism by [De77, Th. finitude, Coroaire 1.11]. Remark 5.2. Let X be a Noetherian scheme and et τ {h, qfh, ét}. We observed in the proof of Lemma 5.1 that the pro-space Π Xτ is the imit of a cofitered diagram of spaces whose integra homoogy groups are finitey generated. It foows from Proposition 3.3 that L F Π τ X L F Π τ X for any fied F. Now et C be a sma fu subcategory of Sch/S endowed with a topoogy σ and suppose that τ is finer than σ. The functor Π τ : C Pro(S) takes vaues in a cocompete -category and is σ-oca according to Lemma 1.5, so it automaticay ifts to a eft adjoint functor C Π τ Pro(S) Shv σ (C) Here Shv σ (C) is the -topos of sheaves of spaces on C for the σ-topoogy. Remark 5.3. If C is such that C/X contains the sma τ-site of X for any X C, then Π τ is simpy the composition Π τ

8 8 MARC HOYOIS a τ Shv σ (C) Shv τ (C) Top R Π Pro(S), where for X an -topos the incusion X Top R is X X/X. Indeed, this composition preserves coimits (by topos descent), and it restricts to Π τ on C (cf. Remark 1.1). However, the reader shoud be warned that we wi use Π τ in situations where this hypothesis on C is not satisfied. The extension Π τ invoves taking infinite coimits in Pro(S), which are somewhat i-behaved (they are not universa). As we wi see ater, it is sometimes advantageous to consider a variant of Π τ taking vaues in ind-pro-spaces. Theorem 5.4. Let k be a separaby cosed fied, 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 canonica equivaence L Z/ Π τ (S d X) L Z/ S d Π τ (X). Proof. Let C be the category of quasi-projective schemes over k. By Coroary 4.6, the representabe sheaf functor r qfh : C Shv qfh (C) preserves symmetric powers. Using Lemma 5.1 and the fact that L Πqfh Z/ preserves coimits, we deduce that L Z/ Π qfh preserves symmetric powers on C. For τ {h, qfh, ét}, we get L Z/ Π τ (S d X) L Z/ Π qfh (S d X) S d L Z/ Π qfh (X) S d L Z/ Π τ (X), the first and ast equivaences being from Proposition 4.1. The functor L Z/ itsef aso preserves finite products (Proposition 3.3) and hence symmetric powers, so we are done. Remark 5.5. It is possibe to define a natura map Π (S d X) τ S d Π X τ in Pro(S) which induces the equivaence of Theorem 5.4. It suffices to make the square (4) Π (X d ) τ (Π X τ ) d Π (S d X) τ S d Π (X τ ) commute. Using the point-set mode for the τ-homotopy type discussed in Remark 1.3 and the fact that connected components commute with symmetric powers, the task to accompish is the foowing: 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 simpiciay enriched functoria way (i.e., we must define a simpicia functor HC τ (X) HC τ (S d X) and the refinements must be natura). If τ = h or τ = qfh, S d U S d X is itsef a τ-hypercover and we are done, but things get more compicated for τ = ét as symmetric powers of étae maps are not étae anymore. We refer to [Ko97, 4.5] and [Ryd12, 3] for more detais on the foowing ideas. Given a finite group G and quasi-projective G-schemes U and X, a map f : U X is G-equivariant iff it admits descent data for the action groupoid of G on X. The map f is fixed-point refecting if it admits descent data for the Čech groupoid of the quotient map X X/G (this condition can be expressed more expicity using the fact that G X X X/G X is faithfuy fat: f is fixed-point refecting iff it is G-equivariant and induces a fiberwise isomorphism between the stabiizer schemes). Since étae morphisms descend effectivey aong universay open surjective morphisms ([Ryd10, Theorem 5.19]), such as X X/G, the condition that f refects fixed points is equivaent to the induced map U/G X/G being étae and the square U X U/G X/G

9 THE ÉTALE SYMMETRIC KÜNNETH THEOREM 9 being cartesian. If f is G-equivariant, there exists a argest G-equivariant open subset fpr(f) U on which f is fixed-point refecting. Moreover, if f : U X is an étae cover, the restriction of f d to fpr(f d ) is sti surjective. Now given an étae hypercover U X, we can define an étae hypercover V S d X refining S d U S d X as foows. Let W 0 U0 d be the ocus where U0 d X d refects fixed points. If W has been defined up to eve n 1, define W n by the cartesian square W n fpr(u d n (cosk n 1 U d ) n ) (cosk n 1 W ) n (cosk n 1 U d ) n in which the vertica maps are Σ d -equivariant fixed-point refecting étae covers (because fixedpoint refecting morphisms are stabe under base change). Finay, et V n = W n /Σ d. It is then easy to prove that V S d X is an étae hypercover with the desired functoriaity. Using the commutativity of (4), one can aso show that the map induced by Πét S d X S d Πét X in cohomoogy with coefficients in a Z/-modue coincides with the symmetric Künneth map defined in [SGA4, XVII, ]. Thus, for X proper, it is possibe to deduce Theorem 5.4 from [SGA4, XVII, Théorème ]. 6. A 1 -ocaization Let S be a Noetherian scheme of finite Kru dimension and C a fu subcategory of Sch/S such that (1) objects of C are of finite type over S; (2) S C and A 1 C; (3) if X C and U X is étae of finite type, then U C; (4) C is cosed under finite products and finite coproducts. In agreement with [Voe10], we ca such a category C admissibe. Let Shv Nis (C) denote the -topos of sheaves of spaces on C for the Nisnevich topoogy (by the hypothesis on S, this is a hypercompete -topos). Reca that we ve defined the functor and that it ifts to a eft adjoint functor Πét : C Pro(S), Πét : Shv Nis (C) Pro(S). From now on we fix a prime number different from the residua characteristics of S. By [SGA5, VII, Coroaire 1.2], the composition Shv Nis (C) Πét L Z/ Pro(S) Pro(S) Z/ sends any morphism A 1 X X, X C, to an equivaence and therefore factors through the A 1 -ocaization L A 1 : Shv Nis (C) Shv Nis (C) A 1 which inverts exacty those morphisms (as far as coimit-preserving functors are concerned). That is, there is a unique commutative square of eft adjoint functors (5) Shv Nis (C) L A 1 Πét Pro(S) L Z/ The Z/-profinite étae homotopy type functor Shv Nis (C) A 1 Pro(S) Z/. Ét : Shv Nis (C) A 1 Pro(S) Z/ is given by the composition of the dashed arrow in (5) and the ocaization functor L Z/. Note qfh that we coud as we use Π or even Π h instead of Πét in the above diagram, according to Proposition 4.1, and hence the functor Ét factors through Shv h (C) A 1.

10 10 MARC HOYOIS Of course, if C = Sm/S is the category of smooth schemes of finite type over S, then Shv Nis (C) A 1 is the standard -category of motivic spaces over S. The functor Ét is in this case the -categorica incarnation of the étae reaization functor defined by Isaksen in [Isa04] as a eft Quien functor. Suppose now that S = Spec k where k is a separaby cosed fied. Denote by Shv fin Nis(C) A 1 the cosure of the image of C under finite coimits (this is equivaenty the subcategory of compact objects in Shv Nis (C) A 1, by the usua characterization of Nisnevich sheaves). Lemma 6.1. The restriction of Ét to Shv fin Nis(C) A 1 preserves finite products. Proof. By Lemma 5.1, the functor L Z/ Πét preserve finite products on C. By Coroary 3.5, the restriction of L Πét Z/ to the cosure Shv fin Nis(C) of C under finite coimits preserves finite products. We observed in Remark 5.2 that, for any X Shv fin Nis(C), the pro-space Πét X is a cofitered imit of spaces with finite Z/-homoogy groups. It foows from Proposition 3.3 (2) that the canonica transformation L Πét Z/ Z/ L Πét is an equivaence on Shv fin Nis(C), and the resut foows. Let p q 0. We define the Z/-profinite mixed spheres S p,q Pro(S) Z/ by S 1,0 = L Z/ S 1 = K(Z, 1), S 1,1 = K(T µ, 1), S p,q = (S 1,0 ) p q (S 1,1 ) q, where µ is the group of roots of unity in k and T µ = im n µ n is its -adic Tate modue. Of course, S p,q L Z/ S p, but if q > 0 this equivaence depends on infinitey many noncanonica choices (viz., an isomorphism Z T µ). Note that the functor Ét preserves pointed objects, since Πét (Spec k). Proposition 6.2. Let p q 0. Then Ét S p,q S p,q. Proof. This is obvious if q = 0. By Lemma 6.1, it remains to treat the case p = q = 1. The étae Čech cover of the mutipication by n map G m G m has Bµ n as simpicia set of connected components. This defines a morphism of pro-spaces ϕ: Πét G m K(T µ, 1). If M is a Z/-modue, then by [SGA5, VII, Proposition 1.3 (i) (c)], M if i = 0, Het(G í m, M) = Hom(µ, M) if i = 1, 0 if i 2. One checks easiy that these cohomoogy groups are reaized by any of the previousy considered Čech covers. It foows that ϕ is a Z/-homoogica equivaence. 7. Group competions Let C be an -category with finite products. Reca from [HA, 2.4.2] that a commutative monoid in C is a functor M : Fin C such that for a n 0 the canonica map M( n ) M( 1 ) n is an equivaence. We et E (C ) denote the fu subcategory of Fun(Fin, C) spanned by commutative monoids in C. A commutative monoid M in C has an underying simpicia object, namey its restriction aong the functor Cut: op Fin sending [n] to the finite set of cuts of [n] pointed at the trivia cut, which can be identified with n. The commutative monoid M is caed groupike if its underying simpicia object is a groupoid object in the sense of [HTT, Definition ]. We denote by E gr (C ) E (C ) the fu subcategory of groupike objects. Since a commutative monoid M is such that M( 0 ) is fina, [HTT, Proposition (4 )] shows that M is groupike iff for every pair of subsets S, T [n] such that S T = [n] and S T has a singe eement, the canonica map M( n ) M(Cut(S)) M(Cut(T )) is an equivaence. If f : C D preserves finite products (and C and D admit finite products), then it induces a functor E (C ) E (D ) by postcomposition; this functor ceary preserves groupike objects and hence restricts to a functor E gr (C ) E gr (D ). We wi continue to use f to denote either induced functor. Lemma 7.1. Suppose that f : C D preserves finite products and has a right adjoint g. Then the functors E (C ) E (D ) and E gr (C ) E gr (D ) induced by f are eft adjoint to the corresponding functors induced by g.

11 THE ÉTALE SYMMETRIC KÜNNETH THEOREM 11 Proof. It suffices to observe that postcomposing Fin -diagrams with the unit and counit of the adjunction preserves the property of being a commutative monoid. The statement for groupike objects foows since E gr is a fu subcategory of E. Ca an -category C distributive if it is presentabe and if finite products in C distribute over sma coimits. A functor f : C D between distributive -categories is a distributive functor if it preserves sma coimits and finite products. For exampe, for any -topos X, the truncation functors τ n : X X n are distributive, and for any admissibe category C Sch/S and any topoogy τ on C, the ocaization functor L A 1 : Shv τ (C) Shv τ (C) A 1 is distributive (by [Mor12, Lemma 6.47]). If C is distributive, then the -category E (C ) is itsef presentabe by [HA, Coroary ], and the subcategory E gr (C ) is strongy refective by [HTT, Proposition ] and [HTT, Proposition ]. That is, there exists a group competion functor ( ) + : E (C ) E gr (C ) which exhibits E gr (C ) as an accessibe ocaization of E (C ). Lemma 7.2. Let f : C D be a distributive functor. Then for M E (C ), f(m + ) f(m) +. Proof. By Lemma 7.1, the square f E (C ) E (D ) + + E gr (C ) f E gr (D ) has a commutative right adjoint and hence is commutative. We can define a generaized free Z-modue functor in any distributive -category C. Given X C, the object NX = d 0 S d X has a structure of commutative monoid in C, i.e., there exists a functor M : Fin C with M( n ) (NX) n, easiy defined using distributivity. Moreover, the functor N: C E (C ) preserves coimits. Indeed, it preserves sifted coimits by definition of S d and the fact that the forgetfu functor E (C ) C detects sifted coimits ([HA, Coroary ]), and again by definition of S d it transforms finite coproducts into finite products, which coincide with finite coproducts in the pointed category E (C ) ([HA, Proposition ]). The group competion of NX wi be denoted by ZX. Thus, Z: C E gr (C ) is a coimit-preserving functor which commutes with any distributive functor, by Lemma 7.2. Note that the functors N and Z factor through coimit-preserving functors Ñ and Z on the -category C of pointed objects in C. Further, if A E gr (S ) is a connective HZ-modue, we can define a groupike commutative monoid object ZX A as foows. Note that if R is a commutative ring, any connective HRmodue A can be obtained from R in the foowing steps: (1) take finite products of copies of R to get finitey generated free R-modues; (2) take fitered coimits of finitey generated free R-modues to get arbitrary fat R-modues; (3) take coimits of simpicia diagrams of projective R-modues to get arbitrary connective HR-modues; Define ZX A using the same steps. A map of finitey generated free abeian groups A B ceary induces a map ZX A ZX B in E gr (C ) natura in X, and an arbitrary map of connective HZ-modues can be obtained from such maps using steps (2) and (3), so that ZX A varies functoriay with the pair (X, A). As a bifunctor to E gr (C ), ZX A preserves coimits in either variabe. Moreover, since a these steps ony invove (sifted) coimits and finite products, for any distributive functor f : C D we have f(zx A) Zf(X) A in E gr (D ). In the distributive -category S, ZX A has its usua meaning. For instance, if A is an abeian group, then ZS p A is an Eienberg Mac Lane space K(A, p).

12 12 MARC HOYOIS 8. Étae homotopy type of motivic Eienberg Mac Lane spaces We start by recaing the definition of the motivic Eienberg Mac Lane spaces. Let C Sch/S be an admissibe category and et R be a commutative ring. We denote by PSh (C) the -category of pointed presheaves on C, by PSh tr (C, R) the -category of presheaves with R-transfers (see [Hoy12, 1] for a more detaied exposition of categories of presheaves with transfers), and by R tr : PSh (C) PSh tr (C, R) : u tr the free forgetfu adjunction. The functor u tr preserves imits and sifted coimits and factors through the -category E gr (PSh(C) ). Since finite products and finite coproducts coincide in E gr (PSh(C) ), the induced functor u tr : PSh tr (C, R) E gr (PSh(C) ) preserves a coimits. We denote by Shv tr Nis(C, R) A 1 the -category of homotopy invariant Nisnevich sheaves with R-transfers on C, defined by the cartesian square Shv tr Nis(C, R) A 1 PSh tr (C, R) u tr Shv Nis(C) A 1 u tr PSh (C), and by R tr : Shv Nis(C) A 1 Shv tr Nis(C, R) A 1 the eft adjoint to u tr (it wi aways be cear from the context which category of R tr defined on). The -categories PSh tr (C, R) and Shv tr Nis(C, R) A 1 are tensored over connective HR-modues. For p q 0 and A a connective HR-modue, the motivic Eienberg Mac Lane space K(A(q), p) C Shv Nis(C) A 1 is defined by K(A(q), p) C = u tr (R tr S p,q R A), where S p,q Shv Nis(C) A 1 is the motivic p-sphere of weight q. This space depends ony on the HZ-modue structure of A and not on R. When the admissibe category C is not specified, it is understood to be Sm/S. Lemma 8.1. The square PSh tr (C, R) L Shv tr Nis(C, R) A 1 u tr u tr E gr (PSh(C) ) L E gr (Shv Nis (C) A 1 ) commutes, where L and L are eft adjoint to the incusions. Proof. Note that L is induced by the distributive functor L A 1a Nis, by Lemma 7.1. Consider the diagram Shv tr Nis(C, R) A 1 PSh tr (C, R) Shv tr Nis(C, R) A 1 L u tr u tr f E gr (Shv Nis (C) A 1 ) E gr (PSh(C) ) L E gr (Shv Nis (C) A 1 ). We wi show that a functor f exists as indicated. Define E Nis = {R tr U + R tr X + U X is a Nisnevich covering sieve in C}, and E A 1 = {R tr (X A 1 ) + R tr X + X C}, so that Shv tr Nis(C, R) A 1 PSh tr (C, R) is the fu subcategory of (E Nis E A 1)-oca objects. The functor L u tr carries morphisms in E Nis and E A 1 to equivaences: for E Nis, this is because the Nisnevich topoogy is compatibe with transfers (see [Hoy12, 1]), and for E A 1 it is because

13 THE ÉTALE SYMMETRIC KÜNNETH THEOREM 13 u tr R tr (X A 1 ) + u tr R tr X + is an A 1 -homotopy equivaence (see the ast part of the proof of [Voe10, Theorem 1.7]). By [HTT, Proposition ], there exists a functor f making the above diagram commutes. Since the horizonta compositions are the identity, we have f u tr and hence L u tr u tr L. Coroary 8.2. The forgetfu functor u tr : Shv tr Nis(C, R) A 1 coimits. Proof. Foows from Lemma 8.1 since L and L u tr preserve coimits. E gr (Shv Nis (C) A 1 ) preserves sma Suppose now that S is the spectrum of a separaby cosed fied k. One defect of the étae homotopy type functor Ét is that it does not preserve finite products and hence does not preserve commutative monoids. We have seen in Lemma 6.1 that Ét preserves finite products between finite motivic spaces, but motivic Eienberg Mac Lane spaces are certainy not finite in the required sense. We wi fix this probem by constructing a factorization Shv Nis (C) A 1 Ét E Pro(S) Z/ of Ét such that Ét is distributive and E is a cose approximation of Pro(S) Z/ by a distributive -category. Moreover, the functor Ét wi sti be h-oca, i.e., it wi factor through Shv h (C) A 1. Construction 8.3. Let D denote the category of a k-schemes of finite type, et i: C D be the incusion, and et i! : Shv h (C) Shv h (D) be the induced eft adjoint functor. Since C is admissibe and in particuar cosed under finite products, the functor i! is distributive. Note that we have a commuting triange Shv h (C) i! Shv h (D) Π h Π h Pro(S), and that the functor Π h on the right is simpy Π, in the sense of Remark 5.3. For any -topos X, we have a commutative square X τ n X n Π Π n τ n Pro(S) Pro(S n ), where the horizonta maps are given by truncation and Π n = τ n Π. By Proposition 3.9, we can therefore factor the Z/-profinite shape functor L Z/ Π : X Pro(S) Z/ as X τ im X n im Pro(S) Z/ n n n Pro(S)Z/. Since truncations preserve coimits and finite products, the functor τ is distributive. Appying Z/ this procedure to the -topos X = Shv h (D), we obtain a factorization of L Πét as Shv Nis (C) a h Shv h (C) i! Shv h (D) τ im Shv h (D) n im Pro(S) Z/ n n n Pro(S)Z/. Now, et Shv fin h (D) n denote the cosure under finite coimits of the image of D in Shv h (D) n. By Lemma 4.3, this is exacty the subcategory of compact objects in Shv h (D) n and hence Shv h (D) n Ind(Shv fin h (D) n ). Let Pro (S) be the smaest fu subcategory of Pro(S) which contains Π h X for every k-scheme of finite type X and which is cosed under finite products and finite coimits. Note that the tranformation L Z/ L Z/ is an equivaence on Pro (S), by Remark 5.2. Let Pro (S) Z/ (resp. Pro (S) n, Pro (S) Z/ n ) be the image of Pro (S) under L Z/ (resp. τ n, τ n L Z/ ). It foows from Coroary 3.5 that Ind(Pro (S) Z/ ) is a distributive -category. By Lemma 6.1, the functor τ n Ét : Shv fin h (D) n Pro (S) Z/ n

14 14 MARC HOYOIS preserves finite products and finite coimits, so that the induced functor on ind-competions Shv h (D) n Ind(Pro (S) Z/ n ) is distributive. Thus, the composition Shv Nis (C) τ i! a h imn Shv h (D) n im Ind(Pro (S) Z/ n n ) is distributive, and it is ceary A 1 -oca, so that it induces a distributive functor By construction, Ét is the composition of Ét im n Ét : Shv Nis (C) A 1 im n Ind(Pro (S) Z/ n ). Ind(Pro (S) Z/ n ) im n and the coimit functor Pro(S)Z/ n Pro(S)Z/. The next theorem is our étae version of [Voe10, Proposition 3.41]. We point out that the category C in the statement beow need not be cosed under symmetric powers, so the theorem appies directy to C = Sm/k with no need for resoutions of singuarities. Properties (1) and (2) in the theorem are crucia for the appication to the motivic Steenrod agebra in 9 (in fact, simiar compatibiity resuts for the topoogica reaization are impicitey used in [Voe10, 3.4]). Theorem 8.4. Let k be a separaby cosed fied of characteristic exponent c, c a prime number, and C Sch/k an admissibe subcategory consisting of semi-norma schemes. Then for any pointed object X in Shv Nis (C) A 1 and any connective HZ[1/c]-modue A, there is an equivaence e X,A : Z Ét X A Ét u tr (Z tr X A) of groupike commutative monoids in im n Ind(Pro (S) Z/ n ), natura in X and A, with the foowing properties: (1) the triange 1 Ét X Z Ét X Z[1/c] Ét is commutative; (2) given X, Y, A, and B, the square η Ét e X,Z[1/c] u tr Z[1/c] tr X ( Z Ét X A) ( Z Ét Y B) e X,A e Y,B Z Ét (X Y ) (A B) Ét u tr (Z tr X A) Ét u tr (Z tr Y B) e X Y,A B is commutative. Ét u tr (Z tr (X Y ) (A B)) Proof. Any representabe in Shv Nis (C) A 1 is a coimit of quasi-projective representabes, so we can assume that the objects of C are quasi-projective without changing the categories and functors invoved. As X and A vary, the source and target of e X,A are functors taking vaues in E gr (im n Ind(Pro (S) Z/ n ) ), and as such they preserve coimits in either variabe: for the efthand side this was shown in 7, and for the right-hand side it foows from Coroary 8.2. To show the existence of e X,A, it wi therefore suffice to define e X,Z[1/c] for X representabe, i.e.,

15 THE ÉTALE SYMMETRIC KÜNNETH THEOREM 15 X = L A 1r Nis (Z) + for some Z C. The motivic Dod Thom theorem ([Voe10, Theorem 3.7]) says that there is an equivaence u tr Z[1/c] tr Z + a Nis (( d 0 rs d Z) + [1/c]) of pointed presheaves on C, natura in Z (in particuar, the eft-hand side is a Nisnevich sheaf). Note that the vaidity of this formua does not depend on the scheme S d Z beonging to C. Then by Lemma 8.1 and the distributivity of L A 1a Nis, we obtain equivaences u tr Z[1/c] tr X L A 1a Nis (( d 0 r(s d Z)) + [1/c]) ( d 0 L A 1r Nis (S d Z)) + [1/c] in Shv Nis (C) A 1. Since Ét preserves group competions of commutative monoids (Lemma 7.2) and coimits, Ét u tr Z[1/c] tr X ( d 0 Ét L A 1r Nis (S d Z)) + [1/c]. On the other hand, since Ét L A 1 commutes with S d, Z Ét X Z[1/c] ( d 0 Ét L A 1S d r Nis (Z)) + [1/c]. We then define e X,Z[1/c] to be the map induced by the obvious canonica map ϕ: S d r Nis (Z) r Nis (S d Z). To show that e X,Z[1/c] is an equivaence, it suffices to show that Ét L A 1(ϕ) is an equivaence. By definition of Ét, the functor Ét L A 1 factors through a qfh : Shv Nis (C) Shv qfh (C), so it suffices to show that a qfh (ϕ) is an equivaence. This foows from Coroary 4.6. The strategy to prove (1) and (2) is the foowing: we wi first verify the statements in the representabe case, where they foow from properties of the motivic Dod Thom equivaence, and deduce the genera case by functoriaity arguments. If X is represented by Z C, the adjunction map X u tr Z[1/c] tr X corresponds, through the Dod Thom equivaence, to the map Z + S 0 Z S 1 Z d 0 S d Z ( d 0 S d Z) + [1/c]. This shows (1) when X is representabe, whence for a X since the source functor Ét preserves coimits. To prove (2), first assume that X and Y are represented by Z and W, respectivey, and that A = B = Z[1/c]. It wi suffice to show that the square commutes with Ñ instead of Z. The pairing u tr Z[1/c] tr X u tr Z[1/c] tr Y u tr Z[1/c] tr (X Y ) arising from the monoida structure of Z[1/c] tr is induced, via the Dod Thom equivaence, by the obvious maps S d Z S e W S d+e (Z W ). This shows (2) in this case. The source functor in this square of natura transformations preserves sifted coimits in the variabes X and Y, whie the target functor transforms finite coproducts into finite products (this makes sense for pointed functors between pointed categories). It foows that the square commutes for arbitrary X and Y when A = B = Z[1/c]. But again, the source preserves sifted coimits in the variabes A and B and the target preserves finite products in these variabes, so the resut hods for arbitrary A and B. Let A be a connective HZ-modue. We et A be its pro--competion im n A/ n, so that K(A, p) L Z/ K(A, p) for every p 1. Given q Z, et A (q) = A Z T µ q (which is noncanonicay isomorphic to A ). For exampe, Z/ n (q) µ q n.

16 16 MARC HOYOIS Coroary 8.5. Let C Sch/k be an admissibe category consisting of semi-norma schemes. For any connective HZ[1/c]-modue A, any p 1 and p q 0, there is a canonica equivaence Ét K(A(q), p) C K(A (q), p) in Pro(S) Z/, natura in A, and Ét preserves smash products between such spaces. Furthermore, under these equivaences, (1) Ét sends the canonica map to the canonica map (2) Ét sends the canonica map to the canonica map S p,q K(Z[1/c](q), p) C S p,q K(Z (q), p); K(A(q), p) C K(B(s), r) C K((A B)(q + s), p + r) C K(A (q), p) K(B (s), r) K((A B) (q + s), p + r). Proof. By Theorem 8.4 and Proposition 6.2, we have Ét K(A(q), p) C Zτ S p,q A, where S p,q Pro (S) Z/ is considered as a constant ind-z/-profinite pro-space. Now we appy the functor im Ind(Pro (S) Z/ n n ) im n Pro(S)Z/ n Pro(S)Z/ to both sides. The eft-hand side becomes Ét K(A(q), p) C, by definition of Ét. For the righthand side, consider the foowing commutative diagram i j ind L Z/ ind S fin Ind(S fin ) im n Ind(Pro (S) n ) im n Ind(Pro (S) Z/ n ) c c c j L Z/ S im n Pro(S) n im n Pro(S) Z/ Note that τ S p,0 is the image of S p by the top row of this diagram. The functor j ind is ceary distributive and L Z/ ind is distributive by Proposition 3.3. We therefore have equivaences n. c Z(L Z/ ind j indis p ) A cl Z/ ind j ind( ZiS p A) L Z/ jk(a, p), as was to be shown in the case q = 0. The twisting for q > 0 makes the equivaence canonica (when p = q = 1, the canonica map S 1,1 c Zτ S 1,1 is an equivaence, and the genera case foows). A simiar argument shows that Ét (X Y ) Ét (X) Ét (Y ) whenever Ét X and Ét Y beong to the essentia image of L Z/ ind j ind. The remaining statements are easiy deduced from properties (1) and (2) in Theorem Appication to motivic cohomoogy operations Let k be a separaby cosed fied and char k is a prime number. Reca that { H p,q (Spec k, Z/ n µ n(k) q if p = 0 and q 0, ) 0 otherwise. If M is a modue over the ring q 0 µ n(k) q, we wi denote by M 0 Z/ n (resp. by M[τ 1 ]) the modue obtained from M by extending scaars aong the projection q 0 µ n(k) q µ n(k) 0 Z/ n (resp. aong the incusion q 0 µ n(k) q q Z µ n(k) q ). We wi aso denote by τ a chosen generator of H 0,1 (Spec k, Z/ n ), athough the definition of M[τ 1 ] does not depend on such a choice.

17 THE ÉTALE SYMMETRIC KÜNNETH THEOREM 17 Let p 1 and p q 0. By Coroary 8.5, Ét K(Z/ n (q), p) K(µ n(k) q, p). Thus, if X is a homotopy invariant pointed Nisnevich sheaf on Sm/k, Ét induces a map (6) Map(Σ r X, K(Z/ n (q), p)) Map(Ét X, K(µ n(k) q, p r)) whenever p r 0. On connected components this gives a map H p r,q (X, Z/ n ) H p r (Ét X, µ q n ) which a priori depends on r. However, we have commutative squares Map(X, K(Z/ n (q), p)) Σ r,s Ét Proposition 6.2 Map(Ét X, K(µ q n, p)) Σ p,q Map(Σ r,s X, Σ r,s K(Z/ n (q), p)) Map(S r,s Ét X, S r,s K(µ q n, p)) Coroary 8.5 (1) and (2) Map(Σ r,s X, K(Z/ n (q + s), p + r)) Ét Map(S r,s Ét X, K(µ q+s, p + r)) n which show that (6) is compatibe with the bigraded suspension isomorphisms. Lemma 9.1. Let X Shv Nis(Sm/k) A 1. Then (6) is an equivaence whenever p r q. By Re- Proof. Let π : Shvét (Sm/k) Shv Nis (Sm/k) be the canonica geometric morphism. mark 5.3, Ét L A 1 L Z/ Π π, and so (6) is the composition of the two maps (7) (8) π : Map(Σ r X, K(Z/ n (q), p)) Map(Σ r π X, π K(Z/ n (q), p)) and L Z/ Π : Map(Σ r π X, π K(Z/ n (q), p)) Map(Σ r Ét X, L Z/ Π π K(Z/ n (q), p)). By [MVW06, Theorem 10.3], the étae sheaf π K(Z/ n (q), p) is equivaent to the Eienberg Mac Lane sheaf K(µ q, p), and the π n 0 of (7) is the usua map H p r,q (X, Z/ n p r ) H ét (X, µ q ) n from motivic to étae cohomoogy, which by [Voe11, Theorem 6.17] is an isomorphism when X is a pointed simpicia smooth scheme and p r q. Hence (7) is an equivaence in this range for such X, whence for a X since both sides transform coimits into imits. Since k is separaby cosed, K(µ q, p) is in the image of the constant sheaf functor c: S n Shvét (Sm/k) which is fuy faithfu. By definition, Π is pro-eft adjoint to c, and so (8) is aso equivaence. Remark 9.2. The argument in the proof of Lemma 9.1 shows that independenty from Coroary 8.5. Ét K(Z/ n (q), p) Sm/k K(µ n(k) q, p) Lemma 9.3. For every X Shv Nis(Sm/k) A 1, (6) induces an isomorphism H (X, Z/ n )[τ 1 ] H (Ét X, µ n(k) ). Proof. Foows immediatey from Lemma 9.1 since X has no motivic cohomoogy in negative weights (the case X = (Spec k) + shows that the image of τ is invertibe). Remark 9.4. The isomorphism H (X, Z/ n )[τ 1 ] Hét (X, µ ) for X Sm/k is aso proved n in [Lev00]. However, the map there is defined in an ad hoc way and we did not verify that it coincides with the map induced by Ét. Theorem 9.5. Let p 1, p q 0, and et A be a connective HZ[1/c]-modue. Then Ét induces an isomorphism H (K(A(q), p) Sm/k, Z/ n )[τ 1 ] H (K(A (q), p), µ n(k) ). Proof. Combine Lemma 9.3 and Coroary 8.5.