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 CHi(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 Extqfh(z0(X), Z/m), where z0(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 quasipro- jective scheme over an algebraically closed field F of characteristic zero. Assume that i ≥ d = dim X. Then

i 2(d−i)+n # CH (X,n; Z/m)= Hc (X, Z/m(d − i)) where Hc stands for etale cohomology with compact supports. If the scheme X is i 2i−n smooth then this formula simplifies to CH (X,n; Z/m)= Het (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 .

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

Typeset by AMS-TEX 1 2 ANDREI A. SUSLIN

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

(1) Φ|Z×S = ϕ (2) Fibers of the projection Φ−1(V ) → An over points of An−Z have dimension ≤ t

Proof. Notice first that it suffices to treat the special case S = Am. In fact, if this case is already settled then we can proceede as follows. Choose a closed embedding S ֒→ Am. To give an S-morphism ϕ : Z × S → An × S is the same as to give a n ′ m n morphism ψ = pr1 ◦ ϕ : Z × S → A . Extend ψ to a morphism ψ : Z × A → A and denote by ϕ′ the corresponding Am-morphism Z × Am → An × Am. Using the theorem for Am extend ϕ′ to an Am-morphism Φ′ : An × Am → An × Am such that fibers of the projection (Φ′)−1(V ) → An over points of An − Z are of ′ dimension ≤ t and finally set Φ = Φ |An×S. From now on we assume that S = Am, we denote the coordinate fuctions in An m by Y1,...,Yn and by X1,...,Xm we denote the coordinate functions in A . Let m h ∈ F [Y1,...,Yn] be the equation of the divisor Z. The morphism ϕ : Z × A → n m A × A is given by the formula ϕ(y, x) = (ϕ1(y, x),...,ϕn(y, x), x), where ϕi ∈ F [Y,X] are certain polynomials (uniquely defined modh). Define Φ : An × Am → An × Am by means of the formula

Φ(y, x) = (ϕ1(y, x)+ h(y)p1(x),...,ϕn(y, x)+ h(y)pn(x), x) where pi ∈ F [X] are forms of degree N. We’ll show that if N is sufficiently large and pi 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[Y1,...,Yn]. For any h f ∈ R[Y1,...,Yn] we denote by l.f.(f) the leading form of f, we also set f = deg f (Y0) f(Y1/Y0,...,Yn/Y0). Thus l.f.(f) is a form of degree deg f in variables h Y1,...,Yn, while f is a form of degree deg f in variables Y0,...,Yn, moreover h h l.f.(f)= f(0, Y1,...,Yn) and f = f(1, Y1,...,Yn). If A is an ideal in R[Y1,...,Yn] then leading forms of polynomials from A give a homogeneous ideal in R[Y1,...,Yn] k h which we denote l.f.(A). In the same way forms Y0 · f (k ≥ 0,f ∈ A) give a h homogeneous ideal in R[Y0,...,Yn] which we denote A(compare [[]ch. 7,§5]ZS). Note the following evident relations between ideals A, l.f.(A), hA: h (1.2.1). Let g ∈ R[Y0,...,Yn] be a form, then g ∈ A ⇐⇒ g(1, Y1,...,Yn) ∈ A (1.2.2) l.f.(A) coincides with the image of hA under a surjective homomorphism of graded rings s : R[Y0,...,Yn] → R[Y1,...,Yn] sending Y0 to zero.

n n n Identify A = Spec F [Y1,...,Yn] with an open subscheme PY0 of P = n n Proj F [Y0,...,Yn] by means of an open embedding j : A ֒→ P given by = the formula j(y1,...,yn) = [1 : y1 ··· : yn]. In other words j is defined by a trivial n linear bundle OAn over A and (n+1)-tuple of its sections (without common zeros) (1, Y1,...,Yn). HIGHER CHOW GROUPS AND ETALE COHOMOLOGY 3

Lemma 1.3. In the above notations denote the closed subscheme of X × An = = Spec R[Y1,...,Yn] defined by the ideal A by V . Then the closed subscheme of n h Proj R[Y0,...,Yn]= X × P defined by the homogeneous ideal A coincides with n the closure V of V in X × P and the closed subscheme of Proj R[Y1,...,Yn] = = X × Pn−1 defined by the homogeneous ideal l.f.(A) coincides with n− V∞ = V ∩ (X × P 1). Proof. Denote the sheaf of ideals defining V by I and let P be the homogeneous ideal of R[Y0,...,Yn] corresponding to I. Tensoring the exact sequence of sheaves

0 → I → OX×Pn → j∗(OV ) by O(q) and then taking global sections we get exact sequences of R-modules n n 0 → Pq = Γ(X × P , I(q)) → R[Y0,...,Yn]q =Γ(X × P , O(q)) → Γ(V, OV (q)) =

=Γ(V, OV )= R[Y1,...,Yn]/A

The homomorphism R[Y0,...,Yn]q → R[Y1,...,Yn]/A sends the form g to g(1, Y1,...,Yn) mod A. This shows that Pq = {g ∈ R[Y0,...,Yn]q : g(1, Y1,...,Yn) h ∈ A} = ( A)q. To prove the second assertion denote the closed embedding X ×Pn−1 → X ×Pn by i. This embedding corresponds to the epimorphism of graded rings s : R[Y0,...,Yn] → R[Y1,...,Yn] considered in (1.2.2). In these notations we get: ∗ ∗ h ∼ OV∞ = i (OV )= i ((R[Y0 ...,Yn]/ A) )= h ∼ ∼ = (R[Y1,...,Yn] ⊗R[Y0,...,Yn] (R[Y0,...,Yn]/ A)) = (R[Y1,...,Yn]/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 R0 = F and R is generated over F by R1. If a polynomial f ∈ R[Y1,...,Yn] is R-homogenous then l.f.(f) and hf are bihomogenous polynomials. In the same h way if A is an R-homogenous ideal in R[Y1,...,Yn] then the ideals l.f.(A) and 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[Y1, ..., Yn] and let V ⊂ n Proj(R[Y1, ..., Yn]) = X × A be the corresponding closed subscheme. Then the closed subscheme of X × Pn defined by the bihomogenous ideal hA coincides with the closure V of V in X × Pn. Moreover the closed subscheme of X × Pn−1 defined n− by the bihomogenous ideal l.f.(A) coincides with V∞ = V ∩ (X × P 1)

(1.5). Consider the bigraded ring F [Y,X]= F [Y1,...,Yn,X1, ..., Xm]. For any f ∈ F [Y,X] denote by degY f,degX f, l.fY (f), l.fX (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 biho- mogenous 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 = (degY f,degX (l.fY 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 4 ANDREI A. SUSLIN

Lemma 1.5.4. Denote by V the closed subscheme of An × Am defined by the ideal n m n− m A, let V denote the closure of V in P ×A and set V∞ = V ∩(P 1 ×A ). Next n−1 m n−1 m−1 denote V∞ the closure of V∞ in P × P and set V∞,∞ = V∞ ∩ (P × P ). n−1 m−1 Then V∞,∞ coincides with the closed subscheme of P × P defined by the bihomogenous ideal l.f.(A). Corollary 1.5.5. Dimension of the closed subscheme of Pn−1 × Pm−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 ⊂ An × Am ∼ and choose f1, ..., fs ∈ A such that the bihomogenous forms f j = l.f.(fj ) generate the ideal l.f.(A). Assume in the sequel that N > max(degX ϕi,degX fj ). Then one checks easily that for any choice of N-forms pi(X) we have: ∼ ∼ (1) degX fj(ϕi + hp1, ..., ϕn + hpn,X) ≤ NdegY f j + degX f j ∼ ∼ (2) The X-homogenous summand of fj (ϕ+hp,X) of degree NdegY f j +degX f j ∼ degY fj is equal to h · f j (p1, ..., pn,X) Thus denoting by P the ideal defining the closed subschemeΦ−1(V ) and denot- ing by Ph ⊂ F [Y ]h[X] the corresponding localization we come to the following conclusion Lemma 1.6.1. Define a morphism Φ: An × Am → An × Am by the formula Φ(y, x) = (ϕ1(y, x)+ h(y)p1(x), ..., ϕn(y, x)+ h(y)pn(x), x), where pi are forms of degree N > max(degX ϕi,degX fj). Then the ideal l.f.X (Ph) contains the forms ∼ f j (p1, ..., pn,X)

m Consider further the closed subscheme pr2(V ) ⊂ A . It’s defined by the ideal Q = A ∩ F [X]. It’s clear that for any Am-morphism Φ the ideal P contains Q and m−1 hence l.f.X (Ph) ⊃ l.f.X (Q). Denote by W the closed subscheme of P defined by the homogenous ideal l.f.X (Q). Since dim pr2(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 ⊂ Pn−1 × Pm−1 of their common zeros is of dimension ≤ n + t − 2. Assume further that we are given a closed subscheme W ⊂ Pm−1 of dimension ≤ n + t − 1. Then for any N ≥ 0 we can find forms p1, ..., pn ∈ F [X] of degree N such that the dimension of the closed subscheme of W given by the equations ∼ f j (p1, ..., pn,X1, ..., Xm)=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 AM (M = N+m−1 m−1
). In the same way n-tuples of N-forms are in one to one correspon- Mn dence with rational points of the affine space A . Set T1 = {(w,p) ∈ W × Mn A : p1(w) = ··· = pn(w) = 0}. It’s clear that the set T1 is closed and dim T1 = dim W + n(M − 1) ≤ nM + t − 1. Consider now the following morphism Mn n−1 g : W × A − T1 → W × P : (w,p) 7→ (w, [p1(w): ... : pn(w)]) and set −1 n−1 Mn T2 = g (C ∩ (W × P )). The set T2 is closed in W × A − T1 and hence HIGHER CHOW GROUPS AND ETALE COHOMOLOGY 5

Mn T = T1 ∪ T2 is closed in W × A . One checks easily that fibers of g are of dimen- sion Mn − (n − 1) and hence dim T2 ≤ dim C + Mn − (n − 1) ≤ nM + t − 1. This shows that dim T ≤ nM + t − 1. Finally consider the projection q : T → AMn. According to the theorem on the dimension of fibers of a morphism there exists a nonempty open set U ⊂ AMn such that for any p ∈ U dim q−1(p) ≤ t − 1. Now we can take p = (p1, ..., pn) 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 n−1 m−1 the choice of fj, the closed subscheme of P × P 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 p1, ..., pn of degree N > max(degX ϕi,degX fj ) so that the ∼ ∼ dimension of W0 = 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 ) n over any point y ∈ A − Z is contained in W0 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. n n+1 Denote by ∆ the linear subvariety of A given by the equation t0+...+tn = 1. The points pi = (0, ..., 1, ..., 0) (0 ≤ i ≤ n) are called the vertices of the “simplex” i ∆n. They have the following evident property : if X ⊂ Am is any linear subvariety (i.e. X is defined by a sistem of linear equations on coordinates) and if x0, ..., xn is any (n+1)-tuple of rational points of X then there exists a unique linear morphism n ∆ −→ X, taking pi to xi. This shows in particular that any nondecreasing map φ : {0, ..., n}→{0, ..., m} defines a canonical morphism ∆n → ∆m (taking · pi to pφ(i)), which we’ll denote by the same letter φ. In this way ∆ becomes a cosimplicial scheme. The morphisms ∆n → ∆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 ∆m is clearly a linear subvariety and will be called a face of ∆m (of codimension m − n). As always there m−1 m exists m + 1 cofaces of codimension one, they are denoted δi : ∆ → ∆ (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

i X × S −−−−→ X × S × T

 pr1,3 f  y y 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 codimX×S(W ) ≤ codimX×S×T (V × S)= codimX×T (V ). 6 ANDREI A. SUSLIN

When all components of f −1(V ) have correct codimension we can use the T or- ∗ formula to define the cycle f ([V ]) (since the morphism pr1,2 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 zi(X,n) the free abelian group generated by codimension i subvarieties V ⊂ X × ∆n, which intersect properly X × ∆m for any face ∆m ֒→ ∆n. It’s clear from the above definition that if [V ] is a generator of zi(X,n) then cycles ∗ i ∂j ([V ]) = (1X × δj ) ([V ]) are defined and lie in z (X,n − 1). More generally, if φ : ∆m → ∆n is any structure morphism of the cosimplicial scheme ∆∗ then the ∗ i i cycle (1X × φ) ([V ]) is defined and lies in z (X,m). Thus z (X, −) is a simplicial abelian group. The homotopy groups of this simplicial abelian group are denoted CHi(X,n) - see[B1]. In other words CHi(X,n) is the n-th homology group of the complex d d zi(X, 0) ← zi(X, 1) ← ... where, as always, d is the alternating sum of face operators. A morphism of varieties f : X → Y is called equidimensional of relative dimen- sion 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 × ∆n and assume that the projection V → ∆n is an equidimensional morphism of relative dimension t. This m n implies that codimX×∆n (V )= i. Moreover, if φ : ∆ → ∆ is any morphism and −1 if W is any component of (1X × φ) (V ) then codimX×∆m (W ) ≤ codimX×∆n (V ), i.e. dimW ≥ m + t. On the other hand each fiber of the projection W → ∆m is contained in the corresponding fiber of V → ∆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. codimX×∆m (W ) = codimX×∆n (V )) and moreover the projection W → ∆m is equidimensional of relative dimension t. Thus, denoting by i n zeq(X,n) the free abelian group generated by closed subvarieties V ⊂ X × ∆ for which the projection V → ∆n is equidimensional of relative dimension t, we see i i that zequi(X, −) is a simplicial abelian subgroup in z (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 = i i -dimX. Then the embedding of complexes zequi(X, −) ֒→ z (X, −) is a quasiisomor phism. 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 n n are given an X-morphism ϕn : X ×∆ → X ×∆ such that the following diagrams commute for all 0 ≤ j ≤ n ≤ N

ϕ −1 X × ∆n−1 −−−−→n X × ∆n−1

(2.2.1) ×   × 1X δj  1X δj y ϕ y X × ∆n −−−−→n X × ∆n i Define ϕz (X,n) tro be a free abelian group generated by close subvarieties V ⊂ X × ∆n such that HIGHER CHOW GROUPS AND ETALE COHOMOLOGY 7

a) [V ] ∈ zi(X,n) ∗ i b) The cycle (ϕn) ([V ]) is defined and lies in z (X,n) i The commutativity of (2.2.1) shows that for any generator [V ] of ϕz (X,n) and ∗ i any j (0 ≤ j ≤ n) the cycle ∂j ([V ]) = (1X × δj ) ([V ]) lies in ϕz (X,n − 1), so that i i setting ϕz (X,n) = 0 for n>N we get a subcomplex ϕz (X, −) of the complex zi(X, −). Moreover the commutativity of (2.2.1) implies that the assignment [V ] 7→ ∗ ∗ i i ϕn([V ]) defines a homomorphism ϕ : ϕz (X, −) → z (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.

ϕ∗ i i Proposition 2.3. The homomorphisms ϕz (X, −) ⇒ z (X, −) are weakly homo- inc topic. Proof. Assume that for each n (0 ≤ n ≤ N) we have fixed a finite number of closed n n n i n−1 subvarieties Vk ⊂ X × ∆ such that [Vk ] ∈ ϕz (X,n) and the family {V∗ } con- n tains all components of all cycles ∂j ([Vk ]). Denote by Cn the free abelian group n i generated by [Vk ]. It’s clear that C∗ is a finitely generated subcomplex in ϕz (X, −) and that subcomplexes of this type are cofinal among all finitely generated subcom- i ∗ plexes of ϕz (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 n 1 n 1 Φn : X × ∆ × A → X × ∆ × A with the following properties (2.3.1). The following diagram commutes (here i0 and i1 denote closed embed- dings defined by points 0, 1 ∈ A1

X × ∆n −−−−→id X × ∆n (2.3.1.1)   i0 i0 y y Φ X × ∆n × A1 −−−−→n X × ∆n × A1

ϕ X × ∆n −−−−→n X × ∆n

(2.3.1.2) i1 i1   y y Φ X × ∆n × A1 −−−−→n X × ∆n × A1

Φ −1 X × ∆n−1 × A1 −−−−→n X × ∆n−1 × A1

(2.3.1.2) × × 1   × × 1 1X δj 1A  1X δj 1A y y Φ X × ∆n × A1 −−−−→n X × ∆n × A1

−1 n 1 n 1 (2.3.2) The fibers of the projection Φn (Sk Vk × A ) → ∆ × A over points n n n n−1 1 n−1 not lying on the divisor Z = (∆ × 0) + (∆ × 1) + Pj=0 ∆j × A (here ∆j is the j-th face of ∆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 8 ANDREI A. SUSLIN

define the morphism Φn on X × Z. The existence of extension of this morphism to X × ∆n × A1 satisfying the additional requirement (2.3.2) is garanteed by the theorem 1.1.

n 1 The vertices of ∆ ×A are the points pi ×0 and pi ×1. We introduce the partial ordering on these vertices, taking the standart order on pi’s (pi ≤ pj ⇔ i ≤ j) and taking the standart order on the set {0, 1}. For any nondecreasing system of m n 1 vertices q0 ≤ q1 ≤ ···≤ qm denote by θq the linear morphism ∆ → ∆ × A , taking pi to qi.

Lemma 2.4. Let q0 < q1 < ··· < qm be a strictly increasing sequence of vertices n 1 −1 n 1 of ∆ × A (thus m ≤ n +1). Then dim[Φn ◦ (1X × θq)] (Sk Vk × A ) ≤ m + t. Proof. We proceed by induction on n. Consider first the case n = m − 1. In this case θq is an isomorphism and our statement reduces to the following formula: −1 n 1 dim Φn (Sk Vk × A ) ≤ n +1+ t. Using the induction hypothes, the properties n of the varieties Vk and (2.3.1) we prove immediately the following statement −1 n 1 (2.4.1) The dimension of the part of Φn (S Vk × A lying over Z does not exceed n + t.

−1 n 1 n 1 The fibers of the projection Φn (S Vk × A ) → ∆ × A over the points not lying in Z are of dimension ≤ t. This implies that the dimension of the part of −1 n 1 n 1 Φn (S Vk × A ) lying over ∆ × A − Z is not more than n +1+ t. Finally −1 n 1 dim Φn (S Vk × A ) ≤ max(n + t,n +1+ t)= n +1+ t. Consider next the case n = m. Note that θq is always a closed embedding. ′ Denote by Z the inverse image of Z under θq. Using (2.4.1) we conclude im- −1 n 1 mediately that the dimension of the part of [Φn ◦ (1X × θq)] (S Vk × A ) = −1 −1 n 1 ′ (1X ◦ θq) Φn (Sk Vk × A ) lying over Z does not exceed n + t. The fibers of −1 −1 n 1 n n ′ the projection (1X × θq) Φn (S Vk × A ) → ∆ over the points of ∆ − Z are of dimension not more than t and hence the dimension of the part of [Φn ◦ −1 n 1 n ′ (1X × θq)] (S Vk × A ) which lies over ∆ − Z does not exceed n + t. Finally −1 n 1 dim([Φn ◦ (1X × θq)] (S Vk × A )) ≤ max(n + t,n + t)= n + t. Finally suppose that m

i We also need a version of (2.3) for the complex zequi(X, −). In the above situation i n denote by ϕzequi(X,n) the free abelian group generated by subvarieties V ⊂ X ×∆ such that i (1) [V ] ∈ zequi(X,n) ∗ i (2) The cycle ϕn([V ]) is defined and lies in zequi(X,n) HIGHER CHOW GROUPS AND ETALE COHOMOLOGY 9

i i In the same way as above we see that ϕzequi(X, −) is a subcomplex in zequi(X, −) ∗ i i and we have a canonical homomorphism ϕ : ϕzequi(X, −) → zequi(X, −).

ϕ∗ i i Proposition 2.6. The homomorphisms ϕzequi(X, −) ⇒ zequi(X, −) are weekly inc 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).

−1 n 1 n 1 Lemma 2.7. All fibers of the projection Φn (Sk Vk × A ) → ∆ × A are of dimension ≤ t.

∗ n 1 This lemma shows that the cycles [Φn ◦ (1X × θj )] ([Vk × A ]) are defined and i lie in zequi(X,n + 1) so that we can define the homotopy by the same formula as above.

i Proposition 2.8. Let C∗ be a finitely generated subcomplex in z (X, −). Then we i ∗ i can find N and ϕ as above such that C∗ ⊂ ϕz (X, −) and ϕ (C∗) ⊂ zequi(X, −). n Proof. We may suppose that C∗ is generated by a system of subvarieties Vk ⊂ X × ∆n (0 ≤ n ≤ N) with the corresponding properties (see the proof of (2.3)). n n Using induction on n we construct X-morphisms ϕn : X × ∆ → X × ∆ with the following properties: (1) The diagrames (2.2.1) commute −1 n n (2) The fibers of the projection ϕn (Sk Vk ) → ∆ over the points not lying on n n−1 the divisor Z = Pj=0 ∆j 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 −1 n n ∗ n ϕn (S Vk ) → ∆ are of dimension ≤ t. Thus the cycles ϕn([Vk ]) are defined and i lie in zequi(X, −). (2.9) Proof of the theorem 2.1. Let v be any n-dimensional cycle of the complex zi(X, −). Using proposition 2.8 i ∗ i find ϕ such that v ∈ ϕz (X,n) and ϕ (v) ∈ zequi(X,n). According to proposition ∗ i 2.3 v−ϕ (v) is a boundary. This shows that the homomorphism Hn(zequi(X, −)) → i Hn(z (X, −)) is surjective. i Assume now that v is an n-dimensional cycle in zequi(X, −) such that v = d(w) for a certain w ∈ zi(X,n + 1). Using proposition 2.8 find ϕ such that v, w ∈ i ∗ ∗ i ∗ ∗ ϕz (X, −) and ϕ (v), ϕ (w) ∈ zequi(X, −). Since ϕ (v) = d(ϕ (w)) we see that ∗ i ∗ ϕ (v) is a boundary in zequi(X, −). On the other hand v −ϕ (v) is also a boundary i i in zequi(X, −) according to (2.6). Thus v is a boundary in zequi(X, −), i.e. the i i homomorphism Hn(zequi(X, −)) → Hn(z (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 zt(X) which is characterized by the following property: 10 ANDREIA.SUSLIN

for any normal connected scheme S ∈ Sch/F the group of sections of zt(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 singu- n n lar complex - see [SV1], i.e. Cn(F) = F(∆ ) and the differential d : C (F) → sing Cn−1(F) is the alternating sum of face operators. We’ll use the notation H∗ (F), sing ∗ H∗ (F, Z/n),Hsing (F, Z/n) for the homology groups of the complexes C∗(F), L C∗(F) ⊗ 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 sing H∗ (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 quasipro- jective scheme X ∈ Sch/F and any i ≤ d = dim X the canonical embedding of complexes

i i (− ,C∗(zd−i(X)) = zequi(X, −) ֒→ z (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 −→ zd−i(Y ) −→ zd−i(X) −→ zd−i(U) is exact and the h-sheaf associated with the presheaf zd−i(U)/zd−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 sing using the induction hypothesis and theorem 2.1 we conclude that Hn (zd−i(X) → CHi(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. i def i Theorem 3.2 shows that for any i ≤ d the group CH (X,n; Z/m) = Hn(z (X, −)⊗ sing Z/m) coincides with Hn (zd−i(X), Z/m). For any periodical abelian group A we denote by A# its dual, i.e. A# = Hom(A, Q/Z). Thus CHi(X,n; Z/m)# = sing # n = Hn (zd−i(X), Z/m) = Hsing(zd−i(X), Z/m) and the last group according n to the main result of [SV1] is canonically isomorphic to Extqfh(zd−i(X), Z/m). Consider now special case i = d. In this case we may use the following result. HIGHER CHOW GROUPS AND ETALE COHOMOLOGY 11

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

n n ∼ n Extqfh(z0(X), Z/m)= Exth(z0(X)h , Z/m)= Hc (X, Z/m) n where Hc 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 ֒→ X, where X is a complete separated scheme, set Y = X − X and let i denote the closed embedding Y ֒→ X. According to [SV2] we have an exact sequence of h-sheaves

∼ ∼ ∼ 0 → z0(Y )h → z0(X)h → z0(X)h → 0.

For any complete separated scheme Z the sheaf z0(Z) coincides with the free qfh- 1 sheaf Z[ p ]qfh(Z) generated by Z (see [SV1, theorem 6.3]). Taking associated h- ∼ 1 sheaves we see that z0(Z)h = Z[ p ]h(Z). Thus the above sequence takes the form 1 1 ∼ 0 → Z[ p ]h(Y ) → Z[ p ]h(X) → z0(X)h → 0 This exact sequence shows that for any 1 h-sheaf of Z[ p ]-modules F we have the formula

∼ Hom(z0(X)h , F)= Ker(Γ(X, F) → Γ(Y, F))

∗ 1 Choose an injective resolution I of the h-sheaf Z/m, consisting of Z[ p ]-modules. n ∼ n ∗ The previous remarks show that Exth(z0(X)h , Z/m) = H (Ker(Γ(X, I ) → → Γ(Y, I∗))). Moreover the homomorphism of complexes Γ(X, I∗) → Γ(Y, I∗) is surjective. Consider the natural morphism of sites γ : (Sch/F )h → (Sch/F )et. i The comparison theorem for h-cohomology [SV1, §10] shows that R γ∗(Z/m)=0 for i > 0. This means that I∗ is an injective resolution of Z/m not only in h- topology, 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 FZ on small etale site Zet for each Z ∈ (Sch/F ) and to give homomorphisms FZ → f∗(FT ) defined for each morphism f : T → Z and satisfying evident compatibility properties. It’s clear that functors Shv(Sch/F )et → Shv(Zet) (F 7→ FZ ) are exact and preserve injectivity. Consider the bicomplex of sheaves on Xet

∗ ∗ ∗ IX → i (IY )

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

n n n ∗ ∗ n ∗ ∗ Hc (X, Z/m)= H (X, j!(Z/m)) = H (Γ(X, (IX → i (IY ))) = H ((Γ(X, I ) → ∗ n ∗ ∗ n ∼ → Γ(Y, I ))) = H (Ker(Γ(X, I ) → Γ(Y, I ))) = Exth(z0(X)h , Z/m).

Theorem 4.2. Let X be an equidimensional quasiprojective scheme over an alge- braically closed field F of characteristic zero. Assume that i ≥ d = dim X. Then i 2(d−i)+n # for any m> 0 CH (X,n; Z/m)= Hc (X, Z/m(d − i)) . Proof. Assume first that i = d. In this case according to proposition 4.2 we have: 12 ANDREIA.SUSLIN

d # n n CH (X,n; Z/m) = Extqfh(z0(X), Z/m)= Hc (X, Z/m) This group being finite we conclude that CHd(X,n; Z/m) is also finite and hence d d # # n # CH (X,n; Z/m) = [CH (X,n; Z/m) ] = Hc (X, Z/m) . The general case follows now from the homotopy invariance of higher Chow groups - see[B1,§2]): i i i−d n i−d # CH (X,n; Z/m) = CH (X × A ,n; Z/m) = Hc (X × A , Z/m) = n−2(i−d) # = Hc (X, Z/m(d − i)) . Corollary 4.3. In assumptions and notations of theorem 4.2 assume further that i 2i−n X is smooth. Then CH (X,n; Z/m)= Het (X, Z/m(i)). Proof. This follows immediately from theorem 4.2 and the Poincar´eduality 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