LI TING

(Mathematical College,Sichuan University,Chengdu,610064)

TheˊEtale Homology and the Cycle Maps in Adic Coefficients*

LI TING

(Mathematical College,Sichuan University,Chengdu,610064)

Communicated by Du Xian-kun

In this article,we def i ne the ?-adic homology for a morphism of schemes satisfying certain f i niteness conditions.This homology has these functors similar to the Chow groups:proper push-forward,f l at pull-back,base change,cap-product, etc.In particular,on singular varieties,this kind of ?-adic homology behaves much better than the classical ?-adic cohomology.As an application,we give a much easier approach to construct the cycle maps for arbitrary algebraic schemes over f i elds.And we prove that these cycle maps kill the algebraic equivalences and commute with the Chern action of locally free sheaves.

?-adic cohomology,cycle map,derived category

1 Introduction

Theˊetale cohomology,especially the ?-adic cohomology,is one of the most important tools of modern algebraic and arithmetic geometry,which allows us to construct a“good”cohomology theory for varieties over f i elds of arbitrary characteristic.More specif i cally,people use the ?-adic cohomology H*(Xˊet,Z?)to substitute for singular cohomology on varieties of arbitrary characteristic.On a nonsingular varieties,the cohomology H*(Xˊet,Z?)has very good properties and produces rich results.But on singular varieties or more generally on arbitrary schemes,the cohomology H*(Xˊet,Z?)behaves not so good,and many important constructions and results are not valid.So on singular varieties,theˊetale homology is more suitable than theˊetale cohomology.

In this paper,we generalize theˊetale homology def i ned in[1]in the following three facets.First,we def i ne theˊetale homology in adic coefficients,which we call the ?-adic homology.Secondly,our theory of ?-adic homology is def i ned over schemes separated and offi nite type over base schemes satisfying certain fi niteness conditions,not just the algebraic schemes over separably closed fi elds as in[1].In particular,algebraic schemes over fi elds which are not necessarily separably closed,are considered by us.Since our theory is based on the adic formalism created by Ekedahl[2],the ?-adic homology over base schemes of certain fi niteness conditions shares almost the same good functorial properties,with that over separably closed base fi elds.Thirdly,the ?-adic homology groups H*(X,F)de fi ned by us take value in arbitrary bounded complex F,not just Z?,Q?or Z/nZ as in[1].And almost all functors and properties are preserved when extending to complexes.Basing on this homology,we also extend the cycle maps de fi ned in[1],from separably closed base fi elds,to arbitrary base fi elds of fi nite ?-adic cohomological dimension.

In Section 2,we brie fl y reiterate the category Dc(Xˊet,R·)together with the Grothendieck's six operations in[2].

In Section 3,we recite the properties of the functor Rf!and use the language of[2]to rewrite the trace morphisms introduced in[3,4].

In Section 4,we de fi ne the ?-adic homology groups Hn(X/Y,N)and Hn(X/Y,N)for a morphism X→Y of schemes satisfying certain fi niteness conditions.These homology groups behave similarly in many facets to the bivariant Chow groups A-n(X→Y)de fi ned in Ch. 17 of[5].We de fi ne two maps:the push-forward maps f*and the pull-back maps f*,for the?-adic homology groups,which correspond to these maps on Chow groups CH*(X)de fi ned in§1.4 and§1.7 of[5];and most important,we prove that these two maps commute(see Theorem 4.1),which is essential to construct various cycle maps basing on ?-adic homology. Moreover,we de fi ne the base change maps on the ?-adic homology.

In Section 5,we apply the ?-adic homology in Section 4 to de fi ne the cycle map

for arbitrary algebraic scheme X over a fi eld of fi nite cohomological dimension at ?.We prove that the cycle map clX,?commutes with the push-forward map f*and the pull-back map f*.And we prove that the cycle maps kill the algebraic equivalence of algebraic cycles.

In Section 6,we prove that the cycle map clX,?commutes with the Chow action ci(E)∩· by locally free sheaves.

The following notations and conventions would be used.

Let N be the set of natural numbers,Z the domain of integers,and Q the fi eld of the rational numbers.Let Z?and Q?be the ?-adic completions of Z and Q respectively.

A morphism f:X→Y of schemes is said to fl at(resp.smooth)of relative dimension n if f is fl at(resp.smooth)and all fi bers of f are n-equidimensional.

A morphism f:X→Y of Noetherian schemes is said to be compacti fi able if it factors as f=?j where j:X→is an open immersion,and:→Y is a proper morphism. By Theorem 4.1 of[6],f is compacti fi able if and only if it is separated and of fi nite type.

An algebraic scheme over a fi eld k is a scheme separated,of fi nite type over k.A variety over k is an integral algebraic scheme over k.

If A is a Noetherian ring,we use D(A)to denote the derived category of A-modules,and de fi ne Dfg(A)to be the full subcategory of D(A)consisting of complexes cohomologicallyfi nitely generated.

If X is a scheme and R is a ring,we use RXto denote the constant sheaf on Xˊetassociated to R.

If F·is a complex of sheaves on Xˊet,we let F·[r]be the shift of F·to left by r,and F·(r):=F·?μ?nrthe Tate twist,and we write F·〈r〉:=F·(r)[2r].

Let A and B be categories.If F:A→B and G:B→A is a pair of adjoint functors, then we write F?G for the adjunction.We also say that F is left adjoint to G,and G is right adjoint to F.

The notation:=means being de fi ned as;means isomorphism;and the notation?in commutative diagrams means Cartesian square.

2 The ?-adic Sheaves

In this section,we brie fl y reiterate the theory of Ekedahl[2]about the category Dc(Xˊet,R·) together with the Grothendieck's six operations(see[7,8]).

Fix a prime number ?,and let R be the integral closure of Z?in a fi nite extension fi eld of Q?.

Let X be a Noetherian scheme.We denote by S(,R·)the abelian category of inverse systems

such that each Fnis a sheaf of Rn-modules on Xˊet.Set

and let Dc(,R·)be the full subcategory of D(,R·)consisting of complexes cohomologically AR-adic and constructible.Let Dc(,R·)be the quotient of Dc(,R·)by inverting AR-quasi-isomorphisms.

If f:X→Y is a morphism of Noetherian schemes,then we have a triangulated functor

As to other f i ve operations,we must add some restrictions on the underlying schemes. We consider the following condition(?)related to a scheme X:

(?)X is Noetherian,quasi-excellent,of f i nite Krull dimension;? is invertible on X and cd?(X)＜∞.

From the Gabber's f i nitenes theorem forˊetale cohomology in[9],we know the following facts:

(1)If X satisf i es(?),then any scheme of f i nite type over X satisf i es(?).

(2)Let R be a quasi-excellent,Henselian local ring with residue f i eld k such that cd?(k)＜∞.Then SpecR satisf i es(?).

(3)If ?/=2,then the affine scheme SpecZ[1/?]satisf i es(?)(see X,6.1 of[3]).

(4)If f:X→Y is a compactif i able morphism of schemes satisfying(?),then both Rf*and Rf!are of f i nite cohomological amplitude.

In particular,if X is a scheme satisfying(?),then Xˊetsatisf i es the condition A)in[2], and thus we have two bi-triangulated functors

And if f:X → Y is a compactif i able morphism of schemes satisfying(?),then there are triangulated functors

For each scheme X satisfying(?),each object F in Dc(Xˊet,R·),and each n∈N,we def i ne

Note that this de fi nition is compatible with the continuousˊetale cohomology(Xˊet,F) de fi ned in[10].

When we consider the schemes of fi nite type over a separably closed fi eld,the following Theorem is essential.

Theorem 2.1 The right derived functors of(Mn)→and the left derived functors of M →(M?RRn)de fi ne a natural equivalence of categories between Dc(R·)and Dfg(R).

Proof. See Proposition 2.2.8 of[7].

Now we fi x a separably closed fi eld k.Note that

Let X be an algebraic scheme over k,and p:X→Speck the structural morphism.Put

Then for each q∈Z,we have

And we def i ne

Theorem 2.2(The Ku¨nneth Formula) Let X and Y be two algebraic schemes over k, Z:=X×kY,f:Z→X and g:Z→Y the projections.Then for each F ∈(Xˊet,R·) and G∈(Yˊet,R·),there are two natural isomorphisms in(R):

Moreover,there are two exact sequences of R-modules

For convenience to study the cycle map,we introduce the following notation.Let X be a scheme satisfying(?).For F ∈Dc(Xˊet,R·)and n∈Z,we write

3 The Functor Rf!and the Trace Morphisms from SGA 4&

Proposition 3.1 Let f:X→Y be a compacti fi able morphism of schemes satisfying(?) such that all fi bers of f are of dimensions≤d.Then for each a∈Z,Rf!sendsto

Proof. See XVIII,3.1.7 of[3].

Lemma 3.1 Let f:X→Y be a compacti fi able morphism of schemes satisfying(?).Then for every pair of objects F and G in(Yˊet,R·),there is a natural morphism

Proof.First we have a composite morphism

where φ is induced by the projection formula for Rf!,and ψ is induced by the adjunction.

Since Rf!is left adjoint to Rf!,the above morphism induces the required morphism

Proposition 3.2 Let f:X → Y and g:Y → Z be two compacti fi able morphisms of schemes satisfying(?).For every pair of objects F and G in(Zˊet,R·),there is a natural morphism

in Dc(Xˊet,R·)which is functorial in F and G.

Proof.We have

Proposition 3.3 Let

be a Cartesian square of schemes satisfying(?).Assume that f is compacti fi able.

(1)For each object F in Dc(Xˊet,R·),there is a natural morphism in Dc(,R·)

(2)For each object G in Dc(Yˊet,R·),there is a natural morphism in Dc(,R·)

(3)Assume that Y is an algebraic scheme over a fi eld k,and there exists a k-scheme T such that Y′=Y×kT.Then the morphisms in(1)and(2)are both isomorphisms.

(4)For each object F in Dc(Xˊet,R·),there is a natural isomorphism in Dc(,R·)

(5)For each object G in Dc(,R·),there is a natural isomorphism in Dc(Xˊet,R·)

Proof. (1)is induced by the classical base change morphisms.

(2)is from[3],XVIII,3.1.14.2.

(3)is by Th.Finitude,1.9 of[4].

(4)is by XVII,5.2.6 of[3].

(5)is by XVIII,3.1.12.3 of[3].

Now we review the trace morphisms.

De fi nition 3.1 A morphism f:X →Y of schemes is said to be fl at at dimension d if there exists a nonempty open subset U of X satisfying the following conditions:

(1)f:U→Y is fl at;

(2)for each point y∈Y,Uyis either empty or d-dimensional;

(3)every fi ber of XU→Y is of dimension＜d.

By XVIII,2.9 of[3],for every compacti fi able morphism f:X→Y of schemes satisfying (?)which is fl at at dimension d,and for every object G in Dc(Yˊet,R·),we have a trace morphism:

Since Rf!is right adjoint to Rf!,the morphism Trfinduces a canonical morphism in Dc(Xˊet,R·):

Moreover,we have a commutative diagram

By XVIII,3.2.5 of[3],we have

Proposition 3.4 Let f:X → Y be a compactif i able smooth morphism of relative dimension d of schemes satisfying(?).Then for any object G in Dc(Yˊet,R·),the canonical morphism

is an isomorphism in Dc(Xˊet,R·).

The following Propositions 3.5–3.7 are deduced from XVIII,2.9 of[3].

Proposition 3.5 Let

be a Cartesian square of schemes satisfying(?).Assume that f is compactif i able and f l at at dimension d.Then f′is also f l at at dimension d,and for each object G in Dc(Yˊet,R·)we have

(1)the composite morphism

is equal to Trf′,where the isomorphism φ is def i ned in Proposition 3.3(4);

(2)the composite morphism

is equal to tf′,where the last morphism is def i ned in Proposition 3.3(2).

Proposition 3.6 Let f:X → Y and g:Y → Z be two compactif i able morphisms of schemes satisfying(?)which are f l at at dimension d and e respectively,and H be an object in Dc(Zˊet,R·).Then we have

(1)the composite morphism

is equal to Trg?f;

(2)the composite morphism

is equal to tg?f.

Proposition 3.7 Let f:X→Y be a f i nite morphism of schemes satisfying(?)such that f*Xis a locally freeY-module of degree d.Then for each object F in Dc(Yˊet,R·),the composite morphism

is equal to the multiplication by n.

The following proposition shows that the trace morphism is essentially determinated by the generic points.Let A be a Noetherian ring(in particular,A=Rn).

Proposition 3.8 Let X be an n-dimensional algebraic scheme over k,X1,X2,···,Xrbe all irreducible components of dimension n of X,and F be an A-module.For each i,let Yi/=? be an open subset of X containedand regard Yias a reduced subscheme of X.For each i,let xibe the generic point of Xiand put ai:=length(X,xi).Then there is a canonical isomorphism ω of A-modules which makes a commutative diagram

4 ?-adic Homology for Morphisms of Algebraic Schemes

Let f:X→Y be a compactif i able morphism of schemes satisfying(?).For each object N in Dc(Yˊet,R·)and for each n∈Z,we def i ne the n-th ?-adic homology associated to f to be

which is an R-module.

For convenience to def i ne pull-backs along f l at morphisms and cycle maps,we also def i ne

We set

We also use Hn(X/Y,N)(resp.Hn(X/Y,N))to denote(resp.if no confusion arises.

If X is an algebraic schemes over a separably closed f i eld k and N is an object in Dfg(R), we write

By Proposition 3.4 we have

Lemma 4.1 Let f:X→Y be a compacti fi able morphism of schemes satisfying(?)which is fl at at dimension d.Then for each object N in Dc(Yˊet,R·)and for n∈Z,the morphism tfinduces a canonical homomorphism of R-modules:

Moreover,if f is smooth of relative dimension d,then the above homomorphism is an isomorphism.

Proposition 4.1 Let f:X→S be a compacti fi able morphism of schemes satisfying(?), Y be a closed subscheme of X and U:=XY.Then we have a long exact sequence

Proof.Put M:=Rf!N.Then the proposition follows from the distinguished triangle

where i:Y→X and j:U→X are the inclusions.

Proposition 4.2(Mayer-Vietoris Sequence) Let f:X→S be a compacti fi able morphism of schemes satisfying(?),X1and X2be two closed subschemes of X such that X=X1∪X2(as sets).Then we have a long exact sequence

Proof.Put M:=Rf!N.Then the proposition follows from the distinguished triangle

where i:X1×XX2→X,i1:X1→X,i2:X2→X are the inclusions.

Proposition 4.3(Vanishing) Let f:X → Y be a compacti fi able morphism of schemes satisfying(?)and such that all fi bers of f are of dimensions≤d,and N be an object inThen Hn(X/Y,N)=0 whenever n＞2d-a.

Proposition 4.4 Let f:X→S be a compacti fi able morphism of schemes satisfying(?), Y be a closed subscheme of X such that dimYs≤d for all s∈S,and X′:=XY,N be an object in(Sˊet,R·).Then for each integer n＞2d+1-a,there is a canonical isomorphism of R-modules

Proof. Apply Propositions 4.1 and 4.3.

Let f:X→Y be a compacti fi able morphism of schemes satisfying(?).For each object G in Dc(Yˊet,R·),we de fi ne

to be the canonical morphisms induced by the adjunctions f*?Rf*and Rf!?Rf!respectively.

The following map is a kind of variant of the Gysin homomorphism.

De fi nition 4.1(Push-forward) Let p:X→S and q:Y→S be two compacti fi able morphisms of schemes satisfying(?),and f:X→Y a proper S-morphism.For every object N in Dc(Sˊet,R·)and for every n∈Z,we de fi ne a homomorphism of R-modules

as follows.For each α∈Hn(X/S,N),f*(α)is de fi ned to be the composition

Proof.This comes from the following simple lemma.

Lemma 4.2 Let f:X→Y and g:Y→Z be two compacti fi able morphisms of schemes satisfying(?),and H be an object in Dc(Zˊet,R·).Then we have

(1)the following composition is equal to δg?f:

(2)the following composition is equal to θg?f:

De fi nition 4.2(Pull-back) Let p:X→S and q:Y→S be two compacti fi able morphisms of schemes satisfying(?),and f:X→Y an S-morphism which is fl at at dimension d.For every object N in Dc(Sˊet,R·)and for every n∈Z,we de fi ne a homomorphism of R-modules

as follows.For each β∈Hn(Y/S,N),f*(β)is de fi ned to be the composition

Proof. This follows from Proposition 3.6(2).

Theorem 4.1 Let S be a scheme satisfying(?),and r:Y→ S a compacti fi able morphism.Let

be a Cartesian square of schemes such that f is proper and q is compacti fi able and fl at at dimension d,N be an object in Dc(Sˊet,R·)and n∈Z.Then we have

Proof.Put M:=Rr!N.Let α∈Hn(X/S,N).Then q*?f*(α)is equal to the composition

After applying Proposition 3.5(2)to tp,we obtain that the morphism?p*(α)is equal to the composition

Consider the following diagram

where?means commutative square.The commutativity of(a)and(b)are by the following simple Lemma 4.3.So the whole diagram is commutative.Note that the composition along the direction

*in the above diagram is equal to q?f*(α),and the composition alongis equal to?p*(α).Thus

Lemma 4.3 Let

be a Cartesian square of schemes satisfying(?)with all morphisms compacti fi able.Then we have

(1)For each object G in Dc(Yˊet,R·),the diagram

is commutative in Dc(,R·),where φ is de fi ned in Proposition 3.3(5)and ψ is de fi ned in Proposition 3.3(2).

(2)Assume that f is proper.Then for each object G in Dc(Yˊet,R·),the diagram

is commutative in Dc(,R·),where α is de fi ned in Proposition 3.3(5)and β is induced by the composition

De fi nition 4.3(Base Change) Let

be a Cartesian square of schemes satisfying(?)with f compacti fi able.For every object N in Dc(Sˊet,R·)and for every n∈Z,we de fi ne a homomorphism of R-modules

as follows.For each α∈Hn(X/S,N),u*(α)is de fi ned to be the composition

where φ is de fi ned in Proposition 3.3(2).

We have the following three obvious propositions about the base change homomorphisms. Proposition 4.7 Let k?K be two separably closed fi elds,f:X→S be a morphism of algebraic schemes over k,and u:SK→ S be the projection.Then for each object N in Dc(Sˊet,R·)and for each n∈Z,the homomorphism

is an isomorphism.

Proof. It follows from Proposition 3.3(3)and Theorem 2.1.

Proposition 4.8 Let

be a commutative diagram of schemes satisfying(?)with both squares Cartesian,and all three vertical arrows being compacti fi able.Then for all N ∈Dc(Sˊet,R·)and n∈Z,we have

Proposition 4.9 Let

be a commutative diagram of schemes satisfying(?)with both squares Cartesian,and all level arrows being compacti fi able.Let N be an object in Dc(Sˊet,R·)and n∈Z.Then we have

(1) If f is proper,then

(2)If f is fl at at dimension d,then

De fi nition 4.4(Galois action) Let k0be a fi eld,k the separably closed fi eld of k0,G:= Gal(k/k0),X be an algebraic scheme over k,Y0be an algebraic scheme over k0,Y := Y0?k0k,N0be an object in Dc(Y0,ˊet,R·)and N the pull-back of N0on Y.Then there is an action of G on Hn(X/Y,N)de fi ned by

In particular,if N∈Dfg(R)and n∈Z,then there is a Galois action of G on Hn(X,N).

The following theorem is used to prove that cycle maps eliminate algebraic equivalent classes.

Theorem 4.2 Let f:X→Y be a morphism of algebraic schemes over a separably closed fi eld k,Z be a nonsingular variety over k,N be an object in Dc(Yˊet,R·),

For each z∈Z(k),put

Proof. By Proposition 4.7 we may assume that k is algebraically closed.Since every two rational points of Z can be jointed by a series of nonsingular curves,we may further assume that Z is a complete nonsingular curve.First we have a commutative diagram with both squares Cartesian:

By Proposition 3.3,we have

Since Z is a complete nonsingular curve over k,we have

where g is the genus of Z,andR are all free R-modules.Now we apply Theorem 2.2 to obtain an isomorphism:

Let β∈Hn(X/Y,N)be the image of α induced by above isomorphism.Then

5 The Cycle Maps for Chow Groups

In this section,we construct the cycle maps for arbitrary algebraic schemes over k,where k is a f i eld such that

Let f:X→Y be a compactif i able morphism of schemes satisfying(?)which is f l at at dimension d.We def i ne

Proposition 5.1 Let X→S and Y→S be two compactif i able morphisms of schemes satisfying(?),and f:X→Y a morphism of S-schemes.Assume that Y→S and f:X→Y are f l at at dimension n and d respectively.Then we have

Proof. This follows from Proposition 3.6(2).

Proposition 5.2 Let p:X→S and q:Y→S be two morphisms of schemes satisfying (?),both of which are compactif i able and f l at at dimension d,and f:X → Y be a f i nite S-morphism such that f*Xis a locally freeY-module of degree n.Then we have

Proof. By the de fi nition of f*and Proposition 3.6(2),the element f*c?(X/S)is equal to the composite morphism

By Diagram(3.1)and Proposition 3.7,we have a commutative diagram as follows:

Thus we get the proof.

Let X→S be a compacti fi able morphism of schemes satisfying(?),and i:Y→X a closed immersion.Assume that the morphism Y→S is fl at at dimension d.Then we de fi ne

Let X be an algebraic scheme over k.Then for each n∈Z,there is a canonical homomorphism of groups

Proposition 5.3 Let f:X→Y be a proper morphism of algebraic schemes over k.Then for every n∈N,we have a commutative diagram

Proof.Let X′be an n-dimensional subvariety of X,Y′:=f(X′),i:X′→X and j:Y′→Y the inclusion,and g :X′→Y′the induced morphism.By Proposition 4.5 we have

Since f*[X′]=deg(X′/Y′)[Y′](see the section 1.4 of[5]),we have only to prove that

If dimY′＜n,then

And by Proposition 4.3,

If dimY′=n,we apply the result in Example 3.7 of[11].Since the morphism g is generically fi nite and Y′is an integral scheme,there exists an nonempty subscheme V of Y′such that g:g-1(V)→V is a fi nite morphism and g*X′|Vis a locally freeV-module. Now the proposition follows from Propositions 4.4 and 5.2.

Proposition 5.4 Let X be an algebraic scheme over k,and Y be a n-equidimensional closed subscheme of X.Then we have

Proof. This is easily deduced from Proposition 3.8.

Proposition 5.5 Let f:X→Y be a fl at morphism of relative dimension d of algebraic schemes over k.Then for every n∈N,we have a commutative diagram

Proof.Let α∈Zn(Y).We may assume that Y is a variety of dimension n and α=[Y]. Then we have only to apply Proposition 5.1.

Lemma 5.1 Let X be a nonsingular variety of dimension n over k,and D be an e ff ective divisor on X.Then

Proof. See(3.26)of[10].

Theorem 5.1 Let X be an algebraic schemeover k.Then for each n∈N,

Proof. After applying Proposition 1.6 of[5]together with Propositions 5.3 and 5.5,we have only to prove that

This follows from Lemma 5.1.

In the following,we de fi ne the degree map for the homology of degree zero.

De fi nition 5.1 For any proper algebraic scheme X over k,we de fi ne degree map deg?to be the homomorphism

where p:X→Speck is the structural morphism.

Lemma 5.2 Let X be an n-dimensional proper algebraic scheme over k.

(1)We have a commutative diagram

(2)We have a commutative diagram

Proof.(1)can be proved by the commutative diagram(3.1).

(2)can be proved by Proposition 5.3.

Proposition 5.6 Assume that k is separably closed and let X be a nonsingular complete variety over k.Then deg?:is an isomorphism.

Proof.Put dimX=n.By Proposition 3.4,we have only to prove that

is an isomorphism.This is true by VI,11.1(a)of[12].

The following theorem shows that the cycle map clX,?annihilate algebraic equivalence of cycles.

Theorem 5.2 Assume that k is separably closed and let X be an algebraic scheme over k.Then for each n∈N,the cycles in CHn(X()which are algebraically equivalent to zero (in the sense of 10.3 of[5])are contained in kerclX,?).

Proof. By Proposition 4.7 we may assume that k is algebraically closed.Let c1,c2∈CHn(X)be such that c1～ac2,T be a nonsingular curve over k,t1,t2∈T(k),and c∈CHn+1(X×kT)be such that cti=cifor i=1,2.Obviously,we may assume that c=[Y], where Y is an(n+1)-dimensional subvariety of X×kT such that for all t∈T(k),Y is not contained in

Obviously,the induced morphism Y→T is dominant and fl at.Put

By Propositions 4.9 and 3.5(2),we have

So we have only to apply Proposition 4.2.

6 Cap-products and Compatibility with Chern classes

First we de fi ne the cap-products for the ?-adic homology.

De fi nition 6.1(Cap-product) Let f:X→Y and g:Y→Z be compacti fi able morphisms of schemes satisfying(?),M and N two objects in(Zˊet,R·).For every m,n∈Z,there is a cap-product

where φ is de fi ned in Proposition 3.2.

Similarly,we may de fi ne the cap-product for H*as follows:

In particular,if X→S is a compacti fi able morphisms of schemes satisfying(?),and N is an object in(Sˊet,R·),then for every m,n∈Z,there are cap-products

The following Proposition can be directly calculated.

Proposition 6.1(Projection Formula) Let f:X →Y and g:Y→ S be morphisms of schemes satisfying(?)with f proper and g compacti fi able,and N an object in Dc(Yˊet,R·). Then we have

(1)For every α∈Hr(Y,R)and β∈Hn(X/S,N),

(2)For every α∈Hr(Y,R)and β∈Hn(X/S,N),

It may be further showed that the cap-product de fi ned in De fi nition 6.1 has many similar properties to bivariant intersection theory de fi ned in Ch.17 of[5],i.e.,has associativity and is compatible with the Pull-back functor f*,the push-out functor f*and the base change functor u*.Since we need not them here,we leave it to the readers.

Next,we review the cycle maps for locally free sheaves.First by(3.26)a)of[10],we have a homomorphism of groups

for every scheme X satisfying(?).The following two propositions depict the cycle maps for locally free sheaves.

Proposition 6.2 Let X be a scheme satisfying(?),E a locally freeX-module of constant rank r+1,P:=P(E),and p:P → X be the projection.Then there is a canonical isomorphism of Z?-algebras

Proof. See(6.13)of[10]or VII,2.2.6 of[13].

As a direct application of the above proposition,we have

Proposition 6.3 Let X be a scheme satisfying(?),E a loca(lly free)-module of constant rank m,P:=P(E∨),and p:P→X be the projection,ξ:=(1).Then for each r∈N there exists a unique element(E)∈Hr(X,Z?)such that

Now we de fi ne the trace morphisms for regular immersions of codimension 1.Let X be a scheme satisfying(?)and i:D→ X a regular closed immersion of codimension 1.By (3.26)of[10]and its proof,i:D→X determinates an element

Similar to 2.3.1 of the paper(cycle)of[4],we have

Proposition 6.4 Let S be a scheme satisfying(?),f:X → S and g:Y→ S be two compacti fi able morphisms which are fl at at dimension n and n-1 respectively,and i:Y→X be a regular closed immersion of codimension 1 such that f?i=g.Then we have

(1)The composite morphism

is equal to Trg;

(2)The composite morphism

is equal to tg.

Finally,we prove that the cycle maps are compatible with Chern classes.According to Ch.3 of[5],if X is an algebraic scheme over k and E is a locally free-module,then there is an operation of Chern classes on each Chow group

Theorem 6.1 Let X be an algebraic scheme over k,E a locally free-module,and α∈CHr(X).Then we have

Proof.By the the projection formulas(Proposition 6.1 and Theorem 3.2(c)of[5]),we obtain that if f:X′→X is a proper morphism and α′∈CHr(X′)such that f*(α′)=α and the pair(f*E,α′)satis fi es(6.2),then the pair(E,α)also satis fi es(6.2).Thus by the splitting construction(see§3.2 of[5]),we may assume that E=L is an invertible-module and have only to prove that

Moreover we may assume that X is a variety of dimension r and α=[X].After replacing X with its normalization,we may assume that X is normal.Then we may assume that L=(Y)where Y→X is a regular closed immersion of codimension 1.Then we have only to apply Proposition 6.4 to end our proof.

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tion:14F20,14C25

A

1674-5647(2013)01-0068-20

*Received date:Nov.25,2010.

The NSF(10626036)of China.