1 Basics

We will need several simple lemmas about the functorial behavior of pointed simplicial sheaves on the categories of smooth schemes over a base with Nisnevich topology. Many of them have well known analogs for sheaves on “small” sites but we often have to give quite differnet proofs. The main reason for most of the differences is that for sheaves on smooth sites stalks and inverse images do not commute i.e. Theorem 3.2(a) of [?] fails to be true.

Everywhere below we work in the context of pointed sheaves of sets on (Sm/S)_{Nis}. In [?] we called such sheaves “spaces” over S but since the techniques of this paper have much more to do with the usual sheaf theory than with the homotopy theory we do not use this name here. The category of pointed (simplicial) sheaves on (Sm/S)_{Nis} is a pointed category which has all small products and coproducts. Its initial/final object is denoted by pt and the direct sums by ∨_{α}F_{α}. The smash product F ∧ G of two pointed simplicial sheaves is given by the usual formula ([?, ]) and satisfies the standard associativity and commutativity conditions. The unit object of the symmetric monoidal structure defined by ∧ is denoted S^{0}.

For any morphism of schemes f : S_{1} → S_{2} we have the inverse image functor f^{∗} which is caracterized by the properties that it commutes with colimits and that for a smooth scheme X over S_{2} one has f^{∗}(X_{+}) = (X ×_{S_{2}} S_{1})_{+}. It has the right adjoint called the direct image functor f_{∗}. For smooth morphisms f the functor of inverse image also has the left adjoint f_{#} (see []) such that for a smooth scheme X over S_{1} one has f_{#}(X_{+}) = X_{+} where on the right side of the equality X is considered as a smooth scheme over S_{2}.

Remark 1.1 [forgetful]Functors of all three types commute with the functor of free base point F ↦ F_{+} from sheaves to pointed sheaves. Functors f^{∗} and f_{∗} also commute with the forgetful functor from pointed sheaves to sheaves but functors f_{#} do not. If we denote the forgetful functor by ϕ then for a smooth morphism f : S_{1} → S_{2} and a pointed sheaf F over S_{1} one has a push-forward square of sheaves of the form

S_{1} → f_{#}(ϕ(F))
↓  ↓
S_{2} → ϕ(f_{#}(F))

The following lemmas can be seen directly from definitions.

Lemma 1.2 [l0] For a composable pair of morphisms S_{1} →^{f} S_{2} →^{g} S_{3} there is a canonical ismorphism (g ∘ f)^{∗} = f^{∗} ∘ g^{∗} and for a composable triple the usual square commutes.

By adjunction the isomorphisms of Lemma ?? define isomorphisms (g∘f)_{∗} = g_{∗} ∘ f_{∗} and for smooth f, g isomorphisms (g ∘ f)_{#} = g_{#} ∘ f_{#}.

Lemma 1.3 [l1] For any f : S_{1} → S_{2} and any F, G over S_{2} there is a canonical isomorphism f^{∗}(F ∧ G) = f^{∗}(F) ∧ f^{∗}(G).

Lemma 1.4 [l2] For any smooth morphism f : S_{1} → S_{2} any F over S_{1} and G over S_{2} the morphism f_{#}(F ∧ f^{∗}G) → f_{#}F ∧ G defined by the adjunctions and the isomorphisms of Lemma ?? is an isomorphism.

Lemma 1.5 [l3] For any pull-back square

\begin{matrix}S′_{1} & →^{f_{1}} & S_{1} \\ p′ ↓ & & ↓ p \\ S′_{2} & →^{f_{2}} & S_{2} \end{matrix}

such that p is smooth and any F over S_{1} the morphism p′_{#}f_{1}^{∗}(F) → f_{2}^{∗}p_{#}(F) defined by the adjunctions and the isomorphism of Lemma ?? is an isomorphism.

In general neither of the three types of functors considered above preserve simplicial or A^{1}-weak equivalences. In order to define the (left) derived functors for f^{∗} and f_{#} we need the following construction.

[ladm] An object is called left admissible if it is admissible with respect to all f^{∗}’s and f_{#}’s.

Lemma 1.6 [l6] Let a : F → G be a simplicial (resp. A^{1}-) weak equivalence of left admissible objects over S. Then for any morphism f : S′ → S the morphism f^{∗}(a) is a simplicial (resp. A^{1}-) weak equivalence and for any smooth morphism f : S → S′ the morphism f_{#}(a) is a simplicial (resp. A^{1}-) weak equivalence.

Lemma 1.7 [l7] Let F be a left admissible object. Then for any morphism f : S′ → S the object f^{∗}(F) is left admissible and for any smooth morphism f : S → S′ the object f_{#}(F) is left admissible.

Lemma 1.8 [admsm] Let F and G be left admissible objects. Then F ∧ G is left admissible.

Lemma 1.9 [lres] For any S there exists a functor Lres : Δ^{op}Spc_{•} → Δ^{op}Spc_{•} called the left resolution functor and a natural transformation Lres → Id such that the following two conditions hold:

1. for any F the terms of the simplicial sheaf Lres(F) are direct sums of pointed sheaves of the form U_{+} for smooth quasi-projective schemes U over S.

2. for any F and any smooth quasi-projective scheme U over S the morphism of simplicial sets Lres(F)(U) → F (U) is a trivial Kan fibration.

Proof: ???

We define the left derived functors of f^{∗} and f_{#} setting Lf^{∗} = f^{∗} ∘ Lres and Lf_{#} = f_{#} ∘ Lres.

Lemma 1.10 [l5] For any morphism f : S_{1} → S_{2} and a simplicial (resp. A^{1}-) weak equivalence a : F → G over S_{2} the morphism Lf^{∗}(a) is a simplicial (resp. A^{1}-) weak equivalence.

For a smooth morphism f : S_{1} → S_{2} and a simplicial (resp. A^{1}-) weak equivalence a : F → G over S_{1} the morphism Lf_{#}(a) is a simplicial (resp. A^{1}-) weak equivalence.

Proof: ???

Lemma 1.11 [l51] For any morphism f : S_{1} → S_{2} and a left admissible object F over S_{2} the morphism Lf^{∗}(F) → f^{∗}(F) is a simplicial weak equivalence.

For a smooth morphism f : S_{1} → S_{2} and a left admissible object F over S_{1} the morphism Lf_{#}(F) → f_{#}(F) is a simplicial weak equivalence.

We will need to know how functors Lf^{∗} and Lf_{#} behave with respect to homotopy colimits. Let us recall the definition of homotopy colimits first. Let I be a small category and X : I → Δ^{op}Spc_{•} a diagram of pointed (simplicial) sheaves indexed by I. For i ∈ I one usually denotes X(i) by X_{i}. Let I/i be the category of objects in I over i (i.e. the category of arrows which end in i) and let Nerv(I/i) be the nerve of I/i i.e. the simplicial set whose n-simplexes are composable sequences of arrows in I/i of length n. For any morphism γ : i → i′ in I we have a functor I/i′ → I/i and thus a morphism of simplicial sets N_{γ} : Nerv(I/i′) → Nerv(I/i). Following [, p.328] one defines the homotopy colimit hocolim_{i∈I}X_{i} as the coequalizer of two morphisms

⋁_{γ:i→i′} Nerv(I/i′)_{+} ∧ X_{i} \overrightarrow{→} ⋁_{i} Nerv(I/i)_{+} ∧ X_{i}

where the first arrow is given on Nerve(I/i′)_{+}∧X_{i} by Id∧X(γ), the second by N(γ)_{+}∧Id and simplicial sets are considered as constant simplicial sheaves in the usual manner. The following three lemmas describe the main properties of this construction.

Lemma 1.12 [hocolim0] Let X, Y : I → Spc_{•} be two diagrams of pointed simplicial sheaves and a : X → Y a morphism such that for any i ∈ I the morphism a_{i} : X_{i} → Y_{i} is a simplicial (resp. A^{1}-) weak equivalence. Then the morphism hocolim(a) is a simplicial (resp. A^{1}-weak equivalence).

Proof: For the simplicial case see [?, Cor. 2.1.21]. For the A^{1}-case see [?, Lemma 2.2.12].

The following two lemmas are immediate corollaries of the corresponding results for simplicial sets proven in [, Ch.XII, §3].

Lemma 1.13 [hocolim1] Let X : Δ^{op} → Δ^{op}Spc_{•} be a pointed bisimplicial sheaf. Then there is a canonical simplicial weak equivalence hocolim_{Δ^{op}}(X) → diag(X) where diag(X) is the diagonal simplicial sheaf of X. In particular for any pointed simplicial sheaf considered as a functor X : Δ^{op} → Spc_{•} ⊂ Δ^{op}Spc_{•} there is a canonical simplicial weak equivalence hocolim_{Δ^{op}}X_{n} → X where X_{n} are the pointed sheaves of n-simplexes of X.

Lemma 1.14 [hocolim2] For a pushforward square

\begin{matrix}A & →^{i} & X \\ ↓ & & ↓ \\ B & → & Y \end{matrix}

such that i is a monomorphism, the canonical map hocolim( \begin{matrix}A & →^{i} & X \\ ↓ \\ B \end{matrix} ) → Y

is a simplicial weak equivalence.

Since the functor of inverse image is a left adjoint it commutes with colimits which immediately implies that for any small diagram (X_{i})_{i∈I} we have a canonical isomorphism i^{∗}hocolim_{I}X_{i} → hocolim_{I}i^{∗}(X_{i}).

Lemma 1.15 [hocolim3] For any small diagram (X_{i})_{i∈I} over S such that X_{i} are left admissible hocolim_{i∈I}X_{i} is left admissible.

Proof: ???

Lemma 1.16 [Lho] For any morphism f : S′ → S and any small diagram (X_{i})_{i∈I} over S there is a natural (in X) isomorphism in the simplicial homotopy category H_{s}(S′) of the form

Lf^{∗}(hocolim_{i∈I}X_{i}) → hocolim_{i∈I}Lf^{∗}(X_{i})

such that the following square commutes

Lf^{∗}(hocolim_{i∈I}X_{i}) → hocolim_{i∈I}Lf^{∗}(X_{i})
↓  ↓
f^{∗}(hocolim_{i∈I}X_{i}) → hocolim_{i∈I}i^{∗}(X_{i})

Proof: Recall that Lf^{∗} = f^{∗} ∘ Lres and consider the diagram

f^{∗}Lres(hc_{i∈I}(Lres(X_{i}))) → f^{∗}hc_{i∈I}(Lres(X_{i})) → hc_{i∈I}f^{∗}Lres(X_{i})
↓  ↓  ↓
f^{∗}Lres(hc_{i∈I}X_{i})  →  f^{∗}(hc_{i∈I}X_{i})  → hc_{i∈I}i^{∗}(X_{i})

where we abbreviated hocolim to hc. The left vertical arrow is a simplicial weak equivalence by Lemmas ?? and ??, the first upper horizontal one is a simplicial weak equivalence by Lemmas ?? and ?? and the second is the canonical isomorphism. We define our isomorphism as the composition of the inverse to the left vertical arrow with the upper horizontal ones. It is clearly natural with respect to morphisms of diagrams. To prove commutativity of the square it is sufficient to show that the two squares of the diagram from above are commutative. The first one is commutative since Lres → Id is a natural transformation of functors and the second one since f^{∗}hc_{i∈I} → hc_{i∈I}f^{∗} is a natural transformation of functors.

Lemma 1.17 [constproj] For any morphism f : S′ → S one has:

1. For any family (F_{i})_{i∈I} of pointed objects over S′ the canonical morphism ∨_{i}f_{∗}(F_{i}) → f_{∗}(∨_{i}F_{i}) is an isomorphism. In particular f_{∗}(pt) = pt.

2. For any object F over S′ and any pointed simplicial set K the morphism K ∧f_{∗}(F) → f_{∗}(K ∧F) defined by the adjunction and the isomorphism of Lemma ?? is an isomorphism.

Proof: ???

Let f : S′ → S be any morphism and (X_{i})_{i∈I} a diagram over S′. Define the canonical morphism hocolim_{i∈I}f_{∗}(X_{i}) → f_{∗}hocolim_{i∈I}X_{i} by the commutative diagram

⋁_{γ:i→i′} Nerv(I/i′)_{+} ∧ p_{∗}(X_{i}) \overrightarrow{→} ⋁_{i} Nerv(I/i)_{+} ∧ p_{∗}(X_{i}) → hc_{i}p_{∗}(X_{i})
↓ ↓ ↓
p_{∗}(⋁_{γ:i→i′} Nerv(I/i′)_{+} ∧ X_{i}) \overrightarrow{→} p_{∗}(⋁_{i} Nerv(I/i)_{+} ∧ X_{i}) → p_{∗}(hc_{i}X_{i})

where the left and the middle vertical arrows are compositions of isomorphisms from Lemma ??(1) and ??(2) and where we abbreviated hocolim to hc.

Lemma 1.18 [adcom] For any small diagram (X_{i})_{i∈I} over S′ the square of canonical morphisms and adjunctions

f^{∗}hocolim_{i∈I}f_{∗}(X_{i}) → hocolim_{i∈I}f^{∗}f_{∗}(X_{i})
↓  ↓
f^{∗}f_{∗}hocolim_{i∈I}X_{i} →  hocolim_{i∈I}X_{i}

commutes.

So far we were able to avoid mentioning “stalks” of sheaves on (Sm/S)_{Nis} but some of the proofs below require their use. Let X be a smooth scheme over S and x be a point of the Zariski topological space of X. To any such pair we can assign a point F ↦ F_{(X,x)} of the site (Sm/S)_{Nis} setting

F_{(X,x)} = colim_{(U,u)→(X,x)}F (U)

where the colimit is taken over the category of all diagrams of the form

U
u  ↗ ↓
Spec(k_{x}) →^{x} X

with U → X being etale. One verifies easily that this is indeed a point i.e. that the functor (−)_{(X,x)} : Shv((Sm/S)_{Nis}) → Sets commutes with both limits nd colimits. One can also verify that the set of points corresponding to all the pairs (X, x) where X runs through smooth quasi-projective (or affine) schemes over S is a “sufficient” set of points i.e. that the following lemma holds.

Lemma 1.19 [points0] A morphism f : F → G of sheaves on (Sm/S)_{Nis} is an isomorphism (rep. a monomorphism, an epimorphism) if and only if for any smooth quasi-projective X over S and any point x of X the corresponding map of pointed sets F_{(X,x)} → G_{(X,x)} is an isomorphism (resp. a monomorphism, an epimorphism).

The main difference between smooth sites and small sites is that for a closed embedding i : Z → S and a sheaf F on S one has (i^{∗}F)_{(Z,z)} ≠ F_{(S,i(z))}. Indeed if F = X_{+} for a smooth scheme X over S we have

(i^{∗}F)_{(Z,z)} = Hom_{S}(Spec(O_{Z,z}^{h}), X)_{+}

and

F_{(S,i(z))} = Hom_{S}(Spec(O_{S,i(z)}^{h}), X)_{+}

Note that if X is etale over S these two sets are the same which is the reason for the equality (i^{∗}F)_{(Z,z)} = F_{(S,i(z))} on small sites.

Lemma 1.20 [stfin] Let f : S′ → S be a finite morphism and F a sheaf on S′. Then for any (X, x) over S there is a canonical isomorphism

f_{∗}(F)_{(X,x)} = ∏_{x′∈X′_{Zar}, pr(x′)=x} F_{(X′,x′)}

where X′ = X ×_{S} S′ and pr : X′ → X is the projection.

Proof: ???

Lemma 1.21 [stet] Let f : S′ → S be an etale morphism and F a sheaf on S′. Then for any (X, x) over S there is a canonical isomorphism

f_{#}(F)_{(X,x)} = ⋁_{x′} F_{(X′,x′)}

where X′ = X ×_{S} S′ and the sum is taken over the points x′ such that pr(x′) = x and the morphism Spec(k_{x′}) → Spec(k_{x}) is an isomorphism.

Proof: ???

2 The gluing theorem and its corollaries

Let i : Z → S be a closed embedding and j : U → S the complimentary open one. In this section we consider the A^{1}-homotopy theoretical analogs of the classical results relating sheaves on S, Z and U. For pointed sheaves of sets on small sites the standard picture can be summarized as follows:

1. for any sheaf F on Z the adjunction i^{∗}i_{∗}(F) → F is an isomorphism

2. for any sheaf F on U the adjunction F → j^{∗}j_{#}(F) is an isomorphism

3. for any sheaf F on S the adjunctions j_{#}j^{∗}(F) → F and F → i_{∗}i^{∗}(F) fit into a pushforward square

j_{#}j^{∗}(F) → F
↓  ↓
pt  → i_{∗}i^{∗}(F)

These facts have two important corollaries:

1. Projection formula: for a sheaf F on Z and sheaf G on S the morphism F ∧ i_{∗}(G) → i_{∗}(i^{∗}F ∧ G) is an isomorphism

2. Base change: for a pull-back square

\begin{matrix}Z′ & →^{f_{Z}} & Z \\ i′ ↓ & & ↓i \\ S′ & →^{f_{S}} & S \end{matrix}

and a sheaf F on Z the morphism f_{S}^{∗}i_{∗}(F) → i′_{∗}f_{Z}^{∗} is an isomorphism

Proposition 2.1 [nd1] Let p : Z → S be a finite morphism. Then the functor of direct image p_{∗} is right exact i.e. for any diagram (X_{i})_{i∈I} over Z the canonical morphism colim_{i∈I}p_{∗}(X_{i}) → p_{∗}(colim_{i∈I}X_{i}) is an isomorphism.

Proof: Follows from Lemmas ?? and ??.

Corollary 2.2 [dirhc] Let p : Z → S be a finite morphism. Then for any diagram (X_{i})_{i∈I} over Z the canonical morphism hocolim_{i∈I}p_{∗}(X_{i}) → p_{∗}hocolim_{i∈I}X_{i} is an isomorphism.

Proof: Follows from Proposition ?? and the definition of the canonical morphism hocolim_{i∈I}p_{∗}(X_{i}) → p_{∗}hocolim_{i∈I}X_{i}.

Lemma 2.3 [closed2] Let i : Z → S be a closed embedding and X → Z a smooth scheme over Z. Then there exist a finite Zariski covering X = ∪V_{i}, smooth schemes W_{i} over S and isomorphisms V_{i} ≅ W_{i} ×_{S} Z over Z.

Proof: We may assume that S = Spec(R) and Z = Spec(Q) are affine. By [?, Prop. 3.24(b)] we can find a covering X = ∪V_{i} such that V_{i} are etale over A_{Z}^{n}. By [?, Th. 3.4] we can further choose V_{i}’s such that

V_{i} = Spec((A_{i}[T]/P_{i})[1/b_{i}]), A = Q[x_{1}, …, x_{n}][1/f_{i}]

and P′_{i} is a unit in (A_{i}[T]/P_{i})[1/b_{i}]. Let f̃_{i} be a lifting of f_{i} to an element in R, P̃_{i} a lifting of P_{i} to an element in R[x_{1}, …, x_{n}][T] and b̃_{i} a lifting of b_{i} to an element of R[x_{1}, …, x_{n}][T]. Set W_{i} = Spec(Ã_{i}[T]/P̃_{i}[1/b̃_{i}, 1/P̃′_{i}]) where Ã_{i} = R[x_{1}, …, x_{n}](1/f̃_{i}). Then W_{i} is etale over Spec(Ã_{i}) (by [?, Example 3.4]) and thus smooth over S and W_{i} ×_{S} Z ≅ V_{i} by construction.

Lemma 2.4 [nd2] Let i : Z → S be a closed embedding. Then for any G over S the adjunction G → i_{∗}i^{∗}(G) is an epimorphism.

Proof: Any pointed sheaf G over S is a colimit of a digram of sheaves of the form (X_{i})_{+} where X_{i} are smooth schemes over S. The functor i^{∗} commutes with colimits because it is a left adjoint and i_{∗} commutes with colimits by Lemma ??. Thus it is sufficient to show that X_{+} → i_{∗}i^{∗}X_{+} is an epimorphism for a smooth scheme X over S. For any smooth U over S sections of i_{∗}i^{∗}(X_{+}) over U are just sections of X ×_{S} U → U over the closed subscheme Z ×_{S} U → U. Since X is smooth over S for any such section locally (in the Nisnevich topology) extends to a section over U.

Proposition 2.5 [p1] Let i : Z → S be a closed embedding and j : U → S the complimentary open embedding. Then one has:

1. for any object F over Z the adjunction i^{∗}i_{∗}(F) → F is an isomorphism

2. for any object F over U the adjunction F → j^{∗}j_{#}(F) is an isomorphism

Proof: By Lemma ?? any smooth scheme over Z has a Zariski covering by smooth schemes which come from S. Thus any pointed sheaf over Z is a colimit of pointed sheaves of the form i^{∗}((W_{α})_{+}) where W_{α} are smooth schemes over S. The functor i^{∗} commutes with colimits because it is a left adjoint and i_{∗} commutes with colimits by Lemma ??. Thus it is sufficient to prove that i^{∗}i_{∗}i^{∗}G → i^{∗}G is an isomorphism for any G over S. Since i^{∗} and i_{∗} are adjoint functors the composition i^{∗}G → i^{∗}i_{∗}i^{∗}G → i^{∗}G where the first arrow is i^{∗}(G → i_{∗}i^{∗}G) is identity. On the other hand the first arrow is an epimorphism by Lemma ??. Therefore both arrows are isomorphisms.

To prove the second claim represent F as a colimit of a diagram of representable sheaves. Both j^{∗} and j_{#} are left adjoints and therefore commute with colimits. Thus it is sufficient to verify the case F = X_{+} where X is a smooth scheme over U. By construction of j_{#} we have j_{#}(X_{+}) = X_{+} where on the right hand side X is considered as a smooth scheme over S and j^{∗}j_{#}(X_{+}) = X ×_{S} U. Our claim follows now from the fact that the projection X ×_{S} U → X is an isomorphism.

The second part of this proposition togther with Lemma ?? imply:

Corollary 2.6 [c0] For any F, G over U the morphism j_{#}(F ∧ G) → j_{#}F ∧ j_{#}G given by the adjunction and isomorphism of Lemma ?? is an isomorphism.

Let j^{∗}j_{#} → Id be the natural transformation inverse to the isomorphism of Proposition ??(2). By adjunction it defines a natural transformation j_{#} → j_{∗}. One can immediately verify the following fact.

Lemma 2.7 [ves] Let j : U → S be an open embedding such that j(U) is a connected component of S. Then j_{#} → j_{∗} is an isomorphism.

Let now p : U → S be an etale morphism. Define a natural transformation p_{#} → p_{∗} as the adjoint to the natural transformation p^{∗}p_{#} → Id given by the composition

p^{∗}p_{#}(F) = (pr_{2})_{#}pr_{1}^{∗}(F) → (pr_{2})_{#}Δ_{∗}(F) ≅ (pr_{2})_{#}Δ_{#}(F) ≅ F

where the first arrow is the isomorphism of Lemma ?? for the square

U ×_{S} U →^{pr_{1}} U
pr_{2}↓  ↓
U  → S

the second is obtained from the composition pr_{1}^{∗}(F) → Δ_{∗}Δ^{∗}pr_{1}^{∗}(F) ≅ Δ_{∗}(F) where Δ : U → U ×_{S} U is the diagonal and the third is the isomorphism of Lemma ??.

Proposition 2.8 [l4] Let i_{U} : Z → U be a closed embedding and p : U → S an etale morphism such that the composition i_{S} = p∘i_{U} is again a closed embedding. Then for any F over Z the composition p_{#}(i_{U})_{∗}(F) → p_{∗}(i_{U})_{∗}(F) = (i_{S})_{∗}(F) is an isomorphism.

Proof: Follows from Lemmas ??, ?? and ??.

Proposition 2.9 [clconst] Let i : Z → S be a closed embedding and j : U → S the complimentary open embedding. For any pointed simplicial set K considered as an object over S the canonical square

j_{#}j^{∗}(K) → K
↓  ↓
pt  → i_{∗}i^{∗}(K)

is a push-forward square.

Proof: It follows from Lemmas ??, ?? and ??.

Example 2.10 The statements of Propositions ?? and ??(1) would be false if we considered the category Sm/S with Zariski topology instead of the Nisnevich one. Let S be the spectrum of a local non-henselian ring and i : Spec(k) → S the embedding of the closed point. Let further U → S be a local scheme etale over S such that U ×_{S} Spec(k) = ∐_{i=1}^{n} U_{i} where U_{i} are connected and n > 1. Then S^{0}(U) = S^{0} and i_{∗}i^{∗}(S^{0})(U) = ∨_{i=1}^{n}S^{0} and thus the morphism S^{0} → i_{∗}i^{∗}(S^{0}) is not an epimorphism.

In the following examples S = Spec(A) is the spectrum of a henselian local ring A and i : Z → S is the embedding of the closed point Z = Spec(A/m).

Example 2.11 [noncocart]Consider the pointed sheaf of sets (A^{1}, 0) on (Sm/S)_{Nis}. Then the square

j_{#}j^{∗}(F) → F
↓  ↓
pt  → i_{∗}i^{∗}(F)

is not a pushforward square. Indeed j_{#}j^{∗}(F)(S) = pt, (F/j_{#}j^{∗}(F))(S) = A and i_{∗}i^{∗}(F)(S) = A/m.

Example 2.12 [nonbf]An explicit computation shows that for S and Z as above one has Li^{∗}i_{∗}(A^{1}, 0) ≅ (A^{1}, 0)×B_{simpl}G_{a} i.e. the canonical morphism Li^{∗}i_{∗}(F) → F is not a simplicial weak equivalence for general F.

Theorem 2.13 [gluing] Let i : Z → S be a closed embedding, j : U → S the complimentary open embedding and F a left admissible object over S. Then the adjunction j_{#}j^{∗}(F) → F is a monomorphism and the square

j_{#}j^{∗}(F) → F
↓  ↓
pt  → i_{∗}i^{∗}(F)

is A^{1}-homotopy cocartesian i.e. the canonical morphism F/j_{#}j^{∗}(F) → i_{∗}i^{∗}(F) is an A^{1}-weak equivalence.

Proof: This is the pointed version of [?, Th. 3.2.21]. One can verify it using Remark ??.

Proposition 2.14 [p2] Let F be an object over Z. Then the composition Li^{∗}i_{∗}(F) → i^{∗}i_{∗}(F) → F is an A^{1}-weak equivalence.

Proof: Let us consider first the case when F = i^{∗}G for a left admissible object G over S. Consider the commutative diagram

i^{∗}Lres(G/j_{#}j^{∗}G) → i^{∗}Lres(i_{∗}i^{∗}G)
↓  ↓
i^{∗}(G/j_{#}j^{∗}G) →  i^{∗}i_{∗}i^{∗}G
↘  ↓
i^{∗}G

We have to show that the composition of the right vertical arrows is an A^{1}- weak equivalence. By Theorem ?? the canonical morphism G/j_{#}j^{∗}(G) → i_{∗}i^{∗}(G) is an A^{1}-weak equivalence. Thus by Lemma ?? the upper horizontal arrow is an A^{1}-weak equivalence. By Lemma ?? and our definition of left admissible objects the left hand side is left admissible. Thus by Lemma ?? the left vertical arrow is a simplicial weak equivalence. The composition of the slanted arrow with the canonical morphism i^{∗}G → i^{∗}(G/j_{#}j^{∗}G) is identity by the definition of adjoint functors. This morphism is an isomorphism since i^{∗} commutes with coproducts and i^{∗}j_{#} = pt by Lemma ?? and thus the slanted arrow is an isomorphism which finishes the proof for F of the form i^{∗}G. To prove the proposition for all F we need the following two lemmas.

Lemma 2.15 [closed3] Let i : Z → S be a closed embedding. Then for any object F over S there exists a diagram of the form (i^{∗}((W_{i})_{+}))_{i∈Δ^{op}} where W_{i} are smooth schemes over S and a simplicial weak equivalence hocolim_{i∈Δ^{op}}i^{∗}((W_{i})_{+}) → F.

Proof: Define a functor Lres_{S} : Δ^{op}Spc_{•}(Z) → Δ^{op}Spc_{•}(Z) and a natural transformation Lres_{S} → Id in the same way as we did with Lres in the proof of Lemma ?? but starting with smooth schemes of the form W ×_{S} Z for smooth quasi-projective schemes over S. Consider the composition

hocolim_{i∈Δ^{op}}Lres_{S}(F)_{i} → Lres_{S}(F) → F

The first arrow is a simplicial weak equivalence by Lemma ??. To show that the second one is a simplicial weak equivalence one uses the same argument as in the proof of Lemma ?? together with Lemma ??.

Lemma 2.16 [clhoco] Let p : Z → S be a finite morphism and (X_{i})_{i∈I} a small diagram over Z. There exists a natural (in X) isomorphism

Lp^{∗}p_{∗}hocolim_{i∈I}X_{i} → hocolim_{i∈I}Lp^{∗}p_{∗}X_{i}

in the simplicial homotopy category over Z such that the diagram

Lp^{∗}p_{∗}hocolim_{i∈I}X_{i} → hocolim_{i∈I}Lp^{∗}p_{∗}X_{i}
↓  ↙
hocolim_{i∈I}X_{i}

commutes.

Proof: Consider the diagram

Lp^{∗}p_{∗}hocolim_{i∈I}X_{i} ← Lp^{∗}hocolim_{i∈I}p_{∗}(X_{i}) → hocolim_{i∈I}Lp^{∗}p_{∗}(X_{i})
↓ ↓ ↓
p^{∗}p_{∗}hocolim_{i∈I}X_{i} ← p^{∗}hocolim_{i∈I}p_{∗}(X_{i}) → hocolim_{i∈I}p^{∗}p_{∗}(X_{i})
↘  ↙
hocolim_{i∈I}X_{i}

The first upper horizontal arrow is the isomorphism of Corollary ??. The second one is the isomorphism of Lemma ??. We define our isomorphism as the compostion of the inverse to the first one with the second. To prove commutativity of the triangle claimed in the lemma it is sufficient to prove commutativity of three squares in the diagram above. The upper left one is commutative since Lp^{∗} → p^{∗} is a natural transformation. The upper right one by Lemma ?? and the lower one by Lemma ??.

To finish the proof of Proposition ?? consider the simplicial weak equivalence hocolim_{i∈Δ^{op}}i^{∗}((W_{i})_{+}) → F constructed in Lemma ??. We have a commutative square

Li^{∗}i_{∗}hocolim_{i∈Δ^{op}}i^{∗}((W_{i})_{+}) → Li^{∗}i_{∗}F
↓  ↓
hocolim_{i∈Δ^{op}}i^{∗}((W_{i})_{+}) → F

The upper horizontal arrow is a simplicial weak equivalence by [?, Prop. 3.1.27] and Lemma ?? and the lower horizontal one by construction. Thus it is sufficient to show that the left vertical arrow is an A^{1}-weak equivalence. This follows from the first part of the proof, Lemma ?? and Lemma ??.

Corollary 2.17 [p3] Let F be an object over Z. Then the canonical morphism Li^{∗}i_{∗}(F) → i^{∗}i_{∗}(F) is an A^{1}-weak equivalence.

Proposition 2.18 [projform] Let i : Z → S be a closed embedding, F a left admissible object over S and G a pointed simplicial sheaf over Z. Then the morphism F ∧ i_{∗}G → i_{∗}(i^{∗}F ∧ G) defined by the adjunction and the isomorphism of Lemma ?? is an A^{1}-weak equivalence.

Proof: Consider first the case when G = i^{∗}F′ for a left admissible object F′ over S. We have the following commutative diagram of morphisms of sheaves

F ∧ (F′/j_{#}j^{∗}F′)  → F ∧ i_{∗}i^{∗}F′
↓  ↓
(F ∧ F′)/(F ∧ j_{#}j^{∗}F′)  i_{∗}(i^{∗}F ∧ i^{∗}F′)
↓  ↓
(F ∧ F′)/j_{#}j^{∗}(F ∧ F′) → i_{∗}i^{∗}(F ∧ F′)

where all the vertical arrows except for the upper right one are isomorphisms for obvious reasons (Lemmas ?? and ??). The upper horizontal arrow is an A^{1}-weak equivalence by Theorem ?? and the lower one by Lemma ?? and Theorem ??. Thus F ∧i_{∗}i^{∗}F′ → i_{∗}(i^{∗}F ∧i^{∗}F′) is an A^{1}-weak equivalence. To prove the case of an arbitrary G one uses Lemma ??, Corollary ??, Lemma ?? and [?, Prop. 3.1.27].

Corollary 2.19 [p0] Let i : Z → S be a closed embedding, j : U → S the complimentary open embedding and F a left admissible object over S. Then the morphism F ∧ i_{∗}(S^{0}) → i_{∗}i^{∗}(F) defined by the adjunction and isomorphism of Lemma ?? is an A^{1}-weak equivalence.

Proposition 2.20 [clbasechange] For a pull-back square

\begin{matrix}Z′ & →^{f_{Z}} & Z \\ i′ ↓ & & ↓i \\ S′ & →^{f_{S}} & S \end{matrix}

such that i is a closed embedding and a left admissible F on Z the composition Lf_{S}^{∗}i_{∗}(F) → f_{S}^{∗}i_{∗}(F) → i′_{∗}f_{Z}^{∗}(F) is an A^{1}-weak equivalence.

Proof: Consider first the case when F = i^{∗}G for a left admissible object G over S. Then we have a diagram

Lf_{S}^{∗}(G/j_{#}j^{∗}G) → f_{S}^{∗}(G/j_{#}j^{∗}G) → f_{S}^{∗}(G)/f_{S}^{∗}j_{#}j^{∗}G → f_{S}^{∗}(G)/j′_{#}(j′)^{∗}f_{S}^{∗}(G)
↓  ↓  ↓
Lf_{S}^{∗}i_{∗}i^{∗}(G) →  f_{S}^{∗}i_{∗}i^{∗}(G) →  i′_{∗}f_{Z}^{∗}i^{∗}G  →  i′_{∗}(i′)^{∗}f_{S}^{∗}(G)

where the left square is commutative because Lf_{S}^{∗} → f_{S}^{∗} is a natural transformation and the commutativity of the right hexagon can be easily verified from definitions. The left vertical arrow is an A^{1}-weak equivalence by Theorem ?? and Lemma ??. The first upper horizontal arrow is a simplicial weak equivalence by Lemma ?? and Lemma ??. Two other upper horizontal arrows and the right lower horizontal one are isomorphisms for obvious reasons. The right vertical arrow is an A^{1}-weak equivalence by Theorem ?? and Lemma ??. Thus the composition of the first two lower horizontal arrows is an A^{1}-weak equivalence.

To prove the case of a general F one uses Lemma ?? in a way similar to how it is used in the proof of Proposition ??.

Remark 2.21 It can be shown that in the notations of Proposition ?? the the morphism f_{S}^{∗}i_{∗}(F) → i′_{∗}f_{Z}^{∗} is an isomorphism for any F but since Lf_{S}^{∗}i_{∗}(F) → f_{S}^{∗}i_{∗}(F) is not generally a simplicial weak equivalence this has little use.

3 Duality for smooth quasi-projective morphisms

3.1 Formulation of the main theorem

Definition 3.1 Let p : X → S be a smooth morphism. The dualizing object of X over S is the pointed sheaf D_{X/S} = (X ×_{S} X)/(X ×_{S} X − Δ(X)) considered over X with respect to the projection to the second component.

Note that by Lemma ?? the dualizing object can be written as D_{X/S} = (pr_{2})_{#}Δ_{∗}(S^{0}) where pr_{2} : X ×_{S} X → X is the projection to the second component and Δ : X → X ×_{S} X is the diagonal.

Let p : X → S be a smooth morphism and p = p̄ ∘ j be a decomposition of p such that j : X → X̄ is an open embedding and p̄ : X̄ → S is any morphism. For any left admissible F over X we define a natural (in F) morphism in the A^{1}-homotopy category

β_{F} = β_{F}^{(j,p̄)} : p̄^{∗}p_{#}(F) → j_{#}(F ∧ D_{X/S})

as follows. Consider the following diagram

[dmain] \begin{matrix}X ×_{S} X & →^{p′′} & X \\ j′ ↓ & ↙^{Γ} & ↓j \\ X ×_{S} X̄ & →^{p′} & X̄ \\ p̄′ ↓ & & ↓p̄ \\ X & →^{p} & S \end{matrix}  (1)

where both squares are Cartesian and Γ is the closed embedding of the graph of j. We define β_{F} as the morphism in the A^{1}-homotopy category represented by the following diagram of morphisms of pointed simplicial sheaves

Lp′_{#}(p̄′)^{∗}F  Lp′_{#}j′_{#}((p′′)^{∗}F ∧ Δ_{∗}S^{0})
↓  ↘  ↓  ↘
p̄^{∗}p_{#}(F)  Lp′_{#}Γ_{∗}F  j_{#}(F ∧ D_{X/S})

where:

1. the left vertical arrow is the composition Lp′_{#}(p̄′)^{∗}F → p′_{#}(p̄′)^{∗}F → p̄^{∗}p_{#} where the second arrow is the isomorphism of Lemma ?? and the first one is a simplicial weak equivalence by Lemmas ?? and ??;

2. the left slanted arrow is obtained by applying Lp′_{#} to the morphism adjoint to the ismorphism Γ^{∗}(p̄′)^{∗} ≅ Id of Lemma ??

3. the middle vertical arrow is obtained by applying Lp′_{#} to the composition

j′_{#}((p′′)^{∗}F ∧ Δ_{∗}S^{0}) → j′_{#}(Δ_{∗}Δ^{∗}((p′′)^{∗}(F))) → j′_{#}Δ_{∗}F → Γ_{∗}F

where the first arrow is an A^{1}-weak equivalence by Corollary ?? and Lemma ??, the second is the isomorphism of Lemma ?? and the third is the isomorphism of Proposition ??;

4. the right slanted arrow is the composition

Lp′_{#}j′_{#}((p′′)^{∗}F ∧ Δ_{∗}S^{0}) → p′_{#}j′_{#}((p′′)^{∗}F ∧ Δ_{∗}S^{0})
↓
j_{#}p′′_{#}((p′′)^{∗}F ∧ Δ_{∗}S^{0}) → j_{#}(F ∧ D_{X/S})

where the first arrow is the canonical morphism, the second is an isomorphism of Lemma ?? and the third is the isomorphism of Lemma ??.

The following duality theorem is the main result of this paper.

Theorem 3.2 [main] For any smooth morphism p : X → S and any decomposition p = p̄∘ j such that j : X → X̄ is an open embedding and p̄: X̄ → S is a projective morphism there exists n ≥ 0 such that for any left admissible F on X the n-th T-suspension of the morphism

δ_{F} : p_{#}(F) → Rp̄_{∗}j_{#}(F ∧ D_{X/S})

adjoint to β_{F} is an isomorphism in H_{A^{1}}(S).

Consider the diagram ?? in the case when p is an open embedding and define for any F over X a natural morphism of sheaves ϵ_{F}^{(j,p̄)} : p̄^{∗}p_{#}(F) → j_{#}(F) as the composition

p̄^{∗}p_{#}(F) ≅ p′_{#}(p̄′)^{∗}(F) → p′_{#}Γ_{∗}Γ^{∗}(p̄′)^{∗}(F) ≅ p′_{#}Γ_{∗}(F) ≅ p′_{#}Γ_{#}(F) ≅ j_{#}(F) where the first isomorphism is from Lemma ??, the arrow is the adjunction, the second isomorphism is from Lemma ??, the third from Lemma ?? and the last one again from Lemma ??. Consider a diagram of the form

X̄′
↗^{k} ↓ q̄
X →^{j} X̄
↘^{p} ↓ p̄
S

and the composition p̄q̄ by r̄. Then for any left admissible F over X we have a square of morphisms in H_{A^{1}}(S)

q̄^{∗}p̄^{∗}p_{#}(F)  ⟶  r̄^{∗}p_{#}(F)
↓  ↓
q̄^{∗}j_{#}(F ∧ D_{X/S}) ⟶ k_{#}(F ∧ D_{X/S})

where the upper horizontal arrow is the isomorphism of Lemma ??, the left vertical arrow is q̄^{∗}(β_{F}^{(j,p̄)}), the right vertical arrow is β_{F}^{(j′,r̄)} and the lower horizontal arrow is ϵ_{(F ∧D_{X/S})}^{(k,q̄)}.

Proposition 3.3 [long] This square commutes.

Proof: Consider the diagrams of the form ?? for (j, p̄) and (k, r̄)

\begin{matrix}X ×_{S} X & →^{p′′} & X \\ j′ ↓ & ↙^{Γ_{j}} & ↓j \\ X ×_{S} X̄ & →^{p′_{1}} & X̄ \\ p̄′ ↓ & & ↓p̄ \\ X & →^{p} & S \end{matrix} \begin{matrix}X ×_{S} X & →^{p′′} & X \\ k′ ↓ & ↙^{Γ_{k}} & ↓k \\ X ×_{S} X̄′ & →^{p′_{2}} & X̄′ \\ r̄′ ↓ & & ↓r̄ \\ X & →^{p} & S \end{matrix}  (2)

and the pull-back square

X ×_{S} X̄′ →^{p′_{2}} X̄′
q̄′ ↓  ↓ q̄′
X ×_{S} X̄ →^{p′_{1}}  X̄

which connects them. We have:

q̄^{∗}p̄^{∗}p_{#}  ⟶  r̄^{∗}p_{#}
↑  ↑
q̄^{∗}(p′_{1})_{#}(p̄′)^{∗}  ←  (p′_{2})_{#}(q̄′)^{∗}(p̄′_{1})^{∗}  →  (p′_{2})_{∗}(r̄′)^{∗}
↑  ↑  ↑
q̄^{∗}L(p̄′_{1})_{#}(p̄′)^{∗}  ← L(p′_{2})_{#}L(q̄′)^{∗}(p̄′)^{∗} →  L(p′_{2})_{#}(r̄′)^{∗}
↓  ↓  ↓
q̄^{∗}L(p̄′_{1})_{#}(Γ_{j})_{∗}  ← L(p′_{2})_{#}L(q̄′)^{∗}(Γ_{j})_{∗} →  L(p′_{2})_{#}(Γ_{k})_{∗}
↑  ↑  ↑
q̄^{∗}L(p̄′_{1})_{#}j′_{#}((p′′)^{∗}∧Δ_{∗}S^{0}) ← L(p′_{2})_{#}L(q̄′)^{∗}j′_{#}((p′′)^{∗}∧Δ_{∗}S^{0}) → L(p′_{2})_{#}k′_{#}((p′′)^{∗}∧Δ_{∗}S^{0})
↓  ↓
q̄^{∗}j_{#}p′′_{#}((p′′)^{∗} ∧ Δ_{∗}S^{0}) ⟶ k_{#}p′′_{#}((p′′)^{∗} ∧ Δ_{∗}S^{0})
↓  ↓
q̄^{∗}j_{#}(F ∧ D_{X/S})  ⟶  k_{#}(F ∧ D_{X/S})

Corollary 3.4 In the notations of the proposition the following square of morphisms in H_{A^{1}} commutes

\begin{aligned}& p_{#}(F) & ⟶^{δ_{F}^{(j,p̄)}} & Rp̄_{∗}j_{#}(F ∧ D_{X/S}) \\ δ_{F}^{(k,r̄)} & ↓ & & ↓ \\ & Rr̄_{∗}k_{#}(F ∧ D_{X/S}) & → & Rp̄_{∗}Rq̄_{∗}k_{#}(F ∧ D_{X/S}) \end{aligned}

(here the right vertical arrow is Rp̄_{∗} of the composition j_{#}(F ∧ D_{X/S}) → q̄_{∗}k_{#}(F ∧ D_{X/S}) → Rq̄_{∗}k_{#}(F ∧ D_{X/S}) where the first arrow is the morphism adjoint to ϵ_{F ∧D_{X/S}}^{(k,q̄)}).

Proof: ???

For a pull-back square

\begin{matrix}X′ & →^{q} & S′ \\ g ↓ & & ↓ f \\ X & →^{p} & S \end{matrix}

and any F over S′ define a morphism Lp^{∗}Rf_{∗}(F) → Rg_{∗}Lq^{∗}(F) in the simplicial homotopy category over X by the following diagram

p^{∗}Lres(f_{∗}Rres(F)) ← p^{∗}Lres(f_{∗}Rres(Lres(F))) → p^{∗}f_{∗}(Rres(Lres(F))) →

→ g_{∗}q^{∗}(Rres(Lres(F))) → g_{∗}Rres(q^{∗}(Rres(Lres(F)))) ← g_{∗}Rres(q^{∗}Lres(F)) where the first left arrow is a simplicial weak equivalence by Lemmas ?? and ?? and the second left arrow is a simplicial weak equivalence by Lemmas ?? and ??.

Lemma 3.5 [l5d] For any morphism f : S_{1} → S_{2} and a simplicial (resp. A^{1}-) weak equivalence a : F → G over S_{2} the morphism Rf_{∗}(a) is a simplicial (resp. A^{1}-) weak equivalence.

Lemma 3.6 [mix] For a left admissible object F over S the object Rres(F) is left admissible.

Lemma 3.7 [smbc] Consider a pull-back square

\begin{matrix}X′ & →^{q} & S′ \\ g ↓ & & ↓ f \\ X & →^{p} & S \end{matrix}

such that p is a smooth morphism. Then for any F over S′ the morphism p^{∗}f_{∗}(F) → g_{∗}q^{∗}(F) defined by the adjunctions and Lemma ?? is an isomorphism.

Proof: ???

Lemma 3.8 [n1] In the notations of Diagram ?? assume that p is an open embedding. The for any F over X the morphism F → p^{∗}p̄_{∗}j_{#}(F) adjoint to ϵ_{F}^{(j,p̄)} is an isomorphism.

Proof: ???

Lemma 3.9 [n11] In the notations of Lemma ?? the canonical morphism p^{∗}p̄_{∗}j_{#}(F) → p^{∗}Rp̄_{∗}j_{#}(F) is a simplicial weak equivalence.

Proof: Consider the diagram

p^{∗}p̄_{∗}j_{#}(F)  →  p̄′_{∗}(p′)^{∗}j_{#}(F)
↓  ↓
p^{∗}p̄_{∗}Rres(j_{#}(F)) → p̄′_{∗}(p′)^{∗}Rres(j_{#}(F))

The horizontal arrows are ismorphisms by Lemma ?? and therefore it is sufficient to check that the right vertical arrow is a simplicial weak equivalence.

Assume first that F is right admissible. Then (p′)^{∗}j_{#}(F) = j′_{#}(p′′)^{∗}(F) = j′′_{∗}(p′′)^{∗}(F) is right admissible by Lemma ??, (p′)^{∗}Rres(j_{#}(F)) is right admissible by Lemma ?? and the morphism (p′)^{∗}j_{#}(F) → (p′)^{∗}Rres(j_{#}(F)) is a simplicial weak equivalence. Thus the right vertical arrow is a simplicial weak equivalence in this case by Lemma ??. On the other hand the morphism p̄′_{∗}(p′)^{∗}j_{#}(F) → p̄′_{∗}(p′)^{∗}j_{#}(Rres(F)) is a simplicial weak equivalence since p̄′_{∗}(p′)^{∗}j_{#} ≅ Id and the morphism p̄′_{∗}(p′)^{∗}Rres(j_{#}(F)) → p̄′_{∗}(p′)^{∗}Rres(j_{#}(Rres(F))) is a simplicial weak equivalence because j_{#} preserves simplicial weak equivalences by [?, Prop. 3.1.26], (p′)^{∗} takes right admissible objects to right admissible objects by Lemma ?? and p̄′_{∗} preserves simplicial weak equivalences bewteen right admissible objects by Lemma ??.

Lemma 3.10 [l6d] Let a : F → G be a simplicial (resp. A^{1}-) weak equivalence of right admissible objects over S. Then for any morphism f : S′ → S the morphism f_{∗}(a) is a simplicial (resp. A^{1}-) weak equivalence.

Lemma 3.11 [rtsm] Let F be a right admissible object and f : S′ → S be a smooth morphism. Then f^{∗}(F) is right admissible.

Lemma 3.12 [l7d] Let F be a right admissible object. Then for any morphism f : S′ → S the object f_{∗}(F) is right admissible.

Lemma 3.13 [n2] In the notations of Lemma ?? let i : Z → S be the reduced closed subscheme S − p(X) of S. Assume that for any F over X̄ the base change morphism Li^{∗}Rp̄_{∗}(F) → R(p̄_{Z})_{∗}L(i′)^{∗}(F) associated with the pull-back square

\begin{matrix}Z′ & →^{i′} & X̄ \\ p̄_{Z} ↓ & & ↓ p̄ \\ Z & →^{i} & S \end{matrix}

is an A^{1}-weak equivalence. Then for any left admissible F the first simplicial suspension of the composition p_{#}(F) → p̄_{∗}j_{#}(F) → Rp̄_{∗}j_{#}(F) where the first arrow is the morphism adjoint to ϵ_{F}^{(j,p̄)} is an A^{1}-weak equivalence.

Proof: Consider the diagram

p_{#}p^{∗}p_{#}(F) → p_{#}p^{∗}p̄_{∗}j_{#}(F) → p_{#}p^{∗}Rp̄_{∗}j_{#}(F)
↓ ↓ ↓
p_{#}(F) → p̄_{∗}j_{#}(F) → Rp̄_{∗}j_{#}(F)

where the upper line is obtained from the lower one by applying functor p_{#}p^{∗} and the vertical arrows come from the natural transformation p_{#}p^{∗} → Id. The left vertical arrow is an isomorphism by Proposition ??(2) and the composition of the inverse with the left upper horizontal arrow is p_{#} of the isomorphism of Lemma ??. The right upper horizontal arrow is a simplicial weak equivalence by Lemma ?? and [?, Prop. 3.1.26]. It remains to show that the first simplicial suspension of the right vertical arrow is an A^{1}-weak equivalence.

By [?, Prop. 3.1.26] and [?, Cor. 3.1.24] the functor p_{#}p^{∗} preserves simplicial weak equivalences. Thus it is sufficient to prove that the first simplicial suspension of the morphism p_{#}p^{∗}LresRp̄_{∗}j_{#}(F) → LresRp̄_{∗}j_{#}(F) is an A^{1}-weak equivalence. By Theorem ?? the canonical morphism

LresRp̄_{∗}j_{#}(F)/p_{#}p^{∗}LresRp̄_{∗}j_{#}(F) → i_{∗}i^{∗}LresRp̄_{∗}j_{#}(F)

is an A^{1}-weak equivalence. This by Lemma ?? it is sufficient to show that the right hand side is A^{1}-weakly equivalent to pt.

By the assumption of the Lemma there exists an isomorphism in the A^{1}-homotopy category of the form

i_{∗}i^{∗}LresRp̄_{∗}j_{#}(F) → i_{∗}R(p̄_{Z})_{∗}L(i′)^{∗}j_{#}(F)

By Lemma ?? j_{#}(F) is left admissible and thus the canonical morphism L(i′)^{∗}j_{#}(F) → (i′)^{∗}j_{#}(F) is a simplicial weak equivalence. By Lemma ?? (i′)^{∗}j_{#}(F) = pt since X ×X̄′ Z′ = ∅. Lemma is proven.

Lemma 3.14 [cone] Let i : F → G be a monomorphism such that the canonical morphism G/F → pt is an A^{1}-weak equivalence. Then the first simplicial suspension Id_{S_{s}^{1}} ∧ i is an A^{1}-weak equivalence.

Proof: ???