Twisted product functors

Vladimir Voevodsky^{1}^{,}^{2}

Preliminary version, October 2000

Contents

1 Introduction 1

2 Standard cd-structures on G-schemes 4

3 The functor X ↦ X/G 9 3.1 Geometric lemmas . . . . . . . . . . . . . . . . . . . . . . . . 9

4 Smash powers 18 4.1 Geometric lemmas . . . . . . . . . . . . . . . . . . . . . . . . 19 4.2 Smash powers in the upper topologies . . . . . . . . . . . . . . 23 4.3 Smash powers in the upper topologies on smooth schemes … 32 4.4 Smash powers in the lower topologies . . . . . . . . . . . . . . 32 4.5 Smash powers in the cdh-topology . . . . . . . . . . . . . . . . 33

5 Twisted product functors 33 5.1 Twisted products of sheaves and presheaves . . . . . . . . . . 37 5.2 Twisted product functors on smooth schemes . . . . . . . . . 42

6 Appendix: solid sheaves 45

1 Introduction

Functors of the form

[f]X ↦ (X^{∧n} ∧ E_{+})/Σ_{n}  (1)

1Supported by the NSF grants DMS-97-29992 and DMS-9901219, Sloan Research Fellowship and Veblen Fund

2School of Mathematics, Institute for Advanced Study, Princeton NJ, USA. e-mail:

vladimir@ias.edu

where Σ_{n} is the symmetric group and E is a space with a Σ_{n} action, play an important role in the classical homotopy theory. When E is the configuration space of points in R^{n} they provide building blocks for the spaces Ω^{n}Σ^{n}(X). When E is the point one gets the symmetric products which are the building blocks of the zero space of the spectrum Σ^{∞}X ∧ H_{Z} whose homotopy groups are the homology of X (this is the Dold-Thom theorem). When E is a contractible space with a free Σ_{n}-action the space ((X_{+})^{∧n} ∧E_{+})/Σ_{n} appears in the construction of the reduced power operations in all oriented theories from ordinary cohomology to complex cobordism.

The goal of this paper is to define analogous functors for schemes and show that they extend to functors on the motivic homotopy category. Let us outline first the level of generality we work in. Everywhere below a scheme means a scheme of finite type over a Noetherian, finite dimensional base scheme S. In topology, the group Σ_{n} plays a distinguished role because any functor of the form X ↦ (X^{∧A} ∧ E_{+})/G where G is a finite group, E is a Gspace and A is a finite G-set can be expressed in terms of functors of the form (1). This does not seem to be the case in the motivic context. Therefore, we consider the general case of twisted product functors of the form

[falg]X ↦ (X^{W} × E)/G  (2)

where G is a finite flat group scheme, W is a finite flat G-scheme and E is a quasi-projective G-scheme. In applications to Steenrod operations we will be especially interested in the case when G = μ_{p} is the group of p-th roots of unity. Since we do not want to exclude the case when p divides the characteristic of S we are forced to consider groups with nilpotents.

In the first section ?? of the paper we construct, for any G, W and E as above, a functor

[fnis]T̃ = T̃_{G,W,E} : H_{A^{1},•}(S) → H_{A^{1},•}(S)  (3)

such that for a smooth quasi-projective scheme X over S the value of T̃ on X_{+} is the pointed sheaf on Sm/S represented by the scheme ((X^{W} × E)/G)_{+}^{3}. Unless the action of G on E is free this scheme is not smooth but we can nevertheless consider the sheaf represented by it on smooth schemes as an object of the motivic homotopy category. The functor T̃ has the property that for any smooth, quasi-projective pointed simplicial scheme X = (X_{n}), such that for any n one has X_{n} = (X′_{n})_{+} for some schemes X′_{n}, one has

3The existence of X^{W} and the quotient scheme under our assumptions is proved in ??

and ?? below.

[fnis2]T̃(X)_{n} = (((X′_{n})^{W} × E)/G)_{+}  (4)

Let X be a quasi-projective scheme (or simplicial scheme) over S which is not smooth. As noted above, the Nisnevich sheaf ρ(X) represented by X on the category of smooth schemes over S defines an object of H_{A^{1}}(S) and we may ask for a formula expressing T̃(ρ(X)_{+}) in terms of X. It is easy to see that there is a canonical morphism of the form

[themorphism]T̃(ρ(X)_{+}) → ρ((X^{W} × E)/G)_{+}  (5)

In the second section ?? we prove that if S = Spec(k) where k is a field with resolution of singularities, W is etale and the quotients with respect to G take closed embeddings to closed embeddings then the first simplicial suspension of (5) is an A^{1}-weak equivalence (these conditions hold in particular when k is a field of characteristic zero, G is a finite group and W is a finite G-set). This result allows us to commpute in certian cases the composition of two twisted product functors (??).

The proof is based on the construction of the twisted product functors with respect to the cdh-topology and the comparison theorem [11, Theorem 4.4]. Ideally, we would like to construct a functor

[fcdh]T̃_{cdh} : H_{•}((Sch/S)_{cdh}, A^{1}) → H_{•}((Sch/S)_{cdh}, A^{1})  (6)

which has the same property as the functor (3). We do not know how to do this in general because we can not show that the functor (4) on simplicial schemes preserves weak equivalences in the cdh-topology (see ??). Our methods are sufficient however to construct (6) for etale W and finite groups G of order prime to the characteristic of S. In particular we prove that the symmetric product functors in characteristic zero preserve cdh-weak equivalences of simplicial schemes. Our failure to prove this in positive characteristic suggests an interesting possibility that these functor do indeed fail to preserve cdh-weak equivalences and that the corresponding derived functors take a scheme X to an object equivalent to the space of finite cycles on X of a given degree. More precisely, one may expect that the quotient functors X ↦ X/G from the category of G-schemes to the category of schemes do not preserve cdh-weak equivalences and that, if we denote by Q_{G} the corresponding derived functor, the object Q_{Σ_{n}}(X^{n}) is equivalent to the sheaf of finite cycles of degree n on X. If this were true it would clarify the misterious diffrence which exist in positive characteristic between finite cycles and morphisms to symmetric products (see [, ]). Unfortunately, without resolution of singularities, it is not clear how this can be used in the motivic homotopy theory.

The constructions of functors (3) and (6) and the proofs of their main properties are contained in the first two sections of the paper. The later sections contain the proofs of the technical results necessary for these constructions.

Everywhere below a scheme means a Noetherain scheme.

2 Standard cd-structures on G-schemes

Let S be a Noetherain scheme and G be a group scheme over S. We define the standard cd-structures on the category of G-schemes through the forgetful functor to the category of schemes. That is, a square of G-schemes is called upper distinguished, lower distinguished, plain upper distinguished or plain lower distinguished if it is of the corresponding type when considered as a square of schemes without a G-action. Since the forgetful functor commutes with the fiber products [10, Lemma 4.3]implies that the standard cd-structures are complete and [10, Lemma 7.3]that they are regular. Define the standard density structure on G-schemes through the forgetful functor as well. The equivariant dimension of a G-scheme is always less or equal to the non equivariant one. If Z is an invariant subset of a G-scheme X and p : X → Y is a G-morphism then the image of Z is invariant. If Z is an invariant subset of a G-scheme X then the closure cl(Z) of Z is invariant. Finally if Z is an invariant closed subset of a G-scheme X then there exists a unique G-action on the complement X − Z such that X − Z → X is a morphism of G-schemes. These statements imply that the proof of [11, Proposition 2.7]can be transfered without change to the equivariant case. To transfer the proof of [11, Proposition 2.8]we need to know that for an equivariant morphism f : Y → X of finite type the scheme-theoretic image Spec(ker(O_{X} → f_{∗}O_{Y})) of f has a canonical structure of a G-scheme. This follows immediately from the definition if G is flat over S. We proved the following result.

Proposition 2.0.1 [eqcase1] The standard cd-structures on the category of G-schemes over S are complete and regular. The upper and plain upper cdstructures are bounded by the standard density structure for any G. The lower and plain lower cd-structures are bounded by the standard density structure if G is flat over S.

Remark 2.0.2 We do not know of any example of a groups scheme G such that the lower cd-structure on G-schemes is not bounded by the standard density structure.

We will need a criterion for an etale G-morphism to be a covering in the upper cd-topology and for a proper morphism to be a covering in the lower cd-topology.

Definition 2.0.3 [uppersplit] Let f : X̃ → X be a morphism of G-schemes. A splitting sequence for f is a sequence of closed embeddings of G-schemes

∅ = Z_{n+1} → Z_{n} → … → Z_{1} → Z_{0} = X

such that for any i = 0, …, n the projection

(Z_{i} − Z_{i+1}) ×_{X} X̃ → (Z_{i} − Z_{i+1})

has a section (in the category of G-schemes).

Proposition 2.0.4 [upperchar] Let G be a flat group scheme. Then an etale G-morphism f : X̃ → X is a covering in the upper cd-topology if and only if it has a splitting sequence.

Proof: We start with the following lemma. Its proof was suggested by P. Deligne.

Lemma 2.0.5 [complement] Let G be a flat group scheme and Z be an invariant closed subset of a G-scheme X. Then there exists a G-scheme Z′ and a closed embedding i : Z′ → X such that Im(i) = Z.

Proof: Consider Z as a scheme with the reduced structure. The morphism

f : G ×_{S} Z → G ×_{S} X → X

where the last arrow is the action is a G-equivariant morphism with respect to the action of G on G × Z through the first factor. We can now take Z′ to be the scheme-theoretic image of f i.e. Z′ = Spec(ker(O_{X} → f_{∗}O_{G×_{S}Z})). Since G is flat Z′ has a canonical action of G.

The proof of the “only if” part is parallel, modulo Lemma 2.0.5, to the proof of the similar result in the non equivariant case given in [6, Lemma 3.1.5].

Lemma 2.0.6 [hasdense] If f : X̃ → X is an upper cd-covering then there exists an open embedding j : U → X with dense image and a section of f over U.

Proof: We work in the category of G-schemes. Using the fact that the upper cd-structure is complete we may assume that f = ∐ f_{i} where {X̃_{i} → X} is a simple covering. By induction we may assume further that our covering is of the form {Ỹ_{i} →^{p_{i}} Y → X, Ã_{j} →^{q_{j}} A → X} where ∐ p_{i} and ∐ q_{j} have sections over dense open subschemes Y_{0}, A_{0} of Y and A respectively and Y → X and A → X are two sides of an upper distinguished square of the form

[eq1]\begin{matrix}B & ⟶ & Y \\ ↓ & & ↓ \\ A & ⟶ & X \end{matrix}  (7)

An open subset of X is dense if it belongs to D_{1}(X) defined by the standard density structure. Using the fact that any upper square is reducing and applying the definition of a reducing square for Y_{0}, A_{0} and B_{0} = ∅ we conclude that there is a dense open subset U of X such that f has a section over U.

Let f : X̃ → X be an upper cd-covering. To find a splitting sequence for f take a dense open subset U of X such that f has a section over U. Let Z_{1} → Z_{0} = X be a closed embedding of G-schemes whose image is X − U. Consider the pull-back of f to Z_{1} and apply again Lemma 2.0.6. One gets a sequence of closed embeddings Z_{i+1} → Z_{i} → … → X such that Z_{i+1} − Z_{i} is dense in Z_{i+1} and in particular non empty. Since X is Noetherain this sequence must stabilize giving a finite splitting sequence for f. This proofs the “only if” part.

To prove the “if” part consider an etale morphism f : X̃ → X with a splitting sequence Z_{n} → … → Z_{0} = X. We will construct an upper distinguished square of the form (7) based on X such that the pull-back of f to Y has a section and the pull-back of f to A has a splitting sequence of length less than n. The result then follows by induction on n. We take A = X − Z_{n}. To define Y consider the section s of f_{n} : X̃ ×_{X} Z_{n} → Z_{n} which exists by definition of a splitting sequence. Since f is etale and in particular unramified the image of s is an open subscheme. Let W be its complement. The morphism X̃ ×_{X} Z_{n} → X̃ is a closed embedding thus the image of W is closed in X̃. We take Y = X̃ − W. One verifies immediately that the pull-back square defined by A → X and Y → X is upper distinguished. The pull-back of f to Y has a section and the pull-back of f to A has a splitting sequence of length n − 1. This finishes the proof of the proposition.

Proposition 2.0.7 [lowerchar] Let G be a flat group scheme. A proper morphism of G-schemes f : X̃ → X is a lower cd-covering if and only if it has a splitting sequence.

Proof: The proof of the “only if” part is parallel to the proof given for the upper case in Proposition 2.0.4 with the following lemma replacing Lemma 2.0.6.

Lemma 2.0.8 [hasdenselow] Let f : X̃ → X be a lower cd-covering. Then there exists an open embedding U → X with a dense image and a section of f over U.

Proof: Same argument as in the proof of Lemma 2.0.6.

To prove the “if” part consider a proper morphism f : X̃ → X with a splitting sequence Z_{n} → … → Z_{0} = X. We will construct a lower distinguished square of the form (7) based on X such that the pull-back of f to Y has a section and the pull-back of f to A has a splitting sequence of length less than n. The result then follows by induction on n. We take A = Z_{1}. To define Y consider the section s of f_{n} : X̃ ×_{X} (X − Z_{1}) → (X − Z_{1}) which exists by definition of a splitting sequence. Since f is proper and in particular separated, the image of s is a closed subscheme. Let W be its complement. The morphism X̃ ×_{X} (X − Z_{1}) → X̃ is an open embedding thus the image of W is open in X̃. We take Y = X̃ − W. One verifies immediately that the pull-back square defined by A → X and Y → X is lower distinguished. The pull-back of f to Y has a section and the pull-back of f to A has a splitting sequence of length n − 1. This finishes the proof of the proposition.

Corollary 2.0.9 [corlow] A proper morphism f : X̃ → X is a lower cdcovering in the category of schemes if and only if for any point x of X there exists a point x̃ of X̃ such that f(x̃) = x and kx̃ = k_{x}.

Proof: The “only if” part follows from the fact that any lower cd-covering over the spectrum of a field has a section. To prove the “if” part it is sufficient, by the “if” part of Proposition 2.0.7 to show that if f satisfies the condition of the corollary then it has a splitting sequence. This follows easily by Noetherian induction as in the proof of [6, Lemma 3.1.5].

Lemma 2.0.10 [opsat] The class of open embeddings is saturated (in the sense of Definition 6.0.17) in any of the standard topologies.

Proof: The only condition of Definition 6.0.17 which is not obvious is the third one. We have to show that any morphism of sheaves of the form F′ → ρ(X) which is locally representable by open embeddings is representable by an open embedding. Let p : Y → X be a covering such that the pull-back of F′ to Y is representable by an open embedding j : V → Y. Two pull-backs of U to Y ×_{X} Y coincide which implies that V = p^{−1}(p(V)). A covering in any of the standard topologies has the property that a subset in X is open if and only if its preimage under p is open. We conclude that U = p(V) is an open subset in X and one can easily see that the subsheaf of ρ(X) represented by U coincides with F′.

Lemma 2.0.11 [clsat] The class of closed embeddings is saturated (in the sense of Definition 6.0.17) in any of the standard topologies.

Proof: As in the proof of Lemma 2.0.10, the only condition of Definition 6.0.17 which is not obvious is the third one. We have to show that any morphism of sheaves of the form F′ → ρ(X) which is locally representable by closed embeddings is representable by a closed embedding. To prove it for all standard topologies it is clearly sufficient to prove it for the upper, plain upper, lower and plain lower ones. Consider the upper case first. Let p : Y → X be an upper covering such that the pull-back of F′ to Y is representable by a closed embedding i : Z → Y. Two pull-backs of Z to Y ×_{X} Y coincide. Since an upper covering is flat the flat descent for closed embeddings implies that Z = p^{−1}(T) for a closed embedding T → X and one can easily see that the subsheaf of ρ(X) represented by T coincides with F′. The plain upper case is exactly similar.

Consider now the lower case. Let p : Y → X be a lower covering such that the pull-back of F′ to Y is representable by a closed embedding i : Z → Y. Two pull-backs of Z to Y ×_{X} Y coincide as closed subsets (by Lemma 2.0.12) and therefore Z = p^{−1}(p(Z)). Since p is proper p(Z) is a closed subset in X. Let T be any closed subscheme in X supported in p(Z) (it exists by Lemma 2.0.5). Lemma 2.0.12 implies now that F′ coincides with the subsheaf represented by T. The proof in the plain lower case is is exactly similar.

Lemma 2.0.12 [defcl] Let f : X → Y be a morphism and i : T → Y a closed embedding of G-schemes of finite type over S. Then the morphism of sheaves ρ(f), in any topology which contains the plain lower topology, factors throught the morphism ρ(i) if and only if the image of f lies in the closed subset i(T).

Proof: ???

3 The functor X ↦ X/G

3.1 Geometric lemmas

Let G be a finite flat group scheme over S. Let us recall the following general results about quotients with respect to such groups G (see [2, Theorem V.7.1]).

Theorem 3.1.1 [m15l3] For any quasi-projective G-scheme X over S the quotient X/G in the category of ringed spaces exists and is the quotient in the category schemes of finite type over S. The morphism X → X/G is finite, surjective and open. For any flat morphism Y → Y/G one has Y = (Y ×_{X/G} X)/G.

Theorem 3.1.2 [t9.14.1] If under the assumptions of Theorem 3.1.1 the action of G on X is free, i.e. the morphism X × G → X × X is a monomorphism, then the morphism X → X/G is flat and X is a principal homogenious G-space over X/G.

Corollary 3.1.3 [c9.14.2] Let X be a smooth quasi-projective scheme over S with a free action of a finite flat group scheme G. Then X/G is smooth over S.

Proposition 3.1.4 [quasi] Let X be a quasi-projective G-scheme. Then X/G is quasi-projective.

Proof: The fact that the morphism X → X/G is finite and surjective implies easily that X/G is separated. Since any finite morphism is projective and any quasi-finite separated morphism is a composition of an open embedding and a finite morphism it is enough to construct a quasi-finite morphism X/G → Y such that Y is projective. Using the universal property of quotients and the fact that X → X/G is surjective we see that to prove the proposition it is sufficient to find a G-invariant quasi-finite morphism X → Y such that Y is projective. We may obviously assume that S is connected. Then G has a constant degree over S which we denote by d. Consider the scheme X × G over X with respect to the action morphism. It is flat over X and the standard construction (see e.g. [7, p.81]) shows that we have a canonical morphism from X to Symm^{d}(X × G). This morphism is invariant under automorphisms of X × G over X and commutes, in the obvious sense, with the flat pull-backs. These two facts imply that the composition

[m9.22.2]X → Symm^{d}(X × G) → Symm^{d}(X)  (8)

of our morphism with the morphism Symm^{d}(X × G) → Symm^{d}(X) defined by the projection to X is G-invariant. Since G is finite the morphism (8) is finite. Let now X → P^{n} be a quasi-finite morphism which exists by our assumption that X is quasi-projective. Its d-th symmetric power is again quasi-finite (because morphisms of the form Y^{d} → Symm^{d}(Y) are finite by Theorem 3.1.1) and, composing it with (8), we get a G-invariant quasi-finite morphism from X to Symm^{d}(P^{n}). The statement of the proposition follows now from Lemma 3.1.5.

Lemma 3.1.5 [arepr] For any S, any d and any n the symmetric product Symm^{d}(P_{S}^{n}) is projective (over S).

Proof: ???

Proposition 3.1.4 implies that the correspondence X ↦ X/G can be considered as a functor from the category G−QP/S of quasi-projective G-schemes over S to the category QP/S of quasi-projective schemes over S.

Lemma 3.1.6 [optoop2] The functor X ↦ X/G takes open embeddings to open embeddings.

Proof: It follows from the fact that X/G is the equalizer of the projection X × G → X and the action X × G → X in in the category of ringed spaces.

Example 3.1.7 [et0] For an etale morphism U → X the morphism U/G → X/G need not be etale. Indeed let G = Z/2, X = A^{1} and U = A^{1} ∐ A^{1} such that G acts on X by z ↦ −z and on U by the composition of the sign map on each component with the permutation of the components. Then U/G = A^{1}, X/G = A^{1} and the map U/G → X/G is z ↦ z^{2}.

Lemma 3.1.8 [aftoaf2] The functor X ↦ X/G takes affine schemes to affine schemes and quasi-affine schemes to quasi-affine schemes.

Proof: The affine case is shown in [2, Th. V.4.1(ii)]. The quasi-affine case follows from the affine one and Lemma 3.1.6.

Lemma 3.1.9 [proptoprop2] The functor X ↦ X/G takes proper morphisms to proper morphisms.

Proof: Since X → X/G is proper and surjective and X is separated X/G is separated. Since any morphism of separated schemes is separated it remains to show that for a universally closed f : X → Y the morphism X/G → Y/G is universally closed. It follows easily from the fact that X → X/G is universally surjective and Y → Y/G is universally closed.

Example 3.1.10 [clnot]For a closed embedding X → Y the morphism X/G → Y/G need not be a closed embedding. Let S = Spec(k) where k is a field of characteristic 2 and G = Z/2. Let X = A^{1} with the trivial Z/2-action, Y = X^{2} with the permutation action and Y → X is the diagonal embedding. Then the morphism X = X/G → X^{2}/G factors through the Frobenious morphism z ↦ z^{2} and in particular is not a closed embedding.

Lemma 3.1.11 [clto1] Let X → Y be a closed embedding of schemes with free G-actions. Then X/G → Y/G is a closed embedding.

Proof: Let X′ be the fiber product (X/G) ×_{Y/G} Y. It is a G-scheme and we have a G-equivariant morphism over X/G of the form X → X′. Since the action of G on X is free Theorem 3.1.2 implies that X is a principal homogenious G-space over X/G. Since the action on Y is free the same is true for X′ over X. Since a G-equivariant morphism of principal homogenious spaces is an isomorphism we conclude that X = X′ and, therefore, the the morphism (X/G) ×_{Y/G} Y → Y is a closed embedding. Using again Theorem 3.1.2 we see that the morphism Y → Y/G is faithfully flat. We conclude that X/G → Y/G is a closed embedding by flat descent ([4, Th. 2.7.1(xii)]).

Lemma 3.1.12 [clto2] Let G be an etale group scheme over S of order prime to the characteristic of S and X → Y a closed embedding of Gschemes. Then X/G → Y/G is a closed embedding.

Proof: Replacing S by Y/G we may assume that S = Y/G. Since quotients commute with flat base change (by Theorem 3.1.1) it is sufficient to consider the case when S is a strictly henselian local scheme. Since Y is finite over S it is affine and the same holds for X and X/G. It remains to check that the surjection O(Y) → O(X) defines a surjection on the subrings of G-invariants. An elale group scheme over a strictly henselian S is just a discreet group and our result follows from our assumption that |G| is invertible in O(X) through the use of the averaging map x ↦ (∑_{g∈G} x^{g})/|G|.

Recall, that a group scheme G is called diagonalizable if it represents a functor of the form S ↦ Hom_{S}(A, G_{m}) where A is a finitely generated abelian group. If A is finite then the corresponding functor is reprersentable by a finite flat group scheme. In particular, the group μ_{n} corresponds in this way to the cyclic group Z/n. A group G over S is called a group of multiplicative type if there exists a flat covering {S_{i} → S} of S such that each group G_{i} = G×_{S}S_{i} is diagonalizable.

Lemma 3.1.13 [clto3] Let G be a finite group scheme over S of multiplicative type and X → Y a closed embedding of G-schemes. Then X/G → Y/G is a closed embedding.

Proof: ???

Lemma 3.1.14 [very] For any X and G as above and any equivariant open subscheme A in X the square

[qg1]\begin{matrix}A & ⟶ & X \\ ↓ & & ↓ \\ A/G & ⟶ & X/G \end{matrix}  (9)

is a pull-back square.

Proof: Let A′ = (A/G) ×_{X/G} X. We have an open embedding A → A′ and A/G = A′/G. Since the morphisms A → A/G and A′ → A′/G are finite we conclude that A → A′ is a closed embedding. Then A′ = A ∐ A′′ and there exists a function f on A′ which is 0 on A and 1 on A′′. Since A is invariant in A′ this function is invariant and thus factors through A′ → A′/G. Since A → A′/G is surjective it implies that f = 0 i.e. A = A′.

Let us analyze now the exactness properties of the functor X ↦ X/G with respect to the standard cd-structures. We consider the upper cd-structure first. For an upper dsitingushed square Q the square Q/G does not have to be an upper dsitinguished square. Indeed let X = A ∐ X_{0} and Y = B ∐ X_{0} where B → A is isomorphic to the map U → X of Example 3.1.7. Then the map Y/G → X/G is not etale and in particular Q/G is not an upper distinguished square. Instead we have the following weaker result.

Proposition 3.1.15 [upqu] Let Q be an upper distinguished square of quasiprojective G-schemes of the form (7). Consider the square

[qg]Q/G = (\begin{matrix}B/G & ⟶ & Y/G \\ ↓ & & ↓ \\ A/G & ⟶ & X/G \end{matrix})  (10)

The the corresponding square ρ(Q/G) of the representable sheaves in the upper cd-topology on QP/S is push-forward.

Proof: By Lemma 3.1.6 the horizontal arrows in (10) are open embeddings. Let us show that this is a pull-back square i.e. that the open subsets (A/G) ×_{X/G} (Y/G) and B/G of Y/G coincide. The inclusion B/G ⊂ (A/G) ×_{X/G} (Y/G) follows from the commuuativity of the square (10). To check the opposite inclusion take a geometric point ỹ of Y/G whose image in X/G lies in A/G. Let y be its lifting to Y which exists since Y → Y/G is surjective. Then y lies in B since the composite square

B ⟶ Y
↓  ↓
A ⟶ X
↓  ↓
A/G ⟶ X/G

is a pull-back square by Lemma 3.1.14.

To prove the proposition it is sufficient now (cf. the proof of [10, Lemma 7.4]) to check that the morphisms

[firstm]A/G ∐ Y/G → X/G  (11)

and

[secondm]Y/G ∐(B/G ×_{A/G} B/G) → Y/G ×_{X} /GY/G  (12)

define epimorphisms of the representable sheaves in the upper cd-topology. [11, Lemma 2.11]implies that a morphism of finite type Z → W defines an epimorphism of the representable upper sheaves if and only if for any henselian local scheme S and any morphism s : S → W there exists a lifting of s to a morphism S → Z.

Let S be a henselian local scheme of a point x of X/G. The morphism

S_{Y} = S ×_{X/G} (Y/G) → S

is quasi-finite and thus S_{Y} is a disjoint union of the form S_{Y} = S_{Y,fin} ∐ S_{Y,0} where S_{Y,fin} is finite over S and the image of S_{Y,0} does not contain the closed point of S.

Lemma 3.1.16 [ll1] If x does not lie in A/G then the map S_{Y,fin} → S is an isomorphism.

Proof: Let

[qg2]\begin{matrix}S̃_{Y} & ⟶ & S̃ \\ ↓ & & ↓ \\ S_{Y} & ⟶ & S \end{matrix}  (13)

be the pull-back of the square

[qg3]\begin{matrix}Y & ⟶ & X \\ ↓ & & ↓ \\ Y/G & ⟶ & X/G \end{matrix}  (14)

along the morphism S → X. The right vertical arrow is a finite morphism and therefore S̃ is the disjoint union of a finite number of henselian local schemes. Let S̃_{Y,fin} be the union of the connected components of S̃_{Y} which are finite over S̃. Since the closed point of S lies outside of A/G the closed points of S̃ lie outside of A. Together with the fact that Q is an upper distinguished square this implies that the morphism s̃_{Y,fin} → S̃ is an isomorphism. On the other hand the fact that the vertical arrows in (13) are finite implies that the square

[qg4]\begin{matrix}S̃_{Y,fin} & ⟶ & S̃ \\ ↓ & & ↓ \\ S_{Y,fin} & ⟶ & S \end{matrix}  (15)

is pull-back. Thus S_{Y,fin} = S̃_{Y,fin}/G = S̃/G = S.

To prove that (11) defines an epimorphism let S_{0} be a henselian local scheme and x : S_{0} → X/G be a morphism. If the image of the closed point of S_{0} lies in A/G then x lifts to A/G since A/G → X/G is an open embedding. If the image of the closed point of S_{0} lies outside of A/G then x lifts to Y/G by Lemma 3.1.16.

To prove that (12) defines an epimorphism let S_{0} be a henselian local scheme and (y_{1}, y_{2}) : S_{0} → Y/G ×_{X/G} Y/G a morphism. Since Q/G is a pull-back square S_{0} factors through (B/G ×_{A/G} B/G) if and only if the corresponding morphism S_{0} → X/G factors through A/G. Thus we may assume that the image of the closed point of S_{0} lies outside of A/G. Then by Lemma 3.1.16 we have y_{1} = y_{2} i.e. (y_{1}, y_{2}) lifts to the diagonal.

Corollary 3.1.17 [upexact] Let {p_{i} : U_{i} → X} be an upper cd-covering of quasi-projective schemes. Then the family of morphisms {U_{i}/G → X/G} is an upper cd-covering.

Proof: The upper cd-structure is complete by [10, Lemma 4.3]. Thus {p_{i}} has a simple refiniment. It remains to show that for any simple covering {p_{i} : U_{i} → X} the family U_{i}/G → X/G is a covering. The class S of simple coverings for which it is true contains isomorphisms. Let us show that it satisfies the second condition of [10, Definition 3.1]. Let Q be an upper distinguished square in C of the form (7) and {p_{i} : Y_{i} → Y } and {q_{j} : A_{j} → A} be simple coverings in S. The families {Y_{i}/G → Y/G} and {A_{j}/G → A/G} are upper cd-coverings by the assumptions. The pair of morphisms {A/G → X/G, Y/G → X/G} is a covering by Proposition 3.1.15. Thus {A_{j}/G → X/G, Y_{i}/G → X/G} is a covering.

Corollary 3.1.18 [upcont] The functor X ↦ X/G defines a continuous map of sites QP/S → G − QP/S in the upper topologies.

Example 3.1.19 [et] The analog of Corollary 3.1.17 is false in the etale topology. For an etale covering U → X the morphism U/G → X/G need not be an etale covering. Indeed, in the situation of Example 3.1.19, the morphism U → X is an etale covering but the morphism U/G → X/G has no section over the strictly henselian local scheme of the point zero and thus is not an etale covering.

Consider now the lower cd-topology.

Proposition 3.1.20 [lowqu] Let Q be a lower distinguished square of quasiprojective G-schemes of the form (7) such that the morphisms A/G → X/G and B/G → Y/G are closed embeddings. Consider the square

[qglow]Q/G = (\begin{matrix}B/G & ⟶ & Y/G \\ ↓ & & ↓ \\ A/G & ⟶ & X/G \end{matrix})  (16)

Then the corresponding square ρ(Q/G) of the representable sheaves in the lower cd-topology on QP/S is push-forward.

Proof: Let Q′ be the pull-back square based on the morphisms A/G → X/G and Y/G → X/G. We have a morphism of squares Q/G → Q′ which is an isomorphism on all verices but, may be, the upper left one where it is given by the morphism B/G → (A/G) ×_{X/G} (Y/G). Our conditions imply that this is a closed embedding and one verifies easily that it defines bijection on the sets of geometric points. Therefore, the morphism of reduced schemes defined by the morphism Q/G → Q′ is an isomorphism. Since the lower sheaf represented by a scheme is isomorphic to the lower sheaf represented by its maximal reduced subscheme we conclude that the square ρ(Q/G) is isomorphic to the square ρ(Q′). Since the lower cd-structure is regular the square of sheaves corresponding to a distinguished square is push-forward by [10, ]. It remains to check that Q′ is a lower distinguished square. This square is a pull-back square by definition. Its vertical sides are proper morphisms by Lemma 3.1.9. Its horizontal sides are closed embeddings by our assumption. Finally, the condition that the morphism Y/G → X/G is an isomorphism outside A/G follows from the fact that X/G − A/G = (X − A)/G.

Remark 3.1.21 If, under the assumptions of Proposition 3.1.20, the square Q/G is a pull-back square then it is a lower distinguished square. I do not know of any example of a lower distinguished square of G-schemes satisfying the conditions of Proposition 3.1.20 such that Q/G is not a pull-back square.

Corollary 3.1.22 [lowexact] Let C be a full subcategory of the category of quasi-projective G-schemes such that the restriction of the lower cd-structure to C is complete and for any closed embedding Z → X the morphism Z/G → X/G is a closed embedding. Then for any lower cd-covering {U_{i} → X} the family {U_{i}/G → X/G} is a lower cd-covering.

Corollary 3.1.23 [lowcont] Let C be a as in Corollary 3.1.22. Then functor X ↦ X/G defines a continuous map of sites QP/S → C in the lower topologies.

The computations in the following example are due to P. Deligne.

Example 3.1.24 [m16ex2] Let X be a smooth surface over a field of characteristic 2. Consider X^{2} as a Z/2-scheme with respect to the permutation action. Let X_{Δ}^{2} be the blow-up of the diagonal in X^{2}. One can easy see that there is a unique action of Z/2 on X_{Δ}^{2} such that the projection X_{Δ}^{2} → X^{2} is equivariant. The fiber square Q

[impsq]\begin{matrix}p^{−1}(X) & ⟶ & X_{Δ}^{2} \\ ↓ & & ↓^{p} \\ X & ⟶^{Δ} & X^{2} \end{matrix}  (17)

is a lower distinguised square in Z/2 − QP/Spec(k). Consider the square Q/Z/2 of quotient schemes. Then it is not a lower distinguished square of schemes. Moreover, the corresponding morphism

[notcover](X ∐ X_{Δ}^{2})/G = X/G ∐ X_{Δ}^{2}/G → X^{2}/G  (18)

is not a lower covering. Indeed, for any lower covering f : Z_{1} → Z_{2} and any point z of Z_{2} the fiber of f over z has a rational point over the residue field k_{z} of z. Let Y be the reduced closed subscheme of X^{2}/G corresponding to the image of the diagonal in X^{2} and let y be its generic point. Then the fiber of (18) over y is a scheme which has no rational points over the residue field of y (in fact over the separable closure of the residue field) which proves that (18) is not a lower covering.

Lemma 3.1.25 [m25l2] The functor X ↦ X/G takes A^{1}-homotopic morphisms to A^{1}-homotopic morphisms.

Proof: Since the quotients commute with flat base change the obvious morphism

[m25eq1](X × A^{1})/G → X/G × A^{1}  (19)

is an isomorphism. If h : X × A^{1} → Y is an A^{1}-homotopy connecting morphisms f and g then the composition

X/G × A^{1} → (X × A^{1})/G → Y/G

where the first arrow is the inverse to (19), is a homotopy connecting f/G and g/G.

4 Smash powers

Let C be a site and Shv_{•}(C) the category of pointed sheaves on C. Let F be a pointed sheaf on C and R a sheaf without a base point. We define the sheaf F^{∧R} as follows. Let F^{R} be the internal Hom-object Hom(R, F). Let F_{R} be the subsheaf in F^{R} such that a morphism G → F^{R} lands in F_{R} if and only if for any non-empty G_{0} and a morphism G_{0} → G the pull-back of the distinguished point pt → F along the adjoint morphism G_{0} × R → F is not empty. Intuitively, F^{R} consists of families of morphisms from R to F and F_{R} consists of families such that the image of each morphism in the family contains the distinguished point. The following two lemmas provide two main examples of the functor (−)^{∧R}.

Lemma 4.0.26 [l9.25.1] Let R = ∐_{n} pt, then F^{∧R} = ∧_{n}F.

Recall that for an unpointed sheaf F one denotes by F_{+} the sheaf F ∐ pt pointed by the inclusion of pt.

Lemma 4.0.27 [l9.25.2] One has (F_{+})^{∧R} = (F^{R})_{+}.

The goal of this section is to prove basic properties of the functor (−)^{∧R} in the case when C is the category of quasi-projective G-schemes over S (where G is a finite flat group scheme) with one of the standard cd-topologies and R = ρ(W) is the sheaf represented by a finite flat G-scheme W. In each subsection we first investigate the functor F ↦ F^{ρ(W)} on free (unpointed) sheaves and then reformulate our results in terms of the functor (−)^{∧W}.

We say below that a morphism is upper solid if it is solid (in the sense of Definition 6.0.6) with respect to the class of open embeddings and that it is lower solid if it is solid with respect to the class of closed embeddings. We say that a morphism is open if it is locally solid with respect to open embeddings (see Definition 6.0.8) and that it is closed if it is locally solid with respect to closed embeddings. Lemma 2.0.10 (resp. Lemma 2.0.11) and Lemma 6.0.18 imply that a morphism of sheaves f : F → F′ is open (resp. closed) if and only if for any morphism of the form ρ(X) → F the projection ρ(X) ×_{F} F′ → ρ(X) is representable by an open (resp. closed) embedding of schemes.

Example 4.0.28 [notsolid] One can easily show that solid sheaves (with respect to open embeddings, closed embeddings or both) in upper topology on Sch/S are etale sheaves. In particular, any upper sheaf which is not a sheaf in the etale topology provides an example of a sheaf wich is not solid. The standard example of such a sheaf is the image of a morphism of the form ρ(p) : ρ(Y) → ρ(X) where p is an etale covering which is not an upper covering.

4.1 Geometric lemmas

Lemma 4.1.1 [m151] Let W be a finite flat scheme over S. Then for any quasi-projective scheme X over S the functor

[m15eq2]S′ ↦ Hom(S′ ×_{S} W, X)  (20)

on the category of schemes over S is representable by a quasi-projective scheme X^{W}.

Proof: Let d be the degree of W over S (the maximum of the degrees in all points of S) and Hilb_{0,≤d}(X ×_{S} W) the Hilbert scheme of subschemes of dimension zero and degree ≤ d on X ×_{S} W. Let Z be the canonical closed subscheme in Hilb_{0,≤d}(X × W) × X × W. Consider the projection

Z → Hilb_{0,≤d}(X × W) × W  (21)

as a morphism over Hilb_{0,≤d}. Both schemes are finite and flat and therefore there is an open subset U of Hilb_{0,≤d} such that the pull-back of (21) with respect to a map Y → Hilb_{0,≤d} is an isomorphism if and only if this map factors through U → Hilb_{0,≤d}. One can easily see now that U represents the functor (20).

Let now G be a finite flat group scheme over S, W a finite flat scheme over S with a G-action and X a quasi-projective G-scheme. The categorical translation of the usual formula defining an action of a group H on the set of maps between two H-sets gives an action of G on the scheme X^{W} and one verifies immediately that the corresponding G-scheme represents the functor Y ↦ Hom(Y ×W, X) from G-schemes to sets. In this section we prove some simple geometric properties of the functor X ↦ X^{W}. To avaoid unnecessary words we work with schemes without a G-action. Since the notions of etale, proper etc. morphisms of G-schemes are defined through the forgetfull functor the corresponding G-equivariant results follow immediately from their non equivariant counterparts.

Lemma 4.1.2 [mon] The functor X ↦ X^{W} takes S to S, monomorphisms to monomorphisms and fiber products to fiber products.

Proof: Follows from the fact that it is the right adjoint of the functor X ↦ X × W.

Lemma 4.1.3 [ettoet] Let f : X̃ → X be an etale (resp. smooth) morphism of quasi-projective schemes. Then f^{W} : X̃^{W} → X^{W} is an etale (resp. smooth) morphism.

Proof: It follows immediately from the fact that a morphism g : X_{1} → X_{2} of finite type is etale (resp. smooth) if and only if for any morphism Z → X_{2} and any closed embedding Z_{0} → Z defined by a nilpotent ideal the map

Hom_{X_{2}}(Z, X_{1}) → Hom_{X_{2}}(Z_{0}, X_{1})

is bijective (resp. surjective). See [, ].

Since open embeddings are etale monomorphisms and Lemma 4.1.3 and Lemma 4.1.2 imply the following result.

Lemma 4.1.4 [optoop] The functor X ↦ X^{W} takes open embeddings to open embeddings.

Lemma 4.1.5 [qfin] Let W be a finite etale scheme. Then the functor X ↦ X^{W} takes quasi-finite morphisms to quasi-finite morphisms.

Proof: A morphism of finite type is quasi-finite if and only if its fibers are of dimension zero. To check that the fibers of X^{W} → Y^{W} are of dimension zero it is sufficient to do so over the strictly henselina local scheme of each point of S. Therefore we may assume that S is strictly henselian. In this case W is a disjoint union of finitely many copies of S and our functor takes X to X^{d} where d = deg(W/S). The result follows now by induction on d from the fact that the pull-back of a quasi-finite morphism is quasi-finite and composition of quasi-finite morphisms is quasi-finite.

Example 4.1.6 [prop2] Let Spec(L) → Spec(k) be the morphism corresponding to a field extension and W = Spec(k[ϵ]/(ϵ^{2})) be the spectrum of the ring of dual numbers. Then Spec(L)^{W} is an affine space of Spec(L) which is quasi-finite over k if and only if L/k is a separable extension.

Lemma 4.1.7 [m15l2] Let X̃ → X be a proper (resp. finite) morphism of quasi-projective schemes over S and W a finite etale scheme over S. Then the morphism X̃^{W} → X^{W} is proper (resp. finite).

Proof: One of the versions of the valuative criterion of properness states that a morphism f : X → Y of finite type is proper if and only if for any S′ over Y such that S′ is the spectrum of a Noetherain regular ring of dimension one, a dense open subscheme U in S′ and a morphism s : U → X over Y, there exist a unique extension of s to a morphism S′ → X over Y. For any such S′ and U and any W which is finite and etale over S the scheme S′ ×_{S} W is again regular and of dimension one and the open subscheme U ×_{S} W is dense. This immediately implies the statement of the lemma for proper morphisms. The finite case follows from the proper case and Lemma 4.1.5.

Example 4.1.6 and the example given below show that in general the functor X ↦ X^{W} does not take proper morphisms to proper morphisms.

Example 4.1.8 [prop1] Let S be the affine line and W be the union of two copies of S intersecting at the point zero and projecting to S in the obvious way. Let X = S ∐ S. Then X^{W} is the union of X and two copies of S −{0}.

Lemma 4.1.9 [clemb] Let W be finite flat scheme. The the functor X^{W} takes closed embeddings to closed embeddings.

Proof: Let Z → X be a closed embedding, X = ∪U_{i} an affine open covering of X and Z_{i} = U_{i} ∩ Z. Then (∐ U_{i})^{W} → X^{W} is an etale surjective morphism and the square

\begin{matrix}(∐ Z_{i})^{W} & ⟶ & (∐ U_{i})^{W} \\ ↓ & & ↓ \\ Z^{W} & ⟶ & X^{W} \end{matrix}  (22)

is pull-back. If we can show that the upper horizontal arrow is a closed embedding then we conclude that the lower horizontal one is a closed embedding ([, ]). Therefore we may assume that X is affine. The functor X^{W} takes monomorphisms to monomorphisms and every proper monomorphism is a closed embedding. It remains to show that X^{W} → Z^{W} is a proper morphism i.e. that for the spectrum of a discrete valuation ring S′ with the generic point η : Spec(k) → S, a morphism S′ → Z^{W} and a morphism Spec(k) → Z^{W} such that the obvious square commutes there is a unique morphism S′ → Z^{W} which makes the obvious two triangles commutative. By definition of X^{W} this means that for any morphism S′ ×_{S} W → X and a lifting Spec(k) ×_{S} W → Z there exists a unique morphism S′ ×_{S} W → Z satisfying the obvious conditions. The uniqueness follows from the fact that Z → X is a monomorphism. To prove the existence consider the corresponding diagram of rings

\begin{aligned}O(Spec(k) ×_{S} W) & ⟵ & O(Z) \\ ↑ & & ↑ \\ O(S′ ×_{S} W) & ⟵ & O(X) \end{aligned}  (23)

From this diagram it is clear that it is enough to check that the left vertical arrow is a monomorphism. This follows from the fact that O(S′) → k is a monomorphism (since S is intergal) and the condition that W is flat.

Lemma 4.1.10 [m29l1] Let W be a finite flat scheme and X an affine (resp. quasi-affine) scheme. Then X^{W} is affine (resp. quasi-affine).

Proof: By Lemma 4.1.9 the functor X^{W} takes closed embeddings to closed embeddings. Therefore for an affine X the scheme X^{W} is a closed subscheme of (A^{n})^{W} = ((A^{1})^{W})^{n} for some n. A simple compuuation shows that (A^{1})^{W} = Spec(Symm_{O(S)}(O(W))) where Symm_{O(S)}(O(W)) is the symmetric algebra of the flat module O(W). In particular it is affine over S. The quasi-affine case follows now from Lemma 4.1.4.

Lemma 4.1.11 [fltofl] Let W be a finite etale scheme. Then the functor X ↦ X^{W} takes flat morphisms to flat morphisms.

Proof: ???

Example 4.1.12 [fltoflnot] The statement of Lemma 4.1.11 is false for general W. Indeed, consider the case when W is the spectrum of the ring of dual numbers such that X^{W} is the tangent space of X. If X = Y = A^{1} and f : X → Y is the morphism z ↦ z^{2} then the corresponding morphism of the tangent spaces is not locally equidimensional and in particular is not flat.

4.2 Smash powers in the upper topologies

In this section we consider the functors F^{ρ(W)} and F^{∧ρ(W)} on sheaves in the upper topologies on the category of G-schemes. We always assume that W is a finite flat G-scheme over S and usually denote ρ(W) simply by W. Our goal is to prove two main results. One is that the functor (−)^{W} takes local weak equivalences of simplicial presheaves with respect to the upper topology to local weak equivalences (Proposition 4.2.9). Another one is that it takes upper solid morphisms of upper solid sheaves to upper solid morphisms of upper solid sheaves (Proposition 4.2.15 and Corollary 4.2.17).

We will use the following geometric lemmas. Let p : T → S be a finite flat morphism of schemes and U an open subset in T. Denote by p^{(i)}(U) the set of points s of S such that

dim_{k_{s}}(O(U ×_{S} Spec(k_{s}))) ≥ i

Lemma 4.2.1 [finmor1] The subsets p^{(i)}(U) are open.

Proof: It folows from [, ].

Lemma 4.2.2 [finmor2] If there exists an integer i such that for every point s of S one has dim_{k_{s}}(O(U ×_{S} Spec(k_{s}))) = i, then U is a union of connected components of T.

Proof: It is sufficient to show that under the assumption of the lemma U is closed. Since it is constructible it is sufficient to show that for any henselian local scheme S′ and a morphism S′ → S the subscheme U′ = U ×_{S} S′ is closed in T′ = T ×_{S} S′. Let g be the generic point of S′ and c the closed point. By our assumption we have

dim_{k_{g}}(O(U′ ×_{S′} Spec(k_{g}))) = dim_{k_{c}}(O(U′ ×_{S′} Spec(k_{c}))) = i

The scheme T′ is finite over S′ and therefore it is a disjoint union of connected components which are henselian local schemes. Let T_{0} be the union of components which are contained in U and T_{1} the union of the rest of the components. Since T′ is flat over S′ all its components are flat over S′ and we have

dim_{k_{g}}(O(T_{0} ×_{S′} Spec(k_{g}))) = dim_{k_{c}}(O(T_{0} ×_{S′} Spec(k_{c})))

and since U = T_{0} ∐(U ×_{S} T_{1}) we have

dim_{k_{g}}(O(U ×_{S′} T_{1} ×_{S′} Spec(k_{g}))) = dim_{k_{c}}(O(U ×_{S′} T_{1} ×_{S′} Spec(k_{c})))

By construction the closed points of T_{1} do not lie in U which implies that the right hand side is zero. Thus the left hand side is zero i.e. U ×_{S′} T_{1} ×_{S′} Spec(k_{g}) = ∅ and since T_{1} ×_{S′} Spec(k_{g}) is dense in T_{1} we conclude that U ∩ T_{1} = ∅ i.e. U = T_{0}.

Let j : U → X be an open embedding of quasi-projective G-schemes. Consider the evaluation morphism ev : X^{W} × W → X and define U_{i,X}^{W} as the open subset pr^{(i)}(ev^{−1}(U)) in X^{W} where pr : X^{W} × W → X^{W} is the projection. One verifies immediately that a morphism Y → X^{W} factors through the embedding U_{i,X}^{W} ⊂ X^{W} if and only if the preimage of U with respect to the corresponding morphism Y × W → X is of degree at least i over all points of Y. If S is connected then W has a constant degree d over S and the open subschemes U_{i,X}^{W} define a filtration on X^{W} of the form

∅ = U_{d+1,X}^{W} ⊂ U^{W} = U_{d,X}^{W} ⊂ … ⊂ U_{0,X}^{W} = X^{W}

Lemma 4.2.3 [mainupge] Consider an upper distinguished square of the form

[eq9.29.2]\begin{matrix}B & ⟶ & Y \\ ↓ & & ↓^{p} \\ A & ⟶^{j} & X \end{matrix}  (24)

Then for any i ≥ 0 the following square is upper distinguished

[eq9.29.1]\begin{matrix}A_{i+1,X}^{W} ×_{A_{i,X}^{W}} A_{i,Y ∐ A}^{W} & ⟶ & A_{i,Y ∐ A}^{W} \\ ↓ & & ↓ \\ A_{i+1,X}^{W} & ⟶ & A_{i,X}^{W} \end{matrix}  (25)

Proof: The right vertical arrow of (25) is the pull-back to A_{i,X}^{W} of the composition of the open embedding A_{i,Y ∐ A}^{W} → (Y ∐ A)^{W} and the morphism (Y ∐ A)^{W} → X^{W}. Therefore, it is etale by Lemma 4.1.3. It remains to show that any morphism f̃: Z → A_{i,X}^{W} whose image does not intersect A_{i+1,X}^{W} has a unique lifting to a morphism Z → A_{i,Y ∐ A}^{W}.

The morphism f̃ corresponds to a morphism f : Z × W → X such that the degree of f^{−1}(A) over each point of Z is exactly i. Therefore, by Lemma 4.2.2, f^{−1}(A) is a union of connected components of Z × W. Let Z_{0} be the complement to f^{−1}(A). Since the image of Z_{0} in X does not intersect A and the square (24) is distinguished there exists a unique lifting of the morphism Z_{0} → X to a morphism Z_{0} → Y. Let g : Z × W → Y ∐ A be the coproduct of this lifting with the restriction of f to f^{−1}(A). The adjoint morphism g̃ : Z → (Y ∐ A)^{W} clearly lands in A_{i,Y ∐ A}^{W} and provides the required lifting of f̃.

Let us show that g̃ is unique. Since the image of Z_{0} in X does not intersect A any lifting of f to a morphism Z × W → Y ∐ A must take Z_{0} to Y. Since (24) is distinguished this determines g on Z_{0} uniquely. On the other hand the restriction of g to f^{−1}(A) must land in A → Y ∐ A in order for g̃ to factor through A_{i,Y ∐ A}^{W}. Together with the fact that A → X is a monomorphism this determines g on f^{−1}(A).

Proposition 4.2.4 [takestocov] Let f : X̃ → X be an upper cd-covering of quasi-projective G-schemes. Then f^{W} : X̃^{W} → X^{W} is an upper cd-covering of G-schemes.

Proof: Since the upper cd-structure is complete f has a simple refiniment. Therefore, the standard induction argument shows that it is sufficient to verify that for an upper distinguished square of the form (24) the morphism q : (Y ∐ A)^{W} → X^{W} is an upper covering. We may clearly assume that S is connected and W has a constant degree d over S. Since A_{0,X}^{W} = X^{W} it is sufficient to show, by decreasing induction on i, that the projection q_{i} : A_{i,X}^{W} ×_{X^{W}} (Y ∐ A)^{W} → A_{i,X}^{W} is an upper covering. For i > d both sides are empty and therefore our statement holds. Assume that q_{i+1} is a covering. Consider the upper distinguished square (25). Since coverings are transitive and the projection q_{i+1} is a covering it is enough to check that the projection A_{i,Y ∐ A}^{W} ×_{X^{W}} (Y ∐ A)^{W} → A_{i,Y ∐ A}^{W} is a covering. This is true because it has a section.

Example 4.2.5 The analog of Proposition 4.2.4 for the plain upper topology is false. Let W = S ∐ S and G = Z/2. Then X^{W} = X^{2} with the permutation action of Z/2. Let X = U_{1} ∪ U_{2} be a covering of X by two open subsets such that f : U_{1} ∐ U_{2} → X is a plain upper covering. Consider f^{2} : (U_{1} ∐ U_{2})^{2} → X^{2}. For f^{2} to be an upper covering in the category of schemes with Z/2action there should exists a collection of invariant open subsets V_{i} of X^{2} such that X^{2} = ∪V_{i} and f^{2} has equivariant sections over each V_{i}. Let z_{1} be a point of X − U_{1} and z_{2} a point of X − U_{2} and V an invariant open neighborhood of (z_{1}, z_{2}). Assume that X^{2} is irreducible. Then V is connected and the section of f^{2} over V must land in one of the connected components of (U_{1} ∐ U_{2})^{2}. But one verifies easily that neither one of the components maps surjectively to V which implies that no such section exists.

Remark 4.2.6 [notamor] Proposition 4.2.4 does not imply that the functor X ↦ X^{W} defines a continuous map of sites QP/S → G − QP/S in the upper topology. The usual definition of a continous map (see [1]) applied in our situation requires that for a sheaf F the presheaf F′ of the form F′(X) = F (X^{W}) should be a sheaf. To see that it is not so consider the case G = 1, W = S ∐ S such that X^{W} = X^{2}. Let X = X_{0} ∐ X_{1} where X_{0} and X_{1} are connected and let F be the constant sheaf Z. For F′ to be a sheaf the morphism F′(X) → F′(X_{0}) × F′(X_{1}) should be an isomorphism. However, we have

F′(X) = F (X^{2}) = F (X_{0}^{2} ∐ X_{1}^{2} ∐ X_{0} × X_{1} ∐ X_{1} × X_{0}) = Z^{4}

and F′(X_{0}) × F′(X_{1}) = Z^{2}. One can also see that the usual sufficient condition of continuity saying that the functor in question takes coverings to coverings does not hold in our case unless we consider only covering which consist of one morphism.

Corollary 4.2.7 [takestocov1] Let p : W → S be a finite flat morphism of G-schemes and W = {W_{i} → W } an upper covering. Then there exists an upper covering U = {U_{j} → S} such that p^{∗}(U) is a refiniment of W.

Corollary 4.2.8 [takestocov2] Let f : F → F′ be a morphism of presheaves such that the associated morphism of (upper) sheaves is an epimorphism. Then the morphism associated with f^{W} is an epimorphism. In particular, the functor F ↦ F^{W} takes epimorphisms of (upper) sheaves to epimorphisms.

Proposition 4.2.9 [locweakup] The functor F^{W} takes local weak equivalences of simplicial presheaves with respect to the upper topology to local weak equivalences.

Proof: In the following proof we identify schemes with the corresponding representable presheaves. Let f : X → Y be a local weak equivalence in the upper topology. Since F^{W} clearly preserves projective weak equivalences we may replace X and Y by their standard simplicial resolutions and assume that the terms of X and Y are coproducts of representable presheaves. Then, by Proposition 2.0.1 and [10] the morphism f belongs to clΔ̄(η(Cov_{up})) where η(Cov_{up}) is the class of morphisms of the form η(p) : Č(p) → U where p : U′ → U runs through all upper coverings of G-schemes. Since (−)^{W} commutes with filtered colimits and the class of local weak equivalences is Δ̄-closed and closed under filtered colimits it is enough to show that for an upper covering p the morphism η(p)^{W} is a local weak equivalence. Since (−)^{W} commutes with fiber products we have η(p)^{W} = η(p^{W}). It remains to use Corollary 4.2.8 and the fact that for a morphism p, such that the morphism of associated sheaves is an epimorphism, the morphism η(p) is a local equivalence.

Corollary 4.2.10 [ashf] Let a_{up} be the associated sheaf functor for the upper topology. Then (a_{up}F)^{W} = a_{up}F^{W}.

Proof: The morphism F → a_{up}F is a local equivalence in the upper topology. Therefore, F^{W} → (a_{up}F)^{W} is an upper equivalence. Since the functor (−)^{W} takes sheaves to sheaves the right hand side is a sheaf and we conclude that a_{up}F^{W} = (a_{up}F)^{W}.

Let F be a presheaf and A → F an open morphism. For an integer n ≥ 0 denote by A_{n,F}^{W} the subpresheaf in F^{W} defined as follows. For an element f in F^{W} (X) = F (X × W), i.e. for a morphism ρ(X) → F^{W}, let U_{f} be the open subset such that ρ(U) → ρ(X × W) is the pull-back of F′ to ρ(X × W). We say that f belongs to A_{n,F}^{W} if the degree of U_{f} over all points of X is at least n. If S is connected and deg(W/S) = d the for n > d one has A_{n,F}^{W} = ∅, for n = d one has A_{n,F}^{W} = A^{W} and for n ≤ 0 one has A_{n,F}^{W} = F^{W}. Note that if F is an upper sheaf and A is a subsheaf in F then A_{i,F}^{W} is a sheaf for any i.

Lemma 4.2.11 [partop] For every i, the embedding A_{i,F}^{W} → A_{i−1,F}^{W} is open.

Proof: We have to show that given an open subset U of X × W and an integer n there exists an open subset V_{n} in X such that a morphism Y → X factors through U if and only if the degree of U ×_{X} Y over Y is greater or equal to i in all points of Y. This follows immediately from Lemma 4.2.1.

Since the morphism A^{W} → F^{W} is a composition of morphisms of the form A_{n,F}^{W} → A_{n−1,F}^{W} and a composition of open morphisms is open we get the following result.

Lemma 4.2.12 [lop] The functor (−)^{W} takes open morphisms to open morphisms.

Proposition 4.2.13 [p9.26.1] Consider a push-forward square of the form

[eq9.26.2]\begin{matrix}B & ⟶^{j} & G \\ ↓ & & ↓ \\ A & ⟶ & F \end{matrix}  (26)

where j is an open embedding. Then for any n ≥ 0 the following square is push-forward

[eq9.26.3]\begin{matrix}A_{n,F}^{W} ×_{A_{n−1,F}^{W}} A_{n−1,A ∐ G}^{W} & ⟶ & A_{n−1,A ∐ G}^{W} \\ ↓ & & ↓ \\ A_{n,F}^{W} & ⟶ & A_{n−1,F}^{W} \end{matrix}  (27)

Proof: Let us start with the following general lemma.

Lemma 4.2.14 [ech] Consider a pull-back square of sheaves of the form

\begin{matrix}X_{1} & ⟶ & X_{2} \\ ↓ & & ↓ \\ X_{3} & ⟶ & X_{4} \end{matrix}

such that its horizontal arrows are monomorphisms. Then it is a pushforward square if and only if the morphisms

[mor1]X_{2} ∐ X_{3} → X_{4}  (28)

and

[mor2](X_{1} ×_{X_{3}} X_{1}) ∐ X_{2} → X_{2} ×_{X_{3}} X_{2}  (29)

are epimorphisms.

Proof: The statement can be easily verified in the category of sets. This implies that it holds for presheaves. Since the associated sheaf functor commutes with both finite limits and colimits the result for presheaves implies the result for sheaves.

To verify that (27) is a push-forward square we will check that morphisms of the form (28) and (29) of Lemma 4.2.14 corresponding to this square are epimorphisms. Consider (29) first. Let Y be a G-scheme and f̃, g̃ be a pair of sections of A_{n−1,A ∐ G}^{W} over Y whose projections to A_{n−1,F}^{W} coincide. Sections f̃ and g̃ are given by sections f, g of A ∐ G over Y × W. Let U_{AA} be the union of components of Y × W on which both f and g land in A, U_{AX} the union of components of Y × W on which f lands in A and g lands in X etc. By our condition we have

deg(U_{AX}) + deg(U_{AA}) ≥ n − 1
deg(U_{XA}) + deg(U_{AA}) ≥ n − 1

if either U_{AX} or U_{XA} is non empty then deg(U_{AX})+deg(U_{XA})+deg(U_{AA}) ≥ n and therefore the image of f and g in F^{W} (Y) lies in A_{F,n}^{W}. Then the pair (f̃, g̃) lifts to the section of the first summand of the left hand side of (29). Assume that now that both U_{AX} and U_{XA} are empty. Since the morphism A → F is a monomorphism the restrictions of f ang g to U_{AA} coincide. Lemma 4.2.14 shows that the morphism G ∐(B ×_{A} B) → G ×_{F} G is surjective. Therefore, by Corollary 4.2.8, we can find a covering of Y of the form {p_{0} : Y_{0} → Y, p_{1} : Y_{1} → Y } such that the restrictions of f and g to (p_{0} ×Id_{W})^{−1}(U_{XX}) coincide and the restrictions of both f and g to (p_{1} × Id_{W})^{−1}(U_{XX}) lift to sections of B. Therefore, the pull-back of (f̃, g̃) to Y_{0} lifts to the second summand in the left hand side of (29) and the pull-back to Y_{1} lifts to the section of the first. We have shown that (29) is an epimorphism.

To prove the surjectivity of (28) in our case consider a section f̃ of A_{n−1,F}^{W} (Y) over Y, and let f be the corresponding element of F (Y × W). Since A ∐ G → F is an epimorphism Corollary 4.2.8 implies that, passing to a covering of Y, we may assume that f lifts to a section of A ∐ G. Let V be the open subscheme of Y × W where f belongs to A(Y × W). We get a commuuative square Q of morphisms of sheaves of the following form

ρ(V ∐ Y × W) ⟶ ρ(Y × W)
↓  ↓
A ∐ G  ⟶  F

and one sees immediately that we have a square of subsheaves in Q^{W} of the following form

ρ(V)_{n−1,ρ(V ∐ Y ×W)}^{W} ⟶ ρ(V)_{n−1,ρ(Y ×W)}^{W}
↓  ↓
A_{n−1,A ∐ G}^{W}  ⟶  A_{n−1,F}^{W}

This gives us a diagram of the form

[covdi]\begin{matrix}ρ(V)_{n,ρ(Y ×W)} & ⟶ & ρ(V)_{n−1,ρ(Y ×W)}^{W} & ⟵ & ρ(V)_{n−1,ρ(V ∐ Y ×W)}^{W} \\ ↓ & & ↓ & & ↓ \\ A_{n,F}^{W} & ⟶ & A_{n−1,F}^{W} & ⟵ & A_{n−1,A ∐ G}^{W} \end{matrix}

(30)

By construction, the morphism ρ(Y) → A_{n−1,F}^{W} corresponding to our section f̃ lifts to a morphism ρ(Y) → ρ(V)_{n−1,ρ(Y ×W)}^{W}. By Lemma 4.2.3 applied to the square

V ⟶ Y × W
↓  ↓
V ⟶ Y × W

the coproduct of the upper horizontal arrows of (30) is an epimorphism. Together with the commuuativity of (30) it implies immediately that there is a covering of Y such that the restriction of f̃ to this covering lifts to a section of A_{n,F}^{W} ∐ A_{n−1,A ∐ G}^{W}.

Proposition 4.2.15 [upsolid] Let t be a topology which contains the upper cd-stopology. Then the functor F ↦ F^{W} takes upper solid sheaves to upper solid sheaves.

Proof: Let X be a quasi-projective G-scheme. If \underline{X} is the presheaf represented by X then we know that \underline{X}^{W} is represented by a quasi-projective G-scheme X^{W} by Lemma 4.1.1. Since a presheaf represented by a scheme is an upper sheaf i.e. \underline{X} = ρ(X) we conclude that ρ(X)^{W} = ρ(X^{W}). Let now F be an upper solid sheaf. Using Lemma 6.0.7 and an obvious inductive argument we may assume that there is a push-forward square of the form (26) where j is representable by an open embedding of G-schemes and A is a sheaf such that (A ∐ ρ(X))^{W} is upper solid for any scheme X. Then, by Proposition 4.2.13, we get a sequence of push-forward squares of the form (27). The codomain of the upper horizontal arrow is an open subsheaf in (A ∐ G)^{W} by Lemma 4.2.11. It is upper solid by Lemma 6.0.22 and the inductive assumption. The lower horizontal arrow is open by Lemma 4.2.11. Therefore, the upper horizontal arrow is open (by Lemma 6.0.9). Therefore, the upper horizontal arrow is upper solid by Lemma 6.0.22 and we conclude that the lower horizontal arrow is upper solid.

Example 4.2.16 [smn] The functor (−)^{W} does not in general take upper solid morphisms to upper solid morphisms. Consider the case when W = pt ∐ pt. Then F^{W} = F^{2}. Let F be a sheaf which is not upper solid (see ??). The morphism F → F ∐ pt is upper solid for any F. Its square is the morphism

[eqnot]F^{2} → pt ∐ F ∐ F ∐ F^{2}  (31)

Since the embedding of a copy of F to the codomain of (31) is open, Lemma ?? implies that if (31) is solid then ∅ → F should be solid which is not the case by our choice of F.

Corollary 4.2.17 [solsol] The functor (−)^{W} takes upper solid morphisms of upper solid sheaves to upper solid morphisms.

Proof: By Lemma 2.0.10 and Lemma 6.0.20 an upper solid morphism is open. By Lemma 4.2.12 and Proposition 4.2.15 our functor takes an open morphism between solid objects to an open morphism between solid objects. By Lemma 2.0.10 and Lemma 6.0.22 any such morphism is solid.

Theorem 4.2.18 [cof] The functor (−)^{W} takes simplicial sheaves which are cofibrant in the Brown-Gersten closed model structure to simplicial sheaves which are cofibrant in the Brown-Gersten closed model structure.

Proof: ???

4.3 Smash powers in the upper topologies on smooth schemes

4.4 Smash powers in the lower topologies

Lemma 4.4.1 [lcl] The functor (−)^{W} takes closed morphisms to closed morphisms.

Proof: Using the same argument as in the proof of Lemma 4.4.1 one sees that it is enough to show that for a closed embedding i : Z → X × W the subpresheaf in the presheaf represented by X, which consists of morphisms Y → X such that Y × W → X × W factors through i, is representable by a closed embedding T → X. This is exactly the statement of [3, Theorem VIII.6.4 p.21]. ???? this lemma is only shown there in non equivariant case

The following example shows that the functors X ↦ X^{W} for finite flat W do not, in general, take lower cd-coverings to lower cd-coverings.

Example 4.4.2 [m15ex1]Let S = Spec(k) where k is a field, G = 1 and W = Speck[ϵ]/ϵ^{2} = 0. Then X^{W} is the “tangent space” of X. Consider the (plain) low cd-covering p : X ∐ Y → Z of the closed subscheme Z in A^{2} given by the equation xy = 0 by its irreducible components. Then Z^{W} is the closed subscheme of A^{3} = (A^{2})^{W} which is the union of three coordinate hyperplanes and the image of the map (X ∐ Y)^{W} → Z^{W} is the union of two of these hyperplanes.

Proposition 4.4.3 [m15p1] Let G be a finite flat group scheme, W a finite etale G-scheme and p : X̃ → X a lower cd-covering of quasi-projective schemes. Then X̃^{W} → X^{W} is a lower cd-covering of G-schemes.

Proof: By Lemma 4.1.7 X^{W} takes proper morphisms to proper morphisms. Since any lower cd-covering has a refiniment which consists of proper morphisms Proposition 2.0.7 implies that it is sufficient to show that for any lower cd-covering f : X̃ → X the morphism X̃^{W} → X has a splitting sequence. By induction we may assume that f has a splitting sequence of length one i.e. that there is an open subscheme U and a closed subscheme Z of X such that X = U ∪ Z and f has a section over U ∐ Z. Let U_{i} be the open subschemes of X^{W} defined by U as in the proof of Proposition 4.2.4. Let Z_{i} be the reduced closed subscheme of X^{W} corresponding to the closed subset X − U_{i}. Since W is finite and etale the automorphisms of W over S are representable by a finite etale group scheme Aut(W). The action of G on W factors through the action of Aut(W) on W and since Aut(W) is etale we conclude that there is an action of G on Z_{i} such that the embedding Z_{i} → X^{W} is equivariant. We can now finish the proof in the same way as for Proposition 4.2.4 with the only change that the reference to f being etale at the end is replaced by the refernce to Z_{i} being reduced.

Remark 4.4.4 [notamor2] Proposition 4.4.3 does not imply that for an etale W the functor X ↦ X^{W} defines a continuous map of sites in the lower topology. See Remark 4.2.6.

4.5 Smash powers in the cdh-topology

5 Twisted product functors

Let S be a Noetherian scheme of finite dimension, G a finite flat group scheme over S, W a finite flat G-scheme and E a quasi-projective G-scheme. Let further Sch/S be the category of schemes of finite type over S. Consider the functor T : QP/S → Sch/S given by the formula

T (X) = (X^{W} × E)/G

Functors of this form are called the twisted product functors. A twisted product functor can be written as the composition

QP/S →^{F_{1}} G − QP/S →^{F_{2}} G − QP/S →^{F_{3}} Sch/S

where one has

F_{1}(X) = X^{W}
F_{2}(X) = X × E
F_{3}(X) = X/G

Lemma 5.0.1 [l9.14.3] If E is smooth and the action of G on E is free then the twisted product functor defined by the data (G, W, E) takes smooth schemes to smooth schemes.

Proof: This is an immediate corollary of Lemma 4.1.3 and Corollary 3.1.3.

Lemma 5.0.2 [l9.14.4] Twisted product functors take open embeddings to open embeddings.

Proof: This is an immediate corollary of Lemmas 4.1.4 and 3.1.6.

Lemma 5.0.3 [l9.14.5] Let E be an affine (resp. a quasi-affine) G-scheme. Then the twisted product functors defined by (G, W, E) take affine schemes (resp. quasi-affine schemes) to affine schemes (resp. to quasi-affine schemes).

Proof: This is an immediate corollary of Lemmas 4.1.10 and 3.1.8.

Lemma 5.0.4 [cltocl] The twisted product functors defined by data (G, W, E) take closed embeddings to closed embeddings in the following cases:

1. when G is an etale group scheme of order prime to the characteristic of S

2. when G is a finite group scheme of multiplicative type

3. when the action of G on E is free

Proof: It follows from Lemma 4.1.9 and Lemma 3.1.12 (for the prime characteristic case) or Lemma 3.1.13 (for groups of multiplicative type) or Lemma 3.1.11 (for the free case).

Let G_{1}, W_{1}, E_{1} and G_{2}, W_{2}, E_{2} be two collections of objects defining twisted product functors T_{1} and T_{2}. We are going to show that under some additional assumptions functors of the form

X ↦ T_{1}(X) × T_{2}(X)

and

X ↦ T_{2}(T_{1}(X))

are again twisted product functors. Consider the product case first.

Proposition 5.0.5 [twprod] Let (G_{i}, W_{i}, E_{i}), i = 1, 2 be two sets of twisted product data and T_{1}, T_{2} the corresponding twisted product functors. Then the functor T_{1} ×T_{2} is isomorphic to the twisted product functor given by the data (G_{1} × G_{2}, W_{1} ∐ W_{2}, E_{1} × E_{2}).

Proof: ???

To describe the composition of twisted product functors we need to introduce first the following general construction. Let G_{1}, G_{2} be finite flat group schemes and let V be a finite etale scheme with G_{2}-action. Then, according to Lemma 4.1.7 and Lemma 4.1.11, G_{1}^{V} is a finite flat scheme with G_{2}-action. Since the functor X ↦ X^{V} commutes with fiber products the scheme G_{1}^{V} has a natural structure of a group scheme and the action of G_{2} is compatible with this structure. We can use this action to define a new group structure on the scheme G_{1}^{V} × G_{2} in the same way one defines semi-direct product structure for usual groups. We get a finite flat group scheme which we denote by G_{1}^{V} ⋊ G_{2}.

Proposition 5.0.6 [compositions] Let (G_{i}, W_{i}, E_{i}), i = 1, 2 be two set data defining twsited product functors T_{i} such that E_{i} are quasi-affine. Assume in addition that the following two conditions hold:

1. G_{1}-quotients are universal

2. W_{2} is etale

Then the composition T_{2} ∘ T_{1}, where we consider T_{i} as functors from the category of quasi-affine schemes of finite type over S to itself, is isomorphic to the twisted product functor T_{(G,W,E)} where:

G = G_{1}^{W_{2}} ⋊ G_{2}
W = W_{1} × W_{2}
E = E_{1}^{W_{2}} × E_{2}

Proof: We have

T_{2}(T_{1}(X)) = (((X^{W_{1}} × E_{1})/G_{1})^{W_{2}} × E_{2})/G_{2} =
= ((X^{W_{1}×W_{2}} × E_{1}^{W_{2}})/G_{1}^{W_{2}} × E_{2})/G_{2} =

= ((X^{W_{1}×W_{2}} × E_{1}^{W_{2}} × E_{2})/G_{1}^{W_{2}})/G_{2} =
= (X^{W_{1}×W_{2}} × E_{1}^{W_{2}} × E_{2})/(G_{1}^{W_{2}} ⋊ G_{2})

where the first isomorphism follows from Lemma 5.0.7, the second follows from our assumptions that G_{1}-quotients are universal and Lemma 5.0.8 and the third one directly from the categorical definition of quotients.

Lemma 5.0.7 [m26l1] Let G be a finite flat group scheme, X a quasiprojective G-scheme and V a finite etale scheme over S. Assume in addition that for any quasi-projective G-scheme Y over S the canonical morphism (Y × X/G)/G → Y/G × X/G is an isomorphism (e.g. X/G is flat over S or G-quotients are universal). Then the canonical morphism

[m26eq2]X^{V} /G^{V} → (X/G)^{V}  (32)

is an isomorphism.

Proof: To verify that (32) is an isomorphism it is sufficient to check that its pull-back to any strictly henselian local scheme S′ over S is an isomorphism. By Lemma 4.1.2 the functor X^{V} commutes with pull-backs and by Theorem 3.1.1 the same holds for the functors X/G and X/G^{V} and flat pull-backs. Therefore, our condition on G is also stable under flat pull-backs and we may assume that S is strictly henselian local. Then V is a disjoint union of finitely many copies of S and X^{V} = X^{n}. The fact that X^{n}/G^{n} → (X/G)^{n} is an isomorphism follows inductively from isomorphisms postulated by our condition on G starting with

X^{n}/G^{n} = ((X^{n−1} × X)/G)/G^{n−1} = (X^{n−1} × X/G)/G^{n−1}

Lemma 5.0.8 [m29l3] Let G be a finite flat group scheme and V a finite etale scheme over S. Assume in addition that G-quotients are universal. Then G^{V} -quotients are universal.

Proof: As in the proof of Lemma 5.0.7 we may reduce the problem to the case when the base scheme S is strictly henselian and therefore V is just a finite number of copies of S. In this case G^{V} = G^{n} and the statement follows by easy induction since X/G^{n} = (X/G^{n−1})/G.

Lemma 5.0.9 [m25l1] The functor X ↦ X^{W} takes A^{1}-homotopic morphisms to A^{1}-homotopic morphisms.

Proof: The morphism A^{1} → (A^{1})^{W} adjoint to the projection A^{1} ×W → A^{1} together with the natural isomorphism (X × A^{1})^{W} = X^{W} × (A^{1})^{W} defines a morphism

X^{W} × A^{1} → (X × A^{1})^{W}

. If h : X × A^{1} → Y is an A^{1}-homotopy connecting morphisms f and g then the composition

T (X) × A^{1} → T (X × A^{1}) →^{T (h)} T (Y)

is a homotopy connecting T (f) and T (g).

5.1 Twisted products of sheaves and presheaves

Any functor F from a category C to a category D defines a functor F^{∗} from the category of presheaves on C to the category of presheaves on D left adjoint to the obvious functor going in the opposite direction. In this section we will prove some properties of the functors

T^{∗} : P reShv(QP/S) → P reShv(Sch/S)

defined in this way by the twisted product functors. Since the twisted product functor is the composition of two functors X ↦ X^{W} and X ↦ (X × E)/G the corresponding functor T^{∗} is the composition of the functors

P_{W}^{∗} : P reShv(QP/S) → P reShv(G − QP/S)

and

Q_{G}^{∗} : P reShv(G − QP/S) → P reShv(QP/S)

defines by X ↦ X^{W} and X ↦ (X × E)/G respectively. We consider the first of these two functors first.

Lemma 5.1.1 [l9.15.1] Let F be a coproduct of representable presheaves. Then P_{Q}^{∗}(F) is a coproduct of representable presheaves.

Proof: ???

Definition 5.1.2 [d9.15.2] A presheaf F on Sm/S is called upper cellular if it is a relative Op-complex (in the sense of [5, Definition 2.1.9]) where Op is the set of open embeddings j : U → X of quasi-projective G-schemes. An upper sheaf is called upper cellular if it is the sheaf associated to an upper cellular presheaf.

Definition 5.1.3 [d9.15.3] A presheaf F on Sm/S is called lower cellular if it is a relative Cl-complex (in the sense of [5, Definition 2.1.9]) where Cl is the set of closed embeddings i : Z → X of quasi-projective G-schemes. A lower sheaf is called lower cellular if it is the sheaf associated to a lower cellular presheaf.

Lemma 5.1.4 [cellup] The functor P_{W} takes upper cellular presheaves to upper cellular presheaves.

Proof: ???

Lemma 5.1.5 [upquot] Let U → X be an open embedding. Then P_{Q}^{∗}(X/U)

Proof: trouble unless things are pointed.

Lemma 5.1.6 [celllow] If W is a finite etale scheme then the functor P_{W} takes lower cellular presheaves to lower cellular presheaves.

Proof: ???

Lemma 5.1.7 [istrans] The functor X ↦ X^{W} is transparent.

Proof: Our functor takes monomorphisms to monomorphisms and etale morphisms to etale morphisms by Lemma 4.1.3. Therefore, it takes open embeddings to open embeddings. On the other hand it takes closed embeddings to closed embeddings by Lemma 4.1.9. Since a morphism of quasi-projective schemes is a coprojection if and only if it is a closed embedding and an open embedding we conclude that it takes coprojections to coprojections. The second condition of Definition ?? follows immediately from the fact that our functor commutes with fiber products.

We extend the functor X ↦ X^{W} to a functor from [QP/S] to Shv_{add}(G − QP/S) by (??). By Lemma ?? we get a functor from [QP/S] to [G − QP/S] for which we use the same notation X ↦ X^{W}.

Lemma 5.1.8 [istrans2] The functor X → X/G is transparent.

Proof: Our functor commutes with coproducts and in particular it takes coprojections to coprojections. Any pull-back square of coprojections is a finite coproduct of squares where one side consists of isomorphisms. Since such squares are preserved by any functor we conclude that X ↦ X/G is transparent.

Lemma 5.1.8 implies that can extend our functor by (??) to a functor [G − QP/S] → Sch/S which we denote by the same symbol. Since a composition of transparent functors is transparent Lemmas 5.1.7 and 5.1.8 together with the obvious fact that F_{2} is transparent imply the following result.

Lemma 5.1.9 [istrans3] The functor T is transparent.

Lemma 5.1.9 implies that we can extend T by (??) to a functor

[T] : [QP/S] → [Sch/S]

and one can easily see that this extension can be written as a composition of the form

[QP/S] →^{[F_{1}]} [G − QP/S] →^{[F_{2}]} [QP/S] →^{[F_{3}]} [Sch/S]

where [F_{1}], [F_{2}] and [F_{3}] are obtained by (??) out of the functors F_{1}, F_{2} and F_{3} respectively.

Lemma 5.1.10 [twistedaf] The extension of [T] to simplicial objects takes elements of clΔ̄(W_{A^{1}}) to elements of clΔ̄(W_{A^{1}}).

Proof: Follows from Lemma 5.1.13, Lemma 5.1.16 and the obvious fact that the functor [F_{2}] takes clΔ̄(W_{G,A^{1}}) to itself.

Lemma 5.1.11 [twistedup] The extension of [T] to simplicial objects takes clΔ̄(W_{up}) to itself.

Proof: This follows immediately from Lemma 5.1.14, Lemma 5.1.17 and the obvious fact that the functor F_{2} takes clΔ̄(W_{G,up}) to itself.

Lemma 5.1.12 [twistedlow] Let G be a group scheme such that for a closed embedding of quasi-projective G-schemes Z → X the morphism Z/G → X/G is a closed embedding and W be an etale G-scheme. Then the extension of [T] to simplicial objects takes clΔ̄(W_{low}) to itself.

Proof: This follows immediately from Lemma 5.1.15, Lemma 5.1.18 and the obvious fact that the functor F_{2} takes clΔ̄(W_{G,low}) to itself.

The cdh-topology is the topology associated with the combined cd-structure i.e. the sum of the upper and lower cd-structures. The results of [11] show that the combined cd-structure is complete, regular and bounded on the category of all schemes of finite type and one can easily see that the same is true for its restriction to quasi-projective schemes. Therefore, by [10], a morphism of simplicial objects in (QP/S)∐ is a cdh-weak equivalence if and only if it belongs to clΔ̄(W_{up} ∪ W_{low}).

Lemma 5.1.13 [takesaf] The functor X ↦ X^{W} takes elements of clΔ̄(W_{A^{1}}) to elements of clΔ̄(W_{A^{1}}).

Proof: By [9, Proposition 9.3]we have

(clΔ̄(W_{A^{1}}))^{W} ⊂ clΔ̄((W_{A^{1}} ∐ Id)^{W})

Since any element of W_{A^{1}} ∐ Id is a strict A^{1}-homotopy equivalence and any strict A^{1}-homotopy equivalence is in clΔ̄(W_{A^{1}}) by [10, Corollary 12.4], our claim follows from Lemma 5.0.9.

Lemma 5.1.14 [takesup] For any G and W as above the functor X ↦ X^{W} takes elements of clΔ̄(W_{up}) to elements of clΔ̄(W_{G,up}).

Proof: Observe first that since the upper cd-structure on G − QP/S is complete regular and bounded a morphism belongs to clΔ̄(W_{G,up}) if and only if it is a weak equivalence in the upper topology. In particular the class clΔ̄(W_{G,up}) is closed under filtered colimits and we conclude by [9, Proposition 9.3]that it is sufficient to check that for any f in W_{up} = W_{up} ∐ Id one has f^{W} ∈ clΔ̄(W_{G,up}). Applying [10, Lemma 11.3]to the category of quasiprojective schemes over S with the upper cd-structure we conclude that

W_{up} ⊂ cl_{Δ,∐_{<∞}}(η(Cov_{up}))

where Cov_{up} is the class of upper coverings. By [9, Lemma 9.1]we conclude that W_{up} ⊂ cl_{Δ}(η(Cov_{up})) (when we consider morphisms in [QP/S] as opposed to (QP/S)∐). Thus it remains to show that for an upper covering p the morphism η(p)^{W} is in clΔ̄(W_{G,up}). Since our functor commutes with fiber products we have η(p)^{W} = η(p^{W}). By Proposition 4.2.4 we conclude that p^{W} is an upper covering of G-schemes. Applying [10, Proposition 11.8]to the category of quasi-projective G-schemes and the upper topology we conclude that η(p^{W}) is in clΔ̄(W_{G,up}).

Lemma 5.1.15 [takeslow] Let G be as above and W be an etale G-scheme. Then the functor X ↦ X^{W} takes elements of clΔ̄(W_{low}) to elements of clΔ̄(W_{G,low}).

Proof: Same as the proof of Lemma 5.1.14 with the reference to Proposition 4.2.4 replaced by the refernce to Proposition 4.4.3.

Lemma 5.1.16 [takesaf2] The functor X ↦ X/G takes elements of clΔ̄(W_{G,A^{1}}) to elements of clΔ̄(W_{A^{1}}).

Proof: Same as the proof of Lemma 5.1.13 with the refernce to Lemma 5.0.9 replaced by the refernce to Lemma 5.0.9.

Lemma 5.1.17 [takesup2] For any G as above the functor X ↦ X/G takes elements of clΔ̄(W_{G,up}) to elements of clΔ̄(W_{up}).

Proof: Denote our functor by F. Since F commutes with coproducts we have

F (clΔ̄(W_{G,up})) ⊂ clΔ̄(F (W_{G,up})).

Since the upper cd-structure on QP/S is complete regular and bounded a morphism belongs to clΔ̄(W_{up}) if and only if it is a weak equivalence in the upper topology. Therefore, it is sufficient to check that for any f in W_{up} the morphism F (f) is a weak equivalence in the upper topology. This follows from Proposition 3.1.15 and [10, Lemma 10.2].

Lemma 5.1.18 [takeslow2] Let G be a group scheme such that for a closed embedding Z → X of quasi-projective G-scheme the morphism Z/G → X/G is a closed embedding. Then the functor X ↦ X/G takes elements of clΔ̄(W_{G,low}) in Δ^{op}[G − QP/S] to elements of clΔ̄(W_{low}).

Proof: Same as the proof of Lemma 5.1.17 with the reference to Proposition 3.1.15 replaced by the reference to Proposition 3.1.20.

Let us finish this section by a conjecture closely related with Example 3.1.24. As this example shows the quotient functor with respect to etale group schemes whose order divides the characteristic of the base scheme is not “exact” in the lower topology. We can define the total right derived functor Q_{G} of the quotient functor setting Q_{G}(X) = (Ex(X))/G. Proposition 3.1.20 implies that this functor is exact with respect to the lower cd-structure. The second conjecture relates this derived quotient functor with the “spaces” of finite cycles. For a quasi-projective scheme X over S let c^{d}(X, 0) be the subsheaf of the sheaf c^{eff}(X, 0) of effective finite cycles on X which corresponds to cycles of degree exactly d (see [8]). There is a natural morphism of sheaves c^{d}(X, 0) → Symm^{d}(X) where Symm^{d}(X) = X^{d}/Σ_{d} is the symmetruc power of X. In charactersitic zero this morphism is an isomorphism (see [, ]) but in positive characteristics it is not (see [, ]).

Conjecture 1 [symprod] For any quasi-projective scheme X there is a canonical weak equivalence Q_{Σ_{d}}(X^{d}) = c^{d}(X, 0).

Lemma 5.1.19 [m29l4] Let E be a quasi-projective G-scheme. Then there is an affine simplicial G-scheme E′ and a morphism E′ → E which is a weak equivalence in the (plain) upper cd-structure on G-schemes.

Proof: Let E = ∪U_{i} be a plain upper covering of E which exists by Lemma 5.1.20. Consider the Cech simplicial scheme Č(p) where p = ∐ j_{i} : ∐ U_{i} → X. The intersection of two affine open subsets is afine which implies that Č(p) is a simplicial affine G-scheme. The morphism η(p) : Č(p) → X is an upper weak equivalence by [10, Proposition 11.8].

Lemma 5.1.20 [m29l5] Let E be a quasi-projective G-scheme where G is finite and flat. Then there exists a plain upper (i.e. Zariski) covering X = ∪U_{i} in the category of G-schemes such that all U_{i} are affine.

Proof: ????

5.2 Twisted product functors on smooth schemes

Let λ : (QP/S)_{Nis} → (QP Sm/S)_{Nis} be the continuous map of sites defined by the inclusion of the underlying categories and let Lλ^{∗} and Rλ_{∗} be the corresponding inverse and direct image functors on the homotopy categories. Let further G, W and E be as above and T be the corresponding twisted product functor. We define the twisted product functor T_{sm} on the category H_{A^{1}}(S) = H((QP Sm/S)_{Nis}, A^{1}) by the formula

T_{sm} = Rλ_{∗}T Lλ^{∗}

The following result follows immediately from this definition.

Lemma 5.2.1 [explform] For a smooth quasi-projective simplicial scheme X the object T_{sm}(X) is given by the Nisnevich sheaf on QP Sm/S represented by the simplicial scheme (X^{W} × E)/G.

Denote by π : (QP/S)_{cdh} → (QP Sm/S)_{Nis} the obvious continuous map of sites (which is the composition of the morphism from the cdh- to the Nisnevich topology and λ).

Proposition 5.2.2 [m19p1] Let S = Spec(k) where k is a field with resolution of singularities, G a group scheme over k such that for any closed embedding Z → X of quasi-projective G-schemes the morphism Z/G → X/G is a closed embedding, let W be an etale G-scheme and E a quasi-projective G-scheme. Let f : X → Y be a morphism of pointed simplicial presheaves on QP Sm/S such that π^{∗}(f_{Nis}) is a weak equivalence in the cdh-topology. Then Σ_{s}^{1}(T_{sm}(f)) is an A^{1}-weak equivalence.

Proof: Since k has resolution of singularities the continuous map π is a morphism of sites and in particular π^{∗} = Lπ^{∗}. On the other hand the same condition implies that the adjunction Id → Lπ^{∗}Rπ_{∗} is an isomorphism. Therefore we have Lπ^{∗}(T_{sm}(f)) = T (π^{∗}(f)). Our assumptions on G, W and E imply that T takes cdh-weak equivalences to cdh-weak equivalences. Therefore Lπ^{∗}(T_{sm}(f)) is a cdh-weak equivalence. By [11, Corollary 4.6]we conlude that Σ_{s}^{1}(T_{sm}(f)) is an A^{1}-weak equivalence.

Let X be a quasi-projective (simplicial) scheme over S and ρ(X) the sheaf in the Nisnevich topology represented by X. Let G(ρ(X)) be the standard simplicial resolution of ρ(X) by coproducts of smooth quasi-projective schemes (see [9, Section 5]). Then one has

T_{sm}(ρ(X)) ≅ T_{sm}(G(ρ(X))) = ρ((G(ρ(X))^{W} × E)/G)

Since the simplicial scheme G(ρ(X)) maps to X we get a natural morphism of simplicial sheaves

[underq]p : T_{sm}(G(ρ(X))) → ρ(X^{W} × E)/G  (33)

Proposition 5.2.3 [firstsus] Let S, G and W be as in Proposition 5.2.2. Then for any quasi-projective simplicial scheme X over k the morphism

Σ_{s}^{1}p_{+} : Σ_{s}^{1}T_{sm}(G(ρ(X)))_{+} → Σ_{s}^{1}ρ((X^{W} × E)/G)_{+}

is an A^{1}-weak equivalence.

Proof: We can rewrite (33) as the morphism

[m26eq3]Rλ_{∗}T Lλ^{∗}(G(ρ(X))) → Rλ_{∗}T (X)  (34)

Applying π^{∗} to (34) we get the morphism of cdh-sheaves associated with the morphism T (λ^{∗}G(ρ(X)))_{cdh} → T (X)_{cdh} where the subscript cdh denotes the associated cdh-sheaf. Lemma 5.2.4 below implies that the morphism λ^{∗}(G(ρ(X))) → X is a cdh-weak equivalence. Since T preserves cdh-weak equivalences the statement of the proposition follows now from [11, Corollary 4.6].

Lemma 5.2.4 [m19l1] For any quasi-projective scheme X over k the morphism λ^{∗}G(ρ(X)) → X is a weak equivalence in the cdh-topology.

Proof: Since we assume resolution of singularities for k the continuous map of sites π is a morphism of sites. In particular π^{∗} preserves weak equivalences. Thus for any X the morphism π^{∗}(G(ρ(X))) → π^{∗}(λ_{∗}(X)) is a weak equivalence. Using again resolution of singularities we further see that the adjunction defines an isomorphism π^{∗}(λ_{∗}(X)) = X_{cdh} where X_{cdh} is the cdhsheaf associated with the presheaf represented by X.

Finally let us prove a smooth analog of Proposition 5.0.6. Note that in the smooth case we only consider homotopy categories with respect to the upper cd-structure.

Proposition 5.2.5 [smcompositions] Let (G_{i}, W_{i}, E_{i}), i = 1, 2 be two sets of objects defining twsited product functors T_{i} and let T_{sm,i} be their smooth analogs. Assume that the following two conditions hold:

1. G_{1}-quotients are universal

2. W_{2} is etale Let further

G = G_{1}^{W_{2}} ⋊ G_{2}
W = W_{1} × W_{2}
E = E_{1}^{W_{2}} × E_{2}

and let T_{sm} be the twisted product functor corresponding to (G, W, E). Then there is a natural transformation of functors

T_{sm,2} ∘ T_{sm,1} → T_{sm}

and the first simplicial susupension of it is an isomorphism.

Proof: By definition we have

T_{sm,2} ∘ T_{sm,1} = Rλ_{∗}T_{2}Lλ^{∗}Rλ_{∗}T_{1}Lλ^{∗}

The adjunction

[m29eq1]Lλ^{∗}Rλ_{∗} → Id  (35)

gives us a morphism to Rλ_{∗}T_{2}T_{1}Lλ^{∗} which, by Proposition 5.0.6, is isomorphic to T_{sm}. Under the resolution of singularities assumption the adjunction (35) is a cdh-weak equivalence. Since W_{2} is etale T_{2} preserves cdh-weak equivalences. Using again resolution of singularities assumption we conclude that our morphism is a cdh-weak equivalence. Therefore, the first simplicial suspension of it is an A^{1}-weak equivalence by [11, Corollary 4.6].

??? Include the case when the action on E is free. Then do not need restrictions ....

6 Appendix: solid sheaves

Let C be a site. For an object X of C we denote by ρ(X) the sheaf associated to the presheaf represented by X.

Definition 6.0.6 [solid] Let O be a class of morphisms in C. A morphism of sheaves on C is called solid (resp. ind-solid) with respect to O if it belongs to the smallest class which is closed under push-forwards and compositions (resp. infinite compositions) and contains morphisms of the form ρ(e) : ρ(A) → ρ(X) for e in O. A sheaf F is called solid (resp. ind-solid) if the morphism ∅ → F is solid (resp. ind-solid).

Lemma 6.0.7 [constr] A morphism f : F → F′ is solid if and only if there exists an integer n ≥ 0 and a sequence of push-forward squares

ρ(U_{i}) ⟶^{ρ(e_{i})} ρ(X_{i})
↓  ↓
F_{i} ⟶^{f_{i}} F_{i+1}

where i = 0, …, n and e_{i} ∈ O, such that f is isomorphic to the composition f_{n} ∘ f_{n−1} ∘ … ∘ f_{0}.

Proof: One verifies immediately that the class of f satisfying the conditions of the lemma contains morphisms of the form ρ(e) for e ∈ O and is closed under compositions and push-forwards.

Definition 6.0.8 [locsol] A morphism F_{0} → F is called locally solid if for any morphism of the form ρ(X) → F there exists a covering {X_{i} → X} of X such that the morphisms ρ(X_{i}) ×_{F} F_{0} → ρ(X_{i}) are representable by morphisms in O of the form A_{i} → X_{i}.

The following lemmas are straightforward.

Lemma 6.0.9 [ls1] The class of locally solid morphisms is closed under pullbacks.

Lemma 6.0.10 [ls15] Assume that the category C has an initial object ∅ which represents the initial sheaf and that the morphisms ∅ → X are in O. Then for any F and G the morphism F → F ∐ G is locally solid.

Lemma 6.0.11 [ls2] Let f : F → F′ be a morphism and {p_{i} : G_{i} → F′} a family of morphisms such that ∐ p_{i} is an epimorphism and for each i the morphism G_{i} ×_{F′} F → G_{i} is locally solid. Then f is locally solid.

Lemma 6.0.11 implies in particular the following fact.

Lemma 6.0.12 [copro] The coproduct of a family of locally solid morphisms is locally solid.

Lemma 6.0.13 [aremon] If elements of O are monomorphisms then any locally solid morphism is a monomorphism.

Proof: Let f : F_{0} → F be a locally solid morphism. Then there exists a family of objects X_{i} of C and an epimorphism of sheaves ∐ ρ(X_{i}) → F such that the pull-back of f along this epimorphism is a coproduct of morphisms of the form ρ(e) for e ∈ O. The morphism of sheaves representable by a monomorphism is a monomorphism and therefore elements of ρ(O) are monomorphisms. A coproduct of monomorphisms of sheaves is a monomorphism. This implies easily that f is a morphism of sheaves associated to a monomorphism of presheaves. Since the functor of associated sheaf is left exact f is a monomorphism.

Lemma 6.0.14 [fb] Any push-forward square in the category of sheaves on a site of the form

\begin{matrix}G & ⟶ & F \\ ↓ & & ↓ \\ G′ & ⟶ & F′ \end{matrix}

such that G → G′ is a monomorphism is a pull-back square.

Proof: Using the fact that the associated sheaf functor commutes with both pull-backs and push-forwards one reduces the problem to the case of presheaves. For presheaves pull-backs and push-forwards are given sectionwise which reduces the problem to the case of the category of sets. For sets it follows immediately from the explicit description of push-forwards.

Lemma 6.0.15 [li1] If elements of O are monomorphisms then the class of locally solid morphisms is closed under push-forwards.

Proof: Consider a push-forward square of the form

[pfsq]\begin{matrix}G & ⟶ & F \\ ↓ & & ↓ \\ G′ & ⟶ & F′ \end{matrix}  (36)

where G → G′ is a locally solid morphism. We have to show that F → F′ is locally solid. The morphism F ∐ G′ → F′ is an epimorphism which, togther with Lemma 6.0.11, implies that it is sufficient to check that the morphisms F ×_{F′} F → F and F ×_{F′} G′ → G′ are locally solid. By Lemma 6.0.13 the morphism F → F′ is a monomorphism. Therefore, the first of our two morphisms is an isomorphism and in particular is locally solid. By Lemma 6.0.14 the second morphism is isomorphic to the morphism G → G′ which is locally solid by the assumption.

Example 6.0.16 [infcomp] The class of locally solid morphisms need not be closed under infinite compositions. In the case of the standard cd-structures on schemes this depends on whether one takes an upper or a lower cdstructure. ??? Finite compositions? Example????

Definition 6.0.17 [sat] A class of morphisms O in C is said to be saturated if the following conditions hold:

1. O is closed under compositions

2. the pull-backs of elements of O exist and belong to O

3. any locally solid with respect to O morphism of the form F → ρ(X) is representable by a morphism e : A → X in O.

The following two lemmas are straightforward.

Lemma 6.0.18 [sat2] Let O be a saturated class. Then a morphism f : F_{0} → F is locally solid if and only if for any morphism of the form ρ(X) → F the morphism ρ(X) ×_{F} F_{0} → ρ(X) is representable by a morphisms in O of the form A → X.

Lemma 6.0.19 [l9.22.1] Let O be a saturated class then the class of locally solid with respect to O morphisms is closed under compositions.

Lemma 6.0.20 [liss0] Let P be a saturated regular cd-structure. Then any solid morphism is locally solid.

Proof: The second condition of Definition 6.0.17 implies that morphisms of the form ρ(e) for e ∈ O are locally solid. Lemma 6.0.15 shows that the class of locally solid morphisms is closed under push-forwards. Lemma 6.0.19 implies that it is closed under compositions.

Example 6.0.21 [indnotlo] Let C be the category of smooth schemes over a field with resolution of singularities and P be the smooth blow-up cdstrucrture introduced in the proof of [11, Theorem 4.2]. This cd-structure is complete, regular and bounded. The class O is the class of closed embeddings of smooth schemes. Since a pull-back of a closed embedding need not be representable by a closed embedding this cd-structure is not saturated and in particular the elements of ρ(O) are not locally solid. ????? why ? in the new definition they probably are locally solid

Lemma 6.0.22 [liss] Let O be a saturated class and assume in addition that for any pull-back square of the form

A_{1} ×_{X} A_{2} ⟶ A_{1}
↓  ↓
A_{2}  ⟶ X

with sides being in O the morphism of sheaves

ρ(A_{1}) ×_{ρ(X)} ρ(A_{2}) → ρ(X)

is representable by a morphism e : A → X in O. Let further F be a solid (resp. ind-solid) sheaf and f : F′ → F a locally solid morphism. Then f is solid.

Proof: By Lemma 6.0.7 there exists a sequence of morphisms F_{0} → … → F_{n} = F such that F_{0} = ρ(X_{−1}) and for any i ≥ 0 there is a push-forward square of the form

ρ(U_{i}) ⟶^{ρ(e_{i})} ρ(X_{i})
↓  ↓
F_{i} ⟶^{f_{i}} F_{i+1}

We proceed by induction on n. For n = 0 the statement follows from the definition of a saturated class. By induction we may assume that there is a push-forward square of the form

ρ(U) ⟶^{ρ(e)} ρ(X)
↓  ↓
G ⟶^{f} F

such that the statement of the lemma holds for any locally solid morphism whose codomain is G. Since the push-forwards in the category of sheaves are universal the pull-back of this square along the morphism F′ → F is again a push-forward square which we denote by

[sq9.21.1]\begin{matrix}ρ(U)′ & ⟶^{ρ(e)′} & ρ(X)′ \\ ↓ & & ↓ \\ G′ & ⟶^{f} & F′ \end{matrix}  (37)

Since F′ → F is locally solid and O is saturated ρ(X)′ = ρ(X′) for an object X′ and since the square

[sq9.21.2]\begin{matrix}ρ(U)′ & ⟶^{ρ(e)′} & ρ(X)′ \\ ↓ & & ↓ \\ ρ(U) & ⟶^{ρ(e)} & ρ(X) \end{matrix}  (38)

is pull-back we have ρ(e)′ = ρ(e′) for e in O. The morphism F′ → F can be written as the composition of the form

F′ → F′ ∐_{G′} G → F

The first of these morphisms is solid as the push-forward of the morphism G′ → G which is solid by the inductive assumption (and Lemma 6.0.9). The second morphism fits into the square

[sq9.21.3]\begin{matrix}ρ(X′) ∐_{ρ(U′)} ρ(U) & ⟶^{j} & X \\ ↓ & & ↓ \\ F′ ∐_{G′} G & ⟶ & F \end{matrix}  (39)

which is a push-forward square because the squares (37) and (38) are pushforward. It remains to note that the upper horizontal arrow j in (39) is solid by the assumption of the lemma.

Lemma 6.0.23 [liss2] Let O be a saturated class, then if in a pull-back square of the form

\begin{matrix}G′ & ⟶^{v′} & F′ \\ ↓ & & ↓^{j} \\ G & ⟶^{v} & F \end{matrix}

the morphism v is solid (resp. ind-solid) and j is locally solid then v′ is solid (resp. ind-solid).

Proof: We only condider the solid case. By Lemma 6.0.7 and Lemma 6.0.9 we may assume that v fits into a push-forward square of the form

ρ(U) ⟶^{ρ(e)} ρ(X)
↓  ↓
G ⟶ F

Since push-forwards in the category of sheaves are universal the pull-back of this square along the morphism F′ → F is a push-forward square. Since j is locally solid and O is saturated the pull-back of the morphism ρ(e) is a morphism of the form ρ(e′) for e′ ∈ O. Therefore, the pull-back of the morphism G → F is solid.

Lemma 6.0.24 [liss21] If, under the assumptions of Lemma 6.0.23, F is solid and F′ → F is a locally solid morphism then F′ is solid.

Proof: It follows from Lemma 6.0.23 applied to the morphism ∅ → F.

Let now C be a category and P be a cd-structure on C. Denote by O_{P} the class of morphisms in C such that e : A → X belongs to O_{P} if and only if there exists a distinguished square of the form

[ds]\begin{matrix}B & ⟶ & Y \\ ↓ & & ↓ \\ A & ⟶^{e} & X \end{matrix}  (40)

Everywhere below we say solid meaning solid with respect to O_{P}. A morphism of simplicial sheaves f : X_{n} → Y_{n} is called termwise solid (resp. ind-solid) if each term f_{n} of f is solid (resp. ind-solid). Let I_{P} and J_{P} be the sets of generating cofibrations and trivial cofibrations respectively for the Brown-Gersten model structrure on simplicial sheaves on C (see [10, p.21]). Elements of I_{P} and J_{P} are termwise solid morphisms.

Lemma 6.0.25 [iscof] Let P be a complete, bounded and regular cd-structure. Then an ind-solid morphism of sheaves is a Brown-Gersten cofibration.

Proof: By [10, Lemma 4.4] our assumptions imply that a morphism is a Brown-Gersten cofibration if and only if it has the left lifting property for all morphisms which have the right lifting property for elements of I_{P}. Since I_{P} contains morphisms of the form ρ(e) for e in O_{P} this implies the statement of the lemma.

Lemma 6.0.26 [isret] Let P be a complete, bounded and regular cd-structure. Then Brown-Gersten cofibration f : X → Y is a retract of a termwise indsolid morphism.

Proof: Since P is complete the representable sheaves are compact objects which implies that the domains and codomains of elements of I_{P} are compact. Therefore, the standard decomposition theorem (see [5]) implies that f can be represented as a composition of an I_{P} -cell f_{0} which is a termwise ind-solid morphism and a trivial fibration p. For f which is a cofibration the lifting property of cofibrations with respect to trivial fibrations implies that f is a retract of f_{0}.

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