BLOCH-KATO CONJECTURE AND MOTIVIC

COHOMOLOGY WITH FINITE COEFFICIENTS

Andrei Suslin and Vladimir Voevodsky

Contents

Introduction

0. Notations, terminology and general remarks.

1. Homotopy invariant presfeaves with transfers.

2. Tensor structure on the category DM^{−}(F).

3. Motivic cohomology.

4. Fundamental distinguished triangles in the category DM^{−}(F).

5. Motivic cohomology of non-smooth schemes and the cdhtopology.

6. Truncated etale cohomology and the Beilinson-Lichtenbaum Conjecture.

7. The Bloch-Kato Conjecture.

8. B_{l}(n)-cohomology with Supports.

9. Shift of degrees.

10. Proof of the Main Theorem.

11. Bloch-Kato Conjecture and vanishing of the Bockstein homomorphisms.

12. Appendix. cdh-cohomological dimension of Noetherian schemes.

Introduction

§ 0. Notations, terminology and general remarks.

Throughout the paper we fix a field F and consider the category Sm/F of smooth separated schemes of finite type over F. We make Sm/F into a site using one of the following three topologies: Zariski topology, Nisnevich topology or etale topology. For any presheaf F on Sm/F we denote by F_{Zar}^{∼}, F_{Nis}^{∼} and F_{et}^{∼} the sheaf associated with F in Zariski, Nisnevich and etale topologies respectively.

For any site C we denote by C^{∼} (resp. C^{∧}) the category of abelian sheaves (resp. the category of abelian presheaves) on C.

We denote by Sch/F the category of all separated schemes of finite type over F. Let F : Sm/F → A be a (covariant) functor from Sm/F to an abelian category

A. Let further X ∈ Sm/F be a smooth scheme and let i : Y ↪ X be a smooth

Typeset by AMS-TEX

closed subscheme provided with a retraction r : X → Y (i.e. ri = 1_{Y}). In this case F(r)F(i) = 1_{F(Y)} and hence F(Y) is canonically a direct summand in F(X). We’ll use the notation F(X/Y) for the complementary direct summand. Thus F(X/Y) may be identified either with the kernel of the homomorphism F(r) : F(X) → F(Y) or with the cokernel of the homomorphism F(i) : F(Y) → F(X). We’ll use similar notations in case F : Sch/F → A is a functor from Sch/F to an abelian category A and Y ⊂ X is a retract of X ∈ Sch/F.

More generally, assume we are given a scheme X and n closed subschemes i_{j} : Y_{j} ↪ X (1 ≤ j ≤ n) (the schemes X, Y_{j} should be smooth in case of the category Sm/F). Assume further that we are given retractions r_{j} : X → Y_{j} (1 ≤ j ≤ n) such that the morphisms ρ_{j} = i_{j}r_{j} : X → X pairwise commute. In this case each of F(Y_{j}) may be identified with a direct summand in F(X). Moreover the sum ∑_{j=1}^{n} F(Y_{j}) ⊂ F(X) is also a canonical direct summand: the corresponding projection is given by the formula

1_{F(X)} − (1_{F(X)}−F(ρ_{1})) · . . . · (1_{F(X)} − F(ρ_{n})) =

= ∑_{s=1}^{n} ∑_{1≤j_{1}<...<j_{s}≤n}(−1)^{s−1}F(ρ_{j_{1}} · ... · ρ_{j_{s}}) : F(X) ⟶ ∑_{j=1}^{n} F(Y_{j}).

The above situation arises in particular in the following case. Let (Z, z_{0}) be a scheme provided with a distinguished rational point. Set X = Z^{×n}. The point z_{0} defines n embeddings of Z^{×(n−1)} in X . The corresponding subschemes Y_{j} ⊂ X consist of all points with j-th coordinate equal to z_{0}, the corresponding morphisms ρ_{j} : X → X are given by the formula

ρ_{j}(z_{1}, ..., z_{n}) = (z_{1}, ..., z_{j}_{0}, ..., z_{n})

and obviously pairwise commute. We’ll use the notation F(Z^{∧n}) for the direct summand of F(Z^{×n}) complementary to ∑_{j=1}^{n} F(Y_{j}). Throughout the paper this kind of notation is constantly used in the special case when Z = 𝔾_{m} = 𝔸^{1} \ 0 and z_{0} = 1 is the identity of the group scheme 𝔾_{m}.

The previous construction has an obvious analogue for contravariant functors. In this case F(X/Y) may be identified either with the kernel of the homomorphism F(i) : F(X) → F(Y) or with the cokernel of the homomorphism F(r) : F(Y) → F(X). Furthermore the direct summand F(Z^{∧n}) ⊂ F(Z^{×n}) may be identified with the intersection of kernels of restriction homomorphisms

F(Z^{×n}) ⟶^{F(i_{j})} F(Z^{×(n−1)}).

The standart cosimplicial scheme Δ_{F}^{•}.

Recall that Δ_{F}^{n} is a closed subscheme in 𝔸_{F}^{n+1} defined by the equation t_{0} + ... + t_{n} = 1. We’ll refer to the points v_{i} = (0, ..., 1_{i}, ..., 0) ∈ Δ^{n} (0 ≤ i ≤ n) as vertices

of Δ_{n}. Each nondecreasing map ϕ : [m] = {0, 1, ..., m} → [n] = {0, 1, ..., n} defines the corresponding morphism of schemes

Δ^{m} ⟶ Δ^{n} : (t_{0}, ..., t_{m}) ↦ t_{0}v_{ϕ(0)} + ... + t_{m}v_{ϕ(m)}.

There are n + 1 coface morphisms d^{i} : Δ^{n−1} → Δ^{n} (0 ≤ i ≤ n) (corresponding to strictly increasing maps [n−1] → [n]). These coface morphisms are obviously closed embeddings, the corresponding closed subschemes of Δ^{n} are defined by equations t_{i} = 0 and are called the (codimension one) faces of Δ^{n}.

For any presheaf of abelian groups F : Sm/F → Ab one can consider a simplicial presheaf C_{∗}(F) defined by the formula C_{n}(F)(U) = F(U × Δ^{n}). It’s easy to see that homology presheaves H_{i} of the complex C_{∗}(F) are homotopy invariant - i.e. H_{i}(U × 𝔸^{1}) = H_{i}(U) for any U ∈ Sm/F (see [S-V § 7]). In the same way one can construct the bisimplicial presheaf C_{∗,∗}(F), defined by the formula C_{p,q}(F)(U) = F(U ×Δ^{p} ×Δ^{q}). Homotopy invariance of the cohomology presheaves of C_{∗}(F) implies easily that the natural embedding i : C_{∗}(F) ↪ T ot C_{∗,∗}(F) is a quasiisomorphism. The quasiisomorphism i has a canonical left inverse called the shuffle map (cf. [D] ch. 6, § 12), whose construction we are going to remind.

Every strictly increasing map (ϕ, ψ) : [p + q] → [p] × [q] defines a linear isomorphism of schemes

Δ^{p+q} → Δ^{p} × Δ^{q} (t_{0}, ..., t_{p+q}) ↦ t_{0} · (v_{ϕ(0)}, v_{ψ(0)}) + ... + t_{p+q} · (v_{ϕ(p+q)}, v_{ψ(p+q)})

and hence gives an isomorphism of presheaves (ϕ, ψ)^{∗} : C_{p,q}(F) → C_{p+q}(F). Note further that strictly increasing maps (ϕ, ψ) : [p + q] → [p] × [q] are in one to one correspondence with (p, q) shuffles: each (p, q)-shuffle σ defines a map (ϕ, ψ) via the formula

ϕ(x) = |{1 ≤ i ≤ p : σ(i) ≤ x}|
ψ(x) = |{p + 1 ≤ i ≤ p + q : σ(i) ≤ x}|.

We’ll use the notation σ^{∗} : C_{p,q}(F) ⟶^{∼} C_{p+q}(F) for the isomorphism of presheaves defined by the strictly increasing map (ϕ, ψ) corresponding to σ. The shuffle map ∇ : T ot C_{∗,∗}(F) → C_{∗}(F) is defined via the formula

∇_{p,q} = ∑_{σ−(p,q)−shuffle} ϵ(σ)σ^{∗} : C_{p,q}(F) → C_{p+q}(F).

A well known and easy computation shows that ∇ is a homomorphism of complexes which is left inverse to i (i.e. ∇i = 1_{C_{∗}(F)}). It’s not difficult to see that moreover the composition i∇ is homotopic to identity (so that i and ∇ are mutually unverse homotopy equivalences), but we won’t need this fact.

Resolution of singularities

Definition 0.1. We’ll be saying that resolution of singularities holds over the field F , provided that the following two conditions are satisfied.

(1) For any integral separated scheme of finite type X over F there exists a proper birational morphism Y → X with Y smooth (over F).

(2) For any smooth integral scheme X over F and any birational proper morphism Y → X there exists a tower of morphisms X_{n} → X_{n−1} → ... → X_{0} = X, each stage of which is a blow up with a smooth center, and such that the composition morphism X_{n} → X may be factored through Y → X.

It’s well known that resolution of singularities holds over fields of characteristic zero, whether it holds over (perfect) fields of positive characteristic remains one of the central problems of algebraic geometry. The following important theorem due to de Jong may be used sometimes as a replacement of the first of the above properties .

Theorem 0.2 [J]. For any field F and any integral separated scheme of finite type X over F there exists a proper surjective morphism Y → X with Y smooth over F.

It should be noted that this theorem implies that all results of [S-V] hold over an algebraically closed field of arbitrary characteristic.

Hypercohomology.

Motivic cohomology are defined as Zariski (or Nisnevich) hypercohomology with coefficients in a complex of sheaves which is not bounded below. We find it necessary to remind the definition and properties of hypercohomology in this generality.

Let A be an abelian category with enough injectives. Let further F : A → Ab be a left exact additive functor. Denote the right derived functors of F by R^{i}F. Let finally A^{•} be a complex of degree +1 in A and let A^{•} → I^{••} be the Cartan-Eilenberg resolution of A^{•}. The hypercohomology groups R^{∗}F(A^{•}) are defined as cohomology of the total complex corresponding to the bicomplex F(I^{••})

R^{∗}F(A^{•}) = H^{∗}(T ot(F(I^{••}))).

The following two results concerning hypercohomology constitute a minor generalization of well-known facts. For the sake of completness we sketch the proof below.

Theorem 0.3. Assume that either the complex A^{•} is bounded below or the functor F is of finite cohomological dimension (i.e. R^{i}(F) = 0 for i >> 0). Then both spectral sequences of the bicomplex F(I^{••}) are strongly convergent. Thus we have two hypercohomology spectrtal sequences

I_{1}^{pq} = R^{p}F(A^{q}) ⇒ R^{p+q}F(A^{•})
II_{2}^{pq} = R^{q}F(H^{p}(A^{•})) ⇒ R^{p+q}F(A^{•}).

Proof. The case when the complex A^{•} is bounded below is well-known so we’ll consider only the case when the functor F is of finite cohomological dimension. Let 0 → A^{•} ⟶^{ϵ} I^{•0} ⟶^{d′} I^{•1} ⟶^{d′} … be the Cartan-Eilenberg resolution of A^{•}. Define the complex A^{•n} as the kernel of the homomorphism d′ : I^{•n} → I^{•n+1}. The following properties of the complex A^{•n} are straightforward from definitions.

(0.3.1). For each i we have exact sequences

0 → A^{i} → I^{i,0} → … → I^{i,n−1} → A^{i,n} → 0
0 → Z^{i}(A^{•}) → Z^{i}(I^{•0}) → … → Z^{i}(I^{•n−1}) → Z^{i}(A^{•n}) → 0
0 → B^{i}(A^{•}) → B^{i}(I^{•0}) → … → B^{i}(I^{•n−1}) → B^{i}(A^{•n}) → 0
0 → H^{i}(A^{•}) → H^{i}(I^{•0}) → … → H^{i}(I^{•n−1}) → H^{i}(A^{•n}) → 0

We conclude immediately from (0.3.1) the following further properties of the complex A^{•n}.

(0.3.2). 0 → A^{•n} ⟶ I^{•n} ⟶^{d′} I^{•n+1} ⟶^{d′} … is the Cartan-Eilenberg resolution of the complex A^{•n}.

(0.3.3). Assume that n ≥ cd F then all objects A^{i,n}, Z^{i}(A^{•n}), B^{i}(A^{•n}), H^{i}(A^{•n}) are F-acyclic.

Lemma 0.3.4. Let F be a functor of finite cohomological dimension d. Let further A^{•}be a complex such that all objects A^{i}, H^{i}(A^{•}) are F-acyclic, and let 0 → A^{•} → I^{••} be the Cartan-Eilenberg resolution of A^{•}, then

(1) The objects Z^{i}(A^{•}), B^{i}(A^{•}) are also F-acyclic for all i.

(2) Z^{i}(F(A^{•})) = F(Z^{i}(A^{•})), B^{i}(F(A^{•})) = F(B^{i}(A^{•})), H^{i}(F(A^{•})) = = F(H^{i}(A^{•})) for all i.

(3) F(A^{•}) ⟶ T ot F(I^{••}) is a quasiisomorphism (F-acyclicity of A^{i} (for all i) alone is enough for this conclusion).

Proof. To prove (1) one shows by inverse induction on n ≥ 1 that R^{n}F(Z^{i}(A^{•})) = R^{n}F(B^{i}(A^{•})) = 0 for all i. The point (2) is immediate from (1). To prove (3) one notes that the i-th coloumn of the bicomplex F(I^{••}) is a resolution of F(A^{i}) which is enough to conclude that F(A^{•}) ⟶ T ot F(I^{••}) is a quasiisomorphism.

To conclude the proof of the Theorem 0.3 we denote by τ_{≤n}I^{••} the subbicomplex 0 → I^{•0} ⟶ I^{•1} ⟶ … ⟶ I^{•n−1} ⟶ A^{•n} → 0 of I^{••}. Taking n ≥ cd F we see immediately from the results proved above that the embedding τ_{≤n}I^{••} ↪ I^{••} induces a quasiisomorphism of complexes T ot F(τ_{≤n}I^{••}) ⟶^{∼} T ot F(I^{••}). Moreover the first (respectively the second) spectral sequences of the bicomplexes F(τ_{≤n}I^{••}) and F(I^{••}) coincide from E_{1}-term on (resp. from E_{2}-term on).

In case the functor F coincides with Hom_{A}(X, −) the corresponding hypercohomology groups are ususlly denoted RHom_{A}^{∗}(X, A^{•}). They may be interpreted as appropriate Hom-groups in the derived category D(A).

Proposition 0.4. Assume that either the complex A^{•} is bounded below or Ext^{i}(X, −) = 0 for i >> 0. Then RHom_{A}^{i}(X, A^{•}) = Hom_{D(A)}(X, A^{•}[i]).

Proof. The case when the complex A^{•} is bounded below is well-known, so we’ll consider only the case when the cohomological dimension of the functor Hom_{A}(X, −) = F is finite. Note first of all that according to definitions

Hom_{D(A)}(X, A^{•}[i]) = \underrightarrow{lim}_{A^{•}⟶^{∼}B^{•}} Hom_{K(A)}(X, B^{•}[i]) = \underrightarrow{lim}_{A^{•}⟶^{∼}B^{•}} H^{i}(Hom_{A}(X, B^{•}))

where direct limit is taken over all quasiisomorphisms A^{•} ⟶^{∼} B^{•}, K(A) denotes the homotopy category of complexes and Hom_{A}(X, B^{•}) is the complex of abelian groups ... → Hom_{A}(X, B^{i}) ⟶ Hom_{A}(X, B^{i+1}) → ... .

Lemma 0.4.1. Let F be a left exact additive functor of finite cohomological dimension d, then

(1) For any complex A^{•} there exists a quasiisomorphism A^{•} → B^{•}, where the terms B^{i} are F-acyclic for all i.

(2) Let B^{•} ⟶^{f} C^{•} be a quasiisomorphism of complexes. Assume that all terms B^{i} and C^{i} are F-acyclic. Then the induced homomorphism of complexes of abelian groups F(f) : F(B^{•}) ⟶ F(C^{•}) is also a quasiisomorphism.

Proof. To prove the point (1) it suffices to note that A^{•} ⟶ T ot τ_{≤n}I^{••} is always a quasiisomorphism. Moreover the terms of the complex T ot τ_{≤n}I^{••} are F-acyclic for any left exact additive functor F of cohomological dimension≤ d provided that n ≥ d.

To prove the second statement denote by co(f) the cone of f. The complex co(f) is acyclic (since f is a quasiisomorphism) and consists of F-acyclic terms. Lemma 0.3.4 shows that the complex co(F(f)) = F(co(f)) is acyclic as well and hence F(f) is a quasiisomorphism.

Lemma 0.4.1 shows that

Hom_{D(A)}(X, A^{•}[i]) = H^{i}(Hom_{A}(X, B^{•}))

for any quasiisomorphism A^{•} → B^{•} from A^{•} to a complex with Hom_{A}(X, −)-acyclic terms. Applying this to the complex T ot τ_{≤n}I^{••} (with n >> 0) we easily conclude the proof of Proposition 0.4.

Ext-groups For any two complexes A^{•}, B^{•} ∈ D(A) we set

Ext_{A}^{i}(A^{•}, B^{•}) = Hom_{D(A)}(A^{•}, B^{•}[i]).

It’s well-known that in case A^{•}, B^{•} ∈ D^{−}(A) (resp. ∈ D^{+}(A), D^{b}(A)) the corresponding Ext-groups coincide with Hom-groups in the category D^{−}(A) (resp. in D^{+}(A), D^{b}(A)). Infinite direct sums in the derived categories.

Let A be an abelian category satisfying the Grothendieck’s axiom Ab-?. Below we’ll discuss infinite direct sums in the category D^{−}(A). A similar discussion applies with minimal modifications to D^{+}(A) and D^{b}(A) as well.

Lemma 0.5. Let {A_{i}^{•}}_{i∈I} be a family of complexes. Any quasiisomorphism C^{•} → ⨁_{i∈I} A_{i}^{•} may be dominated (in the category K(A)) by a quasiisomorphism of the

form ⨁_{i∈I} C_{i}^{•} ⟶^{⊕f_{i}} ⨁_{i∈I} A_{i}^{•} (where each f_{i} : C_{i}^{•} → A_{i}^{•} is a quasiisimorphism).

Proof. A well-known property of the homotopy category K(A) shows that for each i there exists a complex C_{i}^{•} a quasiisomorphism f_{i} : C_{i}^{•} → A_{i}^{•} and a morphism g_{i} : C_{i}^{•} → C^{•} which make the following diagram commute up to homotopy.

C_{i}^{•} ⟶^{g_{i}} C^{•}
f_{i}↓  ↓
A_{i}^{•} ⟶^{in_{i}} ⨁_{i∈I} A_{i}^{•}

A family of morphisms g_{i} : C_{i}^{•} → C^{•} defines a morphism g = (g_{i})_{i∈I} : ⨁ C_{i}^{•} → C^{•} and according to the construction the following diagram commutes up to homotopy

⨁_{i∈I} C_{i}^{•} ⟶^{g} C^{•}
⊕f_{i}↓  ↓
⨁_{i∈I} A_{i}^{•} ⟶^{=} ⨁_{i∈I} A_{i}^{•}

Corollary 0.5.1. For any complex B^{•} we have a natural isomorphism

Hom_{D(A)}(⨁_{i∈I} A_{i}^{•}, B^{•}) = ∏_{i∈I} Hom_{D(A)}(A_{i}^{•}, B^{•}).

In other words ⨁_{i∈I} A_{i}^{•} is a direct sum of complexes A_{i}^{•} in the category D(A) as well.

Proof. In view of definitions and Lemma 0.5 we have the following identifications

Hom_{D(A)}(⨁_{i∈I} A_{i}^{•}, B^{•}) = \underrightarrow{lim} ; C^{•}⟶^{∼}⨁_{i∈I} A_{i}^{•} Hom_{K(A)}(C^{•}, B^{•}) =

= \underrightarrow{lim}_{\{C_{i}^{•}⟶^{∼}A_{i}^{•}\}_{i∈I}} ∏_{i∈I} Hom_{K(A)}(C_{i}^{•}, B^{•}) = ∏_{i∈I} Hom_{D(A)}(A_{i}^{•}, B^{•}).

We’ll be saying that the family of complexes {A_{i}^{•}}_{i∈I} is uniformly bounded above (resp. uniformly homologically bounded above) iff there exists an integer N such that A_{i}^{n} = 0 (resp. H^{n}(A_{i}^{•}) = 0) for all i ∈ I and all n ≥ N.

Corollary 0.5.2. A direct sum of a family of complexes {A_{i}^{•}}_{i∈I} ∈ D^{−}(A) exists in D^{−}(A) if and only if this family is uniformly homologically bounded above.

Let {A_{i}^{•}}_{i∈I} ∈ D^{−}(A) be a family of complexes. We’ll denote by < {A_{i}^{•}} >⊂ D^{−}(A) the minimal full triangulated subcategory closed with respect to taking direct sums of uniformly homologically bounded above families of complexes. We’ll be saying that the family {A_{i}^{•}}_{i∈I} weakly generates the category < {A_{i}^{•}}_{i∈I} >.

Lemma 0.6. Let A^{••} be a bounded above bicomplex in A. Then

T ot(A^{••}) ∈< {A^{•i}}_{i∈ℤ} >⊂ D^{−}(A).

Proof. Denote by σ_{≥n}A^{••} ⊂ A^{••} the subbicomplex of A^{••} consisting of terms A^{ij} with j ≥ n. Denote further by i_{n} : σ_{≥n}A^{••} ⟶ A^{••} and j_{n} : σ_{≥n}A^{••} ⟶ σ_{≥n−1}A^{••} the obvious embeddings. The complexes T ot(σ_{≥n}A^{••}) obviously lie in the triangulated subcategory generated by all A^{•i}. Furthermore we have a short exact sequence of bicomplexes

0 → ⨁_{n} σ_{≥n}A^{••} ⟶^{(1_{σ_{≥n}A^{••}} −j_{n})} ⨁_{n} σ_{≥n}A^{••} ⟶^{(i_{n})} A^{••} → 0

which shows that T ot(A^{••}) lies in the subcategory weakly generated by A^{•i}.

§ 1. Homotopy invariant presheaves with transfers.

For any X, Y ∈ Sm/F define Cor(X, Y) to be the free abelian group generated by closed integral subschemes Z ⊂ X ×_{F} Y which are finite and surjective over a component of X. Let X, Y, W ∈ Sm/F be smooth schemes and let Z ∈ Cor(X, Y), T ∈ Cor(Y, W) be cycles on X × Y and Y × W each component of which is finite and surjective over a component of X (respectively over a component of Y). One checks easily that the cycles Z × W and X × T intersect properly on X × Y × W and each component of the intersection cycle (Z × W)_{•}(X × T) is finite and surjective over a component of X. Thus setting T ∘ Z = (pr_{1,3})_{∗}((Z × Y)_{•}(X × T)) we get a bilinear composition map

Cor(Y, W) × Cor(X, Y) → Cor(X, W).

In this way we get a new category (denoted SmCor/F) whose objects are smooth schemes of finite type over F and Hom_{SmCor/F} (X, Y) = Cor(X, Y) – see [V 1] for details.

The category SmCor/F is clearly additive, the direct sum of two schemes being given by their disjoint union. A presheaf with transfers on the category Sm/F is defined as a contravariant additive functor F : SmCor/F → Ab. Note that associating to each morphism f : X → Y its graph Γ_{f} ∈ Cor(X, Y) we get a canonical map Hom_{Sm/F} (X, Y) → Cor(X, Y). In this way we get a canonical functor Sm/F → SmCor/F, which allows to view presheaves with transfers on Sm/F as presheaves in the usual sence equipped with appropriate additional data.

For any X ∈ Sm/F we’ll denote by ℤ_{tr}(X) the corresponding representable functor, i.e. ℤ_{tr}(X) is a presheaf with transfers defined via the formula

ℤ_{tr}(X)(U) = Cor(U, X).

One checks easily that the pesheaf ℤ_{tr}(X) is actually a sheaf in the etale topology (and a fortiory in Zariski and Nisnevich topologies as well). Direct sums of such sheaves are sometimes called free presheaves with transfer or free Nissnevich sheaves with transfers. This terminology is not very consistent since sheaves ℤ_{tr}(X) are obviously projective objects in the category of presheaves with transfers, but clearly not in the category of Nisnevich sheaves with transfers.

In a similar way we may associate a presheaf with transfers ℤ_{tr}(X) to each scheme of finite type X ∈ Sch/F. To be more precise for each U ∈ Sm/F we define ℤ_{tr}(X)(U) to be the free abelian group generated by closed integral subschemes Z ⊂ U × X which are finite and surjective over a component of U. The same construction as above defines (for any U, V ∈ Sm/F) the bilinear pairing

Cor(V, U) × ℤ_{tr}(X)(U) ⟶ ℤ_{tr}(X)(V)

which makes ℤ_{tr}(X) into a presheaf with transfers. Moreover the presheaf with transfers ℤ_{tr}(X) is an etale sheaf for any X ∈ Sch/F. Obviously ℤ_{tr} is a functor from Sch/F to the category of (etale) sheaves with transfer.

Consider the standart cosimplicial object Δ^{•} = Δ_{F}^{•} in Sm/F. For any presheaf of abelian groups F on Sm/F we get a simplicial presheaf C_{•}(F), by setting C_{n}(F)(U) = F(U × Δ^{n}). We’ll use the same notation C_{•}(F) for the corresponding complex (of degree −1) of abelian presheaves. Usually we’ll be dealing with complexes of degree +1, in particular, we reindex the complex C_{•}(F) (in the standart way), by setting

C^{i}(F) = C_{−i}(F).

Recall that a presheaf F : Sm/F → Ab is said to be homotopy invariant, provided that for any U ∈ Sm/F the natural homomorphism F(U) → → F(U × 𝔸^{1}) is an isomorphism. It’s easy to see ( [S-V 1] Corollary 7.5 ) that cohomology presheaves of the complex C^{•}(F) are homotopy invariant.

The following result sums up some of the basic properties of homotopy invariant presheaves with transfers.

Theorem 1.1 [V 1]. Let F be a homotopy invariant presheaf with transfers on Sm/F, then

(1) The sheaf F_{Zar}^{∼} coincides with F_{Nis}^{∼} and has a natural structure of a homotopy invariant sheaf with transfers.

(2) For any X ∈ Sm/F and any i ≥ 0 H_{Zar}^{i}(X, F_{Zar}^{∼}) = H_{Nis}^{i}(X, F_{Nis}^{∼})

(3) The presheaves X ↦ H_{Zar}^{i}(X, F_{Zar}^{∼}) = H_{Nis}^{i}(X, F_{Nis}^{∼}) are homotopy invariant (and have a natural structure of presheaves with transfers).

Corollary 1.1.1. Let C^{•} be a complex (of degree +1) of Nisnevich sheaves with transfers with homotopy invariant cohomology presheaves. Then for any scheme X ∈ Sm/F we have a natural isomorphism of hypercohomology groups H_{Zar}^{∗}(X, C^{•}) = H_{Nis}^{∗}(X, C^{•}).

Proof. Note that cohomological dimension of X with respect to both Zariski and Nisnevich topology is finite (and equal to dim X). This remark and Theorem 0.3 give us two convergent hypercohomology spectral sequences (in which H^{q} denotes the q-th cohomology presheaf of C^{•})

ZarII_{2}^{pq} = H_{Zar}^{p}(X, (H^{q})_{Zar}^{∼}) ⇒ H_{Zar}^{p+q}(X, C^{•})
NisII_{2}^{pq} = H_{Nis}^{p}(X, (H^{q})_{Nis}^{∼}) ⇒ H_{Nis}^{p+q}(X, C^{•})

Moreover we have a natural homomorphism of spectral sequences _{Zar}II → _{Nis}II. Theorem 1.1 shows that the map on E_{2} terms is an isomorphism, hence the map on the limits is an isomorphism as well.

Nisnevich topology is much more convenient in dealing with presheaves with transfers as one sees from the following Lemma (which fails in case of the Zariski topology).

Lemma 1.2 [V 1]. Let F be any presheaf with transfers on Sm/F. Then there exists the unique structure of a Nisnevich sheaf with transfers on F_{Nis}^{∼} making the homomorphism F → F_{Nis}^{∼} into a homomorphism of presheaves with transfers.

For any Nisnevich sheaf with transfers G and any homomorphism of presheaves with transfers F → G the associated homomorphism F_{Nis}^{∼} → G is compatible with transfers.

Lemma 1.2 implies easily that Nisnevich sheaves with transfers form an abelian category which we denote NSwT/F.

Lemma 1.3. Homotopy invariant Nisnevich sheaves with transfers form a full abelian subcategory in NSwT/F, closed under taking kernels and cokernels of morphisms and under extensions.

Proof. Let f : F → G be a homomorphism of homotopy invariant sheaves with transfers. Denote by K and C the kernel and cokernel of f in the category of presheaves. Then K and C obviously are homotopy invariant presheaves with transfers. Furthermore K is a sheaf and coincides with Ker f, whereas Coker f = C_{Nis}^{∼}. Theorem 1.1 (1) shows now that Coker f is homotopy invariant. Finally let 0 → F → H → G → 0 be an extension of Nisnevich sheaves with transfers and assume that F and G are homotopy invariant. For any U ∈ Sm/F we have a commutative diagram with exact rows

0 →  F(U) ⟶ H(U) ⟶ G(U) ⟶ H_{Nis}^{1}(U, F)
=^{∼}↓  ↓  =^{∼}↓  =^{∼}↓
0 → F(U × 𝔸^{1}) ⟶ H(U × 𝔸^{1}) ⟶ G(U × 𝔸^{1}) ⟶ H_{Nis}^{1}(U × 𝔸^{1}, F)

Theorem 1.1 (3) shows that the right hand side vertical map is an isomorphism so that homotopy invariance of H follows from the five lemma.

Homotopy invariance of the cohomology presheaves of the complexes C^{•}(F) implies immediately the following fact.

Lemma 1.4. Let A^{•} be a bounded above complex of Nisnevich sheaves with transfers. All cohomology presheaves of the complex T ot C^{•}(A^{•}) are homotopy invariant. Hence all cohomology sheaves of T ot C^{•}(A^{•}) are homotopy invariant as well.

A convenient framework for dealing with homotopy invariant sheaves with transfers is provided by the tensor triangulated category DM^{−}(F). Recall (see [V 2]) that DM^{−}(F) is defined as the full subcategory of the derived category D^{−}(NSwT/F) of bounded above complexes of Nisnevich sheaves with transfers comprising the complexes with homotopy invariant cohomology sheaves. Lemma 1.4 shows that for any bounded above copmplex of Nisnevich sheaves with transfers A^{•} the complex C^{•}(A^{•}) is in DM^{−}(F). For any X ∈ Sm/F we define its motive M(X) as the element C^{•}(ℤ_{tr}(X)) ∈ DM^{−}(F). The following theorem relates DM^{−}(F) to Nisnevich cohomology.

Theorem 1.5[V 2]. For any complex A^{•} ∈ DM^{−}(F) and any X ∈ Sm/F we have natural isomorphisms

H_{Nis}^{i}(X, A^{•}) = Hom_{DM^{−}(F)}(M(X), A^{•}[i]).

The proof splits naturally into two parts. In the first part we compare Nisnevich cohomology with Hom-groups in the category D^{−}(NSwT/F). We start with the following Lemma.

Lemma 1.6 [V 1]. Let f : Y → X be a Nisnevich covering of a (not necessarily smooth) scheme X. Then the following sequence of Nisnevich sheaves with transfers is exact

0 ⟵ ℤ_{tr}(X) ⟵^{f_{∗}} ℤ_{tr}(Y) ⟵^{(p_{2})_{∗}−(p_{1})_{∗}} ℤ_{tr}(Y ×_{X} Y) ⟵^{(p_{23})_{∗}−(p_{13})_{∗}+(p_{12})_{∗}} …

Proof. It suffices to show that if S is a smooth henselian scheme then the sequence of abelian groups

A_{•}(S) = (0 ⟵ ℤ_{tr}(X)(S) ⟵^{f_{∗}} ℤ_{tr}(Y)(S) ⟵^{(p_{2})_{∗}−(p_{1})_{∗}} ℤ_{tr}(Y ×_{X} Y)(S) ⟵ …)

is exact. Fix a closed integral subscheme Z ⊂ X × S finite and surjective over S and denote by A_{n}^{Z}(S) the subgroup of A_{n}(S) generated by closed integral subschemes T ⊂ Y ×_{X} ... ×_{X} Y × S (finite and surjective over S) whose set theoretical image in X × S coincides with Z. Obviously A_{•}^{Z}(S) is a subcomplex of A_{•}(S) and moreover A_{•}(S) = ⊕_{Z}A_{•}^{Z}(S). Thus it suffices to show that every complex A_{•}^{Z}(S) is contractible. The scheme Z being finite and surjective over a henselian scheme S is henselian itself, which implies that the projection p_{1} : Z → X admits a lifting f : Z → Y. Let T ⊂ \underbrace{Y ×_{X} ... ×_{X} Y}_{n} ×S be a gener-

ator of A_{n}^{Z}(S) and let g : T → Z be the corresponding morphism. The morphism T ⟶^{pr_{1}×_{X}fg×pr_{2}} \underbrace{Y ×_{X} ... ×_{X} Y}_{n} ×_{X}Y ×S is obviously a closed embedding and hence

defines a closed integral subscheme T′ = u_{n}(T) ⊂ \underbrace{Y ×_{X} ... ×_{X} Y}_{n+1} ×S (finite and

surjective over S). In this way a get a homomorphism u_{n} : A_{n}^{Z}(S) → A_{n+1}^{Z}(S), and one checks immediately that u is a contracting homotopy for the complex A_{•}^{Z}(S).

Corollary 1.7. Let I ∈ NSwT/F be an injective Nisnevich sheaf with transfers. Then for any X ∈ Sm/F H_{Nis}^{i}(X, I) = 0 for all i > 0.

Proof. We first compute Chech cohomology of I with respect to a Nisnevich covering f : Y → X. These cohomology are the cohomology of the complex

I(Y)  ⟶  I(Y ×_{X} Y)  ⟶ …
=↓  ^{=}↓
Hom_{NSwT} (ℤ_{tr}(Y), I) ⟶ Hom_{NSwT} (ℤ_{tr}(Y ×_{X} Y), I) ⟶ …

Injectivity of I in NSwT/F and Lemma 1.6 show that H^{∨}^{i}(Y/X, I) = 0 for i > 0. Now the standard argument involving the Cartan-Leray spectral sequence ends up the proof.

Corollary 1.7.1. For any Nisnevich sheaf with transfers F we have natural identifications

Ext_{NSwT}^{i} (ℤ_{tr}(X), F) = H_{Nis}^{i}(X, F).

In particular Ext_{NSwT}^{i} (ℤ_{tr}(X), −) = 0 for i > dimX.

Proposition 1.8. Let A^{•} ∈ D^{−}(NSwT/F) be a bounded above complex of Nisnevich sheaves with transfers. Then for any X ∈ Sm/F we have a natural isomorphism.

H_{Nis}^{i}(X, A^{•}) = Hom_{D^{−}(NSwT/F)}(ℤ_{tr}(X), A^{•}[i]).

Proof. Let A^{•} ⟶ J^{••} and A^{•} ⟶ I^{••} be the Cartan-Eilenberg resolutions of A^{•} in the categories of Nisnevich sheaves with transfers and all Nisnevich sheaves respectively. It’s easy to see from the defining property of the Cartan-Eilenberg resolution that there exists a unique (up to a homotopy of bidegree (0, −1)) homomorphism of resolutions J^{••} ⟶ I^{••}. Furthermore Proposition 0.4 allows us to identify Hom_{D^{−}(NSwT/F)}(ℤ_{tr}(X), A^{•}[i]) to H^{i}(T ot(J^{••}(X))) whereas H_{Nis}^{i}(X, A^{•}) identifies canonically to H^{i}(T ot(I^{••}(X))). What we have to verify is that the above homomorphism of resolutions induces an isomorphism H^{∗}(T ot(J^{••}(X))) ⟶^{∼} H^{∗}(T ot(I^{••}(X))). Theorem 0.3 provides us with two spectral sequences converging to cohomology in question and Corollary 1.7.1 shows that the induced map on the E_{1}-terms of the first spectral sequences is an isomorphism. Thus the map on limits is an isomorphism as well.

In the second part of the proof we compare Hom_{D^{−}(NSwT/F)}(ℤ_{tr}(X), A^{•}) and Hom_{DM^{−}(F)}(M(X), A^{•}) = Hom_{D^{−}(NSwT/F)}(C^{•}(ℤ_{tr}(X)), A^{•}).

A presheaf (resp. a presheaf with transfers) F is said to be contractible provided that there exists a homomorphism of presheaves (resp. of presheaves with transfers) ϕ : F → C^{−1}(F) = C_{1}(F) with the property that ∂_{0}ϕ = 0, ∂_{1}ϕ = 1_{F}. Here ∂_{0}, ∂_{1} : C_{1}(F) → C_{0}(F) = F are the face operations of the simplicial presheaf C_{•}(F). In other words to each section s ∈ F(U) one can associate in a natural way a section ϕ(s) ∈ C_{1}(F)(U) = F(U × 𝔸^{1}) such that ϕ(s)_{|U×0} = 0, ϕ(s)_{|U×1} = s. It follows easily from the definition that Zariski (Nisnevich, etale) sheaf associated to a contractible presheaf is again contractible. Typical examples of contractible presheavs are given by the following Lemma.

Lemma 1.9. For any presheaf (resp. presheaf with transfers) F the kernel of any iterated face operation ∂ : C_{n}(F) → C_{0}(F) = F is contractible.

Proof. We consider only the special case K = Ker(∂_{0} : C_{1}(F) → F). The proof in the general case is similar. For any X ∈ Sm/F we have:

K(X) = {s ∈ F(X × 𝔸^{1}) : s|_{X×0} = 0}.

Denote by m : 𝔸^{1} × 𝔸^{1} → 𝔸^{1} the multiplication morphism (m(a, b) = ab) and finally set ϕ(s) = (1_{X} × m)^{∗}(s) ∈ F(X × 𝔸^{1} × 𝔸^{1}). One checks immediately that ϕ(s)|_{X×𝔸^{1}×0} = 0, i.e. ϕ(s) ∈ K(X × 𝔸^{1}) and further

∂_{0}ϕ(s) = ϕ(s)|_{X×0×𝔸^{1}} = 0, ∂_{1}ϕ(s) = ϕ(s)|_{X×1×𝔸^{1}} = s.

A Nisnevich sheaf F is said to be strongly homotopy invariant provided that all cohomology presheaves X ↦ H_{Nis}^{i}(X, F) are homotopy invariant. According to Theorem 1.1 each homotopy invariant Nisnevich sheaf with transfers is strongly homotopy invariant.

Proposition 1.10. Let G and F be Nisnevich sheaves ( resp. Nisnevich sheaves with transfers). Assume that G is contractible and F is strongly homotopy invariant. Then Ext_{Nis}^{i}(G, F) = 0 for all i ≥ 0 (resp. Ext_{NSwT}^{i} (G, F) = 0 for all i ≥ 0).

Proof. We’ll give a proof for Nisnevich sheaves, the case of Nisnevich sheaves with transfers is treated similarly. Let 0 → F → I^{•} be an injective resolution of F. The cohomology presheaves of the complex C_{1}(I^{•}) are given by the formula X ↦ H_{Nis}^{i}(X × 𝔸^{1}, F) = H_{Nis}^{i}(X, F). Since the sheaf associated to the presheaf X ↦ H_{Nis}^{i}(X, F) is trivial for i > 0 we conclude that C_{1}(I^{•}) is a resolution of C_{1}(F) = F. The face operations ∂_{0}, ∂_{1} : C_{1}(I^{•}) → I^{•} give two maps of resolutions over the identity endomorphism of F and hence are homotopic. Thus there exists a family of sheaf homomorphisms s_{n} : C_{1}(I^{n}) → I^{n−1} such that ds_{n} + s_{n+1}C_{1}(d) = ∂_{1} − ∂_{0}. Let now ϕ : G → C_{1}(G) be the homomorphism from the definition of a contractible sheaf. One checks immediately now that associating to each homomorphism f ∈ Hom_{Nis}(G, I^{n}) the homomorphism u_{n}(f) = s_{n}C_{1}(f)ϕ ∈ Hom_{Nis}(G, I^{n−1}) we get a contracting homotopy for the complex Hom_{Nis}(G, I^{•}).

Corollary 1.10.1. Let G and F be Nisnevich sheaves with transfers. Assume that F is homotopy invariant (and hence strongly homotopy invariant) and G admits a finite resolution

0 ⟶ G ⟶ G^{0} ⟶ … ⟶ G^{n} → 0

in which all G^{i} are contractible Nisnevich sheaves with transfers. Then Ext_{NSwT/F}^{i} (G, F) = 0 for all i ≥ 0.

Corollary 1.10.2. Let A^{•} be a bounded above complex of Nisnevich sheaves with transfers. Assume that all A^{i} are contractible and all the cohomology sheaves H^{i} = H^{i}(A^{•}) are homotopy invariant. Then the complex A^{•} is acyclic.

Proof. We prove that H^{i} = 0 by inverse induction on i. For i >> 0 H^{i} = 0 since A^{•} is bounded above. Assume now that H^{j} = 0 for all j > i and denote by Z^{i} the kernel of d : A^{i} → A^{i+1}. According to our inductive assumption the sheaf Z^{i} has a finite resolution

0 → Z^{i} → A^{i} → A^{i+1} → …

with contractible terms. Since the sheaf H^{i} is homotopy invariant we conclude from Corollary 1.10.1 that Hom_{NSwT} (Z^{i}, H^{i}) = 0. Thus the natural epimorphism Z^{i} ↠ H^{i} is zero and hence H^{i} = 0.

For any presheaf F and any n ≥ 0 we have a natural monomorphism F → C_{n}(F) split by any of the iterated face maps ∂ : C_{n}(F) → C_{0}(F) = F. Denoting by C_{0•}(F) the constant simplicial presheaf (which has F in all dimensions and all face and degeneracy maps are identities) we get a natural monomorphism (split in each dimension) of simplicial presheaves C_{0•}(F) → C_{•}(F) and hence also the associated monomorphism of compexes (of degree +1) C_{0}^{•}(F) → C^{•}(F). The cokernel of this monomorphism consists of contractible presheaves according to Lemma 1.9. This construction generalizes immediately to complexes of presheaves. In particular, for each bounded above complex A^{•} of Nisnevich sheaves with transfers we have a natural monomorphism of complexes C_{0}^{•}(A^{•}) → C^{•}(A^{•}) whose cokernel consists of contractible sheaves. Note further that we also have a natural embedding A^{•} → C_{0}^{•}(A^{•}) whose cokernel is contractible (in the usual sence of homological algebra) and hence acyclic.

Proposition 1.11. Let A^{•} be a bounded above complex of Nisnevich sheaves with transfers.

(1) If each A^{n} is contractible then the complex C^{•}(A^{•}) is acyclic.

(2) If the cohomology sheaves H^{i}(A^{•}) are homotopy invariant then the natural embedding A^{•} → C^{•}(A^{•}) is a quasiisomorphism.

Proof. If each A^{n} is contractible then each C^{i}(A^{n}) is contractible as well, so that all terms of the complex C^{•}(A^{•}) are contractible. Since the cohomology sheaves of this complex are homotopy invariant we conclude from Corollary 1.10.2 that this complex is acyclic.

Assume now that the cohomology sheaves of A^{•} are homotopy invariant. Lemma 1.3 implies immediately that all the cohomology sheaves of the complex C^{•}(A^{•})/C_{0}^{•}(A^{•}) are homotopy invariant. Since the terms of this complex are contractible we conclude from Corollary 1.10.2 that this complex is acyclic. Thus the natural embedding C_{0}^{•}(A^{•}) → C^{•}(A^{•}) is a quasiisomorphism and hence the embedding A^{•} → C_{0}^{•}(A^{•}) → C^{•}(A^{•}) is a quasiisomorphism as well.

Corollary 1.11.1. Let B^{•}, A^{•} be bounded above complexes of Nisnevich sheaves with transfers. Assume that the sheaves B^{n} are contractible and the cohomology sheaves H^{n}(A^{•}) are homotopy invariant. Then Hom_{D^{−}(NSwT/F)}(B^{•}, A^{•}) = 0.

Proof. It suffices to show that each homomorphism of complexes f : B^{•} → A^{•} gives zero in Hom_{D^{−}(NSwT/F)}(B^{•}, A^{•}). This follows immediately from the commutative diagram

B^{•}  ⟶^{f} A^{•}
↓  ↓
C^{•}(B^{•}) ⟶^{C^{•}(f)} C^{•}(A^{•})

in which the right vertical arrow is a qasiisomorphism and the complex C^{•}(B^{•}) is acyclic.

Corollary 1.11.2. Let B^{•}, A^{•} be bounded above complexes of Nisnevich sheaves with transfers. Assume that the cohomology sheaves H^{n}(A^{•}) are homotopy invariant. Then the natural embedding B^{•} → C^{•}(B^{•}) induces an isomorphism

Hom_{D^{−}(NSwT/F)}(C^{•}(B^{•}), A^{•}) ⟶^{∼} Hom_{D^{−}(NSwT/F)}(B^{•}, A^{•}).

Proof. Once again we decompose the embedding B^{•} → C^{•}(B^{•}) into a composition B^{•} → C_{0}^{•}(B^{•}) → C^{•}(B^{•}). The first arrow is a quasiisomorphism and hence induces isomorphisms on Hom_{D^{−}(NSwT/F)}(−, A^{•}). The second embedding has cokernel consisting of contractible sheaves and hence also induces an isomorphism on Hom_{D^{−}(NSwT/F)}(−, A^{•}) according to corollary 1.11.1.

Corollary 1.11.2 shows that if A^{•} ∈ DM^{−}(F) is a bounded above complex of Nisnevich sheaves with transfers with homotopy invariant cohomology sheaves then for any scheme X ∈ Sm/F the natural embedding ℤ_{tr}(X) → C^{•}(ℤ_{tr}(X)) = M(X) induces an isomorphism

Hom_{D^{−}(NSwT/F)}(M(X), A^{•}[i]) ⟶^{∼} Hom_{D^{−}(NSwT/F)}(ℤ_{tr}(X), A^{•}[i])

which together with Corollary 1.8 ends up the proof of the Theorem 1.5.

Proposition 1.11 shows immediately that if A^{•} is an acyclic bounded above complex of Nisnevich sheaves with transfers then the complex C^{•}(A^{•}) is acyclic as well. This implies further that if f : A^{•} → B^{•} is a quasiisomorphism of bounded above complexes of Nisnevich sheaves with transfers then C^{•}(f) is also a quasiisomorphism. Thus we get a functor

C^{•} : D^{−}(NSwT/F) → DM^{−}(F).

Denote by A the thick triangulated subcategory of D^{−}(NSwT/F) comprising those complexes A^{•} for which the complex C^{•}(A^{•}) is acyclic. The previous discussion proves immediately the following result.

Theorem 1.12 [V 2]. (1) The functor C^{•} takes distinguished triangles to distinguished triangles and commutes with direct sums (of homologically bounded above families).

(2) The functor C^{•} is left adjoint to the embedding functor DM^{−}(F) ↪ ↪ D^{−}(NSwT/F) and establishes an equivalence of DM^{−}(F) with the localization of D^{−}(NSwT/F) with respect to the thick triangulated subcategory A.

We finish this section by a more detailed description of the category A.

Lemma 1.13. A complex A^{•} ∈ D^{−}(NSwT) is in A if and only if it is quasiisomorphic to a (bounded above) complex of contractible Nisnevich sheaves with transfers.

Proof. If all the entries of the complex A^{•} are contractible then the complex C^{•}(A^{•}) is acyclic according to Proposition 1.11. Assume now that the complex C^{•}(A^{•}) is acyclic. The distinguished triangle

C_{0}^{•}(A^{•}) → C^{•}(A^{•}) → C^{•}(A^{•})/C_{0}^{•}(A^{•}) → C_{0}^{•}(A^{•})[1] → …

shows that the complex C_{0}^{•}(A^{•}) (which is quasiisomorphic to A^{•}) is quasiisomorphic to a complex C^{•}(A^{•})/C_{0}^{•}(A^{•})[−1] with contractible entries.

Let X ∈ Sch/F be a scheme with a distinguished rational point x_{0} ∈ X. We’ll be saying that the scheme X is (algebraically) contractible (to the point x_{0}) iff there exists a morphism f : X × 𝔸^{1} → X such that f|_{X×1} = 1_{X}, f|_{X×0} = x_{0} and f|_{x_{0}×𝔸^{1}} = x_{0}.

Proposition 1.14. (1) Assume that the scheme (X, x_{0}) is contractible. Then for any Y ∈ Sch/F the sheaf ℤ_{tr}(Y × X/Y × x_{0}) is contractible.

(2) The category A is weakly generated by the contractible sheaves ℤ_{tr}(Y × 𝔸^{1}/Y × 0) (Y ∈ Sm/F).

Proof. Let f : X ×𝔸^{1} → X be the contraction of X. We define the homomorphism of sheaves ϕ : ℤ_{tr}(Y × X) → C_{1}(ℤ_{tr}(Y × X)) as the composition

ℤ_{tr}(Y × X) ⟶^{Z↦Z×Δ_{𝔸^{1}}} C_{1}(ℤ_{tr}(Y × X × 𝔸^{1})) ⟶^{C_{1}(1_{Y} ×f)} C_{1}(ℤ_{tr}(Y × X))

Here the first arrow sends a section of ℤ_{tr}(Y × X) over a smooth scheme U (i.e. a cycle Z ⊂ Y × X × U each component of which is finite and surjective over a component of U) to the section of ℤ_{tr}(Y × X × 𝔸^{1}) over U × 𝔸^{1} given by the cycle Z × Δ_{𝔸^{1}} ⊂ Y × X × A^{1} × U × 𝔸^{1} (each component of which is obviously finite and surjective over a component of U × 𝔸^{1}). One checks immediately that ϕ takes ℤ_{tr}(Y × x_{0}) to C_{1}(ℤ_{tr}(Y × x_{0})) and hence defines a homomorphism \overline{ϕ} : ℤ_{tr}(Y × X/Y × x_{0}) → C_{1}(ℤ_{tr}(Y × X/Y × x_{0})) and furthermore ∂_{1}\overline{ϕ} = Id, ∂_{0}\overline{ϕ} = 0.

To prove the second statement it suffices (in view of Lemmas 0.6 and 1.13) to show that every contractible sheaf F ∈ NSwT admits a left resolution by the direct sums of sheaves of the form ℤ_{tr}(Y × 𝔸^{1}/Y). We’ll show more generally that such resolutions exist for all sheaves F ∈ NSwT for which the complex C^{•}(F) is acyclic (i.e. F ∈ A). Note first of all that to give a homomorphism of sheaves with transfers ℤ_{tr}(Y × 𝔸^{1}/Y) → F is the same as to give a section s ∈ F(Y × 𝔸^{1}) such that s|_{Y ×0} = 0. Surjectivity of the homomorphism C_{1}(F) ⟶^{∂_{1}−∂_{0}} F implies that locally in the Nisnevich topology each section t ∈ F(Y) may be written in the form t = s|_{Y ×1}, where s ∈ F(Y × 𝔸^{1}) is a section for which s|_{Y ×0} = 0. This remark shows that the natural homomorphism

⨁_{Y ∈Sm/F} ⨁_{s∈F(Y ×𝔸^{1}):s|_{Y ×0}=0} ℤ_{tr}(Y × 𝔸^{1}/Y) ⟶ F

is surjective. The kernel of this homomorphism is again in A, so that the construction may be repeated.

§ 2 Tensor structure on the category DM^{−}(F).

To define the tensor structure on the category DM^{−}(F) we start with the definition of the tensor product of presheaves with transfers. The definition we are about to give is dictated by the following (expected) properties of the tensor product operation: it should commute with any direct sums and moreover should commute with tensoring (in the usual sence) with arbitrary abelian groups, be right exact and satisfy the property that (here the superscript pr stands for the tensor product in the category of presheaves) ℤ_{tr}(X) ⊗_{tr}^{pr} ℤ_{tr}(Y) = ℤ_{tr}(X × Y). Note that for any presheaf with transfers F we have a natural exact sequence:

0 ⟵ F ⟵ ⊕_{X∈Sm/F} F(X) ⊗ ℤ_{tr}(X) ⟵ ⊕_{f∈Cor(X,Y)}F(Y) ⊗ ℤ_{tr}(X)

tensoring such resolutions for F and G we come immediately to the following formula for F ⊗_{tr}^{pr} G

F ⊗_{tr}^{pr} G = Coker( ⨁_{X∈Sm/F,g∈Cor(Y′,Y)} F(X) ⊗ G(Y) ⊗ ℤ_{tr}(X × Y′)⊕

⊕ ⨁_{Y ∈Sm/F,f∈Cor(X′,X)} F(X) ⊗ G(Y) ⊗ ℤ_{tr}(X′ × Y) ⟶

⟶ ⨁_{X,Y ∈Sm/F} F(X) ⊗ G(Y) ⊗ ℤ_{tr}(X × Y)).

In other words the presheaf with transfers F ⊗_{tr}^{pr} G is given by the formula:

(F ⊗_{tr}^{pr} G)(Z) = ⨁_{X,Y ∈Sm/F} F(X) ⊗ G(Y) ⊗ Cor(Z, X × Y)/Λ

where Λ is the subgroup generated by elements of the following form

ϕ ⊗ ψ ⊗ (f × 1_{Y}) · h − f^{∗}(ϕ) ⊗ ψ ⊗ h : \begin{matrix}f ∈ Cor(X′, X), ϕ ∈ F(X) \\ ψ ∈ G(Y), h ∈ Cor(Z, X′ × Y) \end{matrix}

ϕ ⊗ ψ ⊗ (1_{X} × g) · h − ϕ ⊗ g^{∗}(ψ) ⊗ h : \begin{matrix}g ∈ Cor(Y′, Y), ϕ ∈ F(X) \\ ψ ∈ G(Y), h ∈ Cor(Z, X × Y′) \end{matrix}

With this definition one verifies easily that ⊗_{tr}^{pr} has all the expected properties. In particular, the functor ⊗_{tr}^{pr} is right exact in each variable, commutative and associative (up to a natural isomorphism) and ℤ_{tr}(X) ⊗_{tr}^{pr} ℤ_{tr}(Y) = ℤ_{tr}(X × Y). Next one defines the tensor product of Nisnevich sheaves with transfers as the sheaf associated with their tensor product in the category of presheaves. We denote the last operation by ⊗_{tr}. The bifunctor ⊗_{tr} : NSwT/F × NSwT/F → NSwT/F is still right exact, commutative and associative , commutes with arbitrary direct sums and satisfies the identity ℤ_{tr}(X)⊗_{tr} ℤ_{tr}(Y) = ℤ_{tr}(X ×Y). The following universal mapping property of the tensor product of Nisnevich sheaves with transfers is obvious from the definition.

Lemma 2.1. Let F, G, H be Nisnevich sheaves with transfers. To give a homomorphism of sheaves with transfers F ⊗_{tr} G ⟶^{p} H is the same as to give bilinear maps p_{X,Y} : F(X) × G(Y) → H(X × Y) (X, Y ∈ Sm/F ) which satisfy the following properties

(1) For any f ∈ Cor(X′, X) the following diagram commutes

F(X) × G(Y) ⟶^{p_{X,Y}} H(X × Y)
f^{∗}×1_{G(Y )}↓  ^{(f×1_{Y} )^{∗}}↓
F(X′) × G(Y) ⟶^{p_{X′,Y}} H(X′ × Y)

(2) For any g ∈ Cor(Y′, Y) the following diagram commutes

F(X) × G(Y) ⟶^{p_{X,Y}} H(X × Y)
1_{F(X)}×g^{∗}↓  ^{(1_{X}×g)^{∗}}↓
F(X) × G(Y′) ⟶^{p_{X,Y ′}} H(X × Y′)

To extend this operation on the category D^{−}(NSwT/F) we need a few auxiliary results. For any X ∈ Sm/F denote by L_{i}^{X} the left derived functors of the right exact functor

ℤ_{tr}(X) ⊗_{tr}^{pr} − : Presheaves with transfers → Presheaves with transfers.

Note that the sheaves ℤ_{tr}(X) are projective in the category of presheaves with transfer, but we don’t know whether they are flat (probably not).

Lemma 2.2. Let F be a presheaf with transfers such that F_{Nis}^{∼} = 0. Then L_{i}^{X}(F)_{Nis}^{∼} = 0 for all i ≥ 0 and all X ∈ Sm/F.

Proof. We first construct certain special presheaves with transfers F for which F_{Nis}^{∼} = 0. Let f : Y′ → Y be a Nisnevich covering of Y ∈ Sm/F. Denote by H_{i}(Y′/Y) the homology presheaves of the complex

Č_{•}(Y′/Y) : 0 ⟵ ℤ_{tr}(Y) ⟵^{f_{∗}} ℤ_{tr}(Y′) ⟵ ℤ_{tr}(Y′ ×_{Y} Y′) ⟵ ….

Lemma 1.6 shows that H_{i}(Y′/Y)_{Nis}^{∼} = 0 for all i. Moreover it’s clear from the definition that for each presheaf with transfers F for which F_{Nis}^{∼} = 0 there exists an epimorphism onto F from a direct sum of presheaves of the form H_{0}(Y′/Y). We now start proving our statement by induction on i. For i < 0 everything is obvious. Assume now that the statement holds for all i < j. The previous remark shows that to prove the vanishing of L_{j}^{X}(F)_{Nis}^{∼} for any presheaf F with F_{Nis}^{∼} = 0 it suffices to consider the special case F = H_{0}(Y′/Y). Tensoring the complex Č_{•}(Y′/Y) with ℤ_{tr}(X) we get a spectral sequence

E_{pq}^{2} = L_{p}^{X}(H_{q}(Y′/Y)) ⇒ H_{p+q}(ℤ_{tr}(X) ⊗_{tr}^{pr} Č_{•}(Y′/Y)).

Note further that the complex ℤ_{tr}(X)⊗_{tr}^{pr} Č_{•}(Y′/Y)) is equal to Č_{•}(X ×Y′/X ×Y), so that the limit of our spectral sequence coincides with H_{∗}(X × Y′/X × Y). This shows that applying to the above spectral sequence the exact functor _{Nis}^{∼} we get a new spectral sequence which converges to zero:

E_{pq}^{2} = L_{p}^{X}(H_{q}(Y′/Y))_{Nis}^{∼} ⇒ 0

Applying the induction hypothesis we get immediately the desired conclusion.

Corollary 2.3. Let A_{•} be a bounded below complex (of degree -1) of free presheaves with transfers. Assume that H_{i}(A_{•})_{Nis}^{∼} = 0 for all i. Then H_{i}(ℤ_{tr}(X)⊗_{tr}^{pr} A_{•})_{Nis}^{∼} = 0 for all i and all X ∈ Sm/F.

Proof. This statement follows immediately from Lemma 2.2 in view of the spectral sequence

E_{pq}^{2} = L_{p}^{X}(H_{q}(A_{•})) ⇒ H_{p+q}(ℤ_{tr}(X) ⊗_{tr}^{pr} A_{•}).

We return back to complexes of degree +1 and cohomological notations. Corollary 2.3 implies easily the following result

Corollary 2.4. Let A^{•} and B^{•} be bounded above complexes (of degree +1) of free Nisnevich sheaves with transfers. Assume that either A^{•} or B^{•} is acyclic, then the complex A^{•} ⊗_{tr} B^{•} is acyclic as well.

Corollary 2.5. Let A_{1}^{•} → A_{2}^{•} be a quasiisomorphism of bounded above complexes of free Nisnevich sheaves with transfers. Then for any bounded above complex B^{•} of free Nisnevich sheaves with transfers the induced homomorphism A_{1}^{•} ⊗_{tr} B^{•} → A_{2}^{•} ⊗_{tr} B^{•} is a quasiisomorphism as well.

For any Nisnevich sheaf with transfers F we have a natural epimorphism onto F from a free Nisnevich sheaf with transfers:⨁_{X∈Sm/F,ϕ∈F(X)} ℤ_{tr}(X) → F. Repeating this construction we get a canonical free resolution X_{•}(F) → F. As always we reindex X_{•} in cohomological terms, so that X^{•}(F) is a complex of degree +1 concentrated in nonpositive degrees. Applying the functor X^{•} to a bounded above complex A^{•} of Nisnevich sheaves with transfers we get a free bounded above complex X^{•}(A^{•}) and a natural quasiisomorphism X^{•}(A^{•}) → A^{•}. We define the tensor product operation on the category D^{−}(NSwT/F) via the formula

A^{•} ⊗^{L} B^{•} = X^{•}(A^{•}) ⊗_{tr} X^{•}(B^{•}).

Corollary 2.5 shows that this construction is well defined and moreover A^{•} ⊗^{L} B^{•} = A^{•} ⊗_{tr} B^{•} for any complexes A^{•}, B^{•} of free Nisnevich sheaves with transfers (more generally for any complexes whose entries are direct summands in appropriate free sheaves, i.e. are projective in the category of presheaves with transfers). The following properties of the bifunctor ⊗^{L} are now straightforward.

Lemma 2.6. (1) The functor ⊗^{L} is commutative and associative (up to a natural isomorphism)

(2) The functor ⊗^{L} takes distinguished triangles in each variable to distinguished triangles.

(3) The functor ⊗^{L} takes direct sums (of homologically bounded families) to direct sums.

To pass from the category D^{−}(NSwT/F) to the category DM^{−}(F) we have to verify that the tensor product A^{•} ⊗^{L} B^{•} of two complexes, one of which is in the localizing subcategory A is again in A.

Proposition 2.7. Let A^{•}, B^{•} be bounded above complexes of Nisnevich sheaves with transfers. Assume that A^{•} ∈ A, then A^{•} ⊗^{L} B^{•} ∈ A.

Proof. Since the category A is weakly generated by sheaves of the form ℤ_{tr}(Y × 𝔸^{1}/Y) (Y ∈ Sm/F) and the category D^{−}(NSwT) is weakly generated by sheaves ℤ_{tr}(X) (X ∈ Sm/F) it suffices to note that

ℤ_{tr}(Y × 𝔸^{1}/Y) ⊗^{L} ℤ_{tr}(X) = ℤ_{tr}(Y × 𝔸^{1}/Y) ⊗_{tr} ℤ_{tr}(X) =

= ℤ_{tr}(Y × X × 𝔸^{1}/Y × X) ∈ A.

Proposition 2.7 together with Theorem 1.12 show that the tensor structure on the category D^{−}(NSwT/F) induces (via the localizing functor C^{•}) a tensor structure on the category DM^{−}(F). Explicitly the tensor product operation on the category DM^{−}(F) is given by the formula

A^{•} ⊗ B^{•} = C^{•}(A^{•} ⊗^{L} B^{•}).

The following result sums up the main properties of this tensor product operation.

Proposition 2.8. (1) The functor ⊗ : DM^{−}(F) × DM^{−}(F) → DM^{−}(F) is commutative and associative (up to a natural isomorphism).

(2) The functor ⊗ takes distinguished triangles in each variable to distinguished triangles.

(3) The functor ⊗ takes direct sums (of homologically bounded families) to direct sums.

(4) For any A^{•}, B^{•} ∈ D^{−}(NSwT) we have a natural isomorphism C^{•}(A^{•}⊗^{L}B^{•}) ⟶^{∼} C^{•}(A^{•}) ⊗ C^{•}(B^{•}). In particular for all schemes X, Y ∈ Sm/F we have a natural isomorphism M(X × Y) =^{∼} M(X) ⊗ M(Y).

§ 3. Motivic Cohomology

The concept of motivic cohomology (as we understand it today) goes back to the beginning of the 80’s when A. Beilinson conjectured that there should exist complexes of Zariski sheaves ℤ(n) on Sm/F which have (among others) the following properties:

Be1 (Normalization) The complex ℤ(0) is quasiisomorphic to the constant sheaf ℤ positioned in degree 0, the complex ℤ(1) is quasiisomorphic to the sheaf O^{∗} of invertible functions positioned in degree 1.

Be2 (The Beilinson Soulé Vanishing Conjecture) For n > 0 the complex ℤ(n) is acyclic outside the interval [1, n], the n-th cohomology sheaf H^{n}(ℤ(n)) coincides with the sheaf K_{n}^{M} of Milnor K_{n}-groups.

Be3 (Relationship to K-theory) For any X ∈ Sm/F there exists a natural spectral sequence

E_{2}^{p,q} = H_{Zar}^{p}(X, ℤ(q)) ⇒ K_{2q−p}(X)

which rationally degenerates and defines isomorphisms of H_{Zar}^{∗}(X, ℤ(∗))⊗ℚ with subsequent factors of the γ filtration on K_{∗}(X) ⊗ ℚ.

Be4 (The Beilinson-Lichtenbaum Conjecture) For any l prime to charF the complex ℤ/l(n) = ℤ(n) ⊗^{L} ℤ/l is quasiisomorphic to τ_{≤n}Rπ_{∗}(μ_{l}^{⊗n}). Here π is the obvious morphism of sites (Sm/F)_{et} → (Sm/F)_{Zar} and τ_{≤n} denotes trancation at the level n.

In this section we define (following [V 2]) the motivic complexes ℤ(n) and establish some of there elementary properties.

Consider the sheaf ℤ_{tr}(𝔾_{m}^{∧n}). According to definitions (see §0) ℤ_{tr}(𝔾_{m}^{∧n}) coincides with ℤ_{tr}(𝔾_{m}^{×n})/D_{n} (here 𝔾_{m} stands for the multiplicative group scheme 𝔾_{m} = 𝔸^{1} \ {0}) and D_{n} is the sum of images of homomorphisms

ℤ_{tr}(𝔾_{m}^{×n−1}) → ℤ_{tr}(𝔾_{m}^{×n})

given by the embeddings of the form

(x_{1}, …, x_{n−1}) ↦ (x_{1}, …, 1, …, x_{n−1}).

The subsheaf D_{n} is in fact a direct summand of ℤ_{tr}(𝔾_{m}^{×n}) (see §0). The corresponding projection p : ℤ_{tr}(𝔾_{m}^{×n}) → D_{n} is given by the formula

p = ∑_{I}(−1)^{card(I)−1}(p_{I})_{∗},

where I runs through all non empty subsets of {1, ..., n} and p_{I} : 𝔾_{m}^{×n} → 𝔾_{m}^{×n} is the coordinate morphism, replacing all I-entries by 1 ∈ 𝔾_{m}. The sheaf ℤ_{tr}(𝔾_{m}^{∧n}) coincides with the complementary direct summand and may be also identified with the intersection of kernels of homomorphisms (p_{I})_{∗} over all non empty subsets I ⊂ {1, ..., n} (since (p_{I})_{∗}p = (p_{I})_{∗} for any I ≠ ∅). Furthermore, with these notations we have a canonical direct sum decomposition

(3.0)  ℤ_{tr}(𝔾_{m}^{×n}) = ⨁_{m=0}^{n} ⨁_{I⊂\{1,...,n\}_{card I=m}} ℤ_{tr}(𝔾_{m}^{∧m}) = ⨁_{m=0}^{n} ℤ_{tr}(𝔾_{m}^{∧m})^{\binom{n}{m}}

Definition 3.1. The motivic complex ℤ(n) of weight n on Sm/F is the complex C^{∗}(ℤ_{tr}(𝔾_{m}^{∧n}))[−n] = C^{∗}(ℤ_{tr}(𝔾_{m}^{×n})/D_{n})[−n]. For a smooth scheme X over F we define its motivic cohomology groups H_{M}^{i}(X, ℤ(n)) as Zariski hypercohomology H_{Zar}^{i}(X, ℤ(n)).

Note that ℤ(n) is a complex of sheaves with transfers in Nisnevich topology (actually even in the etale topology) with homotopy invariant cohomology presheaves. Thus Corollary 1.2 shows that H_{Zar}^{∗}(X, ℤ(n)) = H_{Nis}^{∗}(X, ℤ(n)). Using further Theorem 1.5 we see that H_{M}^{i}(X, ℤ(n)) = Hom_{DM^{−}(F)}(M(X), ℤ(n)).

The motiv ℤ(1) is called the Tate motiv. For any X ∈ DM^{−}(F) we denote by X(n) the tensor product X ⊗ ℤ(n). In particular for an abelian group A we denote by A(n) the tensor product complex A ⊗ ℤ(n). Proposition 2.8 implies easily, that A(n) may be identified with the complex C^{∗}(ℤ_{tr}(𝔾_{m}^{∧n}) ⊗ A)[−n]. In particular ℤ/l(n) = C^{∗}(ℤ_{tr}(𝔾_{m}^{∧n})/l)[−n].

Proposition 2.8 implies that for each m, n we have a canonical isomorphism ℤ(m) ⊗ ℤ(n) →^{∼} ℤ(m + n) and the following diagrams commute for all m, n, p

ℤ(m) ⊗ ℤ(n) ⊗ ℤ(p) ⟶^{∼} ℤ(m + n) ⊗ ℤ(p)
=^{∼}↓  =^{∼}↓
ℤ(m) ⊗ ℤ(n + p) ⟶^{∼} ℤ(m + n + p).

Lemma 3.2.

(1) The complex ℤ(0) is canonically quasisomorphic to the constant sheaf ℤ, positioned in degree 0, the complex ℤ(1) is canonically quasiisomorphic to the sheaf O^{∗} positioned in degree one.

(2) The complex ℤ(n) is acyclic in degrees > n.

(3) For all n, m ≥ 0 we have a natural isomorphism ℤ_{tr}(𝔾_{m}^{∧n}) ⊗_{tr} ℤ_{tr}(𝔾_{m}^{∧m}) = ℤ_{tr}(𝔾_{m}^{∧(n+m)}) and hence a natural quasiisomorphism ℤ(n) ⊗ ℤ(m) =^{∼} ℤ(n + m).

Proof. Identification of the complex ℤ(0) is straightforward. The complex ℤ(1) is a canonical direct summand in C^{∗}(ℤ_{tr}(𝔾_{m}))[−1]. To identify ℤ(1) we start by computing the cohomology presheaves of the complex C^{∗}(ℤ_{tr}(𝔾_{m}))[−1]. Let U ∈ Sm/F be any smooth affine scheme. Denoting by H^{i} the i-th cohomology presheaf of C^{∗}(ℤ_{tr}(𝔾_{m}))[−1] we have (using the notations introduced in [S-V]) the following identifications

H^{i}(U) = H_{1−i}(C_{∗}(ℤ_{tr}(𝔾_{m}))(U)) = H_{1−i}^{sin}(U × 𝔾_{m}/U).

Computation of singular homology of relative smooth curves - [S-V §3] shows that

H^{i}(U) =0  i ≠ 1
H^{1}(U) =P ic(U × ℙ^{1}, U × 0 ∐ U × ∞).

The exact sequence relating relative Picard group to the absolute ones shows immediately that P ic(U × ℙ^{1}, U × 0 ∐ U × ∞) = O^{∗}(U) ⊕ ℤ. This computation implies easily that all cohomology sheaves of ℤ(1) except for H^{1} vanish and H^{1}(ℤ(1)) = O^{∗}.

Acyclicity of the complex ℤ(n) in degrees > n is obvious from the construction. The last statement follows from the identification ℤ_{tr}(𝔾_{m}^{×n}) ⊗_{tr} ℤ_{tr}(𝔾_{m}^{×m}) = ℤ_{tr}(𝔾_{m}^{×(n+m)}) and right exactness of the functor ⊗_{tr}.

Remark 3.2.0 The proof of the Lemma 3.2 gives the following explicit way to identify the presheaf H^{1}(ℤ(1)) with O^{∗}. To each f ∈ O^{∗}(U) = Hom_{Sm/F} (U, 𝔾_{m}) we associate the class of its graph Γ_{f} ∈ ℤ_{tr}(𝔾_{m})(U) in H^{1}(ℤ(1))(U) = ℤ_{tr}(𝔾_{m}^{∧1})(U)/ / Im(ℤ_{tr}(𝔾_{m}^{∧1})(U × 𝔸^{1})). Vice versa start with an irreducible cycle Z ⊂ U × 𝔾_{m} finite and surjective over U. This cycle defines an invertible function p_{2} ∈ O^{∗}(Z), taking the norm of this function from Z to U gives us an invertible function f_{Z} = N_{Z/U}(p_{2}) ∈ O^{∗}(U). The two above maps define mutually inverse isomorphisms of O^{∗}(U) and H^{1}(ℤ(1))(U) (at least for affine U’s).

Corollary 3.2.1. For any X ∈ Sm/F we have the following formulae:

H_{M}^{i}(X, ℤ(0)) = \{ \begin{matrix}H^{0}(X, ℤ) & if i = 0 \\ 0 & if i ≠ 0 \end{matrix}

H_{M}^{i}(X, ℤ(1)) = \{ \begin{matrix}H^{0}(X, O^{∗}) & if i = 1 \\ P ic(X) & if i = 2 \\ 0 & if i ≠ 1, 2 \end{matrix}

The formula (3.0) gives the following identification of the motiv of 𝔾_{m}^{×n}.

Lemma 3.3. For any n ≥ 0 we have the natural direct sum decomposition

M(𝔾_{m}^{×n}) = ⨁_{m=0}^{n} ℤ(m)[m]^{\binom{n}{m}}.

For any X, Y ∈ Sm/F we get a natural pairing

H_{M}^{i}(X, ℤ(n)) ⊗ H_{M}^{j}(Y, ℤ(m)) = Hom_{DM^{−}(F)}(M(X), ℤ(n)[i])⊗
⊗ Hom_{DM^{−}(F)}(M(Y), ℤ(m)[j]) → Hom_{DM^{−}(F)}(M(X) ⊗ M(Y),
(ℤ(n) ⊗ ℤ(m))[i + j]) = H_{M}^{i+j}(X × Y, ℤ(n + m)).

Using these pairings and the homomorphism in motivic cohomology induced by the diagonal embedding Δ : X → X ×X one makes H_{M}^{∗,∗}(X) = ⨁_{i,n} H_{M}^{i}(X, ℤ(n)) into a bigraded associative ring. Note further that H_{M}^{1,1}(SpecF) = F^{∗}. Thus we get a canonical homomorphism from the tensor algebra T (F^{∗}) of the multiplicative group F^{∗} to the subring ⊕_{n=0}^{∞}H_{M}^{n,n}(SpecF) of the motivic cohomology ring H_{M}^{∗,∗}(SpecF).

Theorem 3.4. The natural homomorphism T (F^{∗}) → ⊕_{n=0}^{∞}H_{M}^{n,n}(SpecF) defines an isomorphism K_{∗}^{M}(F) ⟶^{∼} ⊕_{n=0}^{∞}H_{M}^{n,n}(SpecF) (here K_{∗}^{M}(F) is the Milnor ring of the field F - see [B-T]).

Proof. We’ll use the notation H_{M}^{∗,∗}(F) for H_{M}^{∗,∗}(SpecF). Note first of all that motivic cohomology H_{M}^{∗}(F, ℤ(n)) coincides up to a shift of degrees with homology of the complex C_{∗}(ℤ_{tr}(𝔾_{m}^{∧n}))(F): H_{M}^{i}(F, ℤ(n)) = H_{n−i}(C_{∗}(𝔾_{m}^{∧n})(F)). In particular

H_{M}^{n,n}(F) = Coker (ℤ_{tr}(𝔾_{m}^{∧n})(𝔸_{F}^{1} ) ⟶^{∂_{0}−∂_{1}} ℤ_{tr}(𝔾_{m}^{∧n})(SpecF)) .

Moreover products in the motivic cohomology ring H_{M}^{∗,∗}(F) may be computed as follows. Every pair of cycles Z ⊂ 𝔾_{m}^{×n} × Δ^{p}, T ⊂ 𝔾_{m}^{×m} × Δ^{q} defines a cycle Z × T ⊂ 𝔾_{m}^{×n+m} × Δ^{p} × Δ^{q}. In this way we get a homomorphism of complexes C_{∗}(ℤ_{tr}(𝔾_{m}^{×n}))(F) ⊗ C_{∗}(ℤ_{tr}(𝔾_{m}^{×m}))(F) → C_{∗,∗}(ℤ_{tr}(𝔾_{m}^{×n+m}))(F), which obviously factors to define a homomorphism C_{∗}(ℤ_{tr}(𝔾_{m}^{∧n}))(F)⊗C_{∗}(ℤ_{tr}(𝔾_{m}^{∧m}))(F) → C_{∗,∗}(ℤ_{tr}(𝔾_{m}^{∧n+m}))(F). Composing this homomorphism with the shuffle map (see § 0) we finally get a homomorphism of complexes ℤ(n)(F) ⊗ ℤ(m)(F) → ℤ(n + m)(F) which defines the desired pairings in cohomology.

Every n-tuple (a_{1}, ..., a_{n}) of elements of F^{∗} defines an F -rational point (a_{1}, ..., a_{n}) ∈ 𝔾_{m}^{×n}. We’ll denote by [a_{1}, ..., a_{n}] the class of this rational point in H_{M}^{n,n}(F). According to definitions [a_{1}, ..., a_{n}] = 0 if at least one of a_{i} is equal to 1. Moreover the definition of products in motivic cohomology implies easily that [a_{1}, ..., a_{n}]·[b_{1}, ..., b_{m}] = [a_{1}, ..., a_{n}, b_{1}, ...b_{m}]. This remark shows that the homomorphism (F^{∗})^{⊗n} → H_{M}^{n,n}(F) sends the tensor a_{1} ⊗ ... ⊗ a_{n} to [a_{1}, ..., a_{n}]). We first check that the homomorphism (F^{∗})^{⊗n} → H^{n,n}(F) factors through K_{n}^{M}(F). To do so it suffices to verify that that the class of the rational point (a, 1−a) ∈ 𝔾_{m}^{×2} dies in H_{M}^{2,2}(F). The proof of this fact requires the use of the transfer homomorphisms in motivic cohomology. We start by reminding the definition and properties of these homomorphisms.

Let j : F ↪ E be a field extension. Extending scalars in cycles we get a canonical homomorphism of complexes C_{∗}(ℤ_{tr}(𝔾_{m}^{∧n}))(F) → C_{∗}(ℤ_{tr}(𝔾_{m}^{∧n}))(E) and the induced homomorphism in motivic cohomology j_{∗} : H_{M}^{i}(F, ℤ(n)) → H_{M}^{i}(E, ℤ(n)). We often use the notation x_{E} for the image j_{∗}(x) ∈ H_{M}^{i}(E, ℤ(n)) of the element x ∈ H_{M}^{i}(F, ℤ(n)). The homomorphism j_{∗} : H_{M}^{∗,∗}(F) → H_{M}^{∗,∗}(E) is obviously compatible with products in motivic cohomology.

Assume now that E/F is a finite field extension. Taking direct images of cycles we get a homomorphism of complexes C_{∗}(𝔾_{m}^{∧n})(E) → C_{∗}(𝔾_{m}^{∧n})(F) and hence the induced homomorphisms in homology groups

N_{E/F} : H_{i}(C_{∗}(𝔾_{m}^{∧n})(E)) = H_{M}^{n−i,n}(E) → H_{i}(C_{∗}(𝔾_{m}^{∧n})(F)) = H_{M}^{n−i,n}(F).

Lemma 3.4.1. Let E/F be a finite field extension and let further K/F be a finite normal field extension such that Hom_{F} (E, K) ≠ ∅ . For any x ∈ H_{M}^{n,n}(E) and y ∈ H_{M}^{m,m}(F) we have:

(1) N_{E/F} (y_{E} · x) = y · N_{E/F} (x), N_{E/F} (x · y_{E}) = N_{E/F} (x) · y. In particular N_{E/F} (y_{E}) = [E : F] · y.

(2) (N_{E/F} (x))_{K} = [E : F]_{i} · ∑_{j∈Hom_{F} (E,K)} j_{∗}(x) (here [E/F]_{i} stands for the inseparable degree of the field extension E/F).

(3) The homomorphism N_{E/F} : E^{∗} = H_{M}^{1,1}(E) → F^{∗} = H_{M}^{1,1}(F) coincides with the usual norm homomorphism in field extensions.

Proof. The first two properties follow immediately from the corresponding properties of the direct image homomorphism for cycles. The last property follows from (2), since the homomorphism F^{∗} = H_{M}^{1,1}(F) → K^{∗} = H_{M}^{1,1}(K) is injective.

Lemma 3.4.2. Assume that there exists an integer N > 0 such that N·[a, 1−a] = 0 for any field F and any element a ∈ F^{∗} \ {1} (resp. N · [a, −a] = 0 for any F and any a ∈ F^{∗}). Then [a, 1−a] = 0 ∀F ∀a ∈ F^{∗}\{1} (resp. [a, −a] = 0 ∀F ∀a ∈ F^{∗}).

Proof. The proof is the same in both cases so we’ll consider only the first one. It suffices to show that if all symbols of the form [a, 1 − a] are killed by an integer N = Mp, where p is a prime then all such symbols are killed by M already. The case a ∈ (F^{∗})^{p} is easy so we’ll assume that a is not p-th power. Set α = a^{1/p}, E = F (α). According to our assumptions 0 = Mp · [α, 1 − α] = M · [a, 1 − α]. Applying to this equation the homomorphism N_{E/F} and using Lemma 3.4.1 we get:

0 = N_{E/F} (M·[a, 1−α]) = M·[a]·N_{E/F} ([1−α]) = M·[a]·[N_{E/F} (1−α)] = M·[a, 1−a].

Proposition 3.4.3. For any field F and any a ∈ F^{∗} \ {1} the element [a, 1 − a] ∈ H_{M}^{2,2}(F) is trivial.

Proof. For any a, b ∈ F^{∗} \ {1} consider the closed subscheme Y ⊂ 𝔸^{1} × 𝔾_{m}, given by the equation

X^{2} − (t · (a + b) + (1 − t) · (1 + ab))X + ab = 0

(here t is the coordinate in 𝔸^{1} and X is the coordinate in 𝔾_{m}). One checks immediately that the projection p_{2} : Y → 𝔾_{m} is an isomorphism so that in particular Y is integral. Moreover it’s clear that Y is finite and surjective over 𝔸^{1} and hence defines an element y ∈ ℤ_{tr}(𝔾_{m}^{∧1})(𝔸^{1}). Note that ∂_{0}(y) = [ab] + [1] = [ab] and ∂_{1}(y) = [a]+[b] (this shows again that [ab] = [a]+[b]). Next consider the closed embedding Y ↪ 𝔸^{1} × 𝔾_{m}^{×2} obtained by means of the diagonal embedding 𝔾_{m} ↪ 𝔾_{m}^{×2} and denote by y′ ∈ ℤ_{tr}(𝔾_{m}^{∧2})(𝔸^{1}) the corresponding element. The faces of y′ are given by the formulae ∂_{0}(y′) = [ab, ab] + [1, 1] = [ab, ab] ,∂_{1}(y′) = [a, a] + [b, b]. This gives us the relation [ab, ab] = [a, a] + [b, b]. Taking here a = b and using bimultiplicativity of the symbol [−, −] we see that 2 · [a, −a] = 0 ∀a ∈ F^{∗}. Using Lemma 3.4.2 we conclude that in fact [a, −a] = 0 ∀a ∈ F^{∗}. The relation [a, −a] = 0 implies immediately that [a, 1 − a] + [a^{−1}, 1 − a^{−1}] = 0 ∀a ∈ F^{∗} \ {1}. Assume now that a ∈ F^{∗} is an element such that a^{6} ≠ 1. Consider a closed subscheme Z ⊂ 𝔸^{1} × (𝔾_{m} \ {1}) given by the equation

X^{3} − t · (a^{3} + 1) · X^{2} + t · (a^{3} + 1) · X − a^{3} = 0.

Once again one checks immediately that the projection p_{2} : Z → 𝔾_{m} \ {1} is an isomorphism, so that Z is, in particular, integral. The fiber of Z over 0 ∈ 𝔸^{1} consists of all cubical roots of a^{3} and the fiber of Z over 1 ∈ 𝔸^{1} consists of a^{3} and two roots x_{1}, x_{2} of the equation X^{2} − X + 1 = 0. We embed Z into 𝔸^{1} × 𝔾_{m}^{×2} using the embedding 𝔾_{m} \ {1} ↪ 𝔾_{m}^{×2} x ↦ (x, 1 − x) and denote by z′ the corresponding element in ℤ_{tr}(𝔾_{m}^{∧2})(𝔸^{1}). Assume for the moment that F contains all roots (in the algebraic closure of F ) of the equation X^{6} − 1 = 0. In this case ∂_{0}(z′) = [a, 1 − a] + [ξa, 1 − ξa] + [ξ^{2}a, 1 − ξ^{2}a] = [a, 1 − a^{3}] + [ξ, (1 − ξa)(1 − ξ^{2}a)^{2}], where ξ is the generator of the group μ_{3} (ξ = 1 if char(F) = 3). At the same time ∂_{1}(z′) = [a^{3}, 1 − a^{3}] + [x_{1}, 1 − x_{1}] + [x_{2}, 1 − x_{2}] = [a^{3}, 1 − a^{3}]. Multiplying the relation ∂_{0}(z′) = ∂_{1}(z′) by 3 and keeping in mind that ξ^{3} = 1 we get the relation 2[a^{3}, 1 − a^{3}] = 0. Using once again the transfer argument we conclude further that 6[a, 1 − a] = 0 for any a ∈ F^{∗} \ {1}. Lemma 3.4.2 shows finally that [a, 1 − a] = 0 ∀a ∈ F^{∗} \ {1}.

Proposition 3.4.2 shows that the homomorphism T (F^{∗}) → ⊕_{n=0}^{∞}H_{M}^{n,n}(F) factors through the Milnor ring K_{∗}^{M}(F). We denote by λ = λ_{F} the resulting ring homomorphism K_{∗}^{M}(F) → ⊕_{n=0}^{∞}H_{M}^{n,n}(F).

Lemma 3.4.4. The homomorphism λ : K_{∗}^{M}(F) → ⊕_{n=0}^{∞}H_{M}^{n,n}(F) is compatible with transfer maps in Milnor K-theory and motivic cohomology respectively. In other words for any finite field extension E/F the following diagram commutes

K_{∗}^{M}(E) ⟶^{λ_{E}} ⊕_{n=0}^{∞}H_{M}^{n,n}(E)
N_{E/F} ↓  ^{N_{E/F}} ↓
K_{∗}^{M}(F) ⟶^{λ_{F}} ⊕_{n=0}^{∞}H_{M}^{n,n}(F).

Proof. The general case is reduced easily to the case where the field F has no extensions of degree prime to p (for a certain prime integer p) and [E : F] = p. In this case the Milnor ring K_{∗}^{M}(E) is additively generated by symbols of the form {a_{1}, ..., a_{n−1}, b} (a_{i} ∈ F^{∗}, b ∈ E^{∗}) - see [B-T]. Using Lemma 3.4.1 we easily conclude that both images of such a symbol in H_{M}^{n,n}(F) are equal to {a_{1}, ..., a_{n−1}, N_{E/F} (b)}.

Corollary 3.4.5. The homomorphism λ_{F} : K_{n}^{M}(F) → H_{M}^{n,n}(F) is surjective.

Proof. The group H_{M}^{n,n}(F) is generated by classes of closed points in 𝔾_{m}^{×n}. Each closed point x ∈ 𝔾_{m}^{×n} defines a finite field extension E = F (x) of F and a rational point x′ = (x′_{1}, ..., x′_{n}) ∈ 𝔾_{m}^{×n}(E). Moreover according to the definition of the transfer maps in motivic cohomology we have : [x] = N_{E/F} ([x′]). Lemma 3.4.4 shows now that [x] = λ_{F} (N_{E/F} ({x′_{1}, ..., x′_{n}})).

End of the proof of the Theorem 3.4. To finish the proof we define a homomorphism H_{M}^{n,n}(F) → K_{n}^{M}(F) inverse to λ_{F}. Let v ∈ 𝔾_{m}^{×n} be any closed point. The residue field F (v) is a finite extension of F and we set θ(v) = N_{F (v)/F} ({X_{1}(v), ..., X_{n}(v)}) ∈ K_{n}^{M}(F). The usual argument involving the Weil reciprocity formula (see [N-S]) shows that the homomorphism θ : ℤ_{tr}(𝔾_{m}^{×n})(F) → K_{n}^{M}(F) killes the image of ∂_{0} − ∂_{1} and thus defines a homomorphism θ : H_{M}^{n,n}(F) → K_{n}^{M}(F) inverse to λ_{F}.

§ 4. Fundamental distinguished triangles in the category DM^{−}(F).

In this section we construct (following [V 2]) several fundamental distinguished triangles in the category DM^{−}(F), which lead to various exact sequences of motivic cohomology. We also identify motives of affine and projective vector bundles and of blow ups (with smooth center).

Lemma 4.1. Let X = U ∪V be a (Zariski) open covering of a scheme X ∈ Sm/F. Then we have a natural distinguished triangle

M(U ∩ V) → M(U) ⊕ M(V) → M(X) → M(U ∩ V)[1]

in the category DM^{−}(F).

Proof. One checks easily that the following sequence of Nisnevich sheaves is exact (cf. Lemma 1.6)

0 → ℤ_{tr}(U ∩ V) → ℤ_{tr}(U) ⊕ ℤ_{tr}(V) → ℤ_{tr}(X) → 0.

This gives us a distinguished triangle

ℤ_{tr}(U ∩ V) → ℤ_{tr}(U) ⊕ ℤ_{tr}(V) → ℤ_{tr}(X) → ℤ_{tr}(U ∩ V)[1]

in the category D^{−}(NSwT/F). Applying to this triangle the functor C^{∗} we get the triangle in question.

The proof of the following Proposition gives a typical example of the use of the above distinguished triangles.

Proposition 4.2. Let E be a locally free O_{X}-module and let p : Y = SpecS^{∗}(E^{#}) → X be the corresponding affine vector bundle. Then M(p) : M(Y) ⟶^{∼} M(X) is a quasiisomorphism.

Proof. Let X = U ∪ V be an open covering of X. The morphism p induces a map from the distinguished triangle, corresponding to the covering Y = p^{−1}(U)∪p^{−1}(V) to the distinguished triangle, corresponding to the covering X = U ∪V. This shows immediately that if M(p) : M(p^{−1}(U)) → M(U), M(p) : M(p^{−1}(V)) → M(V) and M(p) : M(p^{−1}(U ∩ V)) → M(U ∩ V) are quasiisomorphisms, then M(p) : M(Y) → M(X) is a quasiisomorphism as well. This remark allows to reduce the general case of our statement to the situation when the locally free sheef E is trivial. In the latter case Y = X×𝔸^{n}. Proceeding by induction on n it suffices to consider the case n = 1. Finally the natural morphism M(i_{0}) : M(X) → M(X × 𝔸^{1}) is a quasiisomorphism according to Proposition 1.14 and hence M(p) : M(X × 𝔸^{1}) → M(X) is also a quasiisomorphism (since pi_{0} = 1_{X}).

As a next application of the distinguished triangle of Lemma 4.1 we compute the motives of the affine space without the origin and of the projective space.

Proposition 4.3. There exists a canonical direct sum decomposition

M(𝔸^{n} \ 0) = ℤ(0) ⊕ ℤ(n)[2n − 1].

Proof. We proceed by induction on n. The case n = 1 is covered by Lemma 3.3. In the general case we cover the scheme X = 𝔸^{n} \ 0 by two open pieces U = {x ∈ 𝔸^{n} : x_{1} ≠ 0}, V = {x ∈ 𝔸^{n} : x_{2} ≠ 0 or x_{3} ≠ 0, ..., or x_{n} ≠ 0}. Obviously V = 𝔸^{1} × (𝔸^{n−1} \ 0), U ∩ V = 𝔾_{m} × (𝔸^{n−1} \ 0). Induction hypothesis and Proposition 4.2 show that M(U) = ℤ(0) ⊕ ℤ(1)[1], M(V) = ℤ(0) ⊕ ℤ(n − 1)[2n − 3], M(U ∩ V) = (ℤ(0) ⊕ ℤ(1)[1]) ⊗ (ℤ(0) ⊕ ℤ(n − 1)[2n − 3]) = ℤ(0) ⊕ ⊕ℤ(n − 1)[2n − 3] ⊕ ℤ(1)[1] ⊕ ℤ(n)[2n − 2]. Furthermore it’s clear from the construction that the homomorphism

M(U ∩ V) = ℤ(0) ⊕ ℤ(1)[1] ⊕ ℤ(n − 1)[2n − 3] ⊕ ℤ(n)[2n − 2] →

→ M(U) ⊕ M(V) = ℤ(0) ⊕ ℤ(1)[1] ⊕ ℤ(0) ⊕ ℤ(n − 1)[2n − 3]

sends the second and the third term on the left isomorphically onto the same summand on the right and sends ℤ(0) diagonally to ℤ(0)⊕ℤ(0). Thus the distinguished triangle, corresponding to the covering X = U ∪ V simplifies and takes the form

ℤ(n)[2n − 2] → ℤ(0) → M(X) → ℤ(n)[2n − 1].

Now it suffices to note that the morphism M(SpecF) = ℤ(0) → M(X) is induced by the rational point (1, ..., 0) ∈ X and is split by the obvious morphism M(X) → M(SpecF) = ℤ(0) induced by the projection X → SpecF .

The canonical line bundle O(1) defines an element τ ∈ P ic(ℙ^{n}) = H_{M}^{2}(ℙ^{n}, ℤ(1)) = Hom_{DM^{−}(F)}(M(ℙ^{n}), ℤ(1)[2]). Raising τ to powers 0, 1, ..., n we get the corresponding morphisms τ^{k} : M(ℙ^{n}) → ℤ(k)[2k].

Proposition 4.4. The canonical morphism

(1, τ, ..., τ^{n}) : M(ℙ^{n}) → ℤ(0) ⊕ ℤ(1)[2] ⊕ ... ⊕ ℤ(n)[2n]

is an isomorphism.

Proof. Consider an open covering of ℙ^{n} by two open pieces X = ℙ^{n} = U ∪V, where U = ℙ^{n} \ [0 : ... : 0 : 1], V = (ℙ^{n})_{T_{n}}. It’s well known that the projection with the center [0 : ... : 0 : 1] makes U into an affine line bundle over ℙ^{n−1}. Furthermore V = 𝔸^{n}, U ∩ V = 𝔸^{n} \ 0. Using Propositions 4.2 and 4.3 we conclude that the distinguished triangle, corresponding to the above covering looks as follows:

ℤ(0) ⊕ ℤ(n)[2n − 1] → ℤ(0) ⊕ M(ℙ^{n−1}) → M(ℙ^{n}) → ℤ(0)[1] ⊕ ℤ(n)[2n]

Neglecting the trivial summand ℤ(0), which maps diagonally into ℤ(0) ⊕ M(ℙ^{n−1}) and shifting to the right, this triangle is simplified to

M(ℙ^{n−1}) → M(ℙ^{n}) → ℤ(n)[2n] → M(ℙ^{n−1})[1]

Induction hypothesis implies that the canonical morphism M(ℙ^{n−1}) → M(ℙ^{n}) is split by the morphism (1, τ, ..., τ^{n−1}). Thus we get a direct sum decomposition M(ℙ^{n}) = M(ℙ^{n−1})⊕ℤ(n)[2n]. This basically ends up the proof, the only thing still missing is the verification of the fact that the above morphism M(ℙ^{n}) → ℤ(n)[2n] coincides (at least up to a sign) with τ^{n}. This is not hard to do, we refer the reader to [V 0] for technical details.

Let E be a locally free O_{X}-module of constant rank n + 1. Denote by P = P (E) = P roj S^{∗}(E^{#}) the corresponding projective bundle and by p : P → X the corresponding projection. The same as above the canonical line bundle O_{P} (1) defines a morphism τ : P → ℤ(1)[2].

Theorem 4.5. The natural morphism

M(P) ⟶^{M(Δ)} M(P × P) = M(P) ⊗ M(P) ⟶^{M(p)⊗(1,τ,...,τ^{n})} M(X) ⊗ (ℤ(0)⊕

⊕ ℤ(1)[2] ⊕ ... ⊕ ℤ(n)[2n]) = M(X) ⊕ M(X)(1)[2] ⊕ ... ⊕ M(X)(n)[2n]

is an isomorphism.

Proof. Proceeding in the same way as in the proof of the Proposition 4.2 we easily reduce the general case to the case when the sheaf E is trivial. In the latter case M(P (E)) = M(X) ⊗ M(ℙ^{n}) and our statement follows from Proposition 4.4.

The following important theorem of Voevodsky is the basis for the construction of several distinguished triangles in the category DM^{−}(F).

Theorem 4.6 [V 1]. Assume that resolution of singularities holds over F. Let F be a presheaf with transfers such that for any connected scheme X ∈ Sm/F and any section s ∈ F(X) there exists a proper birational morphism f : Y → X (with Y ∈ Sm/F) such that f^{∗}(s) = 0 ∈ F(Y). Then the complex C^{∗}(F_{Nis}^{∼}) is acyclic.

Theorem 4.7. Let Z ⊂ X be a smooth subscheme in a smooth scheme X. Denote by p : B_{Z}(X) → X the blow up of X with center Z. Then we have a natural distinguished triangle

M(p^{−1}(Z)) → M(Z) ⊕ M(B_{Z}(X)) → M(X) → M(p^{−1}(Z))[1].

Proof. It’s easy to check that the sequence of Nisnevich sheaves with transfers

0 → ℤ_{tr}(p^{−1}(Z)) → ℤ_{tr}(Z) ⊕ ℤ_{tr}(B_{Z}(X)) → ℤ_{tr}(X)

is exact. Denote by C the cokernel of the last homomorphism above. The sheaf C is associated to the presheaf

U ↦ \frac{ℤ_{tr}(X)(U)}{Im(ℤ_{tr}(Z)(U) ⊕ ℤ_{tr}(B_{Z}(X)(U))}.

Our statement would follow from the above exact sequence and Theorem 4.6 as soon as we show that the above presheaf satisfies the conditions of the Theorem 4.6. We start with a section s ∈ ℤ_{tr}(X)(U) (with U ∈ Sm/F - smooth connected scheme) and want to show that it either comes from ℤ_{tr}(Z)(U) or otherwise can be lifted to ℤ_{tr}(B_{Z}(X))(V) after an appropriate base change V → U (with V → U proper and birational). We may clearly assume that s is represented by a closed integral subscheme S ⊂ U × X, finite and surjective over U. The case when S is contained in U × Z is trivial. Thus we may assume that S ⊄ U × Z. Denote by T the closure of (1_{U} × p)^{−1}(S ∩ (U × (X \ Z)) in U × B_{Z}(X). The scheme T is clearly proper and surjective over U, but need not be finite over U. According to the Platification Theorem - see [R-G] there exists a blow up (not necessarily with smooth center) U′ → U such that the proper inverse image T′ of T in U′ × B_{Z}(X) is flat over U′. Since T′ → U′ is proper and finite over the generic point of U′ we conclude that T′ → U′ is finite. Finally there exists (according to the Definition 4.6) a proper birational morphism V → U′ with V smooth over F. Thus we get a proper birational morphism V → U with V smooth, such that the proper inverse image T′′ of T in V × B_{Z}(X) is finite over V. Obviously this T′′ gives us a lifting to ℤ_{tr}(B_{Z}(X))(V) of the inverse image of S to ℤ_{tr}(X)(V).

Theorem 4.8. In conditions and notations of Theorem 4.7 assume further that Z is everywhere of codimension c. Then we have a canonical direct sum decomposition

M(B_{Z}(X)) = M(X) ⊕ ⨁_{i=1}^{c−1} M(Z)(i)[2i].

Proof. Denote by E the conormal sheaf of Z in X. It’s well known that the sheaf E is a locally free O_{Z}-module of rank c and p^{−1}(Z) may be identified with a projective bundle P (E). Using Theorem 4.5 we see that the distinguished triangle of Theorem 4.7 may be rewritten in the following more simple form:

⨁_{i=1}^{c−1} M(Z)(i)[2i] → M(B_{Z}(X)) → M(X) → ⨁_{i=1}^{c−1} M(Z)(i)[2i + 1]

It suffices now to show that the morphism M(B_{Z}(X)) ⟶^{M(p)} M(X) admits a canonical section α_{Z}^{X} : M(X) → M(B_{Z}(X)). To do so we compare the above distinguished triangle to a similar distinguished triangle corresponding to the blow up q : B_{Z×{0}}(X × 𝔸^{1}) → X × 𝔸^{1} of Z × {0} ⊂ X × 𝔸^{1}. It’s easy to see that the morphism i_{0} : X → X × 𝔸^{1} induces a morphism (which we still denote i_{0}) B_{Z}(X) ⟶^{i_{0}} B_{Z×{0}}(X × 𝔸^{1}) (which takes p^{−1}(Z) to q^{−1}(Z × {0})). Moreover it’s clear from the proof of the Theorem 4.7 that we get an induced homomorphism from the distinguished triangle corresponding to the blow up p to the one corresponding to the blow up q. Note further that q^{−1}(Z × {0}) = P (E ⊕ O_{Z}) and the embedding p^{−1}(Z) ⟶^{i_{0}} q^{−1}(Z × {0}) is the obvious one (corresponding to the embedding E → E ⊕ O_{Z}). This implies immediately that the restriction of the canonical bundle O_{P (E⊕O_{Z})}(1) to P (E) coincides with O_{P (E)}(1) and hence the following diagram commutes (for all k ≥ 0)

M(p^{−1}(Z)) = M(P (E))  ⟶^{τ^{k}} ℤ(k)[2k]
M(i_{0})↓  ^{=}↓
M(q^{−1}(Z × {0})) = M(P (E ⊕ O_{Z})) ⟶^{τ^{k}} ℤ(k)[2k]

Thus under the identification of the Theorem 4.5 the canonical morphism M(i_{0}) : M(p^{−1}(Z) → M(q^{−1}(Z × {0}) becomes an obvious embedding ⊕_{i=0}^{c−1}M(Z)(i)[2i] ⟶^{i_{c/c−1}} ⊕_{i=0}^{c}M(Z)(i)[2i]. Denote by pr_{c/c−1} the obvious projection of the direct sum on the right onto the direct sum on the left. Consider next the following commutative diagram with distinguished rows

⨁_{i=1}^{c−1} Z(i)[2i] ⟶  M(B_{Z}(X))  ⟶^{M(p)} M(X) ⟶ …
i_{c/c−1}↓  ^{M(i_{0})}↓  ^{M(i_{0})}↓
⨁_{i=1}^{c} Z(i)[2i] ⟶ M(B_{Z×{0}}(X × 𝔸^{1})) ⟶^{M(q)} M(X × 𝔸^{1}) ⟶ …

The morphism M(i_{0}) : M(X) → M(X × 𝔸^{1}) is an isomorphism with inverse M(pr_{1}) : M(X × 𝔸^{1}) → M(X) and hence coincides with M(i_{1}). The morphism i_{1} : X → X × 𝔸^{1} admits a canonical lifting to B_{Z×{0}}(X × 𝔸^{1}), which we still denote by i_{1}. Thus M(i_{1} · pr_{1}) yields a section for M(q) and hence the bottom triangle splits providing an isomorphism M(B_{Z×{0}}(X × 𝔸^{1})) = M(X × 𝔸^{1}) ⊕

⨁_{i=1}^{c} M(Z)(i)[2i]. Denote by β^{∼} the corresponding projection of M(B_{Z×{0}}(X × 𝔸^{1})) onto ⊕_{i=1}^{c}M(Z)(i)[2i]. We finally note that the homomorphism β_{Z}^{X} = pr_{c/c−1}·

β^{∼} · M(i_{0}) : M(B_{Z}(X)) → ⊕_{i=1}^{c−1}Z(i)[2i] is a left inverse for the canonical morphism ⊕_{i=1}^{c−1}Z(i)[2i] → M(B_{Z}(X)) appearing in the top triangle. Thus the top triangle also splits canonically , which provides us with the section α_{Z}^{X} : M(X) → M(B_{Z}(X)) for M(p).

To define the Gizin morphisms we need a slight generalization of Theorem 4.8. Let U ↪ X be a (smooth) subscheme of X ∈ Sm/F (not necessarily closed). We define the object M(X/U) ∈ DM^{−}(F) as the cone of the homomorphism M(U) → M(X). Thus we always have a canonical distinguished triangle M(U) → M(X) → M(X/U) → M(U)[1]. Whenever f : X → X′ is a morphism of schemes, taking a subscheme U ⊂ X to a subscheme U′ ⊂ X′ we get an evident morphism M(f) : M(X/U) → M(X′/U′). Let Z ⊂ X be a smooth closed subscheme, and let U ⊂ X be a smooth subscheme not intersecting Z. In this case U lifts canonically to a subscheme of B_{Z}(X). The arguments used in the proof of the Theorem 4.7 apply without changes to this relative situation and provide us with a distinguished triangle

M(p^{−1}(Z)) → M(B_{Z}(X)/U) → M(X/U) → M(p^{−1}(Z))[1].

Moreover the proof of the Theorem 4.8 also goes through nearly without changes. This time one has to compare the distinguished triangle in question to the distinguished triangle corresponding to q with motives taken relative to the subscheme U × 𝔸^{1}. We’ll keep the notations introduced in the proof of Theorem 4.8 in this relative situation as well. Thus we get in particular canonical morphisms

M(X/U) ⟶^{α_{Z}^{X}} M(B_{Z}(X)/U), M(B_{Z×{0}}(X × 𝔸^{1})/U × 𝔸^{1}) ⟶^{β^{∼}} ⨁_{i=1}^{c} M(Z)(i)[2i].

The absolute and relative morphisms thus obtained are compatible one with another, so that the following diagram (and a similar diagram involving β^{∼}) commutes:

M(X) ⟶^{α_{Z}^{X}} M(B_{Z}(X))
↓  ↓
M(X/U) ⟶^{α_{Z}^{X}} M(B_{Z}(X)/U).

Let Z ⊂ X be a closed subscheme of a scheme X ∈ Sm/F. Denoting the open subscheme X \ Z by U we define the motive of X with supports in Z M_{Z}(X) as the relative motive M(X/U). Assume that Z is smooth everywhere of codimension c in X. In this case we can apply the previous discussion and get a canonical morphism

M_{Z}(X) ⟶^{α_{Z}^{X}} M(B_{Z}(X)/U) ⟶^{M(i_{0})} M(B_{Z×{0}}(X×𝔸^{1})/U ×𝔸^{1}) ⟶^{β^{∼}} ⨁_{i=1}^{c} M(Z)(i)[2i]

It’s easy to see that the composition of this morphism with the projection pr_{c/c−1} onto the direct sum of the first c−1 summands is trivial. Thus the above morphism may be considered as a morphism from M_{Z}(X) to M(Z)(c)[2c], which is called the Gizin morphism and is denoted G_{Z}^{X} : M_{Z}(X) → M(Z)(c)[2c]). The following naturality properties of the Gizin morphism are straightforward from the definition.

Lemma 4.9.

(1) Let f : X′ → X be a smooth morphism. Set Z′ = Z ×_{X} X′, so that Z′ is a smooth closed subscheme of X′ everywhere of codimension c. Then the following diaggram commutes

M_{Z′}(X′)  ⟶^{M(f)}  M_{Z}(X)
G_{Z′}^{X′} ↓  ^{G_{Z}^{X}} ↓
M(Z′)(c)[2c] ⟶^{M(f|_{Z′})(c)[2c]} M(Z)(c)[2c]

(2) Let Y ∈ Sm/F be a smooth scheme. Then the following diagram commutes

M_{Z×Y} (X × Y) ⟶^{G_{Z×Y}^{X×Y}}  M(Z × Y)(c)[2c]
=↓  ^{=}↓
M_{Z}(X) ⊗ M(Y) ⟶^{G_{Z}^{X}⊗1_{M(Y )}} M(Z)(c)[2c] ⊗ M(Y)

(3) Let X = U ∪ V be a Zariski open covering of a (smooth) scheme X. In this case we have a distinguished triangle

M_{Z∩(U∩V)}(U ∩ V) → M_{Z∩U}(U) ⊕ M_{Z∩V} (V) → M_{Z}(X) → M_{Z∩(U∩V)}(U ∩ V)[1] Moreover the Gizin morphisms

G_{Z∩(U∩V)}^{U∩V} : M_{Z∩(U∩V)}(U ∩ V) → M(Z ∩ (U ∩ V))(c)[2c],
G_{Z∩U}^{U} : M_{Z∩U}(U) → M(Z ∩ U)(c)[2c],
G_{Z∩V}^{V} : M_{Z∩V} (V) → M(Z ∩ V)(c)[2c],
G_{Z}^{X} : M_{Z}(X) → M(Z)(c)[2c]

define a homomorphism from the above distinguished triangle to the distinguished triangle corresponding to the Zariski open covering Z = (Z ∩ U) ∪ (Z ∩ V) (twisted and shifted several times).

Theorem 4.10 (Cohomological purity in motivic cohomology)[V 2]. Let Z ⊂ X be a smooth closed subscheme everywhere of codimension c. Then the Gizin morphism G_{Z}^{X} : M_{Z}(X) → M(Z)(c)[2c] is an isomorphism.

Proof. Lemma 4.9 (3) shows that the question is local with respect to the Zariski topology on X. In perticular we may assume that there exists an etale morphism f : X → 𝔸^{d} such that Z = f^{−1}(𝔸^{d−c} × {0}). Consider the cartesian square

Y ⟶^{pr_{2}} Z × 𝔸^{c}
pr_{1}↓  ↓^{f|_{Z}×1_{𝔸^{c}}}
X ⟶^{f} 𝔸^{d}

Since f|_{Z} : Z → 𝔸^{d−c} is an etale morphism we conclude that the diagonal embedding Δ_{Z} : Z → Z ×_{𝔸^{d−c}} Z is both open and closed. Thus we may find an open subscheme X′ ⊂ Y such that X′ ∩ Z ×_{𝔸^{d−c}} Z = Δ(Z). Denote by p_{1}, p_{2} restrictions of projections pr_{1}, pr_{2} to an open subscheme X′ ⊂ Y. It’s clear that the morphisms p_{1}, p_{2} are etale. Moreover, identifying Z with its image in X′ under Δ_{Z} we see easily, that p_{1}^{−1}(Z) = Z, p_{2}^{−1}(Z × {0}) = Z and the induced morphisms (p_{1})|_{Z} : Z → Z, (p_{2})|_{Z} : Z → Z × {0} are both equal to identity. Using Lemma 4.9 we get two commutative diagrams

M_{Z}(X′) ⟶^{G_{Z}^{X′}} M(Z)(c)[2c]  M_{Z}(X′)  ⟶^{G_{Z}^{X′}} M(Z)(c)[2c]
M(p_{1})↓  ^{=}↓  ^{M(p_{2})}↓  ^{=}↓
M_{Z}(X) ⟶^{G_{Z}^{X}} M(Z)(c)[2c] M_{Z×{0}}(Z × 𝔸^{c}) ⟶^{G_{Z×{0}}^{Z×𝔸^{c}}} M(Z)(c)[2c]

Lemma 4.11 below shows that the vertical arrows in the above diagrams are isomorphisms. Thus to show that G_{Z}^{X} is an isomorphism it suffices to show that G_{Z×{0}}^{Z×𝔸^{c}} = 1_{M(Z)} × G_{{0}}^{𝔸^{c}} is an isomorphism. Thus we are reduced to showing that the Gizin morphism G_{{0}}^{𝔸^{c}} : M_{{0}}(𝔸^{c}) → ℤ(c)[2c] is an isomorphism. Proposition 4.3 implies easily that M_{{0}}(𝔸^{c}) =^{∼} ℤ(c)[2c]. Thus we only have to verify, that the isomorphism given by Proposition 4.3 coincides with the Gizin morphism. We omit the technical details of this verification - see [V 0].

Lemma 4.11 (Etale excision - [V 1]). Let f : X′ → X be an etale morphism of smooth schemes. Let further Z ⊂ X be a smooth closed subscheme such that

Z′ = f^{−1}(Z) ⟶^{f|_{Z′}} Z is an isomorphism. Then the induced morphism M(f) : M_{Z′}(X′) ⟶ M_{Z}(X) is an izomorphism.

Proof. The motiv M_{Z}(X) (resp M_{Z′}(X′)) may be identified with C^{∗}(ℤ_{tr}(X)/ℤ_{tr}(X \ Z)) (resp. with C^{∗}(ℤ_{tr}(X′)/ℤ_{tr}(X′ \ Z′)). Thus it suffices to show that the induced homomorphism of the quotient Nisnevich sheaves

f_{∗} : ℤ_{tr}(X′)/ℤ_{tr}(X′ \ Z′) ⟶ ℤ_{tr}(X)/ℤ_{tr}(X \ Z)

is an isomorphism. To do so we have to compare the sections of these sheaves over a smooth henselian scheme S. Note that (ℤ_{tr}(X)/ℤ_{tr}(X \ Z))(S) is a free abelian group with basis consisting of closed integral subschemes T ⊂ X × S finite and surjective over S, which are not contained in (X \ Z) × S. Any such scheme T is again henselian and its (unique) closed point t_{0} lies in Z × S. Denote by T′ the inverse image of T under the morphism X′ × S ⟶^{f×1_{S}} X × S. The scheme T′ is etale over the henselian scheme T and hence (see [M]) splits into a disjoint sum

T′ = T^{∼}′ ∐ T′_{0} ∐ ... ∐ T′_{n}, where the image of T^{∼}′ in T does not contain t_{0} and the schemes T′_{i} are finite and etale over T and henselian. Since t_{0} ∈ Z × S we conclude easily that the inverse image of t_{0} in X′ × S consists of only one point t′_{0} ∈ Z′ × S and moreover F (t′_{0}) = F (t_{0}). This shows that n = 0 and the induced morphism T′_{0} → T is an isomorphism. We define a map

ϕ : (ℤ_{tr}(X)/ℤ_{tr}(X \ Z))(S) → (ℤ_{tr}(X′)/ℤ_{tr}(X′ \ Z′))(S)

setting ϕ([T]) = [T′_{0}]. The above discussion shows that ϕf_{∗} = Id, f_{∗}ϕ = Id.

We refer the reader to [F-V] for the proof of the following important Theorem.

Theorem 4.12 (Quasiinvertibility of the Tate object). Assume that resolution of singularities holds over F. Then for any complexes A^{•}, B^{•} ∈ DM^{−}(F) tensoring with the identity endomorphism of ℤ(1) defines an isomorphism

Hom_{DM^{−}(F)}(A^{•}, B^{•}) ⟶^{∼} Hom_{DM^{−}(F)}(A^{•}(1), B^{•}(1)).

Corollary 4.12.1. Assume that resolution of singularities holds over F. Then for any scheme X ∈ Sm/F multiplcation by τ ∈ H_{M}^{2}(ℙ^{1}, ℤ(1)) = H_{M}^{2}(ℙ^{1}/SpecF, ℤ(1)) defines isomorphisms

H_{M}^{i}(X, ℤ(n − 1)) ⟶^{∼} H_{M}^{i+2}(X × ℙ^{1}/X, ℤ(n))
H_{M}^{i}(X, ℤ/l(n − 1)) ⟶^{∼} H_{M}^{i+2}(X × ℙ^{1}/X, ℤ/l(n)).

Proof. This follows immediately from Theorem 4.12, since under identification M(ℙ^{1}/SpecF) = ℤ(1)[2] the element τ corresponds to the identity endomorphism of ℤ(1).

§ 5. Motivic cohomology of non smooth schemes and the cdh-topology.

Recall that to each scheme X ∈ Sch/F we have associated in § 1 a Nisnevich sheaf with transfers ℤ_{tr}(X). This allows us to define a motiv M(X) of an arbitrary scheme X as an object C^{∗}(ℤ_{tr}(X)) of the category DM^{−}(F) and to define motivic cohomology of X with coefficients in an arbitrary complex C^{•} ∈ DM^{−}(F) via the formula

H_{M}^{i}(X, C^{•}) = Hom_{DM^{−}(F)}(M(X), C^{•}[i]).

Note that according to the Theorem 1.5 for smooth schemes the new definition coincides with the original one. Many of the properties of motivic cohomology proved in § 4 for smooth schemes ,in particular the Mayer-Vietoris distinguished triangle for open coverings, hold actually for arbitrary schemes. Moreover we also have the Mayer-Vietoris distinguished triangle for the closed coverings.

Lemma 5.1. Let X = U ∪ V be a closed covering of a scheme X ∈ Sch/F. In this case we have the following distinguished triangle in the category DM^{−}(F)

M(U ∩ V) → M(U) ⊕ M(V) → M(X) → M(U ∩ V)[1].

Proof. The result follows immediately since the following sequence of Nisnevich sheaves with transfers is exact (even as a sequence of presheaves).

0 → ℤ_{tr}(U ∩ V) → ℤ_{tr}(U) ⊕ ℤ_{tr}(V) → ℤ_{tr}(X) → 0.

The proof of the following result is identical to that of the Theorem 4.7.

Theorem 5.2. Assume that resolution of singularities holds over F. Let p : X′ → X be a proper morphism of schemes and let T ⊂ X be a closed subscheme such that p is an isomorphism over X \ T. Then the bicomplex

M(p^{−1}(T)) ⟶ M(T) ⊕ M(X′) ⟶ M(X)

is acyclic and hence we have a natural distinguished triangle in DM^{−}(F)

M(p^{−1}(T)) ⟶ M(T) ⊕ M(X′) ⟶ M(X) ⟶ M(p^{−1}(T))[1].

If Z ⊂ X is a closed subscheme we define the motiv of X with supports in Z the same way as it was done for smooth schemes in § 4, i.e. M_{Z}(X) is the cone of the

homomorphism of complexes M(X \ Z) ⟶^{M(i)} M(X).

Corollary 5.3. In conditions and notations of the previous Theorem let Z be a closed subscheme of X. Then the bicomplex

M_{p^{−1}(Z∩T)}(p^{−1}(T)) ⟶ M_{Z∩T} (T) ⊕ M_{p^{−1}(Z)}(X′) ⟶ M_{Z}(X)

is acyclic and hence we have a natural distinguished triangle in DM^{−}(F)

M_{p^{−1}(Z∩T)}(p^{−1}(T)) ⟶ M_{Z∩T} (T) ⊕ M_{p^{−1}(Z)}(X′) ⟶ M_{Z}(X) ⟶
⟶M_{p^{−1}(Z∩T)}(p^{−1}(T))[1].

The following Lemma provides some elementary properties of motives with supports.

Lemma 5.4. (1) Assume that the closed subscheme Z ⊂ X is a disjoint sum of two subschemes : Z = Z_{1} ∐ Z_{2}. Then we have a natural isomorphism of motivs with supports : M_{Z}(X) = M_{Z_{1}}(X) ⊕ M_{Z_{2}}(X).

(2) Let Z′ ⊂ Z ⊂ X be a pair of closed subschemes. In this case we have a natural distinguished triangle in the category DM^{−}(F)

M_{Z\Z′}(X \ Z′) → M_{Z}(X) → M_{Z′}(X) → M_{Z\Z′}(X \ Z′)[1].

Proof. (1) Set U_{i} = X \ Z_{i}, U = X \ Z. Then U = U_{1} ∩ U_{2} and U_{1} ∪ U_{2} = X. The Mayer-Vietoris property for the open coverings implies the acyclicty of the following bicomplex

M(U) → M(U_{1}) ⊕ M(U_{2}) → M(X)
↓ ↓ ^{=}↓
M(X) → M(X) ⊕ M(X) → M(X).

Thus the cone of the left vertical arrow is quasiisomorphic to the cone of the middle vertical arrow.

To prove (2) it suffices to note that the cone of the homomorphism M_{Z\Z′}(X \ Z′) → M_{Z}(X) coincides with the total complex corresponding to the bicomplex

M(X \ Z) ⟶^{=} M(X \ Z)
↓  ↓
M(X \ Z′) ⟶ M(X)

and hence is canonically quasiisomorphic to M_{Z′}(X).

Definition 5.5. We define the notion of a strictly dense open subscheme U in the scheme X by induction on dimX. For schemes of dimension 0 the only strictly dense open subscheme of X is X itself. In general we’ll be saying that U is strictly dense in X iff there exists a proper morphism p : X′ → X with X′ ∈ Sm/F and a closed subscheme T ⊂ X such that

(1) p is an isomorphism over X \ T

(2) dimT < dimX, dim p^{−1}(T) < dimX

(3) p^{−1}(U) is dense in X′, U ∩ T is strictly dense in T, p^{−1}(U ∩ T) is strictly in p^{−1}(T).

Obviously for smooth schemes X strictly dense open subschemes are the same as dense open subschemes, in the general case this is however a more restricting condition. For one thing it’s obvious that each strictly dense open subscheme should have a non-empty intersection with the singular locus of X. The above definition essentially was devised for the sake of the following result.

Theorem 5.6. Assume that resolution of singularities holds over F. Let C^{•} ∈ DM^{−}(F) be a complex such that H_{Z}^{∗}(X, C^{•}) = 0 for any smooth irreducible scheme X and any irreducible closed smooth subscheme Z ≠ X. Then H_{Z}^{∗}(X, C^{•}) = 0 for any X ∈ Sch/F and any closed subscheme Z ⊂ X such that X \ Z is strictly dense in X.

Proof. We first treat the case when X is smooth. The general case is reduced immediately to the case when X is smooth and irreducible. In case Z is smooth and irreducible there’s nothing to prove. The case of an arbitrary smooth Z follows now from the additivity of M_{Z}(X) with respect to Z:

M_{Z}(X) = M_{Z_{1}}(X) ⊕ M_{Z_{2}}(X)

in case Z is a disjoint sum of two subschemes Z_{1} and Z_{2}. In the general case we may assume Z reduced and proceed by induction on dim Z. Since over a perfect field each integral scheme is smooth at the generic point we can find a nonempty open U ⊂ X, which contains generic points of all components of Z and such that Z ∩ U is smooth. Denote by Z′ the closed subscheme Z ∩ (X \ U). According to our construction dim Z′ < dim Z. On the other hand the closed subscheme Z \ Z′ ⊂ X \ Z′ = U is smooth, so that it suffices to use the existence of the distinguished triangle

M_{Z\Z′}(X \ Z′) ⟶ M_{Z}(X) ⟶ M_{Z′}(X) ⟶ M_{Z\Z′}(X \ Z′)[1].

The case of an arbitrary scheme X follows now from Corollary 5.3 in view of the Definition 5.5.

Definition 5.7. The cdh-topology (completely decomposed h-topology) on the category Sch/F is defined as the weakest Grothendieck topology for which the following two families of morphisms are coverings

(1) Every Nisnevich covering is also a cdh-covering.

(2) Let p : X′ → X be a proper morphism of schemes and let T ⊂ X be a closed subscheme such that p is an isomorphism over X \ T. Then the morphism

T ∐ X′ ⟶^{(i,p)} X is a cdh-covering.

Note that covering of a scheme X by its irreducible components (considered as closed reduced subschemes of X) is a cdh-covering. Furthermore it’s easy to see from the definition than whenever X′ → X is a cdh-covering each point x ∈ X admits a lifting x′ ∈ X′ with the same residue field: F (x) = F (x′). The last property characterizes the cdh-coverings in the case of proper morphisms.

Lemma 5.8. Let p : X′ → X be a proper morphism such that each point x ∈ X admits a lifting x′ ∈ X′ for which F (x) = F (x′). Then p is a cdh-covering.

Proof. Proceeding by Noethirian induction we may assume that for any proper closed subscheme T ⊂ X the induced morphism X′ ×_{X} T → T is a cdh-covering. Now the statement for X is obvious unless X is integral. Assume that X is integral.

According to our assumptions there exists a closed integral subscheme X′′ ⊂ X′ such that the morphism p′′ = p|_{X′′} : X′′ → X is birational. Let T ⊂ X be a proper closed subscheme for which p′′ is an isomorphism over X \ T. Accord-

ing to definitions the family X′′ ⟶^{p′′} X, T ↪ X is a cdh-covering. Furthermore X′ ×_{X} X′′ → X′′ is a cdh-covering (since it admits a section) and the morphism X′ ×_{X} T → T is a cdh-covering according to the induction hypothesis. Thus X′ → X is a cdh-covering locally in the cdh-topology and hence is a cdh-covering.

We’ll call the cdh-coverings of Lemma 5.8 the proper cdh-coverings. A proper cdh-covering p : X′ → X of an integral scheme X is called a proper birational cdh-covering in case the morphism p is an isomorphism over an appropriate neighbourhood of the generic point of X. Note that the proof of Lemma 5.8 shows that each proper cdh-covering of an integral scheme X admits a proper birational refinement.

The following Proposition gives a clearer view on the structure of cdh-coverings.

Proposition 5.9. Every cdh-covering of a scheme X ∈ Sch/F has a refinement of the following form

{U_{i} → X′ ⟶ X}_{i=1}^{n}

where X′ → X is a proper cdh-covering and {U_{i} → X′}_{i=1}^{n} is a Nisnevich covering of the scheme X′.

Proof. To abbreviate terminology we call coverings of the above type special cdhcoverings. Note that to show that a given covering of a scheme X admits a special refinement is equivallent to showing that there exists a proper cdh-covering X′ → X such that the induced covering of the scheme X′ admits a Nisnevich refinement. Thus one may always replace the scheme X by any of its proper cdh-coverings (and the given covering of X by the induced covering of X′). Furthermore if X = X_{1} ∐ X_{2} is a disjoint sum of two schemes, then verifying the statement for X (and a given covering of X) is equivalent to verifying it for both X_{i} (and the induced coverings). The above remarks imply in particular that to verify our statement for a scheme X (and a given covering of X) is equivallent to veryfing it for all irreducible components of X, considered as closed integral subschemes of X (and the corresponding induced coverings of these components).

Note further that according to definitions each cdh-covering admits a refinement of the form {X_{n}^{i} → X_{n} → X_{n−1} → ... → X_{0} = X}_{i=1}^{m}, where {X_{n}^{i} → X_{n}}_{i=1}^{m} is a Nisnevich covering of the scheme X_{n} and each stage in the tower X_{n} → ... → X_{0} = X is either a Nisnevich covering or a proper cdh-covering. Proceeding by induction on the length of the tower one sees easily that it suffices to verify that coverings of the form T ⟶^{p} U ⟶^{q} X (where T ⟶^{p} U is a proper covering and U ⟶^{q} X is a Nisnevich covering) admit a special refinement.

Proceeding by the Noetherian induction we may assume that for any proper closed subscheme Z ⊂ X the induced covering of Z admits a special refinement and the scheme X is integral. In this case the scheme U is a disjoint sum of its connected components U = ∐ U_{i} and each U_{i} is integral. Denote the inverse image of U_{i} in T by T_{i}. Replacing T → U by a finer covering we may assume further that each T_{i} → U_{i} is a proper birational covering. Applying the Platification Theorem (see [R-G]) to the morphism T → X we find a proper closed subscheme Z ⊂ X such that for each i the proper inverse image T′_{i} of T_{i} under the morphism X′ = B_{Z}(X) → X is flat over X′. Thus for each i we get a commutative diagram

T′_{i}
↓
T_{i} ×_{X} X′ ⟶ T_{i}
↓  ↓
U_{i} ×_{X} X′ ⟶ U_{i}
↓  ↓
X′  ⟶ X

Note that the scheme X′ is integral and birationally isomorphic to X, from which one concludes easily that the scheme U_{i} ×_{X} X′ is integral as well. The morphism T′_{i} → U_{i} ×_{X} X′ becomes flat being composed with the etale morphism U_{i} ×_{X} X′ → X′ and hence is flat itself. Moreover the scheme T′_{i} is integral and the morphism T′_{i} → U_{i}×_{X} X′ is proper and birational. Since every proper flat birational morphism of integral schemes is an isomorphism we conclude that T_{i} ×_{X} X′ → U_{i} ×_{X} X′ admits a section. Thus the induced covering of the scheme X′ admits a Nisnevich refinement U ×_{X} X′ → X′. Finally according to our induction hypothesis there exists a proper covering Z′ → Z such that the induced covering of Z′ admits a Nisnevich refinement. The morphism X′ ∐ Z′ → X is a proper covering of X and the induced covering of the scheme X′ ∐ Z′ admits a Nisnevich refinement, which ends up the proof.

¿From this point till the end of the § we assume that resolution of singularities holds over F. This assumption implies in particular that for smooth schemes Lemma 5.8 admits the following Improvement.

Lemma 5.10. Let X be a smooth integral scheme. Then every proper birational morphism X′ ⟶^{p} X is a cdh-covering.

Proof. According to definitions p admits a refinement which is a composition of blow up’s with smooth centers. So it suffices to note that a blow up with a smooth center satisfies the conditions of the Lemma 5.8.

Consider an obvious morphism of sites (Sch/F)_{cdh} ⟶^{θ} (Sm/F)_{Nis} and the induced functors on the categories of abelian sheaves

(Sch/F)_{cdh}^{∼} ⟶^{θ_{∗}} ; ⟵_{θ^{∗}}(Sm/F)_{Nis}^{∼}

Proposition 5.11. The functor θ^{∗} : (Sm/F)_{Nis}^{∼} → (Sch/F)_{cdh}^{∼} is exact.

Proof. To prove the result we define a cdh-topology on the category Sm/F. This topology is not defined in terms of covering families (since there is not enough fiber products in the category Sm/F), but in terms of covering sieves. More precisely, every family of morphisms {X_{i} → X}_{i∈I} in Sch/F defines a sieve on Sm/F, consisting of those arrows Y → X in the category Sm/F, which may be factored through one of X_{i} → X. The covering sieves of X ∈ Sm/F for the cdh-topology on Sm/F are those which contain a sieve defined by an appropriate cdh-covering of X in Sch/F. One checks easily (using resolution of singularities) that restricting a cdh-sheaf from the category Sch/F to Sm/F we get again a cdh-sheaf, i.e. the inclusion functor Sm/F ↪ Sch/F is continious (in the terminology of [SGA 4]). All in all we get a morphism of sites ϕ : (Sch/F)_{cdh} → (Sm/F)_{cdh} and a pair of induced functors on the categories of sheaves

(Sch/F)_{cdh}^{∼} ⟶^{ϕ_{∗}} ; ⟵_{ϕ^{∗}}(Sm/F)_{cdh}^{∼}

Here ϕ_{∗} is the restriction functor and the functor ϕ^{∗} is left adjoint to ϕ_{∗}. The functor ϕ_{∗} has (in the present circumstances) also a right adjoint, which we denote by e : (Sm/F)_{cdh}^{∼} ⟶ (Sch/F)_{cdh}^{∼}. In fact, the functor ϕ_{∗} on the category of presheaves ϕ_{∗} : (Sch/F)^{∧} ⟶ (Sm/F)^{∧} has a right adjoint e : (Sm/F)^{∧} ⟶ (Sch/F)^{∧} given by the formula e(F)(Y) = Hom_{F unct(Sm/F,Sets)}(h_{Y} , F), where h_{Y} is the functor on Sm/F represented by the scheme Y ∈ Sch/F. One checks easily (once again using resolution of singularities) that the functor e actually takes cdh-sheaves to cdh-sheaves and moreover ϕ_{∗} and e are mutually inverse equivalences of categories (cf. [S-V § 6]). This implies immediately that ϕ^{∗} and e coincide and, in particular, the functor ϕ^{∗} is exact. Finally the functor θ^{∗} coincides with the composition of ϕ^{∗} and the sheafification functor

(Sm/F)_{Nis}^{∼} ⟶^{F↦F_{cdh}^{∼}} (Sm/F)_{cdh}^{∼}

which is always exact - see [SGA 4].

In the sequell we’ll usually identify the categories (Sch/F)_{cdh}^{∼} and (Sm/F)_{cdh}^{∼} via the quasiinverse equivalences ϕ_{∗} and ϕ^{∗} = e. In particular we’ll usually use the notation (−)_{cdh}^{∼} for the inverse image functor θ^{∗} : (Sm/F)_{Nis}^{∼} → (Sch/F)_{cdh}^{∼}.

Proposition 5.12. Let F be a homotopy invariant Nisnevich sheaf with transfers. Let further G be a Nisnevich sheaf such that G_{cdh}^{∼} = 0. Then Ext_{Nis}^{∗}(G, F) = 0.

Proof. We first verify our statement for certain special sheaves G. Let Z ⊂ X be a closed smooth subscheme of a smooth scheme X. Denote by G(X, Z) the cokernel of the homomorphism ℤ_{Nis}(B_{Z}(X)) ⟶ ℤ_{Nis}(X).

(5.12.1). Ext_{Nis}^{∗}(G(X, Z), F) = 0.

Proof. One checks easily that the sheaf G(X, Z) fits into an exact sequence

0 ⟶ ℤ_{Nis}(p^{−1}(Z)) ⟶ ℤ_{Nis}(Z) ⊕ ℤ_{Nis}(B_{Z}(X)) ⟶ ℤ_{Nis}(X) ⟶ G(X, Z) ⟶ 0 which gives us a spectral sequence converging to Ext_{Nis}^{∗}(G(X, Z), F) and whose E_{1} term loks as follows:

H_{Nis}^{∗}(X, F) ⟶ H_{Nis}^{∗}(Z, F) ⊕ H_{Nis}^{∗}(B_{Z}(X), F) ⟶ H_{Nis}^{∗}(p^{−1}(Z), F).

Since Nisnevich cohomology with coefficients in F satisfies descent for blow-up’s - see Theorems 4.8 and 1.5 one concludes immediately that the E_{2}-term of the above spectral sequence is zero.

To conclude the proof we note that the condition G_{cdh}^{∼} = 0 is equivalent to the fact that for each integral scheme X ∈ Sm/F and each section s ∈ G(X) there exists a cdh-covering sieve of X such that s dies being restricted to this sieve. Furthermore resolution of singularities together with Proposition 5.9 and Lemma 5.10 imply that every cdh-covering sieve of X admits a refinement defined by a covering family of the form {U_{i} → X_{n} → ... → X_{0} = X}_{i=1}^{n} where {U_{i} → X_{n}}_{i=1}^{n} is a Nisnevich covering of X_{n} and X_{n} → ... → X_{0} = X is a tower of blow-up’s with smooth centers. Thus for each s ∈ G(X) there exists a tower of blow-up’s with smooth centers X_{n} → ... → X_{0} = X and a Nisnevich covering {U_{i} → X_{n}}_{i=1}^{n} of X_{n} such that s|_{U_{i}} = 0 ∀i. Since G is a Nisnevich sheaf we conclude further that s|_{X_{n}} = 0. Denote by G_{n} the subsheaf of G generated by sections which can be killed by a tower of ≤ n blow-up’s with smooth centers (in particular G_{0} = 0). According to the construction for each n the quotient sheaf G_{n}/G_{n−1} is generated by sections which can be killed by just one blow-up with a smooth center. Thus for each n there exists an epimorphism onto G_{n}/G_{n−1} from the direct sum of sheaves of the form G(X, Z). We now proceed to show that Ext_{Nis}^{i}(G, F) = 0 by induction on i. Induction hypothesis and (5.12.1) show that Ext_{Nis}^{i}(G_{n}/G_{n−1}, F) = 0 for all n. This implies further that Ext_{Nis}^{i}(G_{n}, F) = 0 for all n. Finally we use the exact sequence (cf. the proof of Lemma 0.6)

0 → ⨁_{n=0}^{∞} G_{n} ⟶ ⨁_{n=0}^{∞} G_{n} ⟶ G → 0

to conclude that Ext_{Nis}^{i}(G, F) = 0.

Corollary 5.12.2. In conditions and notations of Proposition 5.12 the Nisnevich sheaf F is a sheaf in the cdh-topology as well.

Proof. One checks immediately that F is separated in the cdh-topology (to do so it suffices to note that the homomorphism F(X) → F(B_{Z}(X)) is injective for a blow-up with a smooth center Z -see Theorem 4.8). This implies that the canonical homomorphism F → F_{cdh}^{∼} is injective. Denote by G the quotient Nisnevich sheaf, so that G fits into an exact sequence

0 → F → F_{cdh}^{∼} → G → 0

and, in particular G_{cdh}^{∼} = 0. Since Ext_{Nis}^{1}(G, F) = 0 we conclude that the above sequence of Nisnevich shieves splits, so that G is a direct summand in the cdh-sheaf F_{cdh}^{∼} and hence is a cdh-sheaf itself. Thus G = 0.

Denote by α : (Sm/F)_{cdh} ⟶ (Sm/F)_{Nis} the obvious morphism of cites. Note that θ = αϕ, moreover the functor α_{∗} : (Sm/F)_{cdh}^{∼} ⟶ (Sm/F)_{Nis}^{∼} is just the forgetful functor, whereas the functor α^{∗} : (Sm/F)_{NIs}^{∼} ⟶ (Sm/F)_{cdh}^{∼} coincides with the sheafification functor (−)_{cdh}^{∼}.

Corollary 5.12.3. Let F be a homotopy invariant Nisnevich sheaf with transfers. Then R^{i}α_{∗}(F) = 0 for i > 0 and hence H_{cdh}^{∗}(X, F) = H_{Nis}^{∗}(X, F) for any X ∈ Sm/F.

Proof. Let 0 → F ⟶^{ϵ} I^{0} ⟶^{d} I^{1} ⟶^{d} … be the cdh-injective resolution of the cdhsheaf F. Applying the functor α_{∗} to the above resolution we get the same complex of sheaves, but considered now as a complex of Nisnevich sheaves. Thus we have to verify that Nisnevich cohomology sheaves H^{i} of the complex I^{•} are trivial for i > 0. We proceed by induction on i. Note first that (H^{i})_{cdh}^{∼} = 0. Denote by Z^{i} the kernel of the homomorphism I^{i} ⟶^{d} I^{i+1}, so that we get an exact sequence of Nisnevich sheaves

0 ⟶ F ⟶ I^{0} ⟶ … → I^{i−1} ⟶^{d} Z^{i} ⟶^{p} H^{i} → 0

and hence an element in Ext_{Nis}^{i+1}(H^{i}, F). The resulting extension class is trivial according to Proposition 5.12. Since the sheaves I^{0}, ..., I^{i−1} are injective considered as Nisnevich sheaves as well this implies easily that the epimorphism Z^{i} ⟶^{p} H^{i} splits and hence H^{i} is a direct summand of the cdh-sheaf Z^{i} and hence is a cdh-sheaf itself. Thus H^{i} = 0.

We delay the proof of the following result till § 12.

Theorem 5.13. The cdh-cohomological dimension of any scheme X ∈ Sch/F is finite (and ≤ dimX).

Theorem 5.13 together with results of § 0 show that cdh-hypercohomology of a scheme X ∈ Sch/F with coefficients in an arbitrary complex of cdh-sheaves makes perfect sence and has all the expected properties.

Theorem 5.14. For any complex C^{•} ∈ DM^{−}(F) and any scheme X ∈ Sch/F we have natural identifications

H_{M}^{∗}(X, C^{•}) = H_{cdh}^{∗}(X, (C^{•})_{cdh}^{∼}).

Proof. For any X ∈ Sch/F let ℤ_{Nis}(X) : Sm/F → Ab be the Nisnevich sheaf associated to the presheaf U ↦ ℤ[Hom_{Sch/F} (U, X)]. One checks easily that ℤ_{Nis}(X) is a subsheaf in ℤ_{tr}(X) and moreover ℤ_{Nis}(X)_{cdh}^{∼} = ℤ_{cdh}(X). This remark gives us a sequence of natural homomorphisms

H_{M}^{i}(X, C^{•}) = Ext_{NSwT}^{i} (ℤ_{tr}(X), C^{•}) ⟶ Ext_{Nis}^{i}(ℤ_{tr}(X), C^{•}) ⟶

→ Ext_{Nis}^{i}(ℤ_{Nis}(X), C^{•}) ⟶ Ext_{cdh}^{i}(ℤ_{Nis}(X)_{cdh}^{∼}, (C^{•})_{cdh}^{∼}) = H_{cdh}^{i}(X, (C^{•})_{cdh}^{∼}).

We have to show that the composition of these homomorphisms is an isomorphism. We first treat the case of a smooth X. In this case the composition of the first two homomorphisms is an isomorphism according to the Theorem 1.5. The proof of the fact that the last map is also an isomorphism is identical to the proof of Corollary 1.1.1 taking into account Theorem 5.13 and Corollary 5.12.3. In the general case we proceed by induction on dimX. Using additional induction on the number of irreducible components of X and Lemma 5.1 one reduces the general case first to the case of an integral scheme X. In the latter case resolution of singularities gives us a proper birational morphism X′ ⟶^{p} X with X′ smooth and irreducible. Let Z ⊂ X be a proper closed subscheme such that p is an isomorphism over X \ Z. In view of Theorem 5.2 (and Lemma 12.1) we get a commutative diagram with exact rows (in which all motivic cohomology are taken with coefficients in C^{•} and all cdh-cohomology are taken with coefficients in (C^{•})_{cdh}^{∼} ).

⟶ H_{M}^{i−1}(p^{−1}(Z)) ⟶ H_{M}^{i}(X) ⟶ H_{M}^{i}(X′) ⊕ H_{M}^{i}(Z) ⟶
↓  ↓  ↓
⟶ H_{cdh}^{i−1}(p^{−1}(Z)) ⟶ H_{cdh}^{i}(X) ⟶ H_{cdh}^{i}(X′) ⊕ H_{cdh}^{i}(Z) ⟶

Induction hypothesis and the five-Lemma finish the proof.

§ 6. Truncated etale cohomology and the Beilinson-Lichtenbaum Conjecture.

The usual way to formulate the Beilinson-Lichtenbaum Conjecture (cf. § 3) is to say that a complex of Nisnevich sheaves ℤ/l(n) is canonically quasiisomorphic to τ_{≤n}R(π_{0})_{∗}(μ_{l}^{⊗n}), where

π_{0} : (Sm/F)_{et} → (Sm/F)_{Nis}

is the obvious morphism of sites and τ_{≤n} denotes the level n truncation functor. We modify this conjecture slightly so that it would concern an isomorphism of two objects of DM^{−}(F). Throughout this section F is a perfect field and l is an integer prime to charF.

We start by reminding a few elementary properties of h-sheaves to be used later. Recall that h-topology on the category Sch/F is a Grothendieck topology which is stronger than the etale topology and which is essentially characterized by the property that every proper surjective morphism is an h-covering (see [V 0] or [S-V] for the explicit definition). Denote by

(Sch/F)_{h} ⟶^{π} (Sm/F)_{Nis} (Sch/F)_{h} ⟶^{ρ} (Sm/F)_{et}

the obvious morphism of sites (so that π = π_{0}ρ).

Lemma 6.1. The inverse image functors

(Sm/F)_{Nis}^{∼} ⟶^{π^{∗}} (Sch/F)_{h}^{∼} (Sm/F)_{et}^{∼} ⟶^{ρ^{∗}} (Sch/F)_{h}^{∼}

are exact.

Proof. The proof is identical to that of Proposition 5.11, using now Theorem 0.2 (which shows that every scheme X ∈ Sch/F admits a smooth h-covering) instead of resolution of singularities.

The proof of Lemma 6.1 is based (the same as the proof of Proposition 5.11) on the fact that one can define h-topology on the category Sm/F and moreover the categories of sheaves (Sch/F)_{h}^{∼} and (Sm/F)_{h}^{∼} are canonically equivalent. In the sequel we’ll usually identify these categories and often use the notation (−)_{h}^{∼} for both π^{∗} and ρ^{∗}.

Corollary 6.1.1. The direct image functors

π_{∗} : (Sch/F)_{h}^{∼} → (Sm/F)_{Nis}^{∼} ρ_{∗} : (Sch/F)_{h}^{∼} → (Sm/F)_{et}^{∼}

take injective sheaves to injective sheaves.

Lemma 6.2. R^{i}ρ_{∗}(μ_{l}^{⊗n}) = \{ \begin{matrix}μ_{l}^{⊗n} & if i = 0 \\ 0 & if i ≠ 0 \end{matrix}

Proof. The morphism ρ may be factored as a composition

(Sch/F)_{h} ⟶^{α} (Sch/F)_{et} ⟶^{β} (Sm/F)_{et}

Since the restriction functor β_{∗} : (Sch/F)_{et}^{∼} → (Sm/F)_{et}^{∼} is obviously exact we conclude that R^{i}ρ_{∗} = β_{∗}R^{i}α_{∗}. Finally the vanishing of R^{i}α_{∗}(μ_{l}^{⊗n}) for i > 0 was proved in [S-V § 10].

Corollary 6.2.1. For any etale sheaf F ∈ (Sm/F)_{et}^{∼} we have natural isomorphisms

Ext_{et}^{∗}(F, μ_{l}^{⊗n}) = Ext_{h}^{∗}(ρ^{∗}F, μ_{l}^{⊗n})

Lemma 6.3. For any X ∈ Sm/F the natural embedding of etale sheaves ℤ_{et}(X) ↪ ℤ_{tr}(X) induces an isomorphism of h-sheaves ℤ_{h}(X) = (ℤ_{et}(X))_{h}^{∼} ⟶^{∼} (ℤ_{tr}(X))_{h}^{∼}.

Proof. Note that the etale sheaf ℤ_{et}(X) may be identified with a subsheaf of ℤ_{tr}(X), whose sections over U ∈ Sm/F are integral linear combinations of closed integral subschemes Z ⊂ X ×U which are finite surjective and etale over a component of U. In view of Lemma 6.1 it suffices to show that (ℤ_{tr}(X)/ℤ_{et}(X))_{h}^{∼} = 0. To do so we’ll show that for any U ∈ Sm/F and any Z ∈ ℤ_{tr}(X)(U) there exists an h-covering

U^{∼} → U with U^{∼} ∈ Sm/F such that the inverse image of Z in ℤ_{tr}(X)(U^{∼}) belongs

to ℤ_{et}(X)(U^{∼}). Obviously we may suppose that U is irreducible and Z is a closed integral subscheme in X × U finite and surjective over U. We proceed by induction on degree d = [Z : U]. Let U′ → Z be a proper surjective morphism with U′ smooth and irreducible (see Theorem 0.2). The inverse image of Z in ℤ_{tr}(X)(U′) equals Z′ = cycle(U′ ×_{U} Z). One of components of the above cycle is the graph of the section U′ → U′ ×_{U} Z (which belongs to ℤ_{et}(X)(U′)) and all the other components are of degree < d.

Corollary 6.3.1. For any X ∈ Sm/F we have natural isomorphisms

Ext_{et}^{∗}(ℤ_{tr}(X), μ_{l}^{⊗n}) = Ext_{et}^{∗}(ℤ_{et}(X), μ_{l}^{⊗n}) = H_{et}^{∗}(X, μ_{l}^{⊗n}) = H_{h}^{∗}(X, μ_{l}^{⊗n}).

The following Lemma shows that the result of Lemma 6.3 is valid for non smooth schemes as well.

Lemma 6.3.2. For any X ∈ Sch/F we have a natural identification ℤ_{tr}(X)_{h}^{∼} = ℤ_{h}(X).

Proof. Define a presheaf ℤ_{Sm/F} (X) on the category Sm/F via the formula ℤ_{Sm/F} (X)(U) = ℤ[Hom_{Sch/F} (U, X)], i.e. ℤ_{Sm/F} (X) is the restriction to Sm/F of the free presheaf on Sch/F generated by X. One checks easily (using as always Theorem 0.2), that for any h-sheaf G on the category Sch/F we have a canonical identification

Hom_{(Sm/F)^{∧}}(ℤ_{Sm/F} (X), π_{∗}(G)) = G(X).

The above formula shows that (ℤ_{Sm/F} (X))_{h}^{∼} = ℤ_{h}(X). Furthermore we have an obvious embedding ℤ_{Sm/F} (X) ↪ ℤ_{tr}(X) and the arguement used in the proof of Lemma 6.3 shows that the h-sheafification of the quotient ℤ_{tr}(X)/ℤ_{Sm/F} (X) is trivial. This shows that ℤ_{h}(X) = ℤ_{Sm/F} (X)_{h}^{∼} = ℤ_{tr}(X)_{h}^{∼}.

The proof of the following Lemma is identical to that of Proposition 1.10.

Lemma 6.4. Let F be a contractible etale sheaf and let G be a strongly homotopy invariant etale sheaf. Then Ext_{et}^{∗}(F, G) = 0.

Corollary 6.4.1. Let C^{•} be a bounded above complex of contractible etale sheaves and let G be a strongly homotopy invariant etale sheaf. Then Ext_{et}^{∗}(C^{•}, G) =^{def} Hom_{D^{−}((Sm/F)_{et}^{∼})}(C^{•}, G[∗]) = 0.

Proof. Let B be the full triangulated subcategory of D^{−}((Sm/F)_{et}) consisting of complexes with trivial Ext^{∗} to G. The category B is obviously closed with respect to taking arbitrary direct sums (of homologically bounded families) and contains all C^{i} according to Lemma 6.4. Lemma 0.6 shows now that C^{∗} ∈ B.

Corollary 6.4.2. For any X ∈ Sm/F we have natural identifications

Ext_{et}^{∗}(C^{∗}(ℤ_{tr}(X)), μ_{l}^{⊗n}) = Ext_{et}^{∗}(ℤ_{tr}(X), μ_{l}^{⊗n}) = H_{et}^{∗}(X, μ_{l}^{⊗n}).

Proof. Recall that the complex C^{∗}(ℤ_{tr}(X)) contains a subcomplex C_{0}^{∗}(ℤ_{tr}(X)) which is canonically quasiisomorphic to ℤ_{tr}(X) and such that the factorcomplex C^{∗}/C_{0}^{∗} consists of contractible sheaves. Our statement follows now from Corollary 6.4.1, taking into account that the etale sheaf μ_{l}^{⊗n} is strongly homotopy invariant.

Lemma 6.5. For any h-sheaf G the Nisnevich sheaf π_{∗}(G) has a natural structure of a Nisnevich sheaf with transfers. Moreover for any F ∈ NSwT/F we have a natural identification

Hom_{Nis}(F, π_{∗}(G)) = Hom_{NSwT} (F, π_{∗}(G)).

In other words every homomorphism of Nisnevich sheaves F → π_{∗}(G) is compatible with transfers.

Proof. Recall that for any h-sheaf G and any finite surjective morphism Z → X from an integral scheme Z to an irreducible normal scheme X we defined in [S-V] the transfer homomorphism T r_{Z/X} : G(Z) → G(X). Let now X, Y be smooth irreducible schemes and let Z ⊂ X × Y be a closed integral subscheme finite and surjective over X. We define the homomorphism Z^{∗} : G(Y) → G(X) as the composition

G(Y) ⟶^{(p_{2})^{∗}} G(Z) ⟶^{T r_{Z/X}} G(X).

One checks easily, using the results of [S-V § 5], that this defines the structure of a Nisnevich sheaf with transfers on π_{∗}(G) .

In proving the second statement we may assume that F = ℤ_{tr}(X) for some X ∈ Sm/F. In this case

Hom_{NSwT/F} (ℤ_{tr}(X), π_{∗}(G)) = π_{∗}(G)(X) = G(X)

Hom_{Nis}(ℤ_{tr}(X), π_{∗}(G)) = Hom_{h}(π^{∗}(ℤ_{tr}(X)), G) = Hom_{h}(ℤ_{h}(X), G) = G(X).

The complex Rπ_{∗}(μ_{l}^{⊗n}) may be defined as follows. Consider a resolution J^{•}(n)

0 → μ_{l}^{⊗n} → J^{0}(n) → J^{1}(n) → …

of μ_{l}^{⊗n} by injective h-sheaves. Then the complex Rπ_{∗}(μ_{l}^{⊗n}) coincides with

π_{∗}(J^{•}(n)) = (π_{∗}(J^{0}(n)) → π_{∗}(J^{1}(n)) → …).

According to Lemma 6.5 the complex π_{∗}(J^{•}(n)) is actually a complex of Nisnevich sheaves with transfers. Moreover the cohomology presheaves of π_{∗}(J^{•}(n)) are of the form

X ↦ H_{h}^{i}(X, μ_{l}^{⊗n}) = H_{et}^{i}(X, μ_{l}^{⊗n})

and hence are homotopy invariant. This shows that the complex

B_{l}(n) = τ_{≤n}π_{∗}(J^{•}(n))

is a bounded above complex of Nisnevich sheaves with transfers with homotopy invariant cohomology sheaves, i.e. an object of DM^{−}(F).

Sometimes it’s preferable to work with resolutions of μ_{l}^{⊗n} by injective objects in the category of h-sheaves of ℤ/l-modules. For any such resolution 0 → μ_{l}^{⊗n} → I^{•}(n) we have a unique (up to homotopy) homomorphism of resolutions I^{•}(n) → J^{•}(n) and the induced homomorphisms

π_{∗}(I^{•}(n)) → π_{∗}(J^{•}(n)) τ_{≤n}π_{∗}(I^{•}(n)) → τ_{≤n}π_{∗}(J^{•}(n))

are quasiisomorphism since injective sheaves of ℤ/l-modules are acyclic (i.e. H_{h}^{∗}(X, I^{k}) = 0 for any X ∈ Sch/F any k ≥ 0 and any ∗ > 0). Thus we may equally define the complex B_{l}(n) as τ_{≤n}π_{∗}(I^{•}(n)).

Lemma 6.6. For any n ≥ 0 we have natural identifications

H_{et}^{i}(𝔾_{m}^{∧n}, μ_{l}^{⊗n}) = H_{et}^{i−n}(F, ℤ/l).

Proof. We first compute etale cohomology of 𝔾_{m,\overline{F}}^{∧n}. Note that according to the definition H_{et}^{∗}(𝔾_{m,\overline{F}}^{∧n} , μ_{l}^{⊗n}) is a direct summand in H_{et}^{∗}(𝔾_{m,\overline{F}}^{×n} , μ_{l}^{⊗n}), consisting of

cohomology classes which vanish being restricted to any copy of 𝔾_{m,\overline{F}}^{×(n−1)} inside

𝔾_{m,\overline{F}}^{×n}. The well-known computation of the etale cohomology of 𝔾_{m,\overline{F}}

H_{et}^{∗}(𝔾_{m,\overline{F}} , μ_{l}) = \{ \begin{matrix}μ_{l} & if ∗ = 0 \\ ℤ/l & if ∗ = 1 \\ 0 & if ∗ > 1 \end{matrix}

and the Künneth formula for etale cohomology imply easily that

H_{et}^{∗}(𝔾_{m,\overline{F}}^{∧n} , μ_{l}^{⊗n}) = \{ \begin{matrix}ℤ/l & if ∗ = n \\ 0 & if ∗ ≠ n \end{matrix}

We conclude the proof using the Hochschild-Serre spectral sequence

Remark 6.6.1. We’ll denote by α_{n} the canonical generator of H_{et}^{n}(𝔾_{m}^{∧n}, μ_{l}^{⊗n}). The proof of the Lemma shows that α_{n} coincides with the n-th power α_{1} ∈ H_{et}^{1}(𝔾_{m}^{∧1}, μ_{l}) and, in particular α_{n} ∪ α_{m} = α_{n+m} for all n, m ≥ 0.

Corollary 6.6.2. Ext_{et}^{0}(ℤ/l(n), μ_{l}^{⊗n}) = Ext_{et}^{0}(ℤ(n), μ_{l}^{⊗n}) = H_{et}^{n}(𝔾_{m}^{∧n}, μ_{l}^{⊗n}) = ℤ/l.

Proof. This follows immediately from Corollary 6.4.2, Lemma 6.6 and an exact sequence of Ext-groups corresponding to the short exact sequence of complexes

0 → ℤ(n) ⟶^{l} ℤ(n) ⟶ ℤ/l(n) → 0

Proposition 6.7. The complex of etale sheaves π_{0}^{∗}(ℤ/l(n)) is canonically quasiisomorphic to μ_{l}^{⊗n} and hence the same is true for the complex of h-sheaves π^{∗}(ℤ/l(n)).

Proof. Denote by H^{i} the i-th cohomology presheaf of ℤ/l(n). Our statement may be rephrased as follows. The etale sheaves (H^{i})_{et}^{∼} are trivial for i ≠ 0 and the sheaf (H^{0})_{et}^{∼} is canonically isomorphic to μ_{l}^{⊗n}. We first show that (H^{i})_{et}^{∼} = 0 for i ≠ 0 and that the sheaf (H^{0})_{et}^{∼} is a locally constant sheaf of free ℤ/l-modules of rank one. This amounts to showing that for any X ∈ Sm/F and any closed point x ∈ X

H^{i}(O_{X,x}^{sh}) = \{ \begin{matrix}=^{∼} ℤ/l & if i = 0 \\ 0 & if i ≠ 0 \end{matrix}

Note first that H^{i} is a homotopy invariant presheaf of ℤ/l-modules with transfers so that the rigidity theorem ([S-V] Theorem 4.4) shows that H^{i}(O_{X,x}^{sh}) = H^{i}(\overline{F}). Futhermore (in notations of [S-V])

H^{i}(\overline{F}) = H_{n−i}^{sin}(𝔾_{m,\overline{F}}^{∧n}), ℤ/l)

The dual ℤ/l-module H^{i}(\overline{F})^{#} coincides with H_{sin}^{n−i}(𝔾_{m,\overline{F}}^{∧n} , ℤ/l) =

= H_{et}^{n−i}(𝔾_{m,\overline{F}}^{∧n} , ℤ/l) - see [S-V]. This together with the computation of the etale cohomology of 𝔾_{m,\overline{F}}^{∧n} (see Lemma 6.6) proves our claim. To finish the proof of the Proposition we note that vanishing of (H^{i})_{et}^{∼} for i ≠ 0 gives us canonical isomorphisms

Hom_{et}((H^{0})_{et}^{∼}, μ_{l}^{⊗n}) = Ext_{et}^{0}(ℤ/l(n), μ_{l}^{⊗n}) = ℤ/l.

Proposition 6.7 provides us with canonical morphisms

π^{∗}(ℤ/l(n)) → τ_{≥0}π^{∗}(ℤ/l(n)) → I^{•}(n) → J^{•}(n)

Using further the adjunction relation between π^{∗} and π_{∗} we get a canonical morphism

ℤ/l(n) → π_{∗}(I^{•}(n)) ⟶^{∼} π_{∗}(J^{•}(n))

which factors through B_{l}(n) = τ_{≤n}(π_{∗}(I^{•}(n)), since the complex ℤ/l(n) is bounded above at level n. We denote the resulting morphism ℤ/l(n) → B_{l}(n) by α_{n}. Note that according to Lemma 6.5 the homomorphism α_{n} is compatible with transfers i.e. α_{n} ∈ Hom_{DM^{−}(F)}(ℤ/l(n), B_{l}(n)).

Conjecture 6.8 (The Beilinson-Lichtenbaum Conjecture in weight n). The canonical morphism α_{n} : ℤ/l(n) → B_{l}(n) = τ_{≤n}(π_{∗}(I^{•}(n))) is a quasiisomorphism.

Remark 6.8.1 Lemma 6.2 and Corollary 6.1.1 show that the complex ρ_{∗}(J^{•}(n)) is an injective resolution of the etale sheaf μ_{l}^{⊗n} ∈ (Sm/F)_{et}^{∼}. This implies immediately that the complex π_{∗}(J^{•}(n)) = (π_{0})_{∗}(ρ_{∗}(J^{•}(n))) coincides (if we ignore the transfers) with R(π_{0})_{∗}(μ_{l}^{⊗n}) and B_{l}(n) coincides with τ_{≤n}R(π_{0})_{∗}(μ_{l}^{⊗n}), so that the present formulation of the Beilinson-Lichtenbaum Conjecture is equivalent to the usual one.

Lemma 6.9. To prove the Beilinson-Lichtenbaum Conjecture in the weight n it suffices to show that for any finitely generated field extension E/F the homomorphisms

(α_{n})_{∗} : H_{M}^{i}(E, ℤ/l(n)) → H_{M}^{i}(E, B_{l}(n))

are isomorphisms.

Proof. Let H^{i} (resp. H^{∼}^{i}) denote the i-th cohomology sheaf of ℤ/l(n) (resp. of B_{l}(n)). The Beilinson-Lichtenbaum Conjecture asserts that the induced maps

(α_{n})_{∗} : H^{i} → H^{∼}^{i} are isomorphisms for all i. Note further that H^{i} and H^{∼}^{i} are homotopy invariant Nisnevich sheaves with transfers and according to [V 1] a homomorphism of Nisnevich sheaves with transfers is an isomorphism iff it induces isomorphisms on sections over an arbitrary finitely generated field extension of F.

The following Lemma allows to reduce the general case of the Beilinson-Lichtenbaum Conjecture to the case of prime numbers.

Lemma 6.10. Let l, l′ be two integers. If the Beilinson-Lichtenbaum Conjecture in the weight n holds modulo l and l′ then it also holds modulo ll′. Thus to prove the Beilinson-Lichtenbaum Conjecture in the weight n modulo l it suffices to prove it modulo all prime divisors of l.

Proof. Note that in DM^{−}(F) we have a distinguished Bockstein triangle

ℤ/l′(n) ⟶ ℤ/ll′(n) ⟶ ℤ/l(n) ⟶ ℤ/l′(n)[1].

Validity of the Beilinson-Lichtenbaum Conjecture modulo l implies that we have a similar distinguished triangle involving B(n):

B_{l′}(n) ⟶ B_{ll′}(n) ⟶ B_{l}(n) ⟶ B_{l′}(n)[1].

To check this one has only to verify that the induced homomorphism on the n-th cohomology sheaves H^{n}(B_{ll′}(n)) ⟶ H^{n}(B_{l}(n)) is surjective. This follows however from the commutative diagram

H^{n}(ℤ(n)) ⟶^{=} H^{n}(ℤ(n))
↓  ↓
H^{n}(B_{ll′}(n)) ⟶ H^{n}(B_{l}(n))

in which the right vertical homomorphism is surjective. Finally one checks easily that the homomorphisms α_{n} give a map from the first distinguished triangle to the second one and our statement follows in view of the five-Lemma.

We end up this section with a few remarks concerning motivic cohomology with coefficients in B_{l}(n).

Lemma 6.11. For any X ∈ Sch/F and any i ≤ n we have natural isomorphisms

H_{M}^{i}(X, B_{l}(n)) = H_{h}^{i}(X, μ_{l}^{⊗n}) = H_{et}^{i}(X, μ_{l}^{⊗n}).

Proof. According to definitions and Corollary 1.11.2 we have natural identifications

H_{M}^{i}(X, B_{l}(n)) = Ext_{NSwT}^{i} (ℤ_{tr}(X), B_{l}(n)).

The short exact sequence of complexes

0 → B_{l}(n) → π_{∗}(J^{•}(n)) → τ_{≥n+1}π_{∗}(J^{•}(n)) → 0

gives rise to a long exact sequence of Ext-groups The Ext-groups from ℤ_{tr}(X) to τ_{≥n+1}π_{∗}(J^{•}(n)) are trivial in degrees ≤ n by obvious reasons. The complex π_{∗}(J^{•}(n)) consists of injective Nisnevich sheaves with transfers and hence

Ext_{NSwT}^{∗} (ℤ_{tr}(X), π_{∗}(J^{•}(n))) = H^{∗}(Hom_{NSwT} (ℤ_{tr}(X), π_{∗}(J^{•}(n)))) =

H^{∗}(Hom_{Nis}(ℤ_{tr}(X), π_{∗}(J^{•}(n)))) = H^{∗}(Hom_{h}(ℤ_{h}(X), J^{•}(n))) = H_{h}^{∗}(X, μ_{l}^{⊗n}).

Corollary 6.11.1. Hom_{DM^{−}(F)}(ℤ/l(n), B_{l}(n)) = Hom_{DM^{−}(F)}(ℤ(n), B_{l}(n)) = = H_{et}^{n}(𝔾_{m}^{∧n}, μ_{l}^{⊗n}) = ℤ/l. Under this identification the homomorphism α_{n} corresponds to the cohomology class denoted by α_{n} previously.

Proof. The first statement follows from Lemma 6.11 and the long exact cohomology sequence, corresponding to the short exact sequence of complexes

0 → ℤ(n) ⟶^{l} ℤ(n) → ℤ/l(n) → 0

The second statement is obvious from the construction.

Remark 6.11.2 Corollary 6.11.1 shows that Hom_{DM^{−}(F)}(ℤ/l(n), B_{l}(n)) may be identified with Hom_{DM^{−}(F)}(ℤ(n), B_{l}(n)), we’ll use the same notation α_{n} for the corresponding morphism ℤ(n) → B_{l}(n).

§ 7. The Bloch-Kato Conjecture.

We keep notations and assumptions of the previous section.

Proposition 7.1. For any n, m ≥ 0 there are natural pairings B_{l}(n) ⊗ B_{l}(m) → → B_{l}(n + m). Furthermore for any X, Y ∈ Sch/F and any integers i ≤ n, j ≤ m the resulting pairing in cohomology

H_{et}^{i}(X, μ_{l}^{⊗n}) ⊗ H_{et}^{j}(Y, μ_{l}^{⊗m}) = H_{M}^{i}(X, B_{l}(n)) ⊗ H_{M}^{j}(Y, B_{l}(m)) →
→ H_{M}^{i+j}(X × Y, B_{l}(n + m)) = H_{et}^{i+j}(X × Y, μ_{l}^{⊗(n+m)})

coincides with the usual product pairing in etale cohomology.

Proof. Note that for any injective h-sheaf of ℤ/l-modules I all groups of sections I(X) are injective ℤ/l-modules and hence free ℤ/l-modules. This remark implies immediately that the complex I^{•}(n) ⊗_{h} I^{•}(m) is a resolution of μ_{l}^{⊗n} ⊗_{h} μ_{l}^{⊗m} = μ_{l}^{⊗n+m}, which gives us a natural homomorphism of complexes I^{•}(n) ⊗_{h} I^{•}(m) → I^{•}(n+m). Furthermore one checks easily (using Lemma 2.1) that for any h-sheaves F, G there exists a canonical homomorphism of Nisnevich sheaves with transfers π_{∗}(F) ⊗_{tr} π_{∗}(G) → π_{∗}(F ⊗_{h} G). This gives us a homomorphism of complexes

B_{l}(n) ⊗^{L} B_{l}(m) → B_{l}(n) ⊗_{tr} B_{l}(m) → π_{∗}(I^{•}(n)) ⊗_{tr} π_{∗}(I^{•}(m)) →
→ π_{∗}(I^{•}(n) ⊗_{h} I^{•}(m)) → π_{∗}(I^{•}(n + m))

which factors through B_{l}(n + m) since the complex B_{l}(n) ⊗^{L} B_{l}(m) is bounded above at the level n + m. Applying finally the localising functor C^{∗} to both sides we get the desired pairing

B_{l}(n) ⊗ B_{l}(m) = C^{∗}(B_{l}(n) ⊗^{L} B_{l}(m)) → C^{∗}(B_{l}(n + m)) ⟵^{∼} B_{l}(n + m)

The second statement is obvious from the construction.

Corollary 7.1.1. For any n, m ≥ 0 the following diagram in DM^{−}(F) commutes

ℤ/l(n) ⊗ ℤ/l(m) ⟶^{∼} ℤ/l(n + m)
α_{n}⊗α_{m}↓  ↓^{α_{n+m}}
B_{l}(n) ⊗ B_{l}(m) ⟶ B_{l}(n + m)

Proof. We have to show that two resulting maps in DM^{−}(F) from ℤ/l(n) ⊗ ℤ/l(m) = ℤ/l(n + m) to B_{l}(n + m) coincide. To do so we note that Hom_{DM^{−}(F)}(ℤ/l(n + m), B_{l}(n + m)) = H_{et}^{n+m}(𝔾_{m}^{∧(n+m)}, μ_{l}^{⊗(n+m)}) (= ℤ/l) and the resulting maps correspond to the cohomology classes α_{n+m} and α_{n} ∪ α_{m} = α_{n+m} respectively.

Lemma 7.2. For any (finitely generated) field extension E/F we have natural identifications:

H_{M}^{i}(E, ℤ/l(1)) = \{ \begin{matrix}H^{0}(E, μ_{l}) & if i = 0 \\ E^{∗}/E^{∗l} & if i = 1 \\ 0 & if i ≠ 0, 1 \end{matrix}

The homomorphism

(α_{1})_{∗} : H_{M}^{i}(E, ℤ/l(1)) → H_{M}^{i}(E, B_{l}(1)) = \{ \begin{matrix}H_{et}^{i}(E, μ_{l}) & if i = 0, 1 \\ 0 & if i ≠ 0, 1 \end{matrix}

is identity in degree 0 and coincides with the Kümmer isomorphism χ : E^{∗}/E^{∗l} ⟶^{∼} H_{et}^{1}(E, μ_{l}) in degree one. In particular the Beilinson-Lichtenbaum Conjecture is true in weight 1.

Proof. As was noted in § 3 the complex ℤ(1) is naturally quasiisomorphic to O^{∗}[−1] and hence the complex ℤ/l(1) is naturally quasiisomorphic to the complex

O^{∗} ⟶^{l} O^{∗}

which gives immediately the computation of motivic cohomology of any field with coefficients in ℤ/l(1). To compute the action of α_{1} in cohomology we may proceed as follows. Let 0 → O^{∗} → L^{•} be an injective resolution of the etale sheaf O^{∗} ∈ (Sm/F)_{et}^{∼}. The cone J^{•} = Cone(L^{•} ⟶^{l} L^{•}) gives us an injective resolution of the etale sheaf μ_{l}. According to definitions and Remark 5.8.1 the homomorphism α_{1} corresponds (by adjunction) to the natural quasiisomorphism of complexes of etale sheaves

Cone(O^{∗} ⟶^{l} O^{∗}) → Cone(L^{•} ⟶^{l} L^{•}).

Thus homomorphisms induced by α_{1} in motivic cohomology of E may be computed using the homomorphism of complexes

Cone(E^{∗} ⟶^{l} E^{∗}) → Cone(L^{•}(E) ⟶^{l} L^{•}(E)) = J^{•}(E).

Since the cohomology of L^{•}(E) coincides with H_{et}^{∗}(E, O^{∗}) we get (applying Hilbert’s Theorem 90) the following commutative diagram with exact rows

0 ⟶  μ_{l}(E)  ⟶  E^{∗}  ⟶^{l}  E^{∗}  ⟶ E^{∗}/E^{∗l} → 0
↓ (α_{1})_{∗}  ↓=  ↓=  ↓ (α_{1})_{∗}
0 ⟶ H^{0}(J^{•}(E)) ⟶ H^{0}(L^{•}(E)) ⟶^{l} H^{0}(L^{•}(E)) ⟶^{δ} H^{1}(J^{•}(E)) → 0
The  diagram  shows  immediately  that  the  homomorphism

(α_{1})_{∗} : E^{∗} → H^{1}(J^{•}(E)) = H_{et}^{1}(E, μ_{l}) coincides with the connecting homomorphism δ : H_{et}^{0}(E, O^{∗}) → H_{et}^{1}(E, μ_{l}), corresponding to the short exact sequence of etale sheaves

0 → μ_{l} → O^{∗} ⟶^{l} O^{∗} → 0

i.e. E^{∗}/E^{∗l} ⟶^{(α_{1})_{∗}} H_{et}^{1}(E, μ_{l}) coincides with the Kümmer isomorphism.

Corollary 7.2.1. For any n ≥ 0 and any finitely generated field extension E/F the homomorphism in motivic cohomology induced by α_{n}

(α_{n})_{∗} : K_{n}^{M}(E)/l = H_{M}^{n}(E, ℤ/l(n)) → H_{M}^{n}(E, B_{l}(n)) = H_{et}^{n}(E, μ_{l}^{⊗n})

coincides with the norm residue homomorphism.

Proof. Corollary 7.1.1 Lemma 7.2 and Proposition 7.1 show that (α_{n})_{∗} sends the symbol {a_{1}, ..., a_{n}} to χ(a_{1}) ∪ ... ∪ χ(a_{n}).

Corollary 7.2.1 shows that the Beilinson-Lichtenbaum Conjecture includes as a special case the following conjecture due to S. Bloch and K. Kato.

Conjecture 7.3 (Bloch-Kato Conjecture for a field E in weight n). The norm-residue homomorphism χ_{n} : K_{n}^{M}(E)/l → H_{et}^{n}(E, μ_{l}^{⊗n}) is an isomorphism.

We’ll be also considering the following weaker form of the Bloch-Kato Conjecture.

Conjecture 7.3.1 (Weak Bloch-Kato Conjecture for a field E in weight n). The norm-residue homomorphism χ_{n} : K_{n}^{M}(E)/l → H_{et}^{n}(E, μ_{l}^{⊗n}) is an epimorphism.

Our main result shows that the validity of the Weak Bloch-Kato Conjecture alone implies the validity of the Beilinson-Lichtenbaum Conjecture.

Theorem 7.4. Assume that resolution of singularities holds over F and the Weak Bloch-Kato Conjecture (modulo l) holds in weights ≤ n for all finitely generated field extensions E/F. Then the Beilinson-Lichtenbaum Conjecture (modulo any power of l) also holds over F in weights ≤ n.

The proof of this Theorem occupies the rest of this section and the next 3 sections. Proceeding by induction on n we may (and will) assume that the Beilinson-Lichtenbaum Conjecture is already known to be true in weights < n and, in particular, the strong form of the Bloch-Kato Conjecture (for finitely generated field extensions E/F) is also true in weights < n. We start with the following elementary observation.

Lemma 7.5. Assume that l is prime. Then for any finitely generated field extension E/F the induced homomorphisms in motivic cohomology

(α_{n})_{∗} : H_{M}^{i}(E, ℤ/l(n)) ⟶ H_{M}^{i}(E, B_{l}(n)) = H_{et}^{i}(E, μ_{l}^{⊗n})

are surjective for all i ≤ n.

Proof. We may obviously assume that i ≤ n − 1. Furthermore since the degree of the field extension E(μ_{l})/E is prime to l we see, using the standard trick with transfer maps, that it suffices to consider the case when E contains the primitive l-th root of unity. In the latter case our statement follows immediately from the commutativity of the following diagram

H_{M}^{0}(E, ℤ/l(1)) ⊗ H_{M}^{i}(E, ℤ/l(n − 1)) ⟶ H_{M}^{i}(E, ℤ/l(n))
=^{∼}↓  ↓
H_{et}^{0}(E, μ_{l}) ⊗ H_{et}^{i}(E, μ_{l}^{⊗(n−1)})  ⟶^{∼} H_{et}^{i}(E, μ_{l}^{⊗n})

We finish this section with the following important result, which gives the first step towards the proof of Theorem 7.4.

Proposition 7.6. For any affine semilocal scheme S essentially of finite type over F the homomorphism in motivic cohomology induced by α_{n}

(α_{n})_{∗} : H_{M}^{n}(S, ℤ/l(n)) ⟶ H_{M}^{n}(S, B_{l}(n)) = H_{et}^{n}(S, μ_{l}^{⊗n})

is surjective.

Proof. Assume first that S is smooth (and irreducible). Denote by K_{n} (resp. H^{n}) the n-th cohomology sheaf of the complex ℤ/l(n) (resp. of B_{l}(n)). Since all higher dimensional cohomology groups of a smooth semilocal scheme with coefficients in a homotopy invariant sheaf with transfers vanish (see [V 1]) we conclude that H_{M}^{n}(S, ℤ/l(n)) = H_{Zar}^{0}(S, K_{n}), H_{M}^{n}(S, B_{l}(n)) = H_{Zar}^{0}(S, H^{n}). The cokernel L of the homomorphism (α_{n})_{∗} : K_{n} → H^{n} is a homotopy invariant Nisnevich sheaf with transfers which vanish on all finitely generated field extensions of F according to our assumptions and hence is zero ([V 1]). Moreover the sequence

H_{Zar}^{0}(S, K_{n}) → H_{Zar}^{0}(S, H^{n}) → H_{Zar}^{0}(S, L) = 0

is exact ([V 1]), which ends up the proof in the smooth case.

For an arbitrary affine semilocal scheme we proceed as follows. Embed S as a closed subscheme into a smooth affine semilocal scheme T and denote by I the defining ideal of S. Let T_{I}^{h} be the henselization of T along I. The scheme S embeds canonically as a closed subscheme of T_{I}^{h} and according to a Theorem of O. Gabber [G] the restriction homomorphism in etale cohomology H_{et}^{n}(T_{I}^{h}, μ_{l}^{⊗n}) → H_{et}^{n}(S, μ_{l}^{⊗n}) is an isomorphism. On the other hand the scheme T_{I}^{h} is a filtered inverse limit of smooth semilocal schemes essentially of finite type over F (each containing S as a closed subscheme). Since etale cohomology commutes with filtered limits we conclude that for every cohomology class x ∈ H_{et}^{n}(S, μ_{l}^{⊗n}) there exists a smooth affine semilocal scheme T′ essentially of finite type over F, containing S as a closed subscheme such that x is the restriction to S of an appropriate cohomology class x′ ∈ H_{et}^{n}(T′, μ_{l}^{⊗n}). Now it suffices to use the commutativity of the following diagram

H_{M}^{n}(T′, ℤ/l(n)) ⟶^{↠} H_{et}^{n}(T′, μ_{l}^{⊗n})
↓  ↓
H_{M}^{n}(S, ℤ/l(n)) ⟶ H_{et}^{n}(S, μ_{l}^{⊗n}).

§ 8. B_{l}(n)-cohomology with supports.

Lemma 8.1.

H_{M}^{i}(ℙ_{F}^{1} , ℤ/l(1)) = \{ \begin{matrix}H_{M}^{i}(F, ℤ/l(1)) & if i = 0, 1 \\ ℤ/l & if i = 2 \\ 0 & if i ≠ 0, 1, 2 \end{matrix}

Proof. This follows easily from the computation of motivic cohomology with coefficients in ℤ(1) = O^{∗}[−1] - see Corollary 3.2.1.

H_{M}^{i}(ℙ_{F}^{1} , ℤ(1)) = \{ \begin{matrix}F^{∗} & if i = 1 \\ P ic(ℙ_{F}^{1} ) = ℤ & if i = 2 \\ 0 & if i ≠ 1, 2 \end{matrix}

Remark 8.1.1 The complex ℤ/l(1) is naturally qasiisomorphic to B_{l}(1) = τ_{≤1}π_{∗}(I^{•}(1)). It has exactly two non-zero cohomology sheaves: H^{0}(1) and H^{1}(1). Moreover the sheaf H^{0}(1) coincides with μ_{l} and the sheaf H^{1}(1) is the sheaf associated with the preasheaf U ↦ H_{et}^{1}(U, μ_{l}). Higher dimensional Zariski cohomology with coefficients in μ_{l} vanish, from which we conclude that H_{M}^{2}(ℙ_{F}^{1} , ℤ/l(1)) = H_{Zar}^{1}(ℙ_{F}^{1} , H^{1}(1)). Furthermore Zariski cohomology of any smooth scheme with coefficients in H^{1}(1) may be computed using the Gersten complexes - see [B-O]. In particular H_{Zar}^{∗}(ℙ_{F}^{1} , H^{1}(1)) may be computed using the Gersten complex

H_{et}^{1}(F (ℙ^{1}), μ_{l}) ⟶ ⨁_{x∈(ℙ_{F}^{1} )^{1}} ℤ/l

Here and in the sequell we denote by U^{i} the set of points of codimension i on a smooth scheme U. We’ll be using the notation β for the canonical generator of H_{M}^{2}(ℙ_{F}^{1} , ℤ/l(1)) = H_{M}^{2}(ℙ_{F}^{1} , B_{l}(1)) = H_{Zar}^{1}(ℙ_{F}^{1} , H^{1}(1)). In terms of the above Gersten complex β may be represented by the class of any rational point of ℙ_{F}^{1} .

Usually we’ll be thinking of β as represented by the class of the rational point ∞ ∈ ℙ_{F}^{1} .

Remark 8.1.2 We use 0 as a distinguished rational point on ℙ_{F}^{1} . For any X ∈ Sm/F this allows us to identify X with a closed subscheme X × 0 ⊂ X × ℙ_{F}^{1} . The motiv M(X × ℙ_{F}^{1} /X) coincides with M(X)(1)[2] - see § 4. Motivic cohomology H_{M}^{∗}(X × ℙ_{F}^{1} /X, C^{∗}) may be identified with the kernel of the natural split epimorphism H_{M}^{∗}(X × ℙ_{F}^{1} , C^{∗}) → H_{M}^{∗}(X, C^{∗}) induced by the closed embedding X = X × 0 ↪ X × ℙ_{F}^{1} . Since motivic cohomology groups are homotopy invariant H_{M}^{∗}(X × ℙ_{F}^{1} /X, C^{∗}) may be also identified with the kernel of the canonical homomorphism H_{M}^{∗}(X × ℙ_{F}^{1} , C^{∗}) → H_{M}^{∗}(X × 𝔸_{F}^{1} , C^{∗}), where we identify 𝔸_{F}^{1} with the open subscheme ℙ_{F}^{1} \ ∞.

Proposition 8.2. For any X  ∈  Sm/F multiplication by

β ∈ H_{M}^{2}(ℙ_{F}^{1} /Spec F, B_{l}(1)) defines isomorphisms

H_{M}^{i−2}(X, B_{l}(n − 1)) ⟶^{∼^{β}} H_{M}^{i}(X × ℙ_{F}^{1} /X, B_{l}(n)).

Proof. The same construction as in § 7 defines natural pairings

τ_{≤k−1}π_{∗}I^{•}(n − 1) ⊗ τ_{≤1}π_{∗}I^{•}(1) ⟶ τ_{≤k}π_{∗}I^{•}(n).

We’ll show more generally that multiplication by β induces isomorphisms

H_{M}^{i−2}(X, τ_{≤k−1}π_{∗}I^{•}(n − 1)) ⟶^{∼^{β}} H_{M}^{i}(X × ℙ_{F}^{1} /X, τ_{≤k}π_{∗}I^{•}(n))

for all k ≥ 0. Denote by H^{k}(n) the k-th cohomology sheaf of the complex π_{∗}I^{•}(n). Thus H^{k}(n) is the Zariski sheaf associated to the presheaf U ↦ H_{et}^{k}(U, μ_{l}^{⊗n}) (H^{k}(n) = 0 for k > n). Proceeding by induction on k it suffices to show that for each k the multiplication by β ∈ H_{Zar}^{1}(ℙ_{F}^{1} , H^{1}(1)) induces isomorphisms

H_{Zar}^{i−1}(X, H^{k−1}(n − 1)) ⟶^{∼^{β}} H_{Zar}^{i}(X × ℙ_{F}^{1} /X, H^{k}(n)).

Zariski cohomology of any smooth scheme Y with coefficients in H^{k}(n) may be computed using the Gersten complex

⨁_{y∈Y^{0}} H_{et}^{k}(F (y), μ_{l}^{⊗n}) ⟶ ⨁_{y∈Y^{1}} H_{et}^{k−1}(F (y), μ_{l}^{⊗n−1} ⟶ …

Remark 8.1.2 shows further that H_{Zar}^{i}(X × ℙ_{F}^{1} /X, H^{k}(n)) may be computed using the Gersten bicomplex

⨁_{y∈(X×ℙ_{F}^{1} )^{0}} H_{et}^{k}(F (y), μ_{l}^{⊗n}) ⟶ ⨁_{y∈(X×ℙ_{F}^{1} )^{1}} H_{et}^{k−1}(F (y), μ_{l}^{⊗(n−1)}) ⟶
↓  ↓
⨁_{y∈(X×𝔸_{F}^{1} )^{0}} H_{et}^{k}(F (y), μ_{l}^{⊗n}) ⟶ ⨁_{y∈(X×𝔸_{F}^{1} )^{1}} H_{et}^{k−1}(F (y), μ_{l}^{⊗(n−1)}) ⟶

since the vertical arrows of the above bicomplex are split surjective, this bicomplex is naturally quasiiisomorphic to the kernel of the vertical homomorphism of Gersten complexes, which coincides with the Gersten complex

⨁_{x∈X^{0}} H_{et}^{k−1}(F (x), μ_{l}^{⊗(n−1)}) ⟶ ⨁_{x∈X^{1}} H_{et}^{k−2}(F (x), μ_{l}^{⊗(n−2)})

shifted to the right by 1. This gives us the desired isomorphisms

H_{Zar}^{i}(X × ℙ_{F}^{1} /X, H^{k}(n)) = H_{Zar}^{i−1}(X, H^{k−1}(n − 1))

Moreover it’s easy to see that this isomorphism coincides (up to a sign) with multiplication by β.

Corollary 8.3. For any Z ∈ Sm/F we have natural izomorphisms H_{M}^{∗}(Z(d), B_{l}(n)) = H_{M}^{∗}(Z, B_{l}(n − d)). Moreover (assuming that resolution of singularities holds over F) the following diagram commutes

H_{M}^{∗}(Z, ℤ/l(n − d)) ⟶^{∼} H_{M}^{∗}(Z(d), ℤ/l(n))
(α_{n−d})_{∗}↓  ^{(α_{n})_{∗}}↓
H_{M}^{∗}(Z, B_{l}(n − d)) ⟶^{∼} H_{M}^{∗}(Z(d), B_{l}(n))

Proof. Proceeding by induction on d it suffices to consider the case d = 1. The motiv Z(1) may be identified with Z × ℙ_{F}^{1} /Z[−2]. Proposition 8.2 shows that multiplication by β gives the desired isomorphism. Moreover Corollary 4.12.1 shows that multiplication by τ ∈ H_{M}^{2}(ℙ^{1}, ℤ(1)) gives an isomorphism in motivic cohomology with coefficients in ℤ/l(−). Compatibility of products in motivic cohomology with ℤ/l(−) and B_{l}(−) coefficients together with the fact that β coincides with the image of τ under the homomorphism ℤ(1) → ℤ/l(1) = B_{l}(1) proves the commutativity of the above diagram.

Corollary 8.4. Let X ∈ Sm/F be a smooth irreducible scheme and let Z ≠ X be a smooth irreducible closed subscheme. Then the map in motivic cohomology with supports in Z

H_{Z}^{∗}(X, ℤ/l(n)) ⟶ H_{Z}^{∗}(X, B_{l}(n))

is an isomorphism.

Proof. Recall that motivic cohomology of X with supports in Z is defined as the motivic cihimilogy of the cone M_{Z}(X) = (M(X) → M(U)) (U = X\Z). Recall also that in case Z is smooth everywhere of codimension d the motiv M_{Z}(X) is naturally isomorphic to M(Z)(d)[2d] - cf § 4. Thus our statement follows immediately from Corollary 8.3 (and the induction hypothesis).

Using now Theorem 5.6 we get finally the following result.

Theorem 8.5. Let X ∈ Sch/F be a scheme and let Z ⊂ X be a closed subscheme such that U = X \ Z is strictly dense in X. Then the natural map in motivic cohomology with supports in Z

(α_{n})_{∗} : H_{Z}^{∗}(X, ℤ/l(n)) ⟶ H_{Z}^{∗}(X, B_{l}(n))

is an isomorphism.

§ 9. Shift of degrees

Let ∂Δ^{n} denote the closed subscheme of Δ^{n}, given by the equation t_{0} ·...·t_{n} = 0, in other words ∂Δ^{n} is the union of all codimension one faces of Δ^{n}. Usually we’ll identify the scheme Δ^{n−1} with the face t_{n} = 0 of Δ^{n} and denote the union of all other faces by σΔ^{n}. Thus σΔ^{n} is a closed subscheme of Δ^{n} given by the equation t_{0} · ... · t_{n−1} = 0. Obviously

∂Δ^{n} = Δ^{n−1} ∪ σΔ^{n} and Δ^{n−1} ∩ σΔ^{n} = ∂Δ^{n−1}.

Lemma 9.1. The scheme σΔ^{n} is algebraically contractible.

Proof. The linear contraction of Δ^{n} to the vertex v_{n} = (0, ..., 0, 1)

Δ^{n} × 𝔸^{1} ⟶ Δ^{n} (t_{0}, ..., t_{n}) × t ↦ t · (t_{0}, ..., t_{n}) + (1 − t) · (0, ..., 0, 1) obviously maps σΔ^{n} × 𝔸^{1} to σΔ^{n} and defines the desired contraction.

Let X ∈ Sch/F be a scheme of finite type over F. Each of the vertices v_{i} ∈ ∂Δ^{n}

defines a natural morphism M(X) ⟶^{(1_{X}×v_{i})_{∗}} M(X × ∂Δ^{n}) which is right inverse to

the morphism defined by the projection M(X×∂Δ^{n}) ⟶^{(p_{1})_{∗}} M(X). Thus the choice of the vertex v_{i} allows us to identify M(X) with a direct summand in M(X ×∂Δ^{n}). In case n > 1 this identification is really independent of the choice of the vertex. In the case n = 1 we’ll be using the vertex v_{0} to make the above identification. We’ll denote by M(X × ∂Δ^{n}/X) the complimentary direct summand. The motivic cohomology groups H_{M}^{i}(X × ∂Δ^{n}/X, C^{∗}) coincide with the kernel of the natural split epimorphism

H_{M}^{i}(X × ∂Δ^{n}, C^{∗}) ⟶^{(1_{X}×v_{0})^{∗}} H_{M}^{i}(X, C^{∗}).

Lemma 9.2. For any X ∈ Sch/F any n > 0 and any C^{∗} ∈ DM^{−}(F) we have natural isomorphisms

δ^{n−1} : H_{M}^{i−(n−1)}(X, C^{∗}) ⟶^{∼} H_{M}^{i}(X × ∂Δ^{n}/X, C^{∗}).

Proof. We proceed by induction on n. The case n = 1 is trivial, so assume that n > 1. The closed covering X × ∂Δ^{n} = X × Δ^{n−1} ∪ X × σΔ^{n} gives us a long exact Mayer-Vietoris sequence, which (since the schemes Δ^{n−1} and σΔ^{n} are contractible) looks as follows (all motivic cohomology groups are taken with coefficients in C^{∗}):

H_{M}^{i−1}(X × ∂Δ^{n}) ⟶ H_{M}^{i−1}(X) ⊕ H_{M}^{i−1}(X) ⟶ H_{M}^{i−1}(X × ∂Δ^{n−1}) ⟶^{δ} H_{M}^{i}(X × ∂Δ^{n})

One checks easily that the image of the homomorphism H_{M}^{i−1}(X × ∂Δ^{n}) ⟶ H_{M}^{i−1}(X) ⊕ H_{M}^{i−1}(X)

consists of all elements of the form y ⊕ (−y) y ∈ H_{M}^{i−1}(X) and hence the above exact sequence gives rise to the following four term exact sequence

0 ⟶ H_{M}^{i−1}(X) ⟶^{(p_{1})^{∗}} H_{M}^{i−1}(X × ∂Δ^{n−1}) ⟶^{δ} H_{M}^{i}(X × ∂Δ^{n}) ⟶^{(1_{X}×v_{0})^{∗}} H_{M}^{i}(X) ⟶ 0

Thus the homomorphism δ defines an isomorphism

δ : H_{M}^{i−1}(X × ∂Δ^{n−1}/X) ⟶^{∼} H_{M}^{i}(X × ∂Δ^{n}/X)

which concludes the proof.

Proposition 9.3. In conditions and notations of Theorem 7.4 the induced homomorphism in motivic cohomology

(α_{n})_{∗} : H_{M}^{i}(E, ℤ/l(n)) ⟶ H_{M}^{i}(E, B_{l}(n)) = H_{et}^{i}(E, μ_{l}^{⊗n})

is surjective for any finitely generated field extension E/F.

Proof. In the most important case when l is prime this is trivial - see Lemma 7.5. In the general case one has to use a more subtle arguement. Lemma 9.2 shows that motivic cohomology groups in question may be identified with H_{M}^{n}(∂Δ_{E}^{n−i+1}, ℤ/l(n)) and H_{M}^{n}(∂Δ_{E}^{n−i+1}, B_{l}(n)) respectively. For any open subscheme U ⊂ ∂Δ_{E}^{n−i+1} containing all the vertices v_{i} we have an exact sequence of motivic cohomology groups (both with ℤ/l(n) and B_{l}(n)-coefficients)

H_{T}^{n}(∂Δ_{E}^{n−i+1}) ⟶ H_{M}^{n}(∂Δ_{E}^{n−i+1}) ⟶ H_{M}^{n}(U) ⟶ H_{T}^{n+1}(∂Δ_{E}^{n−i+1})

where we denoted by T the closed subscheme ∂Δ_{E}^{n−i+1} \ U. Taking direct limits of these sequences over all U’s we get the following commutative diagram with exact rows

H_{𝕋}^{n}(∂Δ_{E}^{n−i+1}) ⟶ H_{M}^{n}(∂Δ_{E}^{n−i+1}) ⟶ H_{M}^{n}(U) ⟶ H_{𝕋}^{n+1}(∂Δ_{E}^{n−i+1})
=^{∼}↓  ↓  ^{epi}↓  =^{∼}↓
H_{𝕋}^{n}(∂Δ_{E}^{n−i+1}) ⟶ H_{M}^{n}(∂Δ_{E}^{n−i+1}) ⟶ H_{M}^{n}(U) ⟶ H_{𝕋}^{n+1}(∂Δ_{E}^{n−i+1})

Here the top row motivic cohomology are taken with ℤ/l(n)-coefficients, bottom row motivic cohomology are taken with B_{l}(n)-coefficients, U denotes the semilocalization of the scheme ∂Δ_{E}^{n−i+1} at the points v_{i} and H_{𝕋}^{n}(∂Δ_{E}^{n−i+1}) stands for

\underrightarrow{lim}_{T} H_{T}^{n}(∂Δ_{E}^{n−i+1})

where direct limit is taken over all closed reduced subschemes containing no vertices. Note that the first and the last vertical maps in the above diagram are isomorphisms according to Theorem 8.5 (one checks easily that every open subscheme of ∂Δ_{E}^{n−i+1} containing all vertices is strictly dense). Finally the second from the right vertical map is surjective according to Proposition 7.6. An easy diagram chase shows that the homomorphism H_{M}^{n}(∂Δ_{E}^{n−i+1}, ℤ/l(n)) → H_{M}^{n}(∂Δ_{E}^{n−i+1}, B_{l}(n)) is surjective as well.

Lemma 9.2 gives us, in particular, for all i < n and all finitely generated field extensions E/F a natural isomorphism

δ^{n−i} : H_{M}^{i}(E, ℤ/l(n)) ⟶^{∼} H_{M}^{n}(∂Δ_{E}^{n−i+1}/SpecE, ℤ/l(n)) ⊂ H_{M}^{n}(∂Δ_{E}^{n−i+1}, ℤ/l(n)).

We’ll need below an explicit formula for this isomorphism.

Note first of all that for any X ∈ Sch/F and any complex C^{∗} ∈ DM^{−}(F) we have a natural homomorphism H^{i}((C^{∗})_{cdh}^{∼}(X)) → H_{M}^{i}(X, C^{∗}) which in certain cases (in the case of fields for example) is an isomorphism. We’ll show, in particular, that the map δ^{n−i} : H_{M}^{i}(E, ℤ/l(n)) → H_{M}^{n}(∂Δ_{E}^{n−i+1}, ℤ/l(n)) factors through

H^{n}(ℤ/l(n)_{cdh}^{∼}(∂Δ_{E}^{n−i+1})) =
= ℤ_{tr}(𝔾_{m}^{∧n})_{cdh}^{∼}(∂Δ_{E}^{n−i+1})/ Im(C_{1}(ℤ_{tr}(𝔾_{m}^{∧n})_{cdh}^{∼}(∂Δ_{E}^{n−i+1})).

Our reasoning applies more generally to any complex of the form C^{∗}(F)[−n]. The motivic cohomology group H_{M}^{i}(E, C^{∗}(F)[−n]) coincides with the (n − i)-th homology group of the simplicial abelian group F(Δ_{E}^{•}) and may be computed using the corressponding Moore complex, i.e it may be identified with an appropriate quotient of F(Δ_{E}^{n−i})_{0} = {s ∈ F(Δ_{E}^{n−i}) : ∂_{j}(s) = 0 (0 ≤ j ≤ n − i)}. Denote by p : Δ^{n−i} ⟶^{∼} Δ^{n−i} an automorphism of Δ^{n−i} given by the formula (t_{0}, ..., t_{n−i}) ↦ (t_{n−i}, ..., t_{0}). The section p^{∗}(s) (the same as s) vanishes on all faces of Δ_{E}^{n−i} and hence may be glued together with the zero section on σΔ_{E}^{n−i+1} to give a section λ(s) of F_{cdh}^{∼} over ∂Δ_{E}^{n−i+1}. In this way we get a homomorphism

F(Δ_{E}^{n−i})_{0} ⟶^{λ} F_{cdh}^{∼}(∂Δ_{E}^{n−i+1}) ⟶^{can} H^{n}((C^{∗}(F)[−n])_{cdh}^{∼}(∂Δ_{E}^{n−i+1})) ⟶^{can}

⟶^{can} H_{M}^{n}(∂Δ_{E}^{n−i+1}, C^{∗}(F)[−n]).

Lemma 9.4. The natural isomorphism

δ^{n−i} : H_{M}^{i}(E, C^{∗}(F)[−n]) ⟶^{∼} H_{M}^{n}(∂Δ_{E}^{n−i+1}/SpecE, C^{∗}(F)[−n])) ⊂
⊂ H_{M}^{n}(∂Δ_{E}^{n−i+1}, C^{∗}(F)[−n])

takes the homology class of s ∈ F(Δ_{E}^{n−i})_{0} to the canonical image of λ(s) ∈ F_{cdh}^{∼}(∂Δ_{E}^{n−i+1}) in H_{M}^{n}(∂Δ_{E}^{n−i+1}, C^{∗}(F)[−n]) (possibly up to a sign).

Proof. For each 0 ≤ j ≤ n − i we define a (j + i)-dimentional cocycle s^{j} of the complex (C^{∗}(F)[−n])_{cdh}^{∼}(∂Δ_{E}^{j+1}), i.e. a n−(j +i) dimensional cycle of the complex (C_{∗}(F))_{cdh}^{∼}(∂Δ_{E}^{j+1}) so that the following properties hold

(1) s^{0}|_{v_{0}} = 0 s^{0}|_{v_{1}} = s

(2) δ(s^{j}) = + −s^{j+1}

(3) s^{n−i} = λ(s) Define a morphism q_{j} : Δ^{n−i−j} × Δ^{j+1} → Δ^{n−i} via the formula

(t_{0}, ..., t_{n−i−j}) × (t′_{0}, ..., t′_{j+1}) ↦ t′_{0}(t_{0}, ..., t_{n−i−j}, 0, ..., 0) + t′_{j+1}v_{n−i−j} + ... + t′_{1}v_{n−i}

and let p_{j} : Δ^{n−i−j} ×Δ^{j} → Δ^{n−i} be the restriction of q_{j}. One checks easily that p_{j} maps all faces of Δ^{n−i−j} × Δ^{j} to faces of Δ^{n−i} . This implies, in particular, that (p_{j})^{∗}(s) ∈ C_{n−i−j}(F)(Δ_{E}^{j}) vanishes on all faces of Δ_{E}^{j} and hence may be glued together (along ∂Δ_{E}^{j}) with the zero section to give a section s^{j} of C_{n−i−j}(F)_{cdh}^{∼} over ∂Δ_{E}^{j+1}. Moreover all faces of s^{j} are equal to zero and hence s^{j} is a cycle. To compute δ(s^{j}) we note that s^{j} extends to a section q_{j}^{∗}(s) of C_{n−i−j}(F) over Δ_{E}^{j+1} and hence δ(s^{j}) is represented by the cycle in C_{n−i−j−1}(F)(∂Δ_{E}^{j+2}) whose restriction to Δ_{E}^{j+1} is equal to ∑_{k=0}^{n−i−j−1}(−1)^{k}∂_{k}(q_{j}^{∗}(s)) and whose restriction to other faces of ∂Δ_{E}^{j+2} is zero. Finally an immediate verification shows that

∂_{k}(q_{j}^{∗}(s)) = \{ \begin{matrix}0 & if k ≠ n − i − j \\ p_{j+1}^{∗}(s) & if k = n − i − j \end{matrix}

For any presheaf F define a new presheaf C^{∼}_{1}(F) via the formula

C^{∼}_{1}(F)(X) = \underrightarrow{lim}_{X×\{0,1\}⊂U⊂X×𝔸^{1}} F(U)  (X ∈ Sm/F)

Here the direct limit is taken over all open subschemes U ⊂ X × 𝔸^{1} which contain X × {0, 1}. The presheaf C^{∼}_{1}(F)(X) is equipped with two obvious homomorphisms

C^{∼}_{1}(F) ⟶^{i_{0}^{∗}} ; ⟶_{i_{1}^{∗}}F

Moreover a straightforward verification shows that if F is a presheaf with transfers, then C^{∼}_{1}(F) also has a natural structure of a presheaf with transfers and the homomorphisms i_{0}^{∗}, i_{1}^{∗} are compatible with transfers.

Definition 9.5. We’ll be saying that a presheaf with transfers F is rationally contractible iff there exists a homomorphism of presheaves with transfers s : F →

C^{∼}_{1}(F), such that i_{0}^{∗}s = 0, i_{1}^{∗}s = Id.

Proposition 9.6. The presheaves with transfers ℤ_{tr}(𝔾_{m}^{∧n}), ℤ_{tr}(𝔾_{m}^{∧n})/l are rationally contractible.

Proof. Obviously it suffices to establish rational contractibility of the sheaf ℤ_{tr}(𝔾_{m}^{∧n}). This follows easily from the fact that the scheme 𝔾_{m}^{×n} ,being an open subscheme in a contractible scheme 𝔸^{n}, is rationally contractible. To be more precise consider a morphism of schemes

𝔾_{m}^{×n} × 𝔸^{1} ⟶^{ψ} 𝔸^{n} : (t_{1}, ..., t_{n}) × t ↦ t · (t_{1}, ..., t_{n}) + (1 − t) · (1, ..., 1)

and set W = ψ^{−1}(𝔾_{m}^{×n}), T = 𝔾_{m}^{×n} × 𝔸^{1} \ W. Thus W is an open subscheme of 𝔾_{m}^{×n} × 𝔸^{1}, containing 𝔾_{m}^{×n} × {0, 1} and ψ defines a morphism ψ : W → 𝔾_{m}^{×n} such that

ψ|_{𝔾_{m}^{×n}×1} = Id, ψ|_{𝔾_{m}^{×n}×0} = (1, ..., 1)

Let X ∈ Sm/F be a smooth irreducibla scheme and let Z ∈ ℤ_{tr}(𝔾_{m}^{×n})(X) be a closed integral subscheme of X × 𝔾_{m}^{×n} finite and surjective over X. The closed integral subscheme

Z × 𝔸^{1} = Z × Δ_{𝔸^{1}} ⊂ (X × 𝔸^{1}) × (𝔾_{m}^{×n} × 𝔸^{1})

defines an element in ℤ_{tr}(𝔾_{m}^{×n} × 𝔸^{1})(X × 𝔸^{1}). Set

U_{Z} = X × 𝔸^{1} \ pr_{X×𝔸^{1}}(Z × 𝔸^{1} ∩ ((X × 𝔸^{1}) × T)

The scheme U_{Z} is an open subscheme of X × 𝔸^{1}, containing X × {0, 1}. Moreover

Z × 𝔸^{1} ∩ (U_{Z} × (𝔾_{m}^{×n} × 𝔸^{1})) ⊂ U_{Z} × W

and this closed integral subschem defines an element in ℤ_{tr}(W)(U_{Z}). Applying to this element the homomorphism ℤ_{tr}(ψ) : ℤ_{tr}(W) → ℤ_{tr}(𝔾_{m}^{×n}) we get an element s(Z) ∈ ℤ_{tr}(𝔾_{m}^{×n})(U_{Z}). A straightforward verification shows that in this way we get a homomorphism of presheaves with transfers s : ℤ_{tr}(𝔾_{m}^{×n}) →

C^{∼}_{1}(ℤ_{tr}(𝔾_{m}^{×n})). Finally the composition i_{1}^{∗}s is the identity map, whereas the composition i_{0}^{∗}s : ℤ_{tr}(𝔾_{m}^{×n}) → ℤ_{tr}(𝔾_{m}^{×n}) is induced by the constant morphism 𝔾_{m}^{×n} → 𝔾_{m}^{×n} : (t_{1}, ..., t_{n}) ↦ (1, ..., 1). This implies that the factor sheaf ℤ_{tr}(𝔾_{m}^{×n})/ℤ_{tr}((1, ..., 1)) is rationally contractible and hence its direct summand ℤ_{tr}(𝔾_{m}^{∧n}) is also rationally contractible.

Corollary 9.7. For any z ∈ H_{M}^{i}(E, ℤ/l(n)) there exists an open subscheme U ⊂ ∂Δ_{E}^{n−i+1}×𝔸^{1} containing ∂Δ_{E}^{n−i+1}×{0, 1} and an element y ∈ H_{M}^{n}(U, ℤ/l(n)) such that y|_{∂Δ_{E}^{n−i+1}×0} = 0, y|_{∂Δ_{E}^{n−i+1}×1} = δ^{n−i}(z).

Proof. In view of Lemma 9.4 it suffices to show that for any element z ∈ (ℤ_{tr}(𝔾_{m}^{∧n})/l)(Δ^{n−i})_{0} there exists an open subscheme U as above and a section y ∈ (ℤ_{tr}(𝔾_{m}^{∧n})/l)_{cdh}^{∼}(U) such that

y|∂Δ_{E}^{n−i+1} × 0 = 0 y|Δ_{E}^{n−i} × 1 = z y|σΔ^{n−i+1} × 1 = 0

According to Proposition 9.6 there exists an open subscheme Δ_{E}^{n−i} × {0, 1} ⊂ V ⊂ Δ_{E}^{n−i} × 𝔸^{1} and a section s(z) ∈ (ℤ_{tr}(𝔾_{m}^{∧n})/l)(V) such that

s(z)|_{Δ_{E}^{n−i}×0} = 0 s(z)|_{Δ_{E}^{n−i}×1} = z

Moreover for any 0 ≤ j ≤ n − i the restriction of s(z) to V ∩ ∂_{j}Δ_{E}^{n−i} dies in the direct limit

\underrightarrow{lim}_{Δ_{E}^{n−i−1}×\{0,1\}⊂W ⊂Δ_{E}^{n−i−1}×𝔸^{1}} ℤ_{tr}(𝔾_{m}^{∧n})/l(W)

Hence decreasing V if necessary we may assume that this restriction is trivial for all j. This shows immediately that ,denoting Δ_{E}^{n−i} × 𝔸^{1} \ V by T, one may extend the section s(z) to a section y of (ℤ_{tr}(𝔾_{m}^{∧n})/l)_{cdh}^{∼} over ∂Δ_{E}^{n−i+1} × 𝔸^{1} \ T which is zero over σΔ_{E}^{n−i+1} × 𝔸^{1} \ T.

§ 10. Proof of the main Theorem.

Denote by S the scheme 𝔸^{1} with points 0 and 1 identied. We denote by pt ∈ S the unique singular point of S.

Lemma 10.1. Let X ∈ Sch/F be a scheme of finite type over F and let U ⊂ X×𝔸^{1}

be an open subscheme containing X × {0, 1}. Denote by U^{∼} the image of U in X × S

under the canonical projection X × 𝔸^{1} ⟶^{p} X × S. Then U^{∼} is an open subscheme in X × S and for any complex C^{∗} ∈ DM^{−}(F) we have a natural long exact sequence of motivic cohomology groups

... ⟶ H_{M}^{n}(U^{∼}, C^{∗}) ⟶^{p^{∗}} H_{M}^{n}(U, C^{∗}) ⟶^{i_{0}^{∗}−i_{1}^{∗}} H_{M}^{n}(X, C^{∗}) ⟶^{δ^{∼}} H_{M}^{n+1}(U^{∼}, C^{∗}) ⟶ ...

Proof. The first statement is trivial. To prove the second it suffices to note that the sequence of Nisnevich sheaves with transfers

0 ⟶ ℤ_{tr}(X) ⟶^{(i_{0})_{∗}−(i_{1})_{∗}} ℤ_{tr}(U) ⟶^{p_{∗}} ℤ_{tr}(U^{∼}) ⟶ 0

is exact (even as a sequence of presheaves).

Corollary 10.2. For any X ∈ Sch/F we have a natural isomorphism

δ^{∼} : H_{M}^{n}(X, C^{∗}) ⟶^{∼} H_{M}^{n+1}(X × S/X, C^{∗}) ⊂ H_{M}^{n+1}(X × S, C^{∗})

Lemma 8.2 and Corollary 10.2 give us (for any i < n and any finitely generated field extension E/F) a sequence of split monomorphisms of motivic cohomology groups with coefficients in an arbitrary complex C^{∗} ∈ DM^{−}(F)

H_{M}^{i}(E) ⟶^{δ^{n−i}} H_{M}^{n}(∂Δ_{E}^{n−i+1}) ⟶^{δ^{∼}} H_{M}^{n+1}(∂Δ_{E}^{n−i+1} × S)

We denote the composition δ^{∼}δ^{n−i} of the above split monomorphisms by ξ.

Proposition 10.3. For any z ∈ H_{M}^{i}(E, ℤ/l(n)) there exists an open subscheme

U^{∼} ⊂ ∂Δ_{E}^{n−i+1} × S, containing ∂Δ_{E}^{n−i−1} × pt such that the restriction of ξ(z) ∈

H_{M}^{n+1}(∂Δ_{E}^{n−i+1} × S, ℤ/l(n)) to U^{∼} is zero.

Proof. Let X × {0, 1} ⊂ U ⊂ X × 𝔸^{1}, y ∈ H_{M}^{n}(U, ℤ/l(n)) be an open subscheme and a motivic cohomology class such that i_{0}^{∗}(y) = 0, i_{1}^{∗}(y) = δ^{n−i}(z) - see Corollary

8.6. Let further U^{∼} be the image of U in ∂Δ_{E}^{n−i+1} × S. According to Lemma 10.1 we have a commutative diagram with exact rows (all motivic cohomology groups are taken with ℤ/l(n) coefficients)

0 ⟶ H_{M}^{n}(∂Δ_{E}^{n−i+1}) ⟶^{δ^{∼}} H_{M}^{n+1}(∂Δ_{E}^{n−i+1} × S)
↓  ^{=}↓  ↓
H_{M}^{n}(U) ⟶^{i_{0}^{∗}−i_{1}^{∗}} H_{M}^{n}(∂Δ_{E}^{n−i+1}) ⟶^{δ^{∼}}  H_{M}^{n+1}(U^{∼})

The above diagram shows that the image of ξ(z) = δ^{∼}δ^{n−i}(z) in H_{M}^{n+1}(U^{∼}) is zero.

We lieve the proof of the following Lemma as an easy exercise to the reader.

Lemma 10.4. Every open subscheme of ∂Δ^{n−i+1} × S, which contains all vertices v_{i} × pt (0 ≤ i ≤ n) is strictly dense.

Proof of the Theorem 7.4

According to Lemma 6.9 and Proposition 9.3 it suffices to show that for any i ≤ n and any finitely generated field extension E/F the map in motivic cohomology

(α_{n})_{∗} : H_{M}^{i}(E, ℤ/l(n)) ⟶ H_{M}^{i}(E, B_{l}(n))

is injective. Note further that according to Lemma 9.2 and Corollary 10.2 the motivic cohomology groups in question inject canonically to H_{M}^{n+1}(∂Δ_{E}^{n−i+1} × S, ℤ/l(n)) and H_{M}^{n+1}(∂Δ_{E}^{n−i+1} × S, B_{l}(n)) respectively. For any open subscheme U ⊂ ∂Δ_{E}^{n−i+1} × S containing all the vertices v_{i} × pt we have an exact sequence of motivic cohomology groups (both with ℤ/l(n) and B_{l}(n)-coefficients)

H_{M}^{n}(U) ⟶ H_{T}^{n+1}(∂Δ_{E}^{n−i+1} × S) ⟶ H^{n+1}(∂Δ_{E}^{n−i+1} × S) ⟶ H_{M}^{n+1}(U)

where we denoted by T the closed subscheme ∂Δ_{E}^{n−i+1} × S \ U. Taking direct limits of these sequences over all U’s we get the following commutative diagram with exact rows

H_{M}^{n}(U, ℤ/l(n)) ⟶ \underrightarrow{lim}_{T} H_{T}^{n+1}(∂Δ_{E}^{n−i+1} × S, ℤ/l(n)) ⟶
(α_{n})_{∗}↓  ^{(α_{n})_{∗}}↓
H_{M}^{n}(U, B_{l}(n)(n)) ⟶ \underrightarrow{lim}_{T} H_{T}^{n+1}(∂Δ_{E}^{n−i+1} × S, B_{l}(n)) ⟶
⟶ H^{n+1}(∂Δ_{E}^{n−i+1} × S, ℤ/l(n)) ⟶ H_{M}^{n+1}(U, ℤ/l(n))
(α_{n})_{∗}↓  ^{(α_{n})_{∗}}↓
⟶ H^{n+1}(∂Δ_{E}^{n−i+1} × S, B_{l}(n)) ⟶ H_{M}^{n+1}(U, B_{l}(n))

where the direct limit is taken over all closed subsets T of ∂Δ_{E}^{n−i+1}×S containing no vertices and U this time denotes the semilocalization of ∂Δ_{E}^{n−i+1} × S at the points v_{i} × pt. Note finally that the first vertical arrow in the above diagram is surjective according to Proposition 7.6 and the second one is an isomorphism according to Theorem 8.5 and Lemma 10.4. An easy diagram chase shows that the intersection of the kernels of homomorphisms

H^{n+1}(∂Δ_{E}^{n−i+1} × S, ℤ/l(n)) ⟶ H_{M}^{n+1}(U, ℤ/l(n))
H^{n+1}(∂Δ_{E}^{n−i+1} × S, ℤ/l(n)) ⟶ H^{n+1}(∂Δ_{E}^{n−i+1} × S, B_{l}(n))

is trivial. For any motivic cohomology class z ∈ H_{M}^{i}(E, ℤ/l(n)) its image ξ(z) ∈ H^{n+1}(∂Δ_{E}^{n−i+1} ×S, ℤ/l(n)) dies in H_{M}^{n+1}(U, ℤ/l(n)) according to Proposition 10.3. Thus if z dies in H_{M}^{i}(E, B_{l}(n)) (in which case ξ(z) dies in H^{n+1}(∂Δ_{E}^{n−i+1} × S, B_{l}(n))) then ξ(z) is zero and hence z itself is zero.

§ 11. Bloch-Kato conjecture and vanishing of the Bockstein homomorphisms.

In this section we fix a prime integer l different from charF. Note that validity of the Weak Bloch-Kato Conjecture for a field F (modulo l) implies, in particular, that all homomorphisms H_{et}^{n}(F, μ_{l^{k+1}}^{⊗n}) ⟶ H_{et}^{n}(F, μ_{l}^{⊗n}) are surjective and hence all the Bockstein homomorphisms β_{1,k} : H_{et}^{n}(F, μ_{l}^{⊗n}) ⟶ H_{et}^{n+1}(F, μ_{l^{k}}^{⊗n}) corresponding to the short exact sequences of etale sheaves 0 → μ_{l^{k}}^{⊗n} → μ_{l^{k+1}}^{⊗n} → μ_{l}^{⊗n} → 0 are trivial. The aime of this section is to show that vice versa vanishing of the above Bockstein homomorphisms implies the validity of the Bloch-Kato Conjecture (and hence of the Beilinson-Lichtenbaum Conjecture as well). In the special case l = 2 such kind of statement was proved previously by A. Merkurjev [Me] using very different technique. We start with the following elementary Lemma.

Lemma 11.1. The following conditions are equivalent

(1) All Bockstein homomorphisms β_{1,k} : H_{et}^{n}(F, μ_{l}^{⊗n}) ⟶ H_{et}^{n+1}(F, μ_{l^{k}}^{⊗n}) are trivial.

(2) All homomorphisms H_{et}^{n}(F, μ_{l^{k+1}}^{⊗n}) ⟶ H_{et}^{n}(F, μ_{l}^{⊗n}) are surjective.

(3) All Bockstein homomorphisms β_{k,1} : H_{et}^{n}(F, μ_{l^{k}}^{⊗n}) ⟶ H_{et}^{n+1}(F, μ_{l}^{⊗n}) are trivial.

(4) All homomorphisms H_{et}^{n}(F, μ_{l^{k+1}}^{⊗n}) ⟶ H_{et}^{n}(F, μ_{l^{k}}^{⊗n}) are surjective.

(5) All Bockstein homomorphisms β_{k,s} : H_{et}^{n}(F, μ_{l^{k}}^{⊗n}) ⟶ H_{et}^{n+1}(F, μ_{l^{s}}^{⊗n}) are trivial.

(6) The group H_{et}^{n}(F, ℚ_{l}/ℤ_{l}(n)) is l-divisible.

Proof. The equivalence of (1) and (2), the same as equivalence of (3) and (4) is obvious. Furthermore (4) clearly implies (2) and (5) implies (3). To end up the proof we only have to prove the implications (2)⇒(6) and (6)⇒(5). To prove the first implication we note that the composition H_{et}^{n}(F, μ_{l^{k+1}}^{⊗n}) ⟶ H_{et}^{n}(F, μ_{l}^{⊗n}) → H_{et}^{n}(F, μ_{l^{k+1}}^{⊗n}) coincides with the multiplication by l^{k}. Thus (2) implies that the image of H_{et}^{n}(F, μ_{l}^{⊗n}) in H_{et}^{n}(F, ℚ_{l}/ℤ_{l}(n)) consists of infinitly l-divisible elements. Since this image is exactly the l-torsion subgroup of the l-primary group H_{et}^{n}(F, ℚ_{l}/ℤ_{l}(n)) we conclude easily that the latter group is l-divisible. Finally (6) implies vanishing of the Bockstein homomorphisms β_{∞,s} which implies the vanishing of all β_{k,s} in view of the commutative diagram

H_{et}^{n}(F, μ_{l^{k}}^{⊗n}) ⟶^{β_{k,s}} H_{et}^{n+1}(F, μ_{l^{s}}^{⊗n})
↓  ↓^{=}
H_{et}^{n}(F, ℚ_{l}/ℤ_{l}(n)) ⟶^{β_{∞,s}} H_{et}^{n+1}(F, μ_{l^{s}}^{⊗n})

Lemma 11.2. Assume that the field F is infinite. Let s ∈ H_{et}^{n}(F, ℤ/l) (n > 0) be an etale cohomology class. Then there exists an integer N, an open subscheme U ⊂ 𝔸^{N}, an etale Galois covering p : U → V, an etale cohomology class s′ ∈ H_{et}^{n}(V, ℤ/l) and two rational points v_{0}, v_{1} ∈ V such that v_{0}^{∗}(s′) = 0, v_{1}^{∗}(s′) = s (and v_{0} admits a rational lifting u_{0} to U).

Proof. Note first of all that for any Galois covering p : U → V with a Galois group G we have a natural homomorphism γ_{p} : H^{n}(G, ℤ/l) → H^{n}(G, H_{et}^{0}(U, ℤ/l)) → H_{et}^{n}(V, ℤ/l). We can always represent s as a cohomology class of the Galois group G = Gal(E/F) of an appropriate Galois extension E/F, i.e. right s in the form s = γ_{p_{1}}(s_{0}), where p_{1} : SpecE → SpecF is a Galois covering with the group G and s_{0} ∈ H^{n}(G, ℤ/l) is an appropriate cohomology class. Set N = |G|, let G act on 𝔸^{N} by permuting the coordinates and let U ⊂ 𝔸^{N} be the open subscheme consisting of points with all coordinates different. Thus the action of G on U is free and hence p : U → V = U/G is a Galois covering. Finally set s′ = γ_{p}(s_{0}). Let u_{0} ∈ U be any F -rational point and let v_{0} = p(u_{0}) be its image in V. The cohomology class v_{0}^{∗}(s′) coincides with γ_{p_{0}}(s_{0}), where p_{0} : U_{0} = p^{−1}(v_{0}) → SpecF is the induced Galois covering. To show that v_{0}^{∗}(s′) = 0 it suffices to note that the fiber p^{−1}(v_{0}) consists of N different rational points hence H_{et}^{0}(U_{0}, ℤ/l) is the induced G module and H^{∗}(G, H_{et}^{0}(U_{0}, ℤ/l)) = 0 for ∗ > 0. Finally choose a normal basis e_{g} (g ∈ G) for the extension E/F. The family {e_{g}}_{g∈G} defines a point u_{1} ∈ U with the residue field F (u_{1}) = E, one checks immediately that this point is G-invariant (and the induced action of G on F (u_{1}) = E is the original one). This shows that the image p(u_{1}) = v_{1} is an F -rational point and the induced Galois covering p_{1} : U_{1} = p^{−1}(v_{1}) → SpecF coincides with what we denoted by p_{1} above. Thus v_{1}^{∗}(s′) = γ_{p_{1}}(s_{0}) = s.

Proposition 11.3. Assume that resolution of singularities holds over F. Assume further that the Beilinson-Lichtenbaum Conjecture (modulo l) holds over F in weights < n. Let finally U ⊂ 𝔸^{N} be an open subscheme and let u ∈ U be an F -rational point. In this case the canonical homomorphisms

(α_{l^{k}})_{∗} : H_{M}^{∗}(U/u, ℤ/l^{k}(n)) ⟶ H_{M}^{∗}(U/u, B_{l^{k}}(n))

are isomorphisms for all k > 0.

Proof. Set Z = 𝔸^{N} \ U. One checks immediately that for any C^{•} ∈ DM^{−}(F) we have a natural isomorphism H_{M}^{∗}(U/u, C^{•}) = H_{Z}^{∗+1}(𝔸^{N}, C^{•}). Thus our statement follows from Theorem 8.5.

Theorem 11.4. Assume that resolution of singularities holds over F. Assume further that the Bockstein homomorphisms β_{1,k}^{i} : H_{et}^{i}(E, μ_{l}^{⊗i}) ⟶ H_{et}^{i+1}(E, μ_{l^{k}}^{⊗i}) are trivial for all k, all i ≤ n and all finitely generated field extensions E/F. Then the Beilison-Lichtenbaum Conjecture holds over F in weights ≤ n.

Proof. Proceeding by induction on n we may assume that Beilinson-Lichtenbaum Conjecture is known to be true over F in weights < n. Define the complex B_{l^{∞}}(n) the same way the complex B_{l^{k}}(n) was defined in § 6. Thus B_{l^{∞}}(n) = τ_{≤n}(π_{∗}(J^{•})) where this time J^{•} is the injective resolution of the h-sheaf ℚ_{l}/ℤ_{l}(n) = μ_{l^{∞}}^{⊗n} = \underrightarrow{lim}_{k} μ_{l^{k}}^{⊗n}. We first show that the homomorphism α_{n} : ℚ_{l}/ℤ_{l}(n) = \underrightarrow{lim}_{k} ℤ/l^{k}(n) = ℤ(n) ⊗ ℚ_{l}/ℤ_{l} ⟶ B_{l^{∞}}(n) is a quasiisomorphism. The proof repeats word by word that of the Theorem 7.4, provided we can show that for any finitely generated field extension E/F the homomorphism in motivic cohomology

H_{M}^{n}(E, ℚ_{l}/ℤ_{l}(n)) ⟶ H_{M}^{n}(E, B_{l^{∞}}(n)) = H_{et}^{n}(E, ℚ_{l}/ℤ_{l}(n))

is surjective. Note that both l-primary groups above are l-divisible: the one on the left coicides with K_{n}^{M}(E) ⊗ ℚ_{l}/ℤ_{l} and the one on the right is l-divisible according to our assumptions. Thus to prove the surjectivity of the above map it suffices to show that the elements of l-torsion in H_{et}^{n}(E, ℚ_{l}/ℤ_{l}(n)) admit a lifting to H_{M}^{n}(E, ℚ_{l}/ℤ_{l}(n)). Note also that the l-torsion subgroup in H_{et}^{n}(E, ℚ_{l}/ℤ_{l}(n)) coincides with the image of H_{et}^{n}(E, μ_{l}^{⊗n}). As always we may assume that E contains a primitive l-th route of unity ξ (which allows to identify H_{et}^{n}(E, μ_{l}^{⊗n}) with H_{et}^{n}(E, ℤ/l)) and obviously we may assume that E is infinite. Start with a cohomology class s ∈ H_{et}^{n}(E, μ_{l}^{⊗n}). Applying Lemma 12.2 we get an open subscheme U ⊂ 𝔸^{N} a Galois covering p : U → V, a cohomology class s′ ∈ H_{et}^{n}(V, μ_{l}^{⊗n}) and two rational points v_{0}, v_{1} ∈ V such that v_{0}^{∗}(s′) = 0, v_{1}^{∗}(s′) = s. Write N in the form N = l^{k}M where M is prime to l. Surjectivity of the homomorphism H_{et}^{n}(E, μ_{l^{k+1}}^{⊗n}) → H_{et}^{n}(E, μ_{l}^{⊗n}) for all field extensions E/F implies immediately that the same holds for any smooth semilocal scheme over F (this is another general property of homotopy invariant presheaves with transfers - see [V 1]). Thus decreasing if necessary V we may assume that s′ may be lifted to a cohomology class s′′ ∈ H_{et}^{n}(V, μ_{l^{k+1}}^{⊗n}). Let u_{0} ∈ U be any rational point over v_{0}. Applying Proposition 11.3 to the etale cohomology class p^{∗}(s′′)−p^{∗}(s′′)(u_{0}) = p^{∗}(s′′)−s′′(v_{0}) ∈ H_{et}^{n}(U/u_{0}, μ_{l^{k+1}}^{⊗n}) we get a motivic cohomology class t′′ ∈ H_{M}^{n}(U/u_{0}, ℤ/l^{k+1}(n)) whose image in H_{et}^{n}(U, μ_{l^{k+1}}^{⊗n}) equals p^{∗}(s′′) − s′′(u_{0}). Set t′ = p_{∗}(t′′) ∈ H_{M}^{n}(V, ℤ/l^{k+1}(n)). The image of this motivic cohomology class in H_{et}^{n}(V, μ_{l^{k+1}}^{⊗n}) equals N(s′′ − s′′(u_{0})) and hence equals M times the image of s′ − s′(u_{0}) = s′ ∈ H^{n}(V, μ_{l}^{⊗n}) in H_{et}^{n}(V, μ_{l^{k+1}}^{⊗n}). Setting finally t = M′v_{1}^{∗}(t′), where M′M ≡ 1 mod l^{k+1} we get a motivic cohomology class t ∈ H_{M}^{n}(E, ℤ/l^{k+1}(n)) whose image in H_{et}^{n}(E, μ_{l^{k+1}}^{⊗n}) equals the image of s in this group. Thus the image of H^{n}(E, μ_{l}^{⊗n}) in H_{et}^{n}(E, Q_{l}/ℤ_{l}(n)) can be lifted to H_{M}^{n}(E, ℚ_{l}/ℤ_{l}(n)) and hence the homomorphism

H_{M}^{n}(E, ℚ_{l}/ℤ_{l}(n)) ⟶ H_{M}^{n}(E, B_{l^{∞}}(n)) = H_{et}^{n}(E, ℚ_{l}/ℤ_{l}(n))

is surjective. Finally to prove the Beilinson-Lichtenbaum Conjecture modulo l we use the Bockstein distinguished triangles in DM^{−}(F)

ℤ/l(n) ⟶ ℚ_{l}/ℤ_{l}(n) ⟶^{l} ℚ_{l}/ℤ_{l}(n) ⟶ ℤ/l(n)[1]
↓  ↓  ↓  ↓
B_{l}(n) ⟶ B_{l^{∞}}(n) ⟶^{l} B_{l^{∞}}(n) ⟶ B_{l}(n)[1]

- cf. the proof of the Lemma 6.10.

§ 12. Appendix: cdh-cohomological dimension of Noetherian schemes.

For any Noetherian scheme S we denote by Sch/S the category of separated schemes of finite type over S. One defines the cdh-topology on Sch/S in the same way it was done in § 5 in case S = SpecF. The proofs of Lemma 5.8 and Proposition 5.9 were given in such a way that they work without any changes over an arbitrary Noetherian base scheme S.

Lemma 12.1. Let p : Y → X be a proper morphism (of separated schemes of finite type over S) and let Z ⊂ X be a closed subscheme such that p is an isomorphism over X \ Z. In this case we have an exact sequence of cdh-sheaves

0 → ℤ_{cdh}(p^{−1}(Z)) ⟶ ℤ_{cdh}(Z) ⊕ ℤ_{cdh}(Y) ⟶ ℤ_{cdh}(X) → 0

and hence for any cdh-sheaf F ∈ (Sch/S)_{cdh}^{∼} we get an exact sequence of cdhcohomology groups

H_{cdh}^{i}(X, F) ⟶ H_{cdh}^{i}(Z, F)⊕H_{cdh}^{i}(Y, F) ⟶ H_{cdh}^{i}(p^{−1}(Z), F) ⟶ H_{cdh}^{i+1}(X, F) → …

In particular for any X ∈ Sch/S we have a natural isomorphism

H_{cdh}^{∗}(X, F) = H_{cdh}^{∗}(X_{red}, F)

and for any closed covering X = Y ∪ Z we have a long exact Mayer-Vietoris sequence

H_{cdh}^{i}(X, F) ⟶ H_{cdh}^{i}(Z, F) ⊕ H_{cdh}^{i}(Y, F) ⟶ H_{cdh}^{i}(Y ∩ Z, F) ⟶ H_{cdh}^{i+1}(X, F) → …

Proof. Recall that the sheaf ℤ_{cdh}(X) is defined as the cdh-sheaf associated to the presheaf ℤ(X) : U ↦ ℤ[Hom_{S}(U, X)]. An immediate verification shows the exactness of the following sequence of presheaves

0 → ℤ(p^{−1}(Z)) ⟶ ℤ(Z) ⊕ ℤ(Y) ⟶ ℤ(X).

Since the sheafification functor is exact this proves the exactness of the above sequence of sheaves without zero on the right. Finally the homomorphism ℤ_{cdh}(Z)⊕ ℤ_{cdh}(Y) ⟶ ℤ_{cdh}(X) is surjective since Y ∐ Z → X is a cdh-covering of X.

Ue used implicitly the second statement of the following Lemma several times in § 5.

Lemma 12.2. 1) Let f : Y → X be a morphism of finite type. Assume that the point x ∈ X admits a lifting y ∈ Y such that the residue fields at x and y coincide: k(x) = k(y). Then there exists an open neighbourhood of x in \overline{x} each point of which has the same property.

2) Every Nisnevich covering {U_{i} → X}_{i∈I} of a Noetherian scheme X admits a finite subcovering.

Proof. 1) Consider \overline{y} and \overline{x} as closed integral subschemes in Y and X respectively. The induced morphism f : \overline{y} → \overline{x} is still of finite type and is an isomorphism over the generic point x of \overline{x}, hence it is an isomorphism over some open neighbourhood of x in \overline{x}. 2)Proceeding by Noetherian induction we may assume that for any proper closed subscheme Z ⊂ X the induced Nisnevich covering {U_{i} ×_{X} Z → Z}_{i∈I} has a finite subcovering. Let x_{1}, ..., x_{n} ∈ X be generic points of components of X and let U_{i_{1}}, ..., U_{i_{n}} be members of our family such that x_{k} admits a lifting to U_{i_{k}} with the same residue field. According to part 1) there exist open neighbourhoods x_{k} ∈ V_{k} such that each point of V_{k} admits a lifting to U_{i_{k}} with the same residue field. Set Z = X \ ∪_{k=1}^{n}V_{k} and consider Z as a (proper) closed reduced subscheme of X. According to our induction hypothesis there exists a finite subset J ⊂ I such that each point z ∈ Z admits a lifting with the same residue field to one of U_{j} j ∈ J. Obviously the subfamily defined by the finite set J ∪{i_{1}, ..., i_{n}} is a finite Nisnevich subcovering of the original covering.

Corollary 12.2.1. Let I be a filtering partially ordered set and let i ↦ X_{i} be a functor from I to the category of Noetherian schemes such that all the transition morphisms p_{ij} : X_{i} → X_{j} (i ≥ j) are affine. Set X_{∞} = \underleftarrow{lim}_{i∈I} X_{i}, denote by p_{i} : X_{∞} → X_{i} the structure morphisms and assume that the scheme X_{∞} is again Noetherian. Assume further that we are given an index i ∈ I and a morphism of finite type Y_{i} → X_{i} such that each point of X_{∞} admits a lifting with the same residue field to Y_{∞} = Y_{i} ×_{X_{i}} X_{∞}. Then there exists an index j ≥ i for which the morphism Y_{j} = Y_{i} ×_{X_{i}} X_{j} → X_{j} has the same property.

Proof. For any j ≥ i let U_{j} ⊂ X_{j} denote the subset of X_{j}, consisting of points which admit a lifting to Y_{j} with the same residue field. Lemma 12.2 shows that the set U_{j} is ind-constructible in X_{j} - [EGA-4 (1.9.10)]. Note further that a point x_{j} ∈ X_{j} with the image x_{i} = p_{ji}(x_{j}) in X_{i} belongs to U_{j} if and only if the scheme Y_{i} ×_{X_{i}} Speck(x_{i}) of finite type over the field k(x_{i}) has a rational point over the extension field k(x_{i}) ⊂ k(x_{j}). This remark shows easily that for j′ ≥ j we have an inclusion p_{j′j}^{−1}(U_{j}) ⊂ U_{j′} and moreover

X_{∞} = U_{∞} = ⋃_{j≥i} p_{j}^{−1}(U_{j}).

Using finally [EGA-4 (8.3.4)] we conclude that there exists an index j ≥ i such that U_{j} = X_{j}.

Corollary 12.2.2. In conditions and notations of (12.2.1) assume that Y_{∞} → X_{∞} is a Nisnevich covering (resp. cdh-covering) then there exists an index j ≥ i such that Y_{j} → X_{j} has the same property.

Proof. For Nisnevich coverings our statement follows immediately from [EGA-4 (17.7.8)] and Corollary 12.2.1. Assume now that Y_{∞} → X_{∞} is a cdh-covering. According to Proposition 5.9 there exists a refinefent of this covering of the form U_{∞} → X′_{∞} → X_{∞} where U_{∞} → X′_{∞} is a Nisnevich covering and X′_{∞} → X_{∞} is a proper cdh-covering. Furthermore, according to [EGA-4 (8.8.2)] we can find an index j ≥ i and a tower of morphisms U_{j} → X′_{j} → X_{j} of finite type such that U_{∞} = U_{j} ×_{X_{j}} X_{∞}, X′_{∞} = X′_{j} ×_{X_{j}} X_{∞}. Using once again [EGA-4 (8.8.2)] we see, that increasing j, we may assume also that there exists a morphism U_{j} → Y_{j} over X_{j}. Finally we conclude, using Corollary (12.2.1) above and [EGA-4 (8.10.5), (17.7.8)] that increasing j we may assume that U_{j} → X′_{j} is a Nisnevich covering and X′_{j} → X_{j} is a proper cdh-covering. This shows that U_{j} → X_{j} is a cdh-covering and hence Y_{j} → X_{j} is a cdh-covering as well.

The main goal of this Appendix is to show that the cdh-cohomological dimension of a scheme X ∈ Sch/S is finite provided that dimX is finite and moreover cd_{cdh}X ≤ dimX. The proof we are about to give was suggested to us by O. Gabber. We start with a few standard remarks.

For any morphism of (Noetherian) schemes f : S′ → S we denote by the same letter f the associated morphism of sites

(Sch/S′)_{cdh} ⟶^{f} (Sch/S)_{cdh} : X/S ↦ X_{S′}/S′ = X ×_{S} S′/S′

and denote by f_{∗} and f^{∗} the corresponding functors on the categories of abelian sheaves

(Sch/S′)_{cdh}^{∼} ⟶^{f_{∗}} ; ⟵_{f^{∗}}(Sch/S)_{cdh}^{∼}.

Thus the direct image functor f_{∗} is given by the formula (f_{∗}F)(X/S) = F(X_{S′}/S′). This functor is left exact and preserves injectives. The inverse image functor f^{∗} is left adjoint to f_{∗}. This functor is exact (since the sites in question have arbitrary fiber products) and satisfies the following property: f^{∗}(ℤ_{cdh}(X/S)) = ℤ_{cdh}(X_{S′}/S′).

Assume that the morphism f : S′ → S is a separated morphism of finite type (so that S′ ∈ Sch/S). In this case one checks easily that the inverse image functor f^{∗} is given by the formula (f^{∗}F)(X′/S′) = F(X′/S). Furthermoer the functor f^{∗} has an exact left adjoint , which is denoted f_{!} and is called the extension by zero functor. In the above situation the functor f^{∗} takes injectives to injectives, which gives immediately for any scheme X ∈ Sch/S′ the following natural isomorphism in cohomology

H_{cdh}^{∗}(X/S′, f^{∗}F) = H_{cdh}^{∗}(X/S, F).

The above formula shows that cdh-cohomology essentially does not depend on the base scheme S: for any separated scheme f : X → S of finite type over S and any cdh-sheaf F ∈ (Sch/S)_{cdh}^{∼} we have a natural isomorphism H_{cdh}^{∗}(X/S, F) = H_{cdh}^{∗}(X/X, f^{∗}F).

Assume now that I is a filtering partially ordered set (i.e. I ≠ ∅ and for any i_{1}, i_{2} ∈ I there exists i ∈ I such that i ≥ i_{1}, i_{2}). Assume further that we are given a functor i ↦ S_{i} from I to Sch/S, such that all transition morphisms S_{i} → S_{j} (i ≥ j) are affine. Set S_{∞} = \underleftarrow{lim}_{i∈I} S_{i} and assume that the scheme S_{∞} is again Noetherian. Denote by f_{∞} : S_{∞} → S the corresponding structure morphism.

Proposition 12.3. For any abelian sheaf F ∈ (Sch/S)_{cdh}^{∼} we have a natural isomorphism \underrightarrow{lim}_{i∈I} H_{cdh}^{∗}(S_{i}, F) = H_{cdh}^{∗}(S_{∞}, f^{∗}(F)).

Proof. Note that in the above situation the category Sch/S_{∞} is a direct limit of categories Sch/S_{i} in the sence that for any i ∈ I and any schemes X, Y ∈ Sch/S_{i} we have a natural bijection

Hom_{S_{∞}}(X ×_{S_{i}} S_{∞}, Y ×_{S_{i}} S_{∞}) = \underrightarrow{lim}_{j≤i} Hom_{S_{j}}(X ×_{S_{i}} S_{j}, Y ×_{S_{i}} S_{j})

and furthermore for each X_{∞} ∈ Sch/S_{∞} there exists an index i ∈ I and a scheme X_{i} ∈ Sch/S_{i} such that X_{∞} = X_{i} ×_{S_{i}} S_{∞} - see [EGA-4]. Moreover Corollary 12.2.2 shows that given a finite family of morphisms {X_{i}^{k} → X_{i}}_{k=1}^{K} over S_{i}, the corresponding family {X_{i}^{k} ×_{S_{i}} S_{∞} → X_{i} ×_{S_{i}} S_{∞}}_{k=1}^{K} is a cdh-covering iff there exists j ≥ i such that {X_{i}^{k} ×_{S_{i}} S_{j} → X_{i} ×_{S_{i}} S_{j}}_{k=1}^{K} is a cdh-covering. Using these remarks one checks easily that the sheaf f^{∗}(F) is given by the formula

f^{∗}(F)(X_{i} ×_{S_{i}} S_{∞}) = \underrightarrow{lim}_{j≥i} F(X_{i} ×_{S_{i}} S_{j}).

Next one checks that the inverse image of an injective sheaf is at least acyclic.

(12.3.1). Assume that the sheaf F is injective. Then for any X ∈ Sch/S_{∞} the cohomology groups H_{cdh}^{∗}(X, f^{∗}(F)) vanish for ∗ > 0.

Proof. Let p : Y → X be a cdh-covering of X. According to what was said above there exists an index i ∈ I and a cdh-covering p_{i} : Y_{i} → X_{i} in the category Sch/S_{i} such that p_{i} ×_{S_{i}} S_{∞} = p. The above explicit formula for the sheaf f^{∗}(F) shows immediately that the Čech complex of the sheaf f^{∗}(F) corresponding to the covering p coincides with the direct limit over j ≥ i of Čech complexes of F, corresponding to the coverings p_{i} ×_{S_{i}} S_{j} : Y_{i} ×_{S_{i}} S_{j} → X_{i} ×_{S_{i}} S_{j} and hence is acyclic. Vanishing of all Čech cohomology groups of the sheaf f^{∗}(F) implies vanishing of all cohomology groups H_{cdh}^{∗}(X, f^{∗}(F)) in view of the Cartan-Leray spectral sequence.

In the general case pick up an injective resolution 0 → F → J^{•} of the sheaf F. Applying to this resolution the functor f^{∗} we get a resolution f^{∗}(J^{•}) of f^{∗}(F) consisting of acyclic sheaves. Hence the cohomology groups H_{cdh}^{∗}(S_{∞}, f^{∗}(F)) coincide with cohomology of the complex

f^{∗}(J^{•})(S_{∞}) = \underrightarrow{lim}_{i∈I} J^{•}(S_{i}).

So it remains to use again that cohomology commutes with filtering direct limits.

In the situation of Proposition 12.3 we usually use notation H_{cdh}^{∗}(S_{∞}, F) instead of H_{cdh}^{∗}(S_{∞}, f^{∗}(F)).

Lemma 12.4. Let p : S′ → S be a birational morphism of finite type of integral Noetherian schemes of finite Krull dimension. Then dimS′ ≤ dimS.

Proof. Since dimS′ coincides with the maximum of dimensions of local rings O_{S′,s′} (s′ ∈ S′) it suffices to note that for any point s′ ∈ S′ we have, according to the ”Dimensions Formula” - [EGA-4 (5.5.8)], the following inequality (in which we denote p(s′) ∈ S by s and denote by k(S) (resp. k(S′) )the field of rational functions on S (resp. S′))

dimO_{S′,s′} ≤ dimO_{S′,s′} + tr.deg_{k(s)}k(s′) ≤ dimO_{S,s} + tr.deg_{k(S)}k(S′) = dimO_{S,s}.

Theorem 12.5. Let S be a Noetherian scheme of finite Krull dimension. Then cohomological dimension of S with respect to the cdh-topology is finite and ≤ dimS.

Proof. We proceed by induction on dimS. The case dimS = 0 being trivial, assume that dimS = d > 0 and the result is known to be true for schemes of dimension < d. Using the Mayer-Vietoris exact sequences in cdh-cohomology, corresponding to the closed coverings, one reduces easily the general case to the case when S is integral. Denote by α the obvious morphism of sites

α : (Sch/S)_{cdh} → (Sch/S)_{Nis}

and consider the corresponding Leray spectral sequence

E_{2}^{pq} = H_{Nis}^{p}(S, R^{q}α_{∗}(F)) ⇒ H_{cdh}^{p+q}(S, F).

Note that the stalk of the sheaf R^{q}α_{∗}(F) at a point x ∈ S is the direct limit of cohomology groups H_{cdh}^{q}(U, F) over Nisnevich neighbourhoods (U, u) → (S, x) of the point x. Moreover we may assume these neighbourhoods to be affine and connected. Such neighbourhoods form a filtering partially ordered set and the resulting functor from this poset to Sch/S satisfies the conditions of the Proposition 12.3. Thus we conclude that for any x ∈ S R^{q}α_{∗}(F)_{x} = H_{cdh}^{q}(SpecO_{x}^{h}, F). Induction hypothesis gives us now the following conclusion

R^{q}α_{∗}(F)_{x} = 0 in case min(q, d) > dimO_{x}^{h} = codim_{S}x.

Applying the same reasoning to an arbitrary scheme etale over S we see that for q < d the Nisnevich sheaf R^{q}α_{∗}(F) is supported in dimension ≤ d − q and hence H_{Nis}^{p}(S, R^{q}α_{∗}(F)) = 0 for p > d−q -see [Nis]. Furthermore for q ≥ d the Nisnevich sheaf R^{q}α_{∗}(F) is supported in dimension 0 and hence H_{Nis}^{p}(S, R^{q}α_{∗}(F)) = 0 for p > 0. The Leray spectral sequence shows now that for n > d we have an isomorphism H_{cdh}^{n}(S, F) = H_{Nis}^{0}(S, R^{n}α_{∗}(F)) and ,in particular, each cohomology class h ∈ H_{cdh}^{n}(S, F) for which there exists a Nisnevich covering {U_{i} → S}_{i=1}^{n} such that h|_{U_{i}} is trivial for all i is trivial itself.

To finish the proof start with an arbitrary cohomology class h ∈ H_{cdh}^{n}(S, F) (n > d). There exists a cdh-covering {X_{i} → S} such that h dies being restricted to each X_{i}. Furthermore Proposition 5.9 shows that this covering has a refinement of the form {U′_{i} → S′ → S}_{i=1}^{n}, where S′ → S is a proper birational cdh-covering and {U′_{i} → S′}_{i=1}^{n} is a Nisnevich covering of the scheme S′. Let S′′ ⊂ S′ be the closure of the inverse image of the generic point of S and let {U′′_{i} → S′′} be the induced Nisnevich covering of the scheme S′′. Thus S′′ is an integral scheme of dimension ≤ d and the morphism S′′ → S is proper and birational. The cohomology class h|_{S′′} dies being restricted to all U′′_{i} and hence is trivial. Let finally Z ⊂ S be a proper closed subscheme such that the morphism p : S′′ → S is an isomorphism outside Z. The long exact cohomology sequence

0 = H_{cdh}^{n−1}(p^{−1}(Z), F) → H_{cdh}^{n}(S, F) → H_{cdh}^{n}(S′′, F) ⊕ H_{cdh}^{n}(Z, F) = H_{cdh}^{n}(S′′, F)

shows that the restriction homomorphism H_{cdh}^{n}(S, F) → H_{cdh}^{n}(S′′, F) is injective. Thus h = 0.

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