THE ÉTALE SYMMETRIC KÜNNETH THEOREM
MARC HOYOIS
Abstract. Let k be a separably closed field, l 6= char k a prime number, and X a quasiprojective scheme over k. We show that the étale homotopy type of the dth symmetric power
of X is Z/l-homologically equivalent to the dth symmetric power of the étale homotopy type of
X. We deduce that the étale realizations of motivic Eilenberg–Mac Lane spaces are ordinary
Eilenberg–Mac Lane spaces.
Contents
Introduction
1. Homotopy types of schemes
2. Symmetric powers in ∞-categories
3. Homological localizations of pro-spaces
4. The qfh topology
5. Étale homotopy type of symmetric powers
6. A1 -localization
7. Group completions
8. Étale homotopy type of motivic Eilenberg–Mac Lane spaces
9. Application to motivic cohomology operations
References
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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 6= char k is any prime.
Symbolically,
(∗)
d
d ét
LZ/l Πét
∞ (S X) ' LZ/l S Π∞ (X),
d
where Πét
∞ is the étale homotopy type, S is the dth symmetric power, and LZ/l is Z/l-localization
à la Bousfield–Kan.
The étale homotopy type Πét
∞ X of a scheme X is a pro-space originally defined by Artin
and Mazur and later refined by Friedlander to the correct notion. It is characterized by the
property that the (nonabelian) étale cohomology of X with constant coefficients coincides with
ét
the cohomology of Πét
∞ X. In fact, we will find it convenient to define Π∞ X by this property.
ét
In our situation, it is appropriate to think of the pro-space Π∞ X as a pro-sheaf of spaces on
the étale site of Spec k. Such an interpretation is necessary to prove a version of (∗) over an
arbitrary field k, but we will not consider this case here.
The formula is related to a theorem of Deligne about the étale cohomology of symmetric
powers ([SGA4, XVII, Théorème 5.5.21]), but there are three differences:
(1) Deligne’s theorem is about cohomology with proper support, and so does not say anything
about the unsupported cohomology of non-proper schemes over k.
(2) We give a cohomology equivalence at the level of homotopy types, whereas Deligne only
gives an equivalence at the level of cochains.
Date: September 24, 2013.
1
2
MARC HOYOIS
(3) Deligne’s theorem is relative and works over an arbitrary coherent base and with arbitrary Noetherian torsion coefficients; for our result the base must be a field 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 usual
Künneth formula in étale cohomology is only known for schemes of finite type over a field (the
validity of the Künneth formula is in fact the only factor restricting the base and the coefficients
in (3)). Deligne’s proof, after reducing to the case where the base is a field k, uses Witt vectors
to further reduce to the case where k has characteristic zero. In this step it is crucial that X be
proper over k, and Deligne’s argument only generalizes to the non-proper case when the base is
a field of characteristic zero. Our argument is thus completely 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 → Sd 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 A is an abelian group
in which the characteristic exponent of k is invertible, the Z/l-profinite étale homotopy type
of a motivic Eilenberg–Mac Lane space K(A(q), p) is an ordinary Eilenberg–Mac Lane space
K(A, p)∧
l . It follows that, for any n ≥ 0, there is an isomorphism
H̃ ∗∗ (K(A(q), p), Z/ln )[τ −1 ] ' H̃ ∗ (K(A, p), µln (k)⊗∗ ),
where τ ∈ H 0,1 (Spec k, Z/l) ' µl (k) is a primitive lth root of unity. Moreover, if k is any perfect
field admitting resolutions of singularities, work of Voevodsky allows us to completely identify
the motivic Steenrod algebra over k. This gets rid of the explicit assumption of characteristic
zero in [Voe10, Theorem 3.49] and is a first step towards the complete computation of motivic
cohomology operations in positive characteristic. In a subsequent paper we will show how to
remove the hypothesis of resolutions of singularities altogether.
In investigating infinite symmetric powers and their preservation by certain functors, the
notion of commutative monoid up to coherent homotopy arises naturally. For this reason and
others that will be apparent we will use the language of ∞-categories as developed in [HTT]
and [HA]. 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 denote by S the ∞-category of small ∞-groupoids, a.k.a. spaces.
1. Homotopy types of schemes
Let τ be a pretopology on the category of schemes. 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 denote by Xτ the ∞-topos of sheaves of spaces on
the small τ -site of X. The assignment X 7→ Xτ is functorial: a morphism of schemes f : X → Y
induces a geometric morphism of ∞-topoi f∗ : Xτ → Yτ given by f∗ (F)(U ) = F(U ×Y X).
Recall that the functor S → TopR associating to an ∞-groupoid its classifying ∞-topos admits
a pro-left adjoint Π∞ : TopR → Pro(S) associating to any ∞-topos its shape. The τ -homotopy
type Πτ∞ X of a scheme X is the shape of the ∞-topos Xτ :
Πτ∞ X = Π∞ Xτ .
This construction defines a functor Πτ∞ : 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(∗, cK) ' Map(Π∞ 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 Π∞ X. If X is locally connected (i.e., if for every X ∈ X the pro-set π0 Π∞ (X/X)
is constant), we have more generally that the category Fun(Π∞ X, Set) of discrete local systems
on Π∞ X is equivalent to the category of locally constant sheaves of sets on X, and, if A is
such a sheaf of abelian groups, then H ∗ (X, A) coincides with the continuous cohomology of the
pro-space Π∞ X with coefficients in the corresponding local system.
Remark 1.1. In the definition of Πτ∞ X, we could have used any τ -site of X-schemes containing
the small one. For if Xτ0 is the resulting ∞-topos of sheaves, the canonical geometric morphism
THE ÉTALE SYMMETRIC KÜNNETH THEOREM
3
Xτ0 → Xτ is obviously a shape equivalence. It follows that the functor Πτ∞ 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(Sτ ).
Remark 1.3. Let Xτ∧ be the hypercompletion of Xτ . By the generalized Verdier hypercovering
theorem ([DHI04, Theorem 7.6 (b)]), Π∞ Xτ∧ is corepresented by the simplicially enriched diagram Π0 : HCτ (X) → Set∆ where HCτ (X) is the cofiltered simplicial category of τ -hypercovers
of X and Π0 (U• ) is the simplicial set which has in degree n the colimit of the presheaf Un .
∧
Remark 1.4. When τ = ét is the étale pretopology and X is locally Noetherian, Π∞ Xét
is
corepresented by the étale topological type defined by Friedlander in [Fri82, §4]. This follows
easily from Remark 1.3.
Lemma 1.5. Let U be a τ -covering diagram in the small τ -site of a scheme X. Then Πτ∞ X is
the colimit of the diagram of pro-spaces Πτ∞ U .
Proof. By topos descent, the ∞-topos Xτ is the colimit in TopR of the diagram of ∞-topoi Uτ .
Since Π∞ : TopR → Pro(S) is left adjoint, it preserves this colimit.
Remark 1.6. Similarly, if U• → X is a representable τ -hypercover of X, then Π∞ Xτ∧ is the
colimit of the simplicial pro-space Π∞ (U• )∧
τ . The trivial proof of this fact can be compared with
the extremely technical proof of [Isa04, Theorem 3.4], which is the special case τ = ét. This is
a good example of the usefulness of the language of ∞-topoi.
2. Symmetric powers in ∞-categories
If X is a CW complex, its dth symmetric power Sd X is the set of orbits of the action of the
symmetric group Σd on X d , endowed with the quotient topology. Even though the action of
Σd on X d is not free, it is well known that the homotopy type of Sd X is an invariant of the
homotopy type of X. More generally, if G is a group acting on a CW complex X with at least
one fixed point, the orbit space X/G can be written as the homotopy colimit
(1)
X/G ' hocolim X H ,
H∈O(G)
where O(G) is the orbit category of G and X is the subspace of H-fixed points ([Far96, Lemma
A.3]). In the case of the symmetric group Σd acting on X d , if H ≤ Σd 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 {i1 , . . . , ir } is sent diagonally into the corresponding r
factors of X d . Since X d has fixed points (any point of the form (x, . . . , x)), the formula (1) gives
H
(2)
Sd X ' hocolim X o(H) .
H∈O(Σd )
This shows that Sd preserves homotopy equivalences between CW complexes. In particular, it
induces a functor Sd from the ∞-category of spaces to itself.
In a general ∞-category C with finite products and colimits, we define the functor Sd : C → C
by the formula (2) (note that S0 X is a final object of C and that S1 X ' X). For example, in
an ∞-category of sheaves of spaces on a site, Sd is computed by applying Sd objectwise and
sheafifying the result, and in a 1-category it has its usual meaning, namely the coequalizer of
the obvious pair Σd × X d ⇒ X d . We note that any functor which preserves colimits and finite
products commutes with Sd .
Remark 2.1. To define Sd X, it suffices that X be a cocommutative comonoid in a symmetric
monoidal ∞-category. Presheaves with transfers are examples of such objects, in which case
Sd X agrees with the symmetric power defined in [Voe10, §2.2].
Lemma 2.2. Let X be an ∞-topos. The inclusion X≤0 ,→ X preserves symmetric powers.
Proof. This is obviously true if X = S, whence if X is presheaf ∞-topos. It remains to observe
that if a : X → Y is a left exact localization and the result is true in X, then it is true in Y: this
follows from the fact that a preserves 0-truncated objects ([HTT, Proposition 5.5.6.16]).
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MARC HOYOIS
3. 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. Any such functor is equivalent to Y 7→
colimi∈I Map(Xi , Y ) for a small cofiltered diagram X : I → S. Moreover, combining [HTT,
Proposition 5.3.1.16] and the proof of [SGA4, Proposition 8.1.6], we can always find such a
corepresentation where I is a cofiltered poset such that for all i ∈ I, there exist only finitely
many j with i ≤ j; such a poset is called cofinite.
In [Isa07], 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 iff it is isomorphic to a diagram (Xs )s∈I such that I is
a cofinite cofiltered poset and Xs → lims<t Xt 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
Map∆ (X, Y ) = Hom(X × ∆• , Y ).
Denote by Pro0 (S) the ∞-category associated to this model category, and by c : S → Pro0 (S) the
left derived functor of the inclusion. Since Pro0 (S) admits cofiltered limits, there is a unique
functor ϕ : Pro(S) → Pro0 (S) which preserves cofiltered limits and such that ϕ ◦ j ' c, where
j : S → Pro(S) is the Yoneda embedding.
Lemma 3.1. ϕ : Pro(S) → Pro0 (S) is an equivalence of ∞-categories.
Proof. Let X ∈ Pro(Set∆ ) be fibrant. Then X is isomorphic to a diagram (Xs ) indexed by a
cofinite cofiltered poset and such that Xs → lims<t Xt is a Kan fibration for all s, and so, for
all Z ∈ Pro(Set∆ ), Map∆ (Z, Xs ) → lims<t Map∆ (Z, Xt ) is a Kan fibration. It follows that the
limit Map∆ (Z, X) ' lims Map∆ (Z, Xs ) in Set∆ is in fact a limit in S, so that X ' lims c(Xs ) in
Pro0 (S). This shows that ϕ is essentially surjective.
Let X ∈ Pro(S) and choose a corepresentation X : I → S where I is a cofinite colfiltered poset.
Using the model structure on Set∆ , it is easy to show that X can be strictified to a diagram
X 0 : I → Set∆ such that Xs0 → lims<t Xt0 is a Kan fibration for all s ∈ I. By the first part of the
proof, we then have X 0 ' lims c(Xs0 ) in Pro0 (S), whence ϕX ' X 0 . Given also Y ∈ Pro(S), we
have
Map(X 0 , Y 0 ) ' lim Map(X 0 , cYt0 ) ' holim Map∆ (X 0 , Yt0 ) ' lim colim Map(Xs0 , Yt0 ),
t
t
t
s
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
τ<∞ : Pro(S) → Pro(S<∞ )
which preserves cofiltered limits and sends a constant pro-space to its Postnikov tower; it also
preserves finite products since truncations do.
Remark 3.2. The model structure on Pro(Set∆ ) defined in [Isa01] 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 progroups H∗ (X, R) ' H∗ (Y, R) (resp. an equivalence of cohomology groups H ∗ (Y, R) ' H ∗ (X, R)).
By [Isa05, Proposition 5.5], f is an R-homological equivalence iff 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 prospace is called R-profinite 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 ÉTALE SYMMETRIC KÜNNETH THEOREM
5
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(S)R ⊂ Pro(S)R ⊂ Pro(S<∞ ) ⊂ Pro(S).
Proposition 3.3. The inclusions Pro(S)R ⊂ Pro(S) and Pro(S)R ⊂ Pro(S) admit left adjoints
LR and LR with the following properties:
(1) LR preserves finite products.
(2) If R is a field and X ∈ Pro(S) is a cofiltered limit of spaces whose R-cohomology groups
are finite-dimensional R-vector spaces, then the canonical map LR X → LR X is an
equivalence.
Proof. The existence of the localization functors LR and LR follows from the existence of the
corresponding left Bousfield localizations of the strict model structure on Pro(Set∆ ) ([Isa05,
Theorems 6.3 and 6.7]).
(1) Follows from the Künneth formula at the chain level.
(2) The canonical map X → LR X induces an isomorphism on cohomology. If the pro-space
X is a cofiltered limit of spaces with finite-dimensional cohomology, it therefore induces an
isomorphism on cohomology ind-groups, whence on homology pro-groups.
Remark 3.4. It is clear that the class of pro-spaces X satisfying the hypothesis of Proposition 3.3 (2), for given R, is closed under finite products and finite colimits (it suffices to verify
it for pushouts).
The fact that LR preserves finite products is very useful and we will use it often. It implies
in particular that LR preserves commutative monoids and commutes with the formation of
symmetric powers. Here is another consequence:
Corollary 3.5. 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 LR preserves
finite products.
Remark 3.6. If X is an ∞-topos, it is clear that the geometric morphism X∧ → X induces an
equivalence of pro-truncated shapes, and a fortiori also of R-local and R-profinite shapes for
any commutative ring R.
Remark 3.7. Let l be a prime number. The Bockstein long exact sequences show that any Z/lcohomological equivalence is also a Z/ln -cohomological equivalence for all n ≥ 1. In particular,
if X is an ∞-topos, its Z/l-profinite shape LZ/l Π∞ X remembers the cohomology of X with
coefficients in Zl or Ql .
Remark 3.8. Let l be a prime number and let Sl−fc ⊂ S<∞ be the full subcategory of truncated
spaces with finite π0 and whose homotopy groups are finite l-groups. Then the ∞-category
Pro(S)Z/l can be identified with the ∞-category Pro(Sl−fc ) studied in [DAG13, §3]. Indeed, it is
obvious that Pro(Sl−fc ) ⊂ Pro(S)Z/l . Conversely, let X be Z/l-profinite and let (−)∧
l denote the
left adjoint to the inclusion Pro(Sl−fc ) ⊂ Pro(S). Since K(Z/l, n) ∈ Sl−fc for all n, the canonical
map X → Xl∧ is a Z/l-cohomological equivalence, whence an equivalence.
Proposition 3.9. Let l be a prime number. Then Postnikov towers in Pro(S)Z/l are convergent.
That is, the ∞-category Pro(S)Z/l is the limit of the tower of ∞-categories
Z/l
Z/l
· · · → Pro(S)≤n → Pro(S)≤n−1 → · · · .
Proof. By [DAG13, Corollary 3.4.8], the subcategory of n-truncated objects in Pro(Sl−fc ) is
Pro(Sl−fc ∩ S≤n ). The result is now obvious since Sl−fc ⊂ S<∞ .
6
MARC HOYOIS
4. The qfh topology
Let X be a Noetherian scheme. An h covering
` of X is a finite family {Ui → X} of morphisms
of finite type such that the induced morphism i Ui → 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 Ui → X is quasi-finite, it is a qfh covering.
These notions of coverings define pretopologies which we denote by h and qfh, respectively. The
h and qfh topologies are both finer than the fpqc topology, and they are not subcanonical.
Proposition 4.1. Let X be a Noetherian scheme.
ét
(1) The canonical map Πqfh
∞ X → Π∞ X induces an isomorphism on cohomology with any
local system of abelian coefficients. In particular, for any commutative ring R,
ét
LR Πqfh
∞ X ' LR Π∞ X.
(2) The canonical map Πh∞ X → Πqfh
∞ X induces an isomorphism on cohomology with any
local system of torsion abelian coefficients. In particular, for any torsion commutative
ring R,
LR Πh∞ X ' LR Πqfh
∞ X.
Proof. [Voe96, Theorem 3.4.4] and [Voe96, Theorem 3.4.5], respectively.
Remark 4.2. Combining Proposition 4.1 with a result of Jardine ([Jar03, Theorem 12]), one
ét
can also show that the canonical map Πqfh
∞ X → Π∞ X is an equivalence when restricted to
spaces with finite π1 .
Lemma 4.3. Let X be a Noetherian scheme, let C be the category of X-schemes of finite type,
and let τ ∈ {h, qfh, ét}. Then, for any n ≥ 0, the image of the canonical functor C → Shvτ (C)≤n
consists of compact objects.
Proof. The category C has finite limits and the topology τ is finitary, and so the ∞-topos Shvτ (C)
is locally coherent and coherent. The result now follows from [DAG13, Proposition 2.3.9].
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 which 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
d0 : X1 → X0 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 ∈ X0 , d1 (d−1
0 (x)) is contained in an affine open subset of X0 (for
example, X0 is quasi-projective over S). Then X• has a colimiting cone p : X• → Y in the
category of S-schemes. Moreover,
(1) p is integral and surjective, and in particular universally submersive;
(2) the canonical morphism X• → cosk0 (p) is degreewise surjective;
(3) if S is locally Noetherian and X0 is of finite type over S, then Y is of finite type over S.
Proof. The claim in parentheses follows from [EGA2, Corollaire 4.5.4] and the definition of quasiprojective morphism [EGA2, Définition 5.3.1]. An integral and surjective morphism is universally
submersive because integral morphisms are closed ([EGA2, Proposition 6.1.10]). The existence
of p which is integral 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 let S be a Noetherian base scheme and C ⊂ Sch/S a full subcategory of S-schemes of
finite type. For any X ∈ Sch/S, we denote by rX the presheaf on C defined by (rX)(U ) =
MapSch/S (U, X), and if τ is a topology on C we denote by rτ X = aτ rX the τ -sheaf associated
with rX.
Proposition 4.5. Let X• be a groupoid scheme in C as in Lemma 4.4. Then rqfh X• →
rqfh colim X• is a colimiting cone in the category of hypercomplete qfh sheaves on C.
THE ÉTALE SYMMETRIC KÜNNETH THEOREM
7
Proof. By Lemma 4.4, the colimiting cone p : X• → colim X• is a (bounded) qfh hypercover.
Let C0 be the extension of C containing cosk0 (p)n for every n ≥ −1. Any object in C0 is then
qfh-covered by objects of C so that the categories of hypercomplete qfh sheaves on C and C0
are equivalent via the restriction. We may therefore assume that C = C0 . But then if F is a
hypercomplete qfh sheaf, we have F(colim X• ) ' lim F(X• ) by definition.
Corollary 4.6. Suppose that C is closed under finite products and finite coproducts and let
X ∈ C be quasi-projective. Then the functor rqfh : Sch/S → Shvqfh (C) preserves symmetric
powers of X, i.e., it sends the schematic symmetric power Sd X to the sheafy symmetric power
Sd rqfh X.
Proof. We have rqfh Sd X ' Sd rqfh X in Shvqfh (C)≤0 by Proposition 4.5, whence in Shvqfh (C) by
Lemma 2.2.
5. Étale homotopy type of symmetric powers
Lemma 5.1. Let k be a separably closed field, l 6= char k a prime number, and X and Y schemes
of finite type over k. Let τ ∈ {h, qfh, ét}. Then
LZ/l Πτ∞ (X ×k Y ) ' LZ/l Πτ∞ X × LZ/l Πτ∞ Y.
Proof. By Proposition 4.1, it suffices to prove the lemma for τ = ét. Since LZ/l preserves finite
∧
products and Xét and Xét
have the same Z/l-local shape (cf. Remark 3.6), it suffices to show
that the canonical map
(3)
∧
∧
Π∞ (X ×k Y )ét
→ Π∞ Xét
× Π∞ Yét∧
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 (3) are corepresented by colfiltered 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 which 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
C ∗ (K, M ) ' C ∗ (K, Z/l) ⊗ M,
so we may assume that M = Z/l. On the other hand, it implies that the Künneth map
∧
∧
H ∗ (Π∞ Xét
, Z/l) ⊗Z/l H ∗ (Π∞ Yét∧ , Z/l) → H ∗ (Π∞ Xét
× Π∞ Yét∧ , Z/l)
is an isomorphism. The composition of this isomorphism with the map induced by (3) in
cohomology is the canonical map
∗
∗
∗
Hét
(X, Z/l) ⊗Z/l Hét
(Y, Z/l) → Hét
(X ×k Y, Z/l)
which is also an isomorphism by [Del77, Th. finitude, Corollaire 1.11].
Remark 5.2. Let X be a Noetherian scheme and let τ ∈ {h, qfh, ét}. We observed in the
proof of Lemma 5.1 that the pro-space Π∞ Xτ∧ is the limit of a cofiltered diagram of spaces
whose integral homology groups are finitely generated. It follows from Proposition 3.3 that
LF Πτ∞ X ' LF Πτ∞ X for any field F .
Now let C be a small full subcategory of Sch/S endowed with a topology σ and suppose that
τ is finer than σ. The functor Πτ∞ : C → Pro(S) takes values in a cocomplete ∞-category and is
σ-local according to Lemma 1.5, so it automatically lifts to a left adjoint functor
C
Πτ∞
Pro(S)
bτ
Π
∞
Shvσ (C)
Here Shvσ (C) is the ∞-topos of sheaves of spaces on C for the σ-topology.
b τ∞ is
Remark 5.3. If C is such that C/X contains the small τ -site of X for any X ∈ C, then Π
simply the composition
8
MARC HOYOIS
Shvσ (C)
aτ
Shvτ (C)
TopR
Π∞
Pro(S),
where for X an ∞-topos the inclusion X ,→ TopR is X 7→ X/X. Indeed, this composition
preserves colimits (by topos descent), and it restricts to Πτ∞ on C (cf. Remark 1.1). However,
b τ in situations where this hypothesis on C is not
the reader should be warned that we will use Π
∞
satisfied.
b τ∞ involves taking infinite colimits in Pro(S), which are somewhat ill-behaved
The extension Π
(they are not universal). As we will see later, it is sometimes advantageous to consider a variant
b τ taking values in ind-pro-spaces.
of Π
∞
Theorem 5.4. Let k be a separably closed field, l 6= 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
LZ/l Πτ∞ (Sd X) ' LZ/l Sd Πτ∞ (X).
Proof. Let C be the category of quasi-projective schemes over k. By Corollary 4.6, the representable sheaf functor rqfh : C → Shvqfh (C) preserves symmetric powers. Using Lemma 5.1 and
qfh
b qfh
the fact that LZ/l Π
∞ preserves colimits, we deduce that LZ/l Π∞ preserves symmetric powers
on C. For τ ∈ {h, qfh, ét}, we get
d
d
qfh
d
τ
LZ/l Πτ∞ (Sd X) ' LZ/l Πqfh
∞ (S X) ' S LZ/l Π∞ (X) ' S LZ/l Π∞ (X),
the first and last equivalences being from Proposition 4.1. The functor LZ/l itself also preserves
finite products (Proposition 3.3) and hence symmetric powers, so we are done.
Remark 5.5. It is possible to define a natural map
d
∧
Π∞ (Sd X)∧
τ → S Π∞ Xτ
in Pro(S) which induces the equivalence of Theorem 5.4. It suffices to make the square
Π∞ (X d )∧
τ
(Π∞ Xτ∧ )d
Π∞ (Sd X)∧
τ
Sd Π∞ (Xτ∧ )
(4)
commute. Using the point-set model for the τ -homotopy type discussed in Remark 1.3 and the
fact that connected components commute with symmetric powers, the task to accomplish is the
following: associate to any τ -hypercover U• → X a τ -hypercover V• → Sd X refining Sd U• →
Sd X and such that V• ×Sd X X d → X d refines U•d → X d , in a simplicially enriched functorial way
(i.e., we must define a simplicial functor HCτ (X) → HCτ (Sd X) and the refinements must be
natural). If τ = h or τ = qfh, Sd U• → Sd 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 [Kol97, §4.5] and [Ryd12, §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 iff
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 the Čech 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 iff it is G-equivariant and induces a fiberwise isomorphism between the
stabilizer schemes). Since étale morphisms descend effectively along universally open surjective
morphisms ([Ryd10, 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
U
X
U/G
X/G
THE ÉTALE SYMMETRIC KÜNNETH THEOREM
9
being cartesian. If f is G-equivariant, there exists a largest G-equivariant open subset fpr(f ) ⊂ U
on which f is fixed-point reflecting. Moreover, if f : U → X is an étale cover, the restriction
of f d to fpr(f d ) is still surjective. Now given an étale hypercover U• → X, we can define an
étale hypercover V• → Sd X refining Sd U• → Sd X as follows. Let W0 ⊂ U0d be the locus where
U0d → X d reflects fixed points. If W• has been defined up to level n − 1, define Wn by the
cartesian square
Wn
fpr(Und → (coskn−1 U•d )n )
(coskn−1 W• )n
(coskn−1 U•d )n
in which the vertical maps are Σd -equivariant fixed-point reflecting étale covers (because fixedpoint reflecting morphisms are stable under base change). Finally, let Vn = Wn /Σd . It is then
easy to prove that V• → Sd X is an étale hypercover with the desired functoriality.
d
Using the commutativity of (4), one can also show that the map induced by Πét
∞S X →
d ét
S Π∞ X in cohomology with coefficients in a Z/l-module coincides with the symmetric Künneth
map defined in [SGA4, XVII, 5.5.17.2]. Thus, for X proper, it is possible to deduce Theorem 5.4
from [SGA4, XVII, Théorème 5.5.21].
6. A1 -localization
Let S be a Noetherian scheme of finite Krull dimension and C a full subcategory of Sch/S
such that
(1) objects of C are of finite type over S;
(2) S ∈ C and A1 ∈ C;
(3) if X ∈ C and U → X is étale of finite type, then U ∈ C;
(4) C is closed under finite products and finite coproducts.
In agreement with [Voe10], we call such a category C admissible. Let ShvNis (C) denote the
∞-topos of sheaves of spaces on C for the Nisnevich topology (by the hypothesis on S, this is a
hypercomplete ∞-topos). Recall that we’ve defined the functor
Πét
∞ : C → Pro(S),
and that it lifts to a left adjoint functor
b ét
Π
∞ : ShvNis (C) → Pro(S).
From now on we fix a prime number l different from the residual characteristics of S. By [SGA5,
VII, Corollaire 1.2], the composition
ShvNis (C)
b ét
Π
∞
Pro(S)
LZ/l
Pro(S)Z/l
sends any morphism A1 × X → X, X ∈ C, to an equivalence and therefore factors through the
A1 -localization LA1 : ShvNis (C) → ShvNis (C)A1 which inverts exactly those morphisms (as far
as colimit-preserving functors are concerned). That is, there is a unique commutative square of
left adjoint functors
ShvNis (C)
(5)
LA1
ShvNis (C)A1
b ét
Π
∞
Pro(S)
LZ/l
Pro(S)Z/l .
The Z/l-profinite étale homotopy type functor
Étl : ShvNis (C)A1 → Pro(S)Z/l
is given by the composition of the dashed arrow in (5) and the localization functor LZ/l . Note
b qfh
bh
b ét
that we could as well use Π
∞ or even Π∞ instead of Π∞ in the above diagram, according to
Proposition 4.1, and hence the functor Étl factors through Shvh (C)A1 .
10
MARC HOYOIS
Of course, if C = Sm/S is the category of smooth schemes of finite type over S, then
ShvNis (C)A1 is the standard ∞-category of motivic spaces over S. The functor Étl is in this case
the ∞-categorical incarnation of the étale realization functor defined by Isaksen in [Isa04] as a
left Quillen functor.
Suppose now that S = Spec k where k is a separably closed field. Denote by Shvfin
Nis (C)A1 the
closure of the image of C under finite colimits (this is equivalently the subcategory of compact
objects in ShvNis (C)A1 , by the usual characterization of Nisnevich sheaves).
Lemma 6.1. The restriction of Étl to Shvfin
Nis (C)A1 preserves finite products.
Proof. By Lemma 5.1, the functor LZ/l Πét
∞ preserve finite products on C. By Corollary 3.5, the
fin
ét
b
restriction of LZ/l Π∞ to the closure ShvNis (C) of C under finite colimits preserves finite products.
b ét
We observed in Remark 5.2 that, for any X ∈ Shvfin
Nis (C), the pro-space Π∞ X is a cofiltered limit
of spaces with finite Z/l-homology groups. It follows from Proposition 3.3 (2) that the canonical
Z/l b ét
b ét
transformation LZ/l Π
Π∞ is an equivalence on Shvfin
∞ →L
Nis (C), and the result follows.
Z/l
Let p ≥ q ≥ 0. We define the Z/l-profinite mixed spheres Slp,q ∈ Pro(S)∗
Sl1,0
Z/l
=L
1
S = K(Zl , 1),
Sl1,1
= K(Tl µ, 1),
Slp,q
=
(Sl1,0 )∧p−q
by
∧ (Sl1,1 )∧q ,
where µ is the group of roots of unity in k and Tl µ = limn µln is its l-adic Tate module. Of
course, Slp,q ' LZ/l S p , but if q > 0 this equivalence depends on infinitely many noncanonical
choices (viz., an isomorphism Zl ' Tl µ).
Note that the functor Étl preserves pointed objects, since Πét
∞ (Spec k) ' ∗.
Proposition 6.2. Let p ≥ q ≥ 0. Then Étl S p,q ' Slp,q .
Proof. This is obvious if q = 0. By Lemma 6.1, it remains to treat the case p = q = 1. The
étale Čech cover of the multiplication by ln map Gm → Gm has Bµln as simplicial set of
connected components. This defines a morphism of pro-spaces ϕ : Πét
∞ Gm → K(Tl µ, 1). If M is
a Z/l-module, then by [SGA5, VII, Proposition 1.3 (i) (c)],


if i = 0,
M
i
Hét
(Gm , M ) = Hom(µl , M ) if i = 1,


0
if i ≥ 2.
One checks easily that these cohomology groups are realized by any of the previously considered
Čech covers. It follows that ϕ is a Z/l-homological equivalence.
7. Group completions
Let C be an ∞-category with finite products. Recall from [HA, §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 (hni) →
M (h1i)n is an equivalence. We let E∞ (C× ) denote the full subcategory of Fun(Fin∗ , C) spanned
by commutative monoids in C.
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 hni. The commutative monoid M is called grouplike if
its underlying simplicial object is a groupoid object in the sense of [HTT, Definition 6.1.2.7]. We
×
×
denote by Egr
∞ (C ) ⊂ E∞ (C ) the full subcategory of grouplike objects. Since a commutative
monoid M is such that M (h0i) is final, [HTT, Proposition 6.1.2.6 (400 )] shows that M is grouplike
iff for every pair of subsets S, T ⊂ [n] such that S ∪ T = [n] and S ∩ T has a single element, the
canonical map M (hni) → M (Cut(S)) × M (Cut(T )) is an equivalence.
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 clearly preserves grouplike objects
×
gr
×
and hence restricts to a functor Egr
∞ (C ) → E∞ (D ). We will 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
×
gr
×
the functors E∞ (C× ) → E∞ (D× ) and Egr
∞ (C ) → E∞ (D ) induced by f are left adjoint to the
corresponding functors induced by g.
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 grouplike
objects follows since Egr
∞ is a full subcategory of E∞ .
Call an ∞-category C distributive if it is presentable and if finite products in C distribute over
small colimits. A functor f : C → D between distributive ∞-categories is a distributive functor if
it preserves small 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 LA1 : Shvτ (C) → Shvτ (C)A1 is distributive (by [Mor12,
Lemma 6.47]).
If C is distributive, then the ∞-category E∞ (C× ) is itself presentable by [HA, Corollary
×
3.2.3.5], and the subcategory Egr
∞ (C ) is strongly reflective by [HTT, Proposition 6.1.2.9] and [HTT,
Proposition 5.5.4.17]. That is, there exists a group completion functor
×
(−)+ : E∞ (C× ) → Egr
∞ (C )
×
×
which exhibits Egr
∞ (C ) as an accessible localization 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
E∞ (C× )
f
+
×
Egr
∞ (C )
E∞ (D× )
+
f
×
Egr
∞ (D )
has a commutative right adjoint and hence is commutative.
We can define a generalized “free Z-module” functor in any distributive ∞-category C. Given
X ∈ C, the object
a
NX =
Sd X
d≥0
has a structure of commutative monoid in C, i.e., there exists a functor M : Fin∗ → C with
M (hni) ' (NX)n , easily defined using distributivity. Moreover, the functor N : C → E∞ (C× )
preserves colimits. Indeed, it preserves sifted colimits by definition of Sd and the fact that the
forgetful functor E∞ (C× ) → C detects sifted colimits ([HA, Corollary 3.2.3.2]), and again by
definition of Sd it transforms finite coproducts into finite products, which coincide with finite
coproducts in the pointed category E∞ (C× ) ([HA, Proposition 3.2.4.7]). The group completion
×
of NX will be denoted by ZX. Thus, Z : C → Egr
∞ (C ) is a colimit-preserving functor which
commutes with any distributive functor, by Lemma 7.2. Note that the functors N and Z factor
through colimit-preserving functors Ñ and Z̃ on the ∞-category C∗ of pointed objects in C.
×
Further, if A ∈ Egr
∞ (S ) is a connective HZ-module, we can define a grouplike commutative
monoid object ZX ⊗ A as follows. Note that if R is a commutative ring, any connective HRmodule A can be obtained from R in the following steps:
(1) take finite products of copies of R to get finitely generated free R-modules;
(2) take filtered colimits of finitely generated free R-modules to get arbitrary flat R-modules;
(3) take colimits of simplicial diagrams of projective R-modules to get arbitrary connective
HR-modules;
Define ZX ⊗ A using the same steps. A map of finitely generated free abelian groups A → B
×
clearly induces a map ZX ⊗ A → ZX ⊗ B in Egr
∞ (C ) natural in X, and an arbitrary map of
connective HZ-modules can be obtained from such maps using steps (2) and (3), so that ZX ⊗A
×
varies functorially with the pair (X, A). As a bifunctor to Egr
∞ (C ), ZX ⊗ A preserves colimits in
either variable. Moreover, since all these steps only involve (sifted) colimits and finite products,
×
for any distributive functor f : C → D we have f (ZX ⊗ A) ' Zf (X) ⊗ A in Egr
∞ (D ).
In the distributive ∞-category S, ZX ⊗ A has its “usual” meaning. For instance, if A is an
abelian group, then Z̃S p ⊗ A is an Eilenberg–Mac Lane space K(A, p).
12
MARC HOYOIS
8. Étale homotopy type of motivic Eilenberg–Mac Lane spaces
We start by recalling the definition of the motivic Eilenberg–Mac Lane spaces.
Let C ⊂ Sch/S be an admissible category and let R be a commutative ring. We denote by
PSh∗ (C) the ∞-category of pointed presheaves on C, by PShtr (C, R) the ∞-category of presheaves
with R-transfers (see [Hoy12, §1] for a more detailed exposition of categories of presheaves with
transfers), and by
Rtr : PSh∗ (C) PShtr (C, R) : utr
the free–forgetful adjunction. The functor utr preserves limits and sifted colimits and factors
×
through the ∞-category Egr
∞ (PSh(C) ). Since finite products and finite coproducts coincide in
gr
×
E∞ (PSh(C) ), the induced functor
×
utr : PShtr (C, R) → Egr
∞ (PSh(C) )
preserves all colimits.
We denote by Shvtr
Nis (C, R)A1 the ∞-category of homotopy invariant Nisnevich sheaves with
R-transfers on C, defined by the cartesian square
Shvtr
Nis (C, R)A1
PShtr (C, R)
utr
utr
Shv∗Nis (C)A1
PSh∗ (C),
and by Rtr : Shv∗Nis (C)A1 → Shvtr
Nis (C, R)A1 the left adjoint to utr (it will always be clear from
the context which category of Rtr defined on).
The ∞-categories PShtr (C, R) and Shvtr
Nis (C, R)A1 are tensored over connective HR-modules.
For p ≥ q ≥ 0 and A a connective HR-module, the motivic Eilenberg–Mac Lane space K(A(q), p)C ∈
Shv∗Nis (C)A1 is defined by
K(A(q), p)C = utr (Rtr S p,q ⊗R A),
∗
where S p,q ∈ ShvNis (C)A1 is the motivic p-sphere of weight q. This space depends only on the
HZ-module structure of A and not on R. When the admissible category C is not specified, it is
understood to be Sm/S.
Lemma 8.1. The square
L
PShtr (C, R)
Shvtr
Nis (C, R)A1
utr
utr
×
Egr
∞ (PSh(C) )
L0
×
Egr
∞ (ShvNis (C)A1 )
commutes, where L and L0 are left adjoint to the inclusions.
Proof. Note that L0 is induced by the distributive functor LA1 aNis , by Lemma 7.1. Consider
the diagram
Shvtr
Nis (C, R)A1
utr
PShtr (C, R)
L
Shvtr
Nis (C, R)A1
utr
×
Egr
∞ (ShvNis (C)A1 )
×
Egr
∞ (PSh(C) )
f
L0
×
Egr
∞ (ShvNis (C)A1 ).
We will show that a functor f exists as indicated. Define
ENis = {Rtr U+ → Rtr X+ | U ,→ X is a Nisnevich covering sieve in C}, and
EA1 = {Rtr (X × A1 )+ → Rtr X+ | X ∈ C},
tr
so that Shvtr
Nis (C, R)A1 ⊂ PSh (C, R) is the full subcategory of (ENis ∪ EA1 )-local objects. The
functor L0 utr carries morphisms in ENis and EA1 to equivalences: for ENis , this is because the
Nisnevich topology is compatible with transfers (see [Hoy12, §1]), and for EA1 it is because
THE ÉTALE SYMMETRIC KÜNNETH THEOREM
13
utr Rtr (X × A1 )+ → utr Rtr X+ is an A1 -homotopy equivalence (see the last part of the proof
of [Voe10, Theorem 1.7]). By [HTT, Proposition 5.5.4.20], there exists a functor f making the
above diagram commutes. Since the horizontal compositions are the identity, we have f ' utr
and hence L0 utr ' utr L.
×
gr
Corollary 8.2. The forgetful functor utr : Shvtr
Nis (C, R)A1 → E∞ (ShvNis (C)A1 ) preserves small
colimits.
Proof. Follows from Lemma 8.1 since L and L0 utr preserve colimits.
Suppose now that S is the spectrum of a separably closed field k. One defect of the étale
homotopy type functor Étl is that it does not preserve finite products and hence does not preserve
commutative monoids. We have seen in Lemma 6.1 that Étl preserves finite products between
“finite” motivic spaces, but motivic Eilenberg–Mac Lane spaces are certainly not finite in the
required sense. We will fix this problem by constructing a factorization
Ét×
l
E → Pro(S)Z/l
ShvNis (C)A1 −−→
Z/l
of Étl such that Ét×
by a distributive
l is distributive and E is a “close approximation” of Pro(S)
×
∞-category. Moreover, the functor Étl will still be h-local, i.e., it will factor through Shvh (C)A1 .
Construction 8.3. Let D denote the category of all k-schemes of finite type, let i : C ,→ D
be the inclusion, and let i! : Shvh (C) → Shvh (D) be the induced left adjoint functor. Since C
is admissible and in particular closed under finite products, the functor i! is distributive. Note
that we have a commuting triangle
i!
Shvh (C)
Shvh (D)
bh
Π
∞
bh
Π
∞
Pro(S),
b h on the right is simply Π∞ , in the sense of Remark 5.3. For any ∞-topos
and that the functor Π
∞
X, we have a commutative square
τ≤n
X
X≤n
Π≤n
∞
Π∞
Pro(S)
τ≤n
Pro(S≤n ),
where the horizontal maps are given by truncation and Π≤n
∞ = τ≤n ◦ Π∞ . By Proposition 3.9,
we can therefore factor the Z/l-profinite shape functor LZ/l Π∞ : X → Pro(S)Z/l as
τ≤∗
Z/l
X −−→ lim X≤n → lim Pro(S)≤n ' Pro(S)Z/l .
n
n
Since truncations preserve colimits and finite products, the functor τ≤∗ is distributive. Applying
b ét
this procedure to the ∞-topos X = Shvh (D), we obtain a factorization of LZ/l Π
∞ as
a
i
τ≤∗
Z/l
h
!
ShvNis (C) −→
Shvh (C) −
→
Shvh (D) −−→ lim Shvh (D)≤n → lim Pro(S)≤n ' Pro(S)Z/l .
n
n
Now, let Shvfin
h (D)≤n denote the closure under finite colimits of the image of D in Shvh (D)≤n .
By Lemma 4.3, this is exactly the subcategory of compact objects in Shvh (D)≤n and hence
Shvh (D)≤n ' Ind(Shvfin
h (D)≤n ).
Let Pro0 (S) be the smallest full subcategory of Pro(S) which contains Πh∞ X for every k-scheme
of finite type X and which is closed under finite products and finite colimits. Note that the
tranformation LZ/l → LZ/l is an equivalence on Pro0 (S), by Remark 5.2. Let Pro0 (S)Z/l (resp.
Z/l
Pro0 (S)≤n , Pro0 (S)≤n ) be the image of Pro0 (S) under LZ/l (resp. τ≤n , τ≤n LZ/l ). It follows from
Corollary 3.5 that Ind(Pro0 (S)Z/l ) is a distributive ∞-category. By Lemma 6.1, the functor
Z/l
0
τ≤n Étl : Shvfin
h (D)≤n → Pro (S)≤n
14
MARC HOYOIS
preserves finite products and finite colimits, so that the induced functor on ind-completions
Z/l
Shvh (D)≤n → Ind(Pro0 (S)≤n )
is distributive. Thus, the composition
τ≤∗ i! ah
Z/l
ShvNis (C) −−−−−→ lim Shvh (D)≤n → lim Ind(Pro0 (S)≤n )
n
n
is distributive, and it is clearly A1 -local, so that it induces a distributive functor
Z/l
0
Ét×
l : ShvNis (C)A1 → lim Ind(Pro (S)≤n ).
n
By construction, Étl is the composition of Ét×
l and the colimit functor
Z/l
Z/l
lim Ind(Pro0 (S)≤n ) → lim Pro(S)≤n ' Pro(S)Z/l .
n
n
The next theorem is our étale version of [Voe10, Proposition 3.41]. We point out that the
category C in the statement below need not be closed under symmetric powers, so the theorem
applies directly to C = Sm/k with no need for resolutions of singularities. Properties (1) and
(2) in the theorem are crucial for the application to the motivic Steenrod algebra in §9 (in fact,
similar compatibility results for the topological realization are implicitely used in [Voe10, §3.4]).
Theorem 8.4. Let k be a separably closed field of characteristic exponent c, l 6= c a prime
number, and C ⊂ Sch/k an admissible subcategory consisting of semi-normal schemes. Then
for any pointed object X in ShvNis (C)A1 and any connective HZ[1/c]-module A, there is an
equivalence
×
eX,A : Z̃ Ét×
l X ⊗ A ' Étl utr (Ztr X ⊗ A)
Z/l
of grouplike commutative monoids in limn Ind(Pro0 (S)≤n ), natural in X and A, with the following
properties:
(1) the triangle
1
Ét×
l X
Z̃ Ét×
l X ⊗ Z[1/c]
'
Ét×
l η
eX,Z[1/c]
Ét×
l utr Z[1/c]tr X
is commutative;
(2) given X, Y , A, and B, the square
×
(Z̃ Ét×
l X ⊗ A) ∧ (Z̃ Étl Y ⊗ B)
eX,A ∧ eY,B
'
Z̃ Ét×
l (X ∧ Y ) ⊗ (A ⊗ B)
×
Ét×
l utr (Ztr X ⊗ A) ∧ Étl utr (Ztr Y ⊗ B)
'
eX∧Y,A⊗B
Ét×
l utr (Ztr (X ∧ Y ) ⊗ (A ⊗ B))
is commutative.
Proof. Any representable in ShvNis (C)A1 is a colimit of quasi-projective representables, so we
can assume that the objects of C are quasi-projective without changing the categories and
functors involved. As X and A vary, the source and target of eX,A are functors taking values
Z/l ×
0
in Egr
∞ (limn Ind(Pro (S)≤n ) ), and as such they preserve colimits in either variable: for the lefthand side this was shown in §7, and for the right-hand side it follows from Corollary 8.2. To
show the existence of eX,A , it will therefore suffice to define eX,Z[1/c] for X representable, i.e.,
THE ÉTALE SYMMETRIC KÜNNETH THEOREM
15
X = LA1 rNis (Z)+ for some Z ∈ C. The motivic Dold–Thom theorem ([Voe10, Theorem 3.7])
says that there is an equivalence
a
utr Z[1/c]tr Z+ ' aNis ((
rSd Z)+ [1/c])
d≥0
of pointed presheaves on C, natural in Z (in particular, the left-hand side is a Nisnevich sheaf).
Note that the validity of this formula does not depend on the scheme Sd Z belonging to C. Then
by Lemma 8.1 and the distributivity of LA1 aNis , we obtain equivalences
a
a
utr Z[1/c]tr X ' LA1 aNis ((
r(Sd Z))+ [1/c]) ' (
LA1 rNis (Sd Z))+ [1/c]
d≥0
d≥0
in ShvNis (C)A1 . Since Ét×
l preserves group completions of commutative monoids (Lemma 7.2)
and colimits,
a
d
+
Ét×
Ét×
l utr Z[1/c]tr X ' (
l LA1 rNis (S Z)) [1/c].
d≥0
Ét×
l
LA1 commutes with Sd ,
a
d
+
Z̃ Ét×
Ét×
l X ⊗ Z[1/c] ' (
l LA1 S rNis (Z)) [1/c].
On the other hand, since
d≥0
We then define eX,Z[1/c] to be the map induced by the obvious canonical map
ϕ : Sd rNis (Z) → rNis (Sd Z).
To show that eX,Z[1/c] is an equivalence, it suffices to show that Ét×
l LA1 (ϕ) is an equivalence.
×
×
By definition of Étl , the functor Étl LA1 factors through aqfh : ShvNis (C) → Shvqfh (C), so it
suffices to show that aqfh (ϕ) is an equivalence. This follows from Corollary 4.6.
The strategy to prove (1) and (2) is the following: we will first verify the statements in the
representable case, where they follow from properties of the motivic Dold–Thom equivalence,
and deduce the general case by functoriality arguments.
If X is represented by Z ∈ C, the adjunction map X → utr Z[1/c]tr X corresponds, through
the Dold–Thom equivalence, to the map
a
a
Z+ ' S0 Z q S1 Z ,→
Sd Z → (
Sd Z)+ [1/c].
d≥0
d≥0
This shows (1) when X is representable, whence for all X since the source functor Ét×
l preserves
colimits.
To prove (2), first assume that X and Y are represented by Z and W , respectively, and that
A = B = Z[1/c]. It will suffice to show that the square commutes with Ñ instead of Z̃. The
pairing
utr Z[1/c]tr X ∧ utr Z[1/c]tr Y → utr Z[1/c]tr (X ∧ Y )
arising from the monoidal structure of Z[1/c]tr is induced, via the Dold–Thom equivalence, by
the obvious maps Sd Z ×Se W → Sd+e (Z ×W ). This shows (2) in this case. The source functor in
this square of natural transformations preserves sifted colimits in the variables X and Y , while
the target functor transforms finite coproducts into finite products (this makes sense for pointed
functors between pointed categories). It follows that the square commutes for arbitrary X and
Y when A = B = Z[1/c]. But again, the source preserves sifted colimits in the variables A and
B and the target preserves finite products in these variables, so the result holds for arbitrary A
and B.
Let A be a connective HZ-module. We let Al be its pro-l-completion limn A/ln , so that
K(Al , p) ' LZ/l K(A, p)
for every p ≥ 1. Given q ∈ Z, let Al (q) = Al ⊗Zl Tl µ⊗q (which is noncanonically isomorphic to
Al ). For example, Z/ln (q) ' µ⊗q
ln .
16
MARC HOYOIS
Corollary 8.5. Let C ⊂ Sch/k be an admissible category consisting of semi-normal schemes.
For any connective HZ[1/c]-module A, any p ≥ 1 and p ≥ q ≥ 0, there is a canonical equivalence
Étl K(A(q), p)C ' K(Al (q), p)
in Pro(S)Z/l , natural in A, and Étl preserves smash products between such spaces. Furthermore,
under these equivalences,
(1) Étl sends the canonical map
S p,q → K(Z[1/c](q), p)C
to the canonical map
Slp,q → K(Zl (q), p);
(2) Étl sends the canonical map
K(A(q), p)C ∧ K(B(s), r)C → K((A ⊗ B)(q + s), p + r)C
to the canonical map
K(Al (q), p) ∧ K(Bl (s), r) → K((A ⊗ B)l (q + s), p + r).
Proof. By Theorem 8.4 and Proposition 6.2, we have
p,q
Ét×
⊗ A,
l K(A(q), p)C ' Z̃τ≤∗ Sl
where Slp,q ∈ Pro0 (S)Z/l is considered as a constant ind-Z/l-profinite pro-space. Now we apply
the functor
Z/l
Z/l
lim Ind(Pro0 (S)≤n ) → lim Pro(S)≤n ' Pro(S)Z/l
n
n
to both sides. The left-hand side becomes Étl K(A(q), p)C , by definition of Ét×
l . For the righthand side, consider the following commutative diagram
Sfin
i
Ind(Sfin )
'
jind
Z/l
lim Ind(Pro0 (S)≤n )
n
c
S
Lind
Z/l
lim Ind(Pro0 (S)≤n )
n
c
c
j
lim Pro(S)≤n
n
LZ/l
Z/l
lim Pro(S)≤n .
n
Note that τ≤∗ Slp,0 is the image of S p by the top row of this diagram. The functor jind is clearly
Z/l
distributive and Lind is distributive by Proposition 3.3. We therefore have equivalences
Z/l
Z/l
cZ̃(Lind jind iS p ) ⊗ A ' cLind jind (Z̃iS p ⊗ A) ' LZ/l jK(A, p),
as was to be shown in the case q = 0. The twisting for q > 0 makes the equivalence canonical
(when p = q = 1, the canonical map Sl1,1 → cZ̃τ≤∗ Sl1,1 is an equivalence, and the general case
follows). A similar argument shows that Étl (X ∧ Y ) ' Étl (X) ∧ Étl (Y ) whenever Ét×
l X and
Z/l
Ét×
Y
belong
to
the
essential
image
of
L
j
.
The
remaining
statements
are
easily
deduced
l
ind ind
from properties (1) and (2) in Theorem 8.4.
9. Application to motivic cohomology operations
Let k be a separably closed field and l 6= char k is a prime number. Recall that
(
µln (k)⊗q if p = 0 and q ≥ 0,
p,q
n
H (Spec k, Z/l ) '
0
otherwise.
L
If M is a module over the ring q≥0 µln (k)⊗q , we will denote by M ⊗0 Z/ln (resp. by M [τ −1 ])
L
⊗q
→
the module obtained from M by extending scalars along the projection
q≥0 µln (k)
L
L
⊗0
n
⊗q
⊗q
n
n
µln (k)
' Z/l (resp. along the inclusion
µ
(k)
,→
µ
(k)
).
We
will
also
q≥0 l
q∈Z l
denote by τ a chosen generator of H 0,1 (Spec k, Z/ln ), although the definition of M [τ −1 ] does
not depend on such a choice.
THE ÉTALE SYMMETRIC KÜNNETH THEOREM
17
Let p ≥ 1 and p ≥ q ≥ 0. By Corollary 8.5,
Étl K(Z/ln (q), p) ' K(µln (k)⊗q , p).
Thus, if X is a homotopy invariant pointed Nisnevich sheaf on Sm/k, Étl induces a map
(6)
Map(Σr X, K(Z/ln (q), p)) → Map(Étl X, K(µln (k)⊗q , p − r))
whenever p−r ≥ 0. On connected components this gives a map H p−r,q (X, Z/ln ) → H p−r (Étl X, µ⊗q
ln )
which a priori depends on r. However, we have commutative squares
Étl
Map(X, K(Z/ln (q), p))
Map(Étl X, K(µ⊗q
ln , p))
Proposition 6.2
Σr,s
Σp,q
l
Map(Slr,s ∧ Étl X, Slr,s ∧ K(µ⊗q
ln , p))
Map(Σr,s X, Σr,s K(Z/ln (q), p))
Corollary 8.5 (1) and (2)
Map(Σr,s X, K(Z/ln (q + s), p + r))
Étl
Map(Slr,s ∧ Étl X, K(µl⊗q+s
, p + r))
n
which show that (6) is compatible with the bigraded suspension isomorphisms.
Lemma 9.1. Let X ∈ Shv∗Nis (Sm/k)A1 . Then (6) is an equivalence whenever p − r ≤ q.
Proof. Let π : Shvét (Sm/k) → ShvNis (Sm/k) be the canonical geometric morphism. By Remark 5.3, Étl ◦LA1 ' LZ/l ◦ Π∞ ◦ π ∗ , and so (6) is the composition of the two maps
(7)
(8)
π ∗ : Map(Σr X, K(Z/ln (q), p)) → Map(Σr π ∗ X, π ∗ K(Z/ln (q), p)) and
LZ/l Π∞ : Map(Σr π ∗ X, π ∗ K(Z/ln (q), p)) → Map(Σr Étl X, LZ/l Π∞ π ∗ K(Z/ln (q), p)).
By [MVW06, Theorem 10.3], the étale sheaf π ∗ K(Z/ln (q), p) is equivalent to the Eilenberg–Mac
Lane sheaf K(µ⊗q
ln , p), and the π0 of (7) is the usual map
p−r
H̃ p−r,q (X, Z/ln ) → H̃ét
(X, µ⊗q
ln )
from motivic to étale cohomology, which by [Voe11, Theorem 6.17] is an isomorphism when X
is a pointed simplicial smooth scheme and p − r ≤ q. Hence (7) is an equivalence in this range
for such X, whence for all X since both sides transform colimits into limits. Since k is separably
closed, K(µ⊗q
ln , p) is in the image of the constant sheaf functor c : S → Shvét (Sm/k) which is
fully faithful. By definition, Π∞ is pro-left adjoint to c, and so (8) is also equivalence.
Remark 9.2. The argument in the proof of Lemma 9.1 shows that
Étl K(Z/ln (q), p)Sm/k ' K(µln (k)⊗q , p)
independently from Corollary 8.5.
Lemma 9.3. For every X ∈ Shv∗Nis (Sm/k)A1 , (6) induces an isomorphism
H̃ ∗∗ (X, Z/ln )[τ −1 ] ' H̃ ∗ (Étl X, µln (k)⊗∗ ).
Proof. Follows immediately from Lemma 9.1 since X has no motivic cohomology in negative
weights (the case X = (Spec k)+ shows that the image of τ is invertible).
∗
Remark 9.4. The isomorphism H ∗∗ (X, Z/ln )[τ −1 ] ' Hét
(X, µ⊗∗
ln ) for X ∈ Sm/k is also proved
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 Étl .
Theorem 9.5. Let p ≥ 1, p ≥ q ≥ 0, and let A be a connective HZ[1/c]-module. Then Étl
induces an isomorphism
H̃ ∗∗ (K(A(q), p)Sm/k , Z/ln )[τ −1 ] ' H̃ ∗ (K(Al (q), p), µln (k)⊗∗ ).
Proof. Combine Lemma 9.3 and Corollary 8.5.
18
MARC HOYOIS
Let now k be any perfect field and l 6= char k a prime number. For n ≥ 0, let Kn be the
motivic Eilenberg–Mac Lane space
Kn = K(Z/l(n), 2n)Sm/k .
Denote by A∗∗ the algebra of bistable operations in motivic cohomology with Z/l coefficients,
i.e.,
A∗∗ = lim H̃ ∗+2n,∗+n (Kn , Z/l),
n→∞
∗∗
and let A∗∗
generated by H ∗∗ (Spec k, Z/l), the power operations P i ,
0 be the subalgebra of A
and the Bockstein β; this is the algebra studied by Voevodsky in [Voe03]. In the rest of this
section we will prove:
Theorem 9.6. Let k be a perfect field and let l 6= char k be a prime number. Suppose that
∗∗
Z/ltr Kn is a split proper Tate object of weight ≥ n. Then A∗∗
0 =A .
We refer to [Voe10, Definition 2.60] for the definition of split proper Tate object. The hypothesis of Theorem 9.6 is satisfied when k admits appropriate resolutions of singularities (cf.
[Voe10, Corollary 3.33]). In [Hoy12] we show that it always holds. If k has characteristic zero,
this theorem is proved in [Voe10, §3.4]. The proof of the general case is essentially the same, the
main difference being that we must replace the topological realization with the étale realization.
Recall from [Voe10, Lemma 3.50] that the hypothesis that Z/ltr Kn be a split proper Tate motive of weight ≥ n implies that there exist split proper Tate motives A and A0 whose cohomology
tr
is identified with A∗∗ and A∗∗
0 , respectively, and a morphism A → A0 in ShvNis (Sm/k, Z/l)A1
∗∗
∗∗
which induces the inclusion A0 ,→ A and which is invariant under change of base field.
By [Voe10, Corollary 2.70 (2)], we may therefore assume that k is separably closed, which we
do from now on. Theorem 9.6 then follows from the following statements:
∗∗
(1) A∗∗
⊗0 Z/l is injective;
0 ⊗0 Z/l → A
∗∗ −1
∗∗ −1
(2) A0 [τ ] → A [τ ] is surjective.
k
Indeed, let x ∈ A∗∗ . By (2), τ k x belongs to A∗∗
0 for some k ≥ 0. If k > 0, τ x becomes zero
in A∗∗ ⊗0 Z/l, and so by (1), it is already zero in A∗∗
⊗
Z/l.
Thus,
there
exists
y ∈ A∗∗
0
0
0 such
that τ k x = τ y. Since A∗∗ is the cohomology of a split Tate object, it has no τ -torsion. Hence,
∗∗
τ k−1 x = y and so τ k−1 x belongs to A∗∗
0 . By induction on k, x ∈ A0 .
∗∗
∗∗
The injectivity of A0 ⊗0 Z/l → A ⊗0 Z/l is proved in [Voe10, Proposition 3.56], so it
−1
remains to prove that A∗∗
] → A∗∗ [τ −1 ] is surjective. Define Knét ∈ S by
0 [τ
Knét = K(µl (k)⊗n , 2n),
∗∗
and let Aét
denote the algebra
⊗(∗+n)
∗∗
Aét
= lim H̃ ∗+2n (Knét , µl
n→∞
).
The motivic Adem relations ([Voe03, Theorems 10.2 and 10.3]) which characterize A∗∗
0 show
that there is a unique map of algebras
∗∗
A∗∗
0 → Aét
(9)
sending P i to P i , β to β, and τ to τ . On the other hand, by (1) and (2) of Corollary 8.5, we
have commutative squares
H̃ ∗+2,∗+1 (Kn+1 , Z/l)
H̃ ∗∗ (Kn , Z/l)
ét
H̃ ∗+2 (Kn+1
, µl (k)⊗∗+1 )
H̃ ∗ (Knét , µl (k)⊗∗ ),
and hence an induced map
(10)
∗∗
A∗∗ → Aét
in the limit. This is also a map of algebras since it is induced by the functor Étl .
THE ÉTALE SYMMETRIC KÜNNETH THEOREM
19
Lemma 9.7. The triangle
A∗∗
0
A∗∗
(10)
(9)
∗∗
Aét
commutes.
Proof. Since this is a triangle of algebras, it suffices to show that (10) sends P i to P i , β to β,
and τ to τ . The latter holds by definition. Denote by P̃ i the image of P i by (10). Because
the identification of Étl Kn with Knét is compatible with cup products (Corollary 8.5 (2)), the
motivic Cartan formula ([Voe03, Proposition 9.6]) shows that the family of operations P̃ i satisfy
the axioms (1)–(5) of [SE62, VI, §1]. Therefore, by [SE62, VIII, Theorems 3.9 and 3.10], P̃ i = P i .
Denote by α the equivalence of Corollary 8.5. Because α is natural in the coefficients, the
right square in the diagram
Étl β
Étl K(Z/l(n), 2n)
γ
'
'
K(µ⊗n
l , 2n)
Étl K(Z/l2 (n), 2n + 1)
Étl K(Z/l(n), 2n + 1)
'
α
K(µ⊗n
l , 2n + 1)
β
α
K(µ⊗n
l2 , 2n + 1)
is commutative and hence there exists a unique map γ as indicated. For formal reasons, γ also
makes the square
Étl ΩK(Z/l2 (n), 2n + 1)
Étl K(Z/l(n), 2n)
γ
Ωα ◦ can
ΩK(µ⊗n
l2 , 2n + 1)
K(µ⊗n
l , 2n)
commute. By Corollary 8.5 (1) and (2) and an adjunction argument, this square can be identified
with
Étl K(Z/l2 (n), 2n)
Étl K(Z/l(n), 2n)
γ
α
K(µ⊗n
l2 , 2n)
K(µ⊗n
l , 2n).
But there can be at most one map γ making this square commute (by virtue of the equivalence
between the category of Eilenberg–Mac Lane spaces of degree 2n and that of abelian groups).
Since α is natural in the coefficients, we deduce that γ = α, whence that (10) sends β to β. Consider the diagram
−1
A∗∗
]
0 [τ
A∗∗ [τ −1 ]
'
(10)
H̃ ∗+2n,∗+n (Kn , Z/l)[τ −1 ]
'
(9)
∗∗
Aét
H̃ ∗+2n (Knét , µl (k)⊗∗+n ),
where the right-hand map is an equivalence by Theorem 9.5. By Lemma 9.7, the triangle
−1
commutes. Thus, to show that A∗∗
] → A∗∗ [τ −1 ] is surjective, it suffices to show that
0 [τ
∗∗ −1
∗∗
∗∗
A [τ ] → Aét is injective. Suppose that x = (x0 , x1 , . . . ) ∈ A∗∗ becomes 0 in Aét
. Then each
∗+2n,∗+n
−1
∗+2n,∗+n
xn becomes 0 in H̃
(Kn , Z/l)[τ ]. Since Z/ltr Kn is split Tate, H̃
(Kn , Z/l) has
no τ -torsion and so xn = 0 for all n. Hence, x = 0.
This completes the proof of Theorem 9.6.
Remark 9.8. In this remark we outline a more elementary proof of Theorem 9.6 that does not
rely on the comparison result between motivic and étale cohomology (and is closer to the proof
in characteristic zero given in [Voe10, §3.4]). Let ChR
≥0 denote the ∞-category of nonnegatively
20
MARC HOYOIS
tr
graded chain complexes of R-modules, and let Shvét
(C, R) denote the ∞-category of étale sheaves
tr
with R-transfers on an admissible category C. The global section functor Γ : Shvét
(C, R) → ChR
≥0
admits a left adjoint ctr such that the square
∗
Shvét
(C)
c
S∗
u
utr
tr
Shvét
(C, R)
ctr
ChR
≥0
commutes (because the étale topology is compatible with transfers, cf. [Hoy12, §1]). Since the
forgetful functors utr and u preserve and reflect limits and filtered colimits, the functor ctr is
accessible and preserves finite limits. It therefore admits a pro-left adjoint
tr
R
Πtr
∞ : Shvét (C, R) → Pro(Ch≥0 ).
Moreover, the square
∗
Shvét
(C)
Π∞
Pro(S∗ )
C∗ (−, R)
Rtr
tr
Shvét
(C, R)
Πtr
∞
Pro(ChR
≥0 )
is commutative since it has a commutative right adjoint. Clearly, the functor Πtr
∞ factors through
tr
(C, R) if R is a torsion ring whose torsion is prime to char k. Using
the A1 -localization of Shvét
this commutative square and Proposition 6.2, one can easily prove the étale analogue of [Voe10,
Lemma 3.52]: if X is a pointed object in ShvNis (C)A1 such that Z/ltr X splits as a direct sum of
objects of the form Z/ltr S p,q , then Étl induces an isomorphism
H̃ ∗∗ (X, Z/l)[τ −1 ] ' H̃ ∗ (Étl X, µl (k)⊗∗ ).
Of course, Lemma 9.3 above shows that this is true for any X, but this weaker statement is
sufficient for Theorem 9.6.
References
[DAG13]
[DG70]
[DHI04]
[Del77]
[EGA2]
[Far96]
[Fri82]
[HA]
[HTT]
[Hoy12]
[Isa01]
[Isa04]
[Isa05]
[Isa07]
[Jar03]
J. Lurie, Derived Algebraic Geometry XIII: Rational and p-adic Homotopy Theory, 2011,
http://www.math.harvard.edu/~lurie/
M. Demazure and P. Gabriel, Groupes algébriques, North-Holland, 1970
D. Dugger, S. Hollander, and D. C. Isaksen, Hypercovers and simplicial presheaves, Math. Proc.
Cambridge Philos. Soc. 136 (2004), no. 1, pp. 9–51, preprint K-theory:0563
P. Deligne, Cohomologie étale, Lecture Notes in Mathematics, vol. 569, Springer, 1977
A. Grothendieck, Éléments de Géométrie Algébrique II, Publ. Math. I.H.É.S. 8 (1961)
E. D. Farjoun, Cellular Spaces, Null Spaces and Homotopy Localization, Lecture Notes in Mathematics,
vol. 1622, Springer, 1996
E. M. Friedlander, Étale Homotopy of Simplicial Schemes, Ann. Math., vol. 104, Princeton University
Press, 1982
J. Lurie, Higher Algebra, 2012,
http://www.math.harvard.edu/~lurie/
, Higher Topos Theory, 2012,
http://www.math.harvard.edu/~lurie/
M. Hoyois, The motivic Steenrod algebra over perfect fields, 2012,
http://math.northwestern.edu/~hoyois/papers/motivicsteenrod.pdf
D. C. Isaksen, A model structure on the category of pro-simplicial sets, Trans. Amer. Math. Soc. 353
(2001), pp. 2805–2841, preprint arXiv:math/0106152v1 [math.AT]
, Étale realization on A1 -homotopy theory of schemes, Adv. Math. 184 (2004), no. 1, pp. 37–63,
preprint arXiv:math/0106158v2 [math.KT]
, Completions of pro-spaces, Math. Zeit. 250 (2005), pp. 113–143, preprint
arXiv:math/0404303v1 [math.AT]
, Strict model structures for pro-categories, Categorical decomposition techniques in algebraic topology, Progress in Mathematics, vol. 215, Birkhauser, 2007, pp. 179–198, preprint
arXiv:math/0108189v1 [math.AT]
J. F. Jardine, Finite group torsors for the qfh topology, Math. Zeit. 244 (2003), no. 3, pp. 859–871,
preprint K-theory:0461
THE ÉTALE SYMMETRIC KÜNNETH THEOREM
[Kol97]
21
J. Kollár, Quotient spaces modulo algebraic groups, Ann. of Math. (2) 145 (1997), no. 1, pp. 33–79,
preprint arXiv:alg-geom/9503007v1 [math.AG]
[Lev00]
M. Levine, Inverting the motivic Bott element, K-theory 19 (2000), no. 1, pp. 1–28, preprint Ktheory:0281
[MVW06] C. Mazza, V. Voevodsky, and C. Weibel, Lecture Notes on Motivic Cohomology, Clay Mathematics
Monographs, vol. 2, AMS, 2006
[Mor12] F. Morel, A1 -Algebraic topology over a field, 2012,
http://www.mathematik.uni-muenchen.de/~morel/A1TopologyLNM.pdf
[Ryd10] D. Rydh, Submersions and effective descent of étale morphisms, Bull. Soc. Math. France 138 (2010),
no. 2, pp. 181–230, preprint arXiv:0710.2488v3 [math.AG]
[Ryd12]
, Existence and properties of geometric quotients, 2012, arXiv:0708.3333v2 [math.AG]
[SE62]
N. E. Steenrod and D. B. A. Epstein, Cohomology operations, Princeton, 1962
[SGA3]
M. Demazure and A. Grothendieck, Propriétés générales des schémas en groupes, Lecture Notes in
Mathematics, vol. 151, Springer, 1970
[SGA4]
M. Artin, A. Grothendieck, and J.-L. Verdier, Théorie des topos et cohomologie étale des schémas,
Lecture Notes in Mathematics, vol. 269, Springer, 1972
[SGA5]
A. Grothendieck, Cohomologie l-adique et fonctions L, Lecture Notes in Mathematics, vol. 589,
Springer, 1977
[Voe96]
V. Voevodsky, Homology of Schemes, Selecta Math. (N.S.) 2 (1996), no. 1, pp. 111–153, preprint
K-theory:0031
[Voe03]
, Reduced power operations in motivic cohomology, Publ. Math. I.H.É.S. 98 (2003), pp. 1–57,
preprint K-theory:0487
[Voe10]
, Motivic Eilenberg–MacLane spaces, Publ. Math. I.H.É.S. 112 (2010), no. 1, pp. 1–99, preprint
arXiv:0805.4432v3 [math.AG]
[Voe11]
, On motivic cohomology with Z/l-coefficients, Ann. of Math. (2) 174 (2011), no. 1, pp. 401–
438, preprint arXiv:0805.4430v2 [math.AG]