HIGHER CHOW GROUPS AND ETALE COHOMOLOGY

Andrei A. Suslin

Saint Petersburg Branch of the Steklov Math. Institute (POMI) and the Northwestern University

Introduction

The main purpose of the present paper is to relate the higher Chow groups of varieties over an algebraically closed field introduced by S.Bloch [B1] to etale cohomology. We follow the approach suggested by the auther in 1987 during the Lumini conference on algebraic K-theory. The first and most important step in this direction was done in [SV1], where singular cohomology of any qfh-sheaf were computed in terms of Ext-groups. The difficulty in the application of the results of [SV1] to higher Chow groups lies in the fact that a priori higher Chow are not defined as singular homology of a sheaf. To overcome this difficulty we prove that for an affine varietie X higher Chow groups CH^{i}(X, n) of codimension i ≤ dimX may be computed using equidimensional cycles only(this is done in the first two sections of the paper). In section 3 we generalize this result to all quasiprojective varieties over a field of characteristic zero. Together with the main theorem of [SV1] this result shows that for an equidimensional quasiprojective variety X over an algebraically closed field F of characteristic zero the higher Chow groups of codimension d = dimX with finite coefficients Z/m are dual to Ext_{qfh}^{∗}(z_{0}(X), Z/m), where z_{0}(X) is the qfh-sheaf of equidimensional cycles of relative dimension zero, introduced and studied in [SV2]. The above Ext-groups are easy to compute and in this way we come to the following main result

Theorem (Theorem 4.2 and Corollary 4.3). Let X be an equidimensional quasiprojective scheme over an algebraically closed field F of characteristic zero. Assume that i ≥ d = dim X. Then

CH^{i}(X, n; Z/m) = H_{c}^{2(d−i)+n}(X, Z/m(d − i))^{#}

where H_{c} stands for etale cohomology with compact supports. If the scheme X is smooth then this formula simplifies to CH^{i}(X, n; Z/m) = H_{et}^{2i−n}(X, Z/m(i)).

Throughout the paper we work with schemes of finite type over a field F. We denote the category of these schemes by Sch/F. We use the term variety as a synonim for integral scheme. We denote by p the exponential characteristic of F.

Typeset by AMS-TEX

The research was supported in part by Grant MOG 000 from the International Science Foundation

§1. Generic equidimensionality.

Theorem 1.1. Assume that S is an affine scheme, V is a closed subscheme in A^{n} × S and t is a nonnegative integer such that dimV ≤ n + t. Assume further that Z is an effective divisor in A^{n} and φ : Z × S → A^{n} × S is any S-morphism. Then there exists an S-morphism Φ : A^{n} × S → A^{n} × S such that

(1) Φ|_{Z×S} = φ

(2) Fibers of the projection Φ^{−1}(V) → A^{n} over points of A^{n}−Z have dimension ≤ t

Proof. Notice first that it suffices to treat the special case S = A^{m}. In fact, if this case is already settled then we can proceede as follows. Choose a closed embedding S ↪ A^{m}. To give an S-morphism φ : Z × S → A^{n} × S is the same as to give a morphism ψ = pr_{1} ∘ φ : Z × S → A^{n}. Extend ψ to a morphism ψ′ : Z × A^{m} → A^{n} and denote by φ′ the corresponding A^{m}-morphism Z × A^{m} → A^{n} × A^{m}. Using the theorem for A^{m} extend φ′ to an A^{m}-morphism Φ′ : A^{n} × A^{m} → A^{n} × A^{m} such that fibers of the projection (Φ′)^{−1}(V) → A^{n} over points of A^{n} − Z are of dimension ≤ t and finally set Φ = Φ′|_{A^{n}×S}.

From now on we assume that S = A^{m}, we denote the coordinate fuctions in A^{n} by Y_{1}, …, Y_{n} and by X_{1}, …, X_{m} we denote the coordinate functions in A^{m}. Let h ∈ F [Y_{1}, …, Y_{n}] be the equation of the divisor Z. The morphism φ : Z × A^{m} → A^{n} × A^{m} is given by the formula φ(y, x) = (φ_{1}(y, x), …, φ_{n}(y, x), x), where φ_{i} ∈ F [Y, X] are certain polynomials (uniquely defined modh). Define Φ : A^{n} × A^{m} → A^{n} × A^{m} by means of the formula

Φ(y, x) = (φ_{1}(y, x) + h(y)p_{1}(x), …, φ_{n}(y, x) + h(y)p_{n}(x), x)

where p_{i} ∈ F [X] are forms of degree N. We’ll show that if N is sufficiently large and p_{i} are sufficiently generic then the corresponding Φ has the desired property. To prove this we need some auxilliary facts.

(1.2). Let R be a finitely generated F -algebra and let X = Spec R be the corresponding affine scheme. Consider the polynomial ring R[Y_{1}, …, Y_{n}]. For any f ∈ R[Y_{1}, …, Y_{n}] we denote by l.f.(f) the leading form of f, we also set ^{h}f = (Y_{0})^{deg f}f(Y_{1}/Y_{0}, …, Y_{n}/Y_{0}). Thus l.f.(f) is a form of degree deg f in variables Y_{1}, …, Y_{n}, while ^{h}f is a form of degree deg f in variables Y_{0}, …, Y_{n}, moreover l.f.(f) = ^{h}f(0, Y_{1}, …, Y_{n}) and f = ^{h}f(1, Y_{1}, …, Y_{n}). If 𝔄 is an ideal in R[Y_{1}, …, Y_{n}] then leading forms of polynomials from 𝔄 give a homogeneous ideal in R[Y_{1}, …, Y_{n}] which we denote l.f.(𝔄). In the same way forms Y_{0}^{k} · ^{h}f (k ≥ 0, f ∈ 𝔄) give a homogeneous ideal in R[Y_{0}, …, Y_{n}] which we denote ^{h}𝔄(compare [[]ch. 7,§5]ZS). Note the following evident relations between ideals 𝔄, l.f.(𝔄),^{h}𝔄:

(1.2.1). Let g ∈ R[Y_{0}, …, Y_{n}] be a form, then g ∈ ^{h}𝔄 ⟺ g(1, Y_{1}, …, Y_{n}) ∈ 𝔄 (1.2.2) l.f.(𝔄) coincides with the image of ^{h}𝔄 under a surjective homomorphism

of graded rings s : R[Y_{0}, …, Y_{n}] → R[Y_{1}, …, Y_{n}] sending Y_{0} to zero.

Identify A^{n} = Spec F [Y_{1}, …, Y_{n}] with an open subscheme P_{Y_{0}}^{n} of P^{n} = = Proj F [Y_{0}, …, Y_{n}] by means of an open embedding j : A^{n} ↪ P^{n} given by the formula j(y_{1}, …, y_{n}) = [1 : y_{1} ⋯: y_{n}]. In other words j is defined by a trivial linear bundle O_{A^{n}} over A^{n} and (n+1)-tuple of its sections (without common zeros) (1, Y_{1}, …, Y_{n}).

Lemma 1.3. In the above notations denote the closed subscheme of X × A^{n} = = Spec R[Y_{1}, …, Y_{n}] defined by the ideal 𝔄 by V. Then the closed subscheme of Proj R[Y_{0}, …, Y_{n}] = X × P^{n} defined by the homogeneous ideal ^{h}𝔄 coincides with the closure \overline{V} of V in X × P^{n} and the closed subscheme of Proj R[Y_{1}, …, Y_{n}] = = X × P^{n−1} defined by the homogeneous ideal l.f.(𝔄) coincides with V_{∞} = \overline{V} ∩ (X × P^{n−1}).

Proof. Denote the sheaf of ideals defining \overline{V} by I and let 𝔓 be the homogeneous ideal of R[Y_{0}, …, Y_{n}] corresponding to I. Tensoring the exact sequence of sheaves

0 → I → O_{X×P^{n}} → j_{∗}(O_{V})

by O(q) and then taking global sections we get exact sequences of R-modules

0 → 𝔓_{q} = Γ(X × P^{n}, I(q)) → R[Y_{0}, …, Y_{n}]_{q} =Γ(X × P^{n}, O(q)) → Γ(V, O_{V} (q)) =
=Γ(V, O_{V}) = R[Y_{1}, …, Y_{n}]/𝔄

The homomorphism R[Y_{0}, …, Y_{n}]_{q} → R[Y_{1}, …, Y_{n}]/𝔄 sends the form g to g(1, Y_{1}, …, Y_{n}) mod 𝔄. This shows that 𝔓_{q} = {g ∈ R[Y_{0}, …, Y_{n}]_{q} : g(1, Y_{1}, …, Y_{n}) ∈ 𝔄} = (^{h}𝔄)_{q}.

To prove the second assertion denote the closed embedding X ×P^{n−1} → X ×P^{n} by i. This embedding corresponds to the epimorphism of graded rings s : R[Y_{0}, …, Y_{n}] → R[Y_{1}, …, Y_{n}] considered in (1.2.2). In these notations we get:

O_{V_{∞}} = i^{∗}(O\overline{V}) = i^{∗}((R[Y_{0} …, Y_{n}]/^{h}𝔄)^{∼}) =
= (R[Y_{1}, …, Y_{n}] ⊗_{R[Y_{0},...,Y_{n}]} (R[Y_{0}, …, Y_{n}]/^{h}𝔄))^{∼} = (R[Y_{1}, …, Y_{n}]/l.f.(𝔄))^{∼}.

Thus V_{∞} is defined by the homogenous ideal l.f.(𝔄).

(1.4). In the situation of (1.2) assume further that R is a graded F -algebra such that R_{0} = F and R is generated over F by R_{1}. If a polynomial f ∈ R[Y_{1}, …, Y_{n}] is R-homogenous then l.f.(f) and ^{h}f are bihomogenous polynomials. In the same way if 𝔄 is an R-homogenous ideal in R[Y_{1}, …, Y_{n}] then the ideals l.f.(𝔄) and ^{h}𝔄 are bihomogenous. Repeating the arguments used in the proof of Lemma 1.3 we get easily the following fact(where now X stands for Proj R).

Lemma 1.4.1. Let 𝔄 be an R-homogenous ideal in R[Y_{1}, ..., Y_{n}] and let V ⊂ Proj(R[Y_{1}, ..., Y_{n}]) = X × A^{n} be the corresponding closed subscheme. Then the closed subscheme of X × P^{n} defined by the bihomogenous ideal ^{h}𝔄 coincides with the closure \overline{V} of V in X × P^{n}. Moreover the closed subscheme of X × P^{n−1} defined by the bihomogenous ideal l.f.(𝔄) coincides with V_{∞} = \overline{V} ∩ (X × P^{n−1})

(1.5). Consider the bigraded ring F [Y, X] = F [Y_{1}, …, Y_{n}, X_{1}, ..., X_{m}]. For any f ∈ F [Y, X] denote by deg_{Y} f, deg_{X}f, l.f_{Y} (f), l.f_{X}(f), the degree and the leading form of f with respect to the corresponding variables. Introduce the lexicographical order on the set of bidegrees and define degf as the highest bidegree of nonzero bihomogenous summands of f. Finally define l.f.(f) to be the bihomogenous summand of f of bidegree degf. The following relations are obvious from the definitions:

(1.5.1) degf = (deg_{Y} f, deg_{X}(l.f_{Y} f))

(1.5.2) l.f.(f) =l.f._{X}(l.f._{Y} (f))

If 𝔄 is any ideal in F [Y, X] then leading forms of polynomials from 𝔄 give a bihomogenous ideal l.f.(𝔄) and in view of (1.5.2) we have:

(1.5.3) l.f.(𝔄) = l.f._{X}(l.f._{Y} (𝔄))

Using (1.3.1) and (1.4.1) we get now the following lemma Lemma 1.5.4. Denote by V the closed subscheme of A^{n}× A^{m} defined by the ideal 𝔄, let \overline{V} denote the closure of V in P^{n} ×A^{m} and set V_{∞} = \overline{V} ∩(P^{n−1} ×A^{m}). Next denote \overline{V_{∞}} the closure of V_{∞} in P^{n−1} × P^{m} and set V_{∞,∞} = \overline{V_{∞}} ∩ (P^{n−1} × P^{m−1}). Then V_{∞,∞} coincides with the closed subscheme of P^{n−1} × P^{m−1} defined by the bihomogenous ideal l.f.(𝔄).

Corollary 1.5.5. Dimension of the closed subscheme of P^{n−1} × P^{m−1} defined by the bihomogenous ideal l.f.(𝔄) is not more than dim V − 2.

(1.6) Continuation of the proof of theorem 1.1.

Denote by 𝔄 the ideal of F [Y, X] defining the closed subscheme V ⊂ A^{n} × A^{m}

and choose f_{1}, ..., f_{s} ∈ 𝔄 such that the bihomogenous forms f^{∼}_{j} = l.f.(f_{j}) generate the ideal l.f.(𝔄). Assume in the sequel that N > max(deg_{X}φ_{i}, deg_{X}f_{j}). Then one checks easily that for any choice of N-forms p_{i}(X) we have:

(1) deg_{X}f_{j}(φ_{i} + hp_{1}, ..., φ_{n} + hp_{n}, X) ≤ Ndeg_{Y} f^{∼}_{j} + deg_{X} f^{∼}_{j}

(2) The X-homogenous summand of f_{j}(φ+hp, X) of degree Ndeg_{Y} f^{∼}_{j} +deg_{X} f^{∼}_{j}

is equal to h^{deg_{Y} f_{j}} · f^{∼}_{j}(p_{1}, ..., p_{n}, X)

Thus denoting by 𝔓 the ideal defining the closed subschemeΦ^{−1}(V) and denoting by 𝔓_{h} ⊂ F [Y]_{h}[X] the corresponding localization we come to the following conclusion

Lemma 1.6.1. Define a morphism Φ : A^{n} × A^{m} → A^{n} × A^{m} by the formula Φ(y, x) = (φ_{1}(y, x) + h(y)p_{1}(x), ..., φ_{n}(y, x) + h(y)p_{n}(x), x), where p_{i} are forms of degree N > max(deg_{X}φ_{i}, deg_{X}f_{j}). Then the ideal l.f._{X}(𝔓_{h}) contains the forms

f^{∼}_{j}(p_{1}, ..., p_{n}, X)

Consider further the closed subscheme \overline{pr_{2}(V)} ⊂ A^{m}. It’s defined by the ideal 𝔔 = 𝔄 ∩ F [X]. It’s clear that for any A^{m}-morphism Φ the ideal 𝔓 contains 𝔔 and hence l.f._{X}(𝔓_{h}) ⊃ l.f._{X}(𝔔). Denote by W the closed subscheme of P^{m−1} defined by the homogenous ideal l.f._{X}(𝔔). Since dim \overline{pr_{2}(V)} ≤ dimV ≤ n + t we conclude from lemma 1.3.1 that dim W ≤ n + t − 1. To finish the proof of theorem 1.1 we need the following fact

Proposition 1.7. Let f^{∼}_{1}(Y, X), ..., f^{∼}_{s}(Y, X) be bihomogenous forms such that the subscheme C ⊂ P^{n−1} × P^{m−1} of their common zeros is of dimension ≤ n + t − 2. Assume further that we are given a closed subscheme W ⊂ P^{m−1} of dimension ≤ n + t − 1. Then for any N ≥ 0 we can find forms p_{1}, ..., p_{n} ∈ F [X] of degree N such that the dimension of the closed subscheme of W given by the equations

f^{∼}_{j}(p_{1}, ..., p_{n}, X_{1}, ..., X_{m}) = 0 is not more than t − 1.

Proof. Identifying each form with the family of its coefficients we get a bijection between the set of forms and set of rational points of an affine space A^{M} (M = \binom{N+m−1}{m−1}). In the same way n-tuples of N-forms are in one to one correspondence with rational points of the affine space A^{Mn}. Set T_{1} = {(w, p) ∈ W × A^{Mn} : p_{1}(w) = ⋯ = p_{n}(w) = 0}. It’s clear that the set T_{1} is closed and dim T_{1} = dim W + n(M − 1) ≤ nM + t − 1. Consider now the following morphism g : W × A^{Mn} − T_{1} → W × P^{n−1} : (w, p) ↦ (w, [p_{1}(w) : ... : p_{n}(w)]) and set T_{2} = g^{−1}(C ∩ (W × P^{n−1})). The set T_{2} is closed in W × A^{Mn} − T_{1} and hence T = T_{1} ∪ T_{2} is closed in W × A^{Mn}. One checks easily that fibers of g are of dimension Mn − (n − 1) and hence dim T_{2} ≤ dim C + Mn − (n − 1) ≤ nM + t − 1. This shows that dim T ≤ nM + t − 1. Finally consider the projection q : T → A^{Mn}. According to the theorem on the dimension of fibers of a morphism there exists a nonempty open set U ⊂ A^{Mn} such that for any p ∈ U dim q^{−1}(p) ≤ t − 1. Now we can take p = (p_{1}, ..., p_{n}) to be any rational point of U.

(1.8) End of the proof of theorem 1.1.

Note first that we are in the situation considered in (1.7). In fact, according to the choice of f_{j}, the closed subscheme of P^{n−1} × P^{m−1} defined by the equations

f^{∼}_{j} = 0 coincides with the one defined by the bihomogenous ideal l.f.(𝔄) and hence is of dimension ≤ dimV − 2 ≤ n + t − 2 according to (1.5.5). Moreover dim W ≤ dim V − 1 ≤ n + t − 1 as was noted already in (1.6). According to the proposition 1.7 we can choose forms p_{1}, ..., p_{n} of degree N > max(deg_{X}φ_{i}, deg_{X}f_{j}) so that the

dimension of W_{0} = W ∩ {f^{∼}_{1}(p, x) = ... = f^{∼}_{s}(p, x) = 0} is not more than t − 1. It suffices to note now that according to (1.6.1) the infinite part of the fiber of Φ^{−1}(V) over any point y ∈ A^{n} − Z is contained in W_{0} and hence is of dimension ≤ t − 1. This implies that the dimension of the fiber itself is not more than t.

§2. Higher Chow groups and equidimensional cycles.

Denote by Δ^{n} the linear subvariety of A^{n+1} given by the equation t_{0}+...+t_{n} = 1. The points p_{i} = (0, ..., 1_{i}, ..., 0) (0 ≤ i ≤ n) are called the vertices of the “simplex” Δ^{n}. They have the following evident property : if X ⊂ A^{m} is any linear subvariety (i.e. X is defined by a sistem of linear equations on coordinates) and if x_{0}, ..., x_{n} is any (n+1)-tuple of rational points of X then there exists a unique linear morphism Δ^{n} ⟶ X, taking p_{i} to x_{i}. This shows in particular that any nondecreasing map ϕ : {0, ..., n} → {0, ..., m} defines a canonical morphism Δ^{n} → Δ^{m} (taking p_{i} to p_{ϕ(i)}), which we’ll denote by the same letter ϕ. In this way Δ^{·} becomes a cosimplicial scheme. The morphisms Δ^{n} → Δ^{m} corresponding to strictly increasing maps {1, ..., n} → {1, ..., m} are called cofaces (of codimension m − n). Each coface is a closed embedding, the corresponding closed subscheme of Δ^{m} is clearly a linear subvariety and will be called a face of Δ^{m} (of codimension m − n). As always there exists m + 1 cofaces of codimension one, they are denoted δ_{i} : Δ^{m−1} → Δ^{m} (0 ≤ i ≤ m).

Assume that X ∈ Sch/F is any equidimensional scheme, S and T are smooth absolutely irreducible varieties and f : X × S → X × T is any X-morphism. In this situation the schemes X × S and X × T are equidimensional of dimension dim X +dim S and dim X +dim T respectively. Consider the graph decomposition of f

X × S ⟶^{i} X × S × T
f↓  ↓^{pr_{1,3}}
X × T ⟶^{=} X × T

Since T is smooth the embedding i is regular (of codimension dim T). This shows that for any subvariety V of X×T and any component W of f^{−1}(V) = i^{−1}(V ×S) we have the usual inequality codim_{X×S}(W) ≤ codim_{X×S×T} (V × S) = codim_{X×T} (V).

When all components of f^{−1}(V) have correct codimension we can use the T orformula to define the cycle f^{∗}([V]) (since the morphism pr_{1,2} is flat and i is a regular embedding we conclude that f is of finite T or-dimension). Alternatively we could say that f is a local complete intersection morphism and use the machinery developped in [[]ch. 6]F.

Denote by z^{i}(X, n) the free abelian group generated by codimension i subvarieties V ⊂ X × Δ^{n}, which intersect properly X × Δ^{m} for any face Δ^{m} ↪ Δ^{n}. It’s clear from the above definition that if [V] is a generator of z^{i}(X, n) then cycles ∂_{j}([V]) = (1_{X} × δ_{j})^{∗}([V]) are defined and lie in z^{i}(X, n − 1). More generally, if ϕ : Δ^{m} → Δ^{n} is any structure morphism of the cosimplicial scheme Δ^{∗} then the cycle (1_{X} × ϕ)^{∗}([V]) is defined and lies in z^{i}(X, m). Thus z^{i}(X, −) is a simplicial abelian group. The homotopy groups of this simplicial abelian group are denoted CH^{i}(X, n) - see[B1]. In other words CH^{i}(X, n) is the n-th homology group of the complex

z^{i}(X, 0) ←^{d} z^{i}(X, 1) ←^{d} ...

where, as always, d is the alternating sum of face operators.

A morphism of varieties f : X → Y is called equidimensional of relative dimension t if it is dominant and each fiber of f has pure dimension t. In this situation dimX is equal to t + dimY.

Returning to the definition of the higher Chow groups assume that i ≤ d = dimX and set t = d − i. Let V be a closed subvariety in X × Δ^{n} and assume that the projection V → Δ^{n} is an equidimensional morphism of relative dimension t. This implies that codim_{X×Δ^{n}}(V) = i. Moreover, if ϕ : Δ^{m} → Δ^{n} is any morphism and if W is any component of (1_{X} × ϕ)^{−1}(V) then codim_{X×Δ^{m}}(W) ≤ codim_{X×Δ^{n}}(V), i.e. dimW ≥ m + t. On the other hand each fiber of the projection W → Δ^{m} is contained in the corresponding fiber of V → Δ^{n} and hence has dimension ≤ t. Since each component of each fiber is of dimension ≥ dimW − m we conclude that dimW = m + t (i.e. codim_{X×Δ^{m}}(W) = codim_{X×Δ^{n}}(V)) and moreover the projection W → Δ^{m} is equidimensional of relative dimension t. Thus, denoting by z_{eq}^{i}(X, n) the free abelian group generated by closed subvarieties V ⊂ X × Δ^{n} for which the projection V → Δ^{n} is equidimensional of relative dimension t, we see that z_{equi}^{i}(X, −) is a simplicial abelian subgroup in z^{i}(X, n). The main purpose of this section is to prove the following result

Theorem 2.1. Assume that X is an affine equidimensional scheme and i ≤ d = dimX. Then the embedding of complexes z_{equi}^{i}(X, −) ↪ z^{i}(X, −) is a quasiisomorphism.

The proof is based on a certain auxilliary construction.

(2.2). Let N be a positive integer. Assume that for every n (0 ≤ n ≤ N) we are given an X-morphism φ_{n} : X ×Δ^{n} → X ×Δ^{n} such that the following diagrams commute for all 0 ≤ j ≤ n ≤ N

(2.2.1)  \begin{matrix}X × Δ^{n−1} & ⟶^{φ_{n−1}} & X × Δ^{n−1} \\ ^{1_{X}×δ_{j}} ↓ & & ↓^{1_{X}×δ_{j}} \\ X × Δ^{n} & ⟶^{φ_{n}} & X × Δ^{n} \end{matrix}

Define _{φ}z^{i}(X, n) tro be a free abelian group generated by close subvarieties V ⊂ X × Δ^{n} such that a) [V] ∈ z^{i}(X, n)

b) The cycle (φ_{n})^{∗}([V]) is defined and lies in z^{i}(X, n)

The commutativity of (2.2.1) shows that for any generator [V] of _{φ}z^{i}(X, n) and any j (0 ≤ j ≤ n) the cycle ∂_{j}([V]) = (1_{X} × δ_{j})^{∗}([V]) lies in _{φ}z^{i}(X, n − 1), so that setting _{φ}z^{i}(X, n) = 0 for n > N we get a subcomplex _{φ}z^{i}(X, −) of the complex z^{i}(X, −). Moreover the commutativity of (2.2.1) implies that the assignment [V] ↦ φ_{n}^{∗}([V]) defines a homomorphism φ^{∗} : _{φ}z^{i}(X, −) → z^{i}(X−).

We’ll say that two homomorphisms C_{∗} ⇉ D_{∗} of nonnegative complexes are weakly homotopic if their restrictions to any finitely generated subcomplex of C_{∗} are homotopic. It’s clear that weakly homotopic homomorphisms induce the same maps on homology.

Proposition 2.3. The homomorphisms _{φ}z^{i}(X, −) ⇉_{inc}^{φ^{∗}} z^{i}(X, −) are weakly homo-

topic.

Proof. Assume that for each n (0 ≤ n ≤ N) we have fixed a finite number of closed subvarieties V_{k}^{n} ⊂ X × Δ^{n} such that [V_{k}^{n}] ∈ _{φ}z^{i}(X, n) and the family {V_{∗}^{n−1}} contains all components of all cycles ∂_{j}([V_{k}^{n}]). Denote by C_{n} the free abelian group generated by [V_{k}^{n}]. It’s clear that C_{∗} is a finitely generated subcomplex in_{φ}z^{i}(X, −) and that subcomplexes of this type are cofinal among all finitely generated subcomplexes of _{φ}z^{i}(X, −). Thus it suffices to show that (φ^{∗} − inc)|_{C_{∗}} is homotopic to zero. To prove this we construct by induction on n (0 ≤ n ≤ N) X-morphisms Φ_{n} : X × Δ^{n} × A^{1} → X × Δ^{n} × A^{1} with the following properties

(2.3.1). The following diagram commutes (here i_{0} and i_{1} denote closed embeddings defined by points 0, 1 ∈ A^{1}

(2.3.1.1)

X × Δ^{n}  ⟶^{id} X × Δ^{n}
i_{0}↓  ↓^{i_{0}}
X × Δ^{n} × A^{1} ⟶^{Φ_{n}} X × Δ^{n} × A^{1}

(2.3.1.2)  \begin{matrix}X × Δ^{n} & ⟶^{φ_{n}} & X × Δ^{n} \\ ^{i_{1}}↓ & & ↓^{i_{1}} \\ X × Δ^{n} × A^{1} & ⟶^{Φ_{n}} & X × Δ^{n} × A^{1} \end{matrix}

(2.3.1.2)

X × Δ^{n−1} × A^{1} ⟶^{Φ_{n−1}} X × Δ^{n−1} × A^{1}
1_{X}×δ_{j}×1_{A^{1}} ↓  ↓^{1_{X}×δ_{j}×1_{A^{1}}}
X × Δ^{n} × A^{1} ⟶^{Φ_{n}} X × Δ^{n} × A^{1}

(2.3.2) The fibers of the projection Φ_{n}^{−1}(⋃_{k} V_{k}^{n} × A^{1}) → Δ^{n} × A^{1} over points not lying on the divisor Z = (Δ^{n} × 0) + (Δ^{n} × 1) + ∑_{j=0}^{n} Δ_{j}^{n−1} × A^{1} (here Δ_{j}^{n−1} is the j-th face of Δ^{n}) are of dimension ≤ t.

Note that the commutativity of diagrams (2.2.1) together with the induction hypothesis implies that the requirements (2.3.1) are compatible with each other and define the morphism Φ_{n} on X × Z. The existence of extension of this morphism to X × Δ^{n} × A^{1} satisfying the additional requirement (2.3.2) is garanteed by the theorem 1.1.

The vertices of Δ^{n} ×A^{1} are the points p_{i} ×0 and p_{i} ×1. We introduce the partial ordering on these vertices, taking the standart order on p_{i}’s (p_{i} ≤ p_{j} ⇔ i ≤ j) and taking the standart order on the set {0, 1}. For any nondecreasing system of vertices q_{0} ≤ q_{1} ≤ ⋯ ≤ q_{m} denote by θ_{q} the linear morphism Δ^{m} → Δ^{n} × A^{1}, taking p_{i} to q_{i}.

Lemma 2.4. Let q_{0} < q_{1} < ⋯ < q_{m} be a strictly increasing sequence of vertices of Δ^{n} × A^{1} (thus m ≤ n + 1). Then dim[Φ_{n} ∘ (1_{X} × θ_{q})]^{−1}(⋃_{k} V_{k}^{n} × A^{1}) ≤ m + t.

Proof. We proceed by induction on n. Consider first the case n = m − 1. In this case θ_{q} is an isomorphism and our statement reduces to the following formula: dim Φ_{n}^{−1}(⋃_{k} V_{k}^{n} × A^{1}) ≤ n + 1 + t. Using the induction hypothes, the properties of the varieties V_{k}^{n} and (2.3.1) we prove immediately the following statement

(2.4.1) The dimension of the part of Φ_{n}^{−1}(⋃ V_{k}^{n} × A^{1} lying over Z does not exceed n + t.

The fibers of the projection Φ_{n}^{−1}(⋃ V_{k}^{n} × A^{1}) → Δ^{n} × A^{1} over the points not lying in Z are of dimension ≤ t. This implies that the dimension of the part of Φ_{n}^{−1}(⋃ V_{k}^{n} × A^{1}) lying over Δ^{n} × A^{1} − Z is not more than n + 1 + t. Finally dim Φ_{n}^{−1}(⋃ V_{k}^{n} × A^{1}) ≤ max(n + t, n + 1 + t) = n + 1 + t.

Consider next the case n = m. Note that θ_{q} is always a closed embedding. Denote by Z′ the inverse image of Z under θ_{q}. Using (2.4.1) we conclude immediately that the dimension of the part of [Φ_{n} ∘ (1_{X} × θ_{q})]^{−1}(⋃ V_{k}^{n} × A^{1}) = (1_{X} ∘ θ_{q})^{−1}Φ_{n}^{−1}(⋃_{k} V_{k}^{n} × A^{1}) lying over Z′ does not exceed n + t. The fibers of the projection (1_{X} × θ_{q})^{−1}Φ_{n}^{−1}(⋃ V_{k}^{n} × A^{1}) → Δ^{n} over the points of Δ^{n} − Z′ are of dimension not more than t and hence the dimension of the part of [Φ_{n} ∘ (1_{X} × θ_{q})]^{−1}(⋃ V_{k}^{n} × A^{1}) which lies over Δ^{n} − Z′ does not exceed n + t. Finally dim([Φ_{n} ∘ (1_{X} × θ_{q})]^{−1}(⋃ V_{k}^{n} × A^{1})) ≤ max(n + t, n + t) = n + t.

Finally suppose that m < n. In this case one can find a strictly incresing sequence of vertices q′_{0} < q′_{1} < ⋯ < q′_{m} in Δ^{n−1} × A^{1} and an index j (0 ≤ j ≤ n) such that θ_{q} = (δ_{j} × 1_{A^{1}}) ∘ θ_{q′} and hence [Φ_{n} ∘ (1_{X} × θ_{q})]^{−1}(⋃ V_{k}^{n} × A^{1}) = [(1_{X} × δ_{j} × 1_{A^{1}}) ∘ Φ_{n−1} ∘ (1_{X} × θ_{q′})]^{−1}(⋃ V_{k}^{n} × A^{1}) ⊂ [Φ_{n−1} ∘ (1_{X} × θ_{q′})]^{−1}(⋃ V_{∗}^{n−1} × A^{1}) so that the induction hypothesis applies.

(2.5) End of the proof of the proposition 2.3.

Denote by θ_{j} (0 ≤ j ≤ n) the morphisms defined by the strictly increasing sequences of vertices p_{0} ×0 < ⋯ < p_{j} ×0 < p_{j} ×1 < ... < p_{n} ×1. Lemma 2.4 shows that the cycles [Φ_{n} ∘ (1_{X} × θ_{j})]^{∗}([V_{k}^{n} × A^{1}]) are defined and lie in z^{i}(X, n + 1). The homotopy we are seeking may be defined now by means of the formula σ([V_{k}^{n}]) = ∑_{j=0}^{n}(−1)^{j}[Φ_{n} ∘ (1_{X} × θ_{j})]^{∗}([V_{k}^{n} × A^{1}]) (cf.[[]§2]B1).

We also need a version of (2.3) for the complex z_{equi}^{i}(X, −). In the above situation denote by_{φ}z_{equi}^{i}(X, n) the free abelian group generated by subvarieties V ⊂ X ×Δ^{n} such that

(1) [V] ∈ z_{equi}^{i}(X, n)

(2) The cycle φ_{n}^{∗}([V]) is defined and lies in z_{equi}^{i}(X, n)

In the same way as above we see that_{φ}z_{equi}^{i}(X, −) is a subcomplex in z_{equi}^{i}(X, −) and we have a canonical homomorphism φ^{∗} : _{φ}z_{equi}^{i}(X, −) → z_{equi}^{i}(X, −).

Proposition 2.6. The homomorphisms _{φ}z_{equi}^{i}(X, −) ⇉_{inc}^{φ^{∗}} z_{equi}^{i}(X, −) are weekly

homotopic

Proof. Repeat the construction used in the proof of proposition 2.3. Lemma 2.4 is replaced by the following fact (which is proved by a straightforward induction on n).

Lemma 2.7. All fibers of the projection Φ_{n}^{−1}(⋃_{k} V_{k}^{n} × A^{1}) → Δ^{n} × A^{1} are of dimension ≤ t.

This lemma shows that the cycles [Φ_{n} ∘ (1_{X} × θ_{j})]^{∗}([V_{k}^{n} × A^{1}]) are defined and lie in z_{equi}^{i}(X, n + 1) so that we can define the homotopy by the same formula as above.

Proposition 2.8. Let C_{∗} be a finitely generated subcomplex in z^{i}(X, −). Then we can find N and φ as above such that C_{∗} ⊂ _{φ}z^{i}(X, −) and φ^{∗}(C_{∗}) ⊂ z_{equi}^{i}(X, −).

Proof. We may suppose that C_{∗} is generated by a system of subvarieties V_{k}^{n} ⊂ X × Δ^{n} (0 ≤ n ≤ N) with the corresponding properties (see the proof of (2.3)). Using induction on n we construct X-morphisms φ_{n} : X × Δ^{n} → X × Δ^{n} with the following properties:

(1) The diagrames (2.2.1) commute

(2) The fibers of the projection φ_{n}^{−1}(⋃_{k} V_{k}^{n}) → Δ^{n} over the points not lying on the divisor Z = ∑_{j=0}^{n} Δ_{j}^{n−1} are of dimension ≤ t.

The induction step in the construction of φ’s is garanteed again by theorem 1.1. Another induction on n shows immediately that all fibers of the projection φ_{n}^{−1}(⋃ V_{k}^{n}) → Δ^{n} are of dimension ≤ t. Thus the cycles φ_{n}^{∗}([V_{k}^{n}]) are defined and lie in z_{equi}^{i}(X, −).

(2.9) Proof of the theorem 2.1.

Let v be any n-dimensional cycle of the complex z^{i}(X, −). Using proposition 2.8 find φ such that v ∈ _{φ}z^{i}(X, n) and φ^{∗}(v) ∈ z_{equi}^{i}(X, n). According to proposition 2.3 v−φ^{∗}(v) is a boundary. This shows that the homomorphism H_{n}(z_{equi}^{i}(X, −)) → H_{n}(z^{i}(X, −)) is surjective.

Assume now that v is an n-dimensional cycle in z_{equi}^{i}(X, −) such that v = d(w) for a certain w ∈ z^{i}(X, n + 1). Using proposition 2.8 find φ such that v, w ∈ φz^{i}(X, −) and φ^{∗}(v), φ^{∗}(w) ∈ z_{equi}^{i}(X, −). Since φ^{∗}(v) = d(φ^{∗}(w)) we see that φ^{∗}(v) is a boundary in z_{equi}^{i}(X, −). On the other hand v −φ^{∗}(v) is also a boundary in z_{equi}^{i}(X, −) according to (2.6). Thus v is a boundary in z_{equi}^{i}(X, −), i.e. the homomorphism H_{n}(z_{equi}^{i}(X, −)) → H_{n}(z^{i}(X, −)) is injective.

§3. Sheaves of equidimensional cycles

Denote by p the exponential characteristic of F. Recall from [SV2] that for any scheme X ∈ Sch/F and any t ≥ 0 there exists a qfh-sheaf z_{t}(X) which is characterized by the following property: for any normal connected scheme S ∈ Sch/F the group of sections of z_{t}(X) over S is equal to the free Z[1/p]-module generated by closed subvarieties Z ⊂ X × S which are equidimensional of relative dimension t over S.

For any presheaf F on the category Sch/F we’ll denote by C_{∗}(F) its singular complex - see [SV1], i.e. C_{n}(F) = F(Δ^{n}) and the differential d : C^{n}(F) → C_{n−1}(F) is the alternating sum of face operators. We’ll use the notation H_{∗}^{sing}(F), H_{∗}^{sing}(F, Z/n), H_{sing}^{∗}(F, Z/n) for the homology groups of the complexes C_{∗}(F), C_{∗}(F) ⊗^{L} Z/n, RHom(C_{∗}(F), Z/n).The functor C_{∗} is an exact functor from the category of presheaves to the category of nonnegative complexes of presheaves. Furthermore we have the following important theorem

Theorem 3.1 [V1]. Assume that char(F) = 0 and let F be a presheaf with transfers on the category Sch/F. If the h-sheaf associated with F is trivial then H_{∗}^{sing}(F) = 0.

Using this theorem of Voevodsky and the localization exact sequence for higher Chow groups (see [B2]) we can generalize theorem 2.1 to arbitrary quasiprojective schemes

Theorem 3.2. Assume that char(F) = 0. Then for any equidimensional quasiprojective scheme X ∈ Sch/F and any i ≤ d = dim X the canonical embedding of complexes

C_{∗}(z_{d−i}(X)) = z_{equi}^{i}(X, −) ↪ z^{i}(X, −)

is a quasiisomorphism.

Proof. We proceed by induction on d = dim X. The case d = 0 is trivial, so assume that d > 0 and for schemes of dimension d − 1 the theorem is true . We can easily find an effective Cartier divisor Y ⊂ X such that the open suvscheme U = X − Y is affine. The sequence of presheaves

0 ⟶ z_{d−i}(Y) ⟶ z_{d−i}(X) ⟶ z_{d−i}(U)

is exact and the h-sheaf associated with the presheaf z_{d−i}(U)/z_{d−i}(X) is trivial. Thus we get a long exact localization sequence for homology of equidimensional cycles. Comparing it with the localization sequence for higher Chow groups [B2] and using the induction hypothesis and theorem 2.1 we conclude that H_{n}^{sing}(z_{d−i}(X) → CH^{i}(X, n) is an isomorphism.

§4. Higher Chow groups and etale cohomology.

In this section we assume F to be an algebraically closed field of characteristic zero. Let X ∈ Sch/F be an equidimensional quasiprojective scheme of dimension d. Theorem 3.2 shows that for any i ≤ d the group CH^{i}(X, n; Z/m) =^{def} H_{n}(z^{i}(X, −)⊗ Z/m) coincides with H_{n}^{sing}(z_{d−i}(X), Z/m). For any periodical abelian group A we denote by A^{#} its dual, i.e. A^{#} = Hom(A, Q/Z). Thus CH^{i}(X, n; Z/m)^{#} = = H_{n}^{sing}(z_{d−i}(X), Z/m)^{#} = H_{sing}^{n}(z_{d−i}(X), Z/m) and the last group according to the main result of [SV1] is canonically isomorphic to Ext_{qfh}^{n}(z_{d−i}(X), Z/m). Consider now special case i = d. In this case we may use the following result.

Proposition 4.1. Let X be a separated scheme of finite type over an arbitrary field F (of exponential characteristic p). Then for any m prime to p we have canonical isomorphisms

Ext_{qfh}^{n}(z_{0}(X), Z/m) = Ext_{h}^{n}(z_{0}(X)_{h}^{∼}, Z/m) = H_{c}^{n}(X, Z/m)

where H_{c}^{n} stands for etale cohomology with compact supports.

Proof. The first equality is proved in [SV1, §10]. To prove the second one choose an open embedding j : X ↪ \overline{X}, where \overline{X} is a complete separated scheme, set Y = \overline{X} − X and let i denote the closed embedding Y ↪ \overline{X}. According to [SV2] we have an exact sequence of h-sheaves

0 → z_{0}(Y)h^{∼} → z_{0}(\overline{X})h^{∼} → z_{0}(X)h^{∼} → 0.

For any complete separated scheme Z the sheaf z_{0}(Z) coincides with the free qfhsheaf Z[\frac{1}{p}]_{qfh}(Z) generated by Z (see [SV1, theorem 6.3]). Taking associated hsheaves we see that z_{0}(Z)h^{∼} = Z[\frac{1}{p}]_{h}(Z). Thus the above sequence takes the form 0 → Z[\frac{1}{p}]_{h}(Y) → Z[\frac{1}{p}]_{h}(\overline{X}) → z_{0}(X)h^{∼} → 0 This exact sequence shows that for any h-sheaf of Z[\frac{1}{p}]-modules F we have the formula

Hom(z_{0}(X)h^{∼}, F) = Ker(Γ(\overline{X}, F) → Γ(Y, F))

Choose an injective resolution I^{∗} of the h-sheaf Z/m, consisting of Z[\frac{1}{p}]-modules. The previous remarks show that Ext_{h}^{n}(z_{0}(X)h^{∼}, Z/m) = H^{n}(Ker(Γ(\overline{X}, I^{∗}) → → Γ(Y, I^{∗}))). Moreover the homomorphism of complexes Γ(\overline{X}, I^{∗}) → Γ(Y, I^{∗}) is surjective. Consider the natural morphism of sites γ : (Sch/F)_{h} → (Sch/F)_{et}. The comparison theorem for h-cohomology [SV1, §10] shows that R^{i}γ_{∗}(Z/m) = 0 for i > 0. This means that I^{∗} is an injective resolution of Z/m not only in htopology, but in etale topology as well. Note further that to give a sheaf F on big etale site (Sch/F)_{et} is the same as to give a sheaf F_{Z} on small etale site Z_{et} for each Z ∈ (Sch/F) and to give homomorphisms F_{Z} → f_{∗}(F_{T}) defined for each morphism f : T → Z and satisfying evident compatibility properties. It’s clear that functors Shv(Sch/F)_{et} → Shv(Z_{et}) (F ↦ F_{Z}) are exact and preserve injectivity. Consider the bicomplex of sheaves on \overline{X}_{et}

I∗ ; \overline{X} → i_{∗}(I_{Y}^{∗})

This bicomplex has only one nonzero homology ( in degree zero) equal to Ker(Z/m → i_{∗}(Z/m)) = j_{!}(Z/m). Hence

H_{c}^{n}(X, Z/m) = H^{n}(\overline{X}, j_{!}(Z/m)) = H^{n}(Γ(\overline{X}, (I∗ ; \overline{X} → i_{∗}(I_{Y}^{∗}))) = H^{n}((Γ(\overline{X}, I^{∗}) →

→ Γ(Y, I^{∗}))) = H^{n}(Ker(Γ(\overline{X}, I^{∗}) → Γ(Y, I^{∗}))) = Ext_{h}^{n}(z_{0}(X)h^{∼}, Z/m).

Theorem 4.2. Let X be an equidimensional quasiprojective scheme over an algebraically closed field F of characteristic zero. Assume that i ≥ d = dim X. Then for any m > 0 CH^{i}(X, n; Z/m) = H_{c}^{2(d−i)+n}(X, Z/m(d − i))^{#}.

Proof. Assume first that i = d. In this case according to proposition 4.2 we have:

CH^{d}(X, n; Z/m)^{#} = Ext_{qfh}^{n}(z_{0}(X), Z/m) = H_{c}^{n}(X, Z/m)

This group being finite we conclude that CH^{d}(X, n; Z/m) is also finite and hence CH^{d}(X, n; Z/m) = [CH^{d}(X, n; Z/m)^{#}]^{#} = H_{c}^{n}(X, Z/m)^{#}. The general case follows now from the homotopy invariance of higher Chow groups - see[B1,§2]): CH^{i}(X, n; Z/m) = CH^{i}(X × A^{i−d}, n; Z/m) = H_{c}^{n}(X × A^{i−d}, Z/m)^{#} = = H_{c}^{n−2(i−d)}(X, Z/m(d − i))^{#}.

Corollary 4.3. In assumptions and notations of theorem 4.2 assume further that X is smooth. Then CH^{i}(X, n; Z/m) = H_{et}^{2i−n}(X, Z/m(i)).

Proof. This follows immediately from theorem 4.2 and the Poincaré duality theorem - see [M].

References

[B1] S. Bloch, Algebraic cycles and higher K-theory, Adv. in Math. 61 (1986), 267–304.

[B2] S. Bloch, The moving lemma for higher Chow groups, Journal of algebraic geometry 3 (1994), 537-568.

[BL] S. Bloch and S. Lichtenbaum, A spectral sequence for motivic cohomology, Preprint 1994.

[F] W. Fulton, Intersection theory, Ergebaisse der Mathematik und ihrer Grenzgebeite, Springer-Verlag, 1998.

[M] J.S. Milne, Etale cohomology, Princeton Mathematical Series, Vol. 33, Princeton University Press, 1980.

[SV1] A. Suslin and V. Voevodsky, Singular homology of abstract algebraic varieties, Invent. Math. 123 (1996), 61-95.

[SV2] A. Suslin and V. Voevodsky, Relative cycles and Chow sheaves, This volume.

[V1] V. Voevodsky, Cohomological theory of presheaves with transfers, This volume.

[ZS] O. Zariski and Pierre Samuel, Commutative algebra, Graduate texts in mathematics, Vol. 29, Springer-Verlag, 1975.

St.Petersburg Branch of the Steklov Mathmatical Institute (POMI), Fontanka 27, St.Petersburg, 191011, Russia

Northwestern University, dep. of Math., Evanston Il, 60208, USA