Relative cycles and Chow sheaves.

Andrei Suslin, Vladimir Voevodsky

Contents

1 Introduction. 1

2 Generalities. 6 2.1 Universally equidimensional morphisms. . . . . . . . . . . . . 6 2.2 Universally equidimensional closed subschemes. . . . . . . . . 9 2.3 Cycles on Noetherian schemes. . . . . . . . . . . . . . . . . . . 13

3 Relative cycles. 16 3.1 Relative cycles. . . . . . . . . . . . . . . . . . . . . . . . . . . 16 3.2 Cycles associated with flat subschemes. . . . . . . . . . . . . . 21 3.3 Chow presheaves. . . . . . . . . . . . . . . . . . . . . . . . . . 23 3.4 Relative cycles over geometrically unibranch schemes. . . . . . 33 3.5 Multiplicities of components of inverse images of equidimen-

sional cycles over geometrically unibranch schemes. . . . . . . 36 3.6 Functoriality of Chow presheaves. . . . . . . . . . . . . . . . . 43 3.7 Correspondence homomorphisms. . . . . . . . . . . . . . . . . 49

4 Chow sheaves in the h-topologies. 56 4.1 The h-topologies. . . . . . . . . . . . . . . . . . . . . . . . . . 56 4.2 Sheaves in the h-topologies associated with Chow presheaves. . 58 4.3 Fundamental exact sequences for Chow sheaves. . . . . . . . . 68 4.4 Representability of Chow sheaves. . . . . . . . . . . . . . . . . 73

1 Introduction.

Let X be a scheme. A cycle on X is a formal linear combination of points of the Zariski topological space of X. A cycle is called an effective cycle if all points appear in it with non negative coefficients. Suppose that X is a projective scheme over a field k of characteristic zero. Then for any projective embedding i : X → P^{n} the classical construction produces a projective variety C_{r}(X, i) called the Chow variety of effective cycles of dimension r on X such that k-valued points of C_{r}(X, i) are in natural bijection with effective cycles of dimesnion r on X. Moreover, for any field extension E/k an Evalued point of C_{r}(X, i) defines a cycle on X ×_{Spec(k)} Spec(E). In particular for any Noetherian scheme S over k and any morphism ϕ : S → C_{r}(X) we get a cycle Z_{ϕ} on S ×_{Spec(k)} X which lies over generic points of S. For any such cycle Z_{ϕ} and any morphism f : S′ → S of Noetherian schemes over k the composition f ∘ ϕ defines a new cycle cycl(f)(Z) on X ×_{Spec(k)} S′. Thus existence of Chow verieties implies that for any Noetherian scheme S over k there is a natural class of cycles on X ×_{Spec(k)} S which are contravariantly functorial with respect to all morphisms S′ → S of Noetherian schemes over k.

Let now X → S be any scheme of finite type over a Noetherian scheme S. In this paper we introduce a class of cycles on X which are called relative cycles on X over S. Their most important property is the existence of well defined base change homomorphisms for arbitrary morphisms S′ → S of Noetherian schemes. In the case when X = X_{k} ×_{Spec(k)} S where X_{k} is a projective scheme over a field of characteristic zero and S is a semi-normal Noetherian scheme over k the class of effective relative cycles on X over S coincides with the class of cycles of the form Z_{ϕ} considered above.

Informally speaking a relative cycle on X over S is a cycle on X which lies over generic points of S and has a well defined specialization to any fiber of the projection X → S. We denote the group of relative cycles of relative dimension r on X over S by Cycl(X/S, r). Unfortunately if S is not a scheme of characterstic zero the specializations Z_{s} of a relative cycle Z on X over S to points s of S do not have in general integral coefficients - the characteristic of the residue field of s may appear in denominators. We denote by z(X/S, r) the subgroup in Cycl(X/S, r) which consists of relative cycles Z such that for any point s of S the specialization Z_{s} of Z to s has integral coefficients. The groups z(X/S, r) are contravariantly functorial with respect to S, i.e. for any morphism of Noetherian schemes f : S′ → S there is a homomorphism cycl(f) : z(X/S, r) → z(X ×_{S} S′/S′, r) and for any composable pair of morphisms f, g one has cycl(f ∘ g) = cycl(g) ∘ cycl(f). It gives us a presheaf of abelian groups on the category of Noetherian schemes over S which we also denote by z(X/S, r). These Chow presheaves are the main objects of our study.

The paper is organized as follows. We start in sections 2.1, 2.2 with some elementary properties of equidimensional morphisms and equidimensional closed subschemes. Besides the standard theorems of Chevalley (Theorem 2.1.1, Proposition 2.1.7(3)) our main technical tool here as well as in the rest of the paper is the “platification theorem” (Theorem 2.2.2).

In section 2.3 we prove some basic results about cycles on Noetherian schemes. All material here is well known and is included only for reader’s convenience.

In section 3.1 we introduce the groups Cycl(X/S, r) of relative cycles on a scheme X of finite type over a Noetherian scheme S. Relative cycles over general Noetherian schemes demonstrate all kinds of “pathological” behavior. For instance the group Cycl(X/S, r) is not necessarily generated by the corresponding submonoid Cycl^{eff}(X/S, r) of effective relative cycles (see example 3.4.7) and supports of noneffective relative cycles of relative dimension r over S do not have to be equidimensional of relative dimension r over S (example 3.1.9).

We also define in this section different versions of our main object - the groups of relative cycles with proper support and the groups of relative cycles with equidimensional support.

The main theorem 3.3.1 of section 3.3 says that for any morphism S′ → S of Noetherian schemes there is a well defined base change homomorphism Cycl(X/S, r) → Cycl(X ×_{S} S′/S′, r) ⊗ Q. It gives us a construction of presheaves Cycl(X/S, r)_{Q} on the category of Noetherian schemes over S such that for any such scheme S′ the group Cycl(X/S, r)_{Q}(S′) is the group of relative cycles of dimension r on X ×_{S} S′ over S′ with rational coefficients.

An example of a relative cycle with integral coefficients whose specialization does not have integral coefficients which is due to A.Merkurjev is given in 3.5.10(1).

In order to obtain a good definition of Chow presheaves with integral coefficients we consider a formal condition on a cycle to be universally with integral coefficients. It turns out that the corresponding groups z(X/S, r) are subgroups in the groups Cycl(X/S, r) with torsion quotient for all Noetherian schemes S (Proposition 3.3.14). We call the corresponding presheaves of abelian grous z(X/S, r) the Chow presheaves of relative cycles on X over S.

We also define in the similar way the presheaves c(X/S, d) which correspond to proper relative cycles and presheaves z_{equi}(X/S, r), c_{equi}(X/S, r) of relative cycles (resp. proper relative cycles) with equidimensional support.

In section 3.4 we consider relative cycles over geometrically unibranch schemes (in particular over normal schemes). It turns out that over such a scheme S the group of relative cycles with equidimensional support is generated by the cycles of integral closed subschemes of X which are equidimensional over S (Theorem 3.4.2) which implies that in this case our definition coincide with the naive one. On the other hand the corresponding statement does not hold in generally for the groups z(X/S, r) as shown in example 3.5.10(2).

One of the important properties of presheaves z(X/S, r) is that for any regular base scheme S the groups z(X/S, r) coincide with the groups Cycl(X/S, r), i.e. in this case any relative cycle has universally integral coefficients. Our base change homomorphisms over a regular base scheme coincide with base change homomorphisms which one can define by means of the Tor-formula (see the end of the section 3.5). We also show in section 3.5 that our base change homomorphisms coincide in the case of finite cycles over normal schemes with base change homomorphisms defined in [16].

In section 3.6 we study functoriality of Chow presheaves z(X/S, r) and c(X/S, r) with respect to X. We show in particular that for any morphism (resp. any proper morphism) p : X → Y of schemes of finite type over S there is a push-forward homomorphism of presheaves p_{∗} : c(X/S, r) → c(Y/S, r) (resp. p_{∗} : z(X/S, r) → z(Y/S, r)) and for any flat (resp. flat and proper) equidimensional morphism f : X → Y of relative dimension n there is a pull-back homomorphism of presheaves f^{∗} : z(Y/S, r) → z(X/S, r +n) (resp. f^{∗} : c(Y/S, r) → c(X/S, r + n)).

In the next section we define for Chow presheaves the correspondence homomorphisms which were considered in context of Chow varieties by Eric M. Friedlander. In particular we show that there is a well defined homomorphism of external product

z(X/S, r_{1}) ⊗ z(Y/S, r_{2}) → z(X ×_{S} Y/S, r_{1} + r_{2})

for any schemes of finite type X, Y over S.

In the last section we consider Chow presheaves as sheaves in the htopologies. In particular we construct in section 4.3 some exact sequences of Chow sheaves which are important for localization-type theorems in algebraic cycle homology and Suslin homology (see [17]). Finally in section 4.4 we study representability of Chow sheaves. One can easily show that except for some trivial cases the Chow presheaves are not representable as presheaves.

To avoid this difficulty we introduce a notion of the h-representability. We construct then the Chow scheme C_{r,d} of cycles of degree d and dimension r on P^{n} over Spec(Z) using essentially the classical construction due to Chow (see also [14]) and show that it h-represents the Chow sheaf z_{d}^{eff}(P^{n}/Spec(Z), r) of effective relative cycles of relative dimension r and degree d on P^{n}. A rather formal reasoning shows then that for any quasi-projective scheme X over a Noetherian scheme S and any r ≥ 0 the Chow sheaf c(X/S, r) is h-representable by disjoint union of quasi-projective schemes over S. As an application of this representability result we show that for a quasi-projective scheme X over a Noetherian scheme S the group z(X/S, r)_{h} of sections of the h-sheaf associated with the Chow presheaf z(X/S, r) can be described using the notion of “ continuous algebraic maps” introduced by Eric M. Friedlander and O. Gabber ([3]) which generalizes a similar result obtained in [2] for quasi-projective schemes over a field.

Everywhere in this text a scheme means a separated scheme.

2 Generalities.

2.1 Universally equidimensional morphisms.

For a scheme X we denote by dim(X) the dimension of the Zariski topological space of X. By definition dim(X) is either a positive integer of infinity. If x ∈ X is a point of a locally Noetherian scheme X we denote by dim_{x}(X) the limit lim dim(U) taken over the partially ordered set of open neighborhoods of x in X. One can easily see that it is well defined and equals dim(U) for a sufficiently small U (see [8, Ch.0,14.4.1]).

For a morphism p : X → S denote by dim(X/S) the function on the set of points of X of the form dim(X/S)(x) = dim_{x}(p^{−1}(p(x))). The most important property of the dimension functions is given by the following well known theorem.

Theorem 2.1.1 (Chevalley) Let p : X → S be a morphism of finite type. Then for any n ≥ 0 the subset {x ∈ X : dim_{X/S}(x) ≥ n} is closed in X.

Proof: See [8, Th. 13.1.3].

Definition 2.1.2 A morphism of schemes p : X → S is called an equidimensional morphism of dimension r if the following conditions hold:

1. p is a morphism of finite type.

2. The function dim(X/S) is constant and equals r.

3. Any irreducible component of X dominates an irreducible component of S.

A morphism of schemes p : X → S is called universally equidimensional of dimension r if for any morphism S′ → S the projection X ×_{S} S′ → S′ is equidimensional of dimension r.

Finally, we say that p : X → S is a morphism of dimension ≤ r if dim(X/S)(x) ≤ r for all points x of X.

One can easily see that in the definition of equidimensional morphism given above one can replace the condition dim(X/S) = r by the condition that for any point y of S the dimension of all irreducible components of the topological space p^{−1}(p(x)) equals r.

Proposition 2.1.3 Let p : X → S be a morphism of finite type of Noetherian schemes. Then p is equidimensional of dimension r if and only if for any point x of X there is an open neighborhood U in X and a factorization of the morphism p_{U} : U → S of the form U →^{p_{0}} A_{S}^{r} → S such that p_{0} is a quasi-finite morphism and any irreducible component of U dominates an irreducible component of A_{S}^{r}.

Proof: See [8, 13.3.1(b)].

Definition 2.1.4 A morphism of schemes p : X → S is called an open morphism if for any open subset U of X the subset p(U) is open in S. It is called universally open if for any morphism S′ → S the projection X ×_{S} S′ → S′ is an open morphism.

Definition 2.1.5 Let S be a Noetherian scheme. It is called unibranch (resp. geometrically unibranch) if for any point s of S the scheme Spec(O_{s,S}^{h}) where O_{s,S}^{h} is the henselization of the local ring of s in S (resp. the scheme Spec(O_{s,S}^{sh}) where O_{s,S}^{sh} is the strict henselization of the local ring of s in S) is irreducible.

Remark: Our definition is consistent with the one given in [8, 6.15.1] in view of [8, 18.8.15].

Proposition 2.1.6 Let S be a Noetherian geometrically unibranch scheme and f : S′ → S be a proper birational morphism. Then for any point s of S the fiber S′_{s} of f over s is geometrically connected.

Proof: It follows from [7, 4.3.5] and [8, 18.8.15(c)].

Proposition 2.1.7 Let p : X → S be a morphism of finite type of Noetherian schemes. Then the following implications hold:

1. If p is a universally equidimensional morphism then p is universally open.

2. If dim(X/S) = r and p is open (resp. universally open) then p is equidimensional (resp. universally equidimensional) of dimension r.

3. If S is geometrically unibranch and p is equidimensional then p is universally equidimensional (and hence universally open).

Proof: (1) It follows from [8, 14.4.8.1 and 14.4.4].

(2) Obvious.

(3) It is known that any equidimensional morphism over a geometrically unibranch scheme is universally open (see [8, 14.4.4]). It implies immediately that for any morphism S′ → S the projection p′ : X ×_{S} S′ → S′ satisfies the condition (3) of Defenition 2.1.2. Since the conditions (1) and (2) are obviously stable under a base change the morphism p′ is equidimensional.

Remarks:

1. Note that an equidimensional morphism does not have to be open.

2. One can easily see that the inverse statement to the third part of this proposition holds. Namely a Noetherian scheme X is geometrically unibranch if any equidimensional morphism over X is universally equidimensional.

3. Let p : X → S be a morphism of finite type such that S = Spec(k) where k is a field. Proposition 2.1.7(3) implies that p is universally equidimensional of dimension r if and only if all irreducible components of X have dimension r.

Proposition 2.1.8 Let p : X → S be a flat morphism of finite type. Then p is universally equidimensional of dimension r if and only if for any generic point y : Spec(K) → S of S the projection X ×_{S} Spec(K) → Spec(K) is equidimensional of dimension r.

Proof: Since any flat morphism of finite type is universally open ([11, I.2.12]) it is sufficient to verify that under the conditions of the proposition we have dim(X/S) = r. It follows immediately from [8, 12.1.1(iv)] and Theorem 2.1.1.

Proposition 2.1.9 Let p : X → S be an equidimensional morphism of relative dimension r such that X is irreducible. Suppose that p admits a decomposition of the form X →^{p_{0}} W →^{p_{1}} S such that p_{0} is surjective and proper and p_{1} has at least one fiber of dimension r. Then p_{1} is equidimensional of dimension r and p_{0} is finite in the generic point of W.

Proof: It follows easily from Theorem 2.1.1.

Lemma 2.1.10 Let p : X → S be a morphism such that any irreducible component of X dominates an irreducible component of S and i : X_{0} → X be a closed embedding which is an isomorphism over the generic points of S. Then i is defined by a nilpotent sheaf of ideals. In particular, p is a universally equidimensional morphism of dimension r if and only if p_{0} : X_{0} → S is a universally equidimensional morphism of dimension r.

Lemma 2.1.11 Let X → S be a scheme of finite type over a Noetherian scheme S, Z ⊂ X be a closed subscheme universally equidimensional of relative dimension r over S and S′ → S be a blow-up of S. Let Z̃ be the proper transform of Z in X ×_{S} S′. Then Z̃ is a closed subscheme of Z ×_{S} S′ defined by a nilpotent sheaf of ideals.

Proof: Since Z ×_{S} S′ is equidimensional over S′ and hence its generic points lie over generic points of S′ our statement follows from Lemma 2.1.10.

2.2 Universally equidimensional closed subschemes.

Let p : X → S be a morphism of finite type. We denote by Z_{i}(X/S) (resp. by Z_{≤i}(X/S)) the set of closed reduced subschemes Z in X such that the morphism p_{|Z} : Z → S is universally equidimensional of relative dimension i (resp. of dimension ≤ i).

For any morphism f : S′ → S and any reduced closed subscheme Z of X we denote by f^{−1}(Z) the maximal reduced subscheme of the closed subscheme Z′ = Z ×_{S} S′ of X′ = X ×_{S} S′. One can easily see that if Z is an element of Z_{≤i}(X/S) (resp. an element of Z_{i}(X/S)) then f^{−1}(Z) is an element of Z_{≤i}(X′/S′) (resp. an element of Z_{i}(X′/S′)).

Lemma 2.2.1 Let p : X → S be a morphism of schemes and Z_{1}, Z_{2} be a pair of elements in Z_{i}(X/S) (resp. in Z_{≤i}(X/S)). Then Z_{1} ∪ Z_{2} is an element of Z_{i}(X/S) (resp. of Z_{≤i}(X/S)).

Proof: Obvious.

Theorem 2.2.2 Let p : S′ → S be a morphism of Noetherian schemes and U be an open subscheme in S such that p is flat over U. Then there exists a closed subscheme Z in S such that U ∩ Z = ∅ and the proper transform of S′ with respect to the blow-up S_{Z} → S of S with center in Z is flat over S_{Z}.

Proof: See [13, 5.2].

Theorem 2.2.3 Let p : X → S be a morphism of finite type, U be an open subscheme of X, Z be an element of Z_{i}(U/S) and V be an open subscheme of S such that the closure Z̄ of Z in X is flat over V. Then there exists a blow-up f : S′ → S of S such that the closure of f^{−1}(Z) in X′ = X ×_{S} S′ belongs to Z_{i}(X′/S′) and the morphism f^{−1}(V) → V is an isomorphism.

Proof: By Theorem 2.2.2 there is a blow-up f : S′ → S such that the proper transform Z̃ of Z̄ with respect to f is flat over S′. One can easily see now that the closure Z̄′ of Z′ = Z ×_{S} S′ in X′ = X ×_{S} S′ is a closed subscheme in Z̃ and the corresponding closed embedding is an isomorphism over the generic points of S′. Therefore, Z̄′ is universally equidimensional over S′ by Lemma 2.1.10.

Definition 2.2.4 Let f : S′ → S be a morphism of Noetherian schemes. We say that f is an abstract blow-up of S if the morphism f is proper, any irreducible component of S′ dominates an irreducible component of S and there exists a dense open subscheme U in S such that the morphism (f^{−1}(U)_{red}) → U_{red} is an isomorphism (here X_{red} denote the maximal reduced subscheme of X).

Note that any abstract blow-up in the sense of Defenition 2.2.4 is surjective.

The following lemma lists some trivial properties of abstract blow-ups.

Lemma 2.2.5 1. A composition of abstract blow-ups is an abstract blowup.

2. If S is a Noetherian scheme and S′ → S, S′′ → S is a pair of abstract blow-ups of S then the morphism S′ ×_{S} S′′ → S is an abstract blow-up of S.

Corollary 2.2.6 Let p : X → S be a morphism of finite type and U be an open subscheme of X, Z be an element of Z_{i}(U/S). Then there exists an abstract blow-up f : S′ → S of S such that the closure of f^{−1}(Z) in X′ = X ×_{S} S′ belongs to Z_{i}(X′/S′)

Proof: It is sufficient to note that we may replace S by the disjoint union of its reduced irreducible components and that in the case of integral S there is an open nonempty subset V in S such that the closure Z̄ is flat over V.

Corollary 2.2.7 Let p : X → S be a universally eqidimensional quasiprojective morphism of dimension r. Then there is an abstract blow-up S′ → S, a universally equidimensional projective morphism X̄′ → S′ of dimension r and an open embedding i : X ×_{S} S′ → X̄′ over S′.

Proof: Obvious.

Theorem 2.2.8 Let p : X → S be a universally equidimensional morphism and Z be an element in Z_{≤i}(X/S). Then there exists an abstract blow-up f : S′ → S of S and an element W in Z_{i}(X ×_{S} S′/S′) such that f^{−1}(Z) is contained in W.

Proof: We start with the following lemma.

Lemma 2.2.9 Let S be a local Noetherian scheme and p : X → S be an affine flat equidimensional morphism of dimension r. Let further Z be a closed subscheme of the closed fiber of X over S which does not contain any of the generic points of this fiber.

Then there exists a closed subscheme W of X which is flat and equidimensional of dimension r − 1 over S such that Z lies in W.

Proof: There exists a finite set {x_{1}, …, x_{k}} of closed points of X such that the following conditions hold:

1. {x_{1}, …, x_{k}} ∩ Z = ∅.

2. Any irreducible component of X contains at least one of the points x_{1}, …, x_{k}.

3. Any irreducible component of the closed fiber X_{s} of X contains al least one of the points x_{1}, …, x_{k}.

Since X is affine there is a regular function f on X such that f = 0 on Z and f = 1 on the set {x_{1}, …, x_{k}}. The third property of this set implies that the divisor W = (f) of f is flat over S (see [11, I.2.5]) and the second one that this divisor is equidimensional of dimension r − 1 over generic points of S. By Proposition 2.1.8 we conclude that W is equidimensional of dimension r − 1 over S. Lemma is proven.

To prove our theorem it is obviously sufficient to show that if Z is an integral closed subscheme of X of dimension ≤ i over S where i < r then there exist an abstract blow-up S′ → S and an element W ∈ Z_{r−1}(X′/S′) such that Z′ is contained in W.

Assume first that X is affine over S. By Theorem 2.2.2 it is sufficient to consider the case of a flat morphism p : X → S. Let z be the generic point of Z and s = p(z). Consider the fiber X_{s} of X over s. Since Z_{s} is of codimension at least one in X_{s} by Lemma 2.2.9 there is an open subscheme U of S which contains s and a closed subscheme W_{U} in p^{−1}(U) which is flat and equidimensional of dimension r−1 over U and which contains Z ∩p^{−1}(U). By Theorem 2.2.3 there is an abstract blow-up f : S′ → S such that f^{−1}(U) → U is an isomorphism and the closure W̄′ of W′ = W_{U} ×_{S} S′ in X′ = X ×_{S} S′ belongs to Z_{r−1}(X′/S′).

In particular the closure of the pre-image in X′ of the generic point of Z is contained in W̄′. Let Q be the complement to U in S and Z′ be the intesection Z ∩ p^{−1}(Q). By our construction we have dim(Z′) < dim(Z). Using induction on dim(Z) we may assume that there is an abstract blowup S′′ → S such that the Z′′ = Z ×_{S} S′′ is contained in an element W_{1} of Z_{d−1}(X′′/S′′). Considering an abstract blow-up S′′′ → S which dominates both S′ and S′′ (Lemma 2.2.5) we conclude that the statement of our theorem is correct for affine morphisms p : X → S.

Let now p : X → S be an arbitrary universally equidimensional morphism of dimension r. Then there is a finite open covering X = ∪U_{i} such that the morphisms p_{i} : U_{i} → S are affine. Since our statement holds for affine morphisms there is an abstract blow-up S′ → S and a family of elements W_{i} ∈ Z_{d−1}(U′_{i}/S′) such that Z′ ∩ U′_{i} is contained in W_{i}. By Theorem 2.2.3 we may choose S′ such that the closure W̄_{i} of each W_{i} in X′ is universally equidimensional of dimension r − 1. It implies the result we need since one obviously has Z′ ⊂ ∩W̄_{i}. Theorem is proven.

2.3 Cycles on Noetherian schemes.

Let X be a Noetherian scheme. We denote by Cycl(X) (resp. by Cycl^{eff}(X)) the free abelian group (resp. the free abelian monoid) generated by points of the Zariski topological space of X.

For any element Z of Cycl(X) we denote by supp(Z) the closure of the set of points on X which appear in Z with nonzero coefficients. We consider supp(Z) as a reduced closed subscheme of X.

Let Z be a closed subscheme of X and ζ_{i}, i = 1, …, k be the generic points of the irreducible components of Z. We define an element Cycl_{X}(Z) of the abelian monoid Cycl^{eff}(X) as the formal linear combination of the form

cycl_{X}(Z) = ∑_{i=1}^{k} m_{i}ζ_{i}

where m_{i} = length(O_{Z,ζ_{i}}). Each number m_{i} is a positive integer which is called the multiplicity of Z in the point ζ_{i}.

This construction gives us a map from the set of closed subschemes of X to the abelian monoid Cycl^{eff}(X) which can be canonically extended to a homomorphism from the free abelian monoid generated by this set to Cycl^{eff}(X). We denote this homomorphism by cycl_{X}.

Let p : X → S be a flat morphism of Noetherian schemes and let Z = ∑ n_{i}z_{i} be a cycle on S. Denote by Z_{i} the closure of the point z_{i} which we consider as a closed integral subscheme in S and set p^{∗}(Z) = ∑ n_{i}cycl_{X}(Z_{i}×_{S} X). In this way we get a homomorphism (flat pull-back) p^{∗} : Cycl(S) → Cycl(X). The following lemma is straightforward (cf. [4, Lemma 1.7.1]).

Lemma 2.3.1 1. If Z is any closed subscheme of S then p^{∗}(cycl_{S}(Z)) = cycl_{X}(Z ×_{S} X).

2. supp(p^{∗}(Z)) = (p^{−1}(supp(Z)))_{red}. In particular the homomorphism p^{∗} : Cycl(S) → Cycl(X) is injective provided that p is surjective.

Let X → Spec(k) be a scheme of finite type over a field k and let L/k be any field extension. The corresponding morphism p : X_{L} → X is flat and hence defines a homomorphism p^{∗} : Cycl(X) → Cycl(X_{L}). The image of a cycle Z ∈ Cycl(X) under this homomorphism will be usually denoted by Z ⊗_{k} L of Z_{L}.

Lemma 2.3.2 Let X → Spec(k) be a scheme of finite type over a field k and let k′/k be a finite normal field extension with the Galois group G. If Z′ ∈ Cycl(X_{k′})^{G} then there is a unique cycle Z ∈ Cycl(X) such that [k′ : k]_{insep}Z′ = Z_{k′}.

Proof: The uniqueness of Z follows immediately from Lemma 2.3.1(2). To prove the existence note that the group Cycl(X_{k′})^{G} is generated by cycles of the form Z′ = ∑_{τ∈G/H} τ(z′) where z′ is a point of X_{k′} and H = Stab_{G}(z′). Let z be the image of z′ in X and let Z be the closure of z which we consider as a closed integral subscheme of X. The points τ(z′) are precisely the generic points of the scheme Z′ = Z ×_{Spec(k)} Spec(k′) and the multiplicities with which they appear in the cycle Z_{k′} are all equal to the length of the local Artinian ring O_{Z′,z′}. The elementary Galois theory shows that this length is a factor of [k′ : k]_{insep}. Thus the cycle Z = [k′ : k]_{insep}z/length(O_{Z′,z′}) has the required property.

Corollary 2.3.3 In the assumptions and notations of the previous lemma denote by p the exponential characteristic of the field k. Then the homomorphism

Cycl(X)[1/p] → (Cycl(X_{k′})[1/p])^{G}

is an isomorphism.

Let X → Spec(k) be a scheme of finite type over a field k. Then we have a direct sum decomposition Cycl(X) = ∐ Cycl(X, r) where Cycl(X, r) is a subgroup of Cycl(X) generated by points of dimension r. Furthermore one sees easily that for a field extension k′/k the homomorphism

Cycl(X) → Cycl(X_{k′})

preserves this decomposition.

Let S be a Noetherian scheme and p : X → S be a proper morphism of finite type. For any cycle Z = ∑ n_{i}z_{i} ∈ Cycl(X) set

p_{∗}(Z) = ∑ n_{i}m_{i}p(z_{i})

where m_{i} is the degree of the field extension k_{z_{i}}/k_{p(z_{i})} if this extension is finite and zero otherwise. The proof of the following statement is similar to that of Proposition 1.7 of [4] and we omit it.

Proposition 2.3.4 Consider a pull-back square of morphisms of finite type of Noetherian schemes

\begin{matrix}X̃ & →^{f_{X}} & X \\ p̃ ↓ & & ↓ p \\ S̃ & →^{f} & S \end{matrix}

in which f is flat and p is proper. Then for any cycle Z ∈ Cycl(X) we have

f^{∗}(p_{∗}(Z)) = p̃_{∗}(f_{X}^{∗}(Z)).

The following lemma is straighforward.

Lemma 2.3.5 Let f : X → Y be a finite flat morphism of connected Noetherian schemes. Denote by deg(f) the degree of f. Then f_{∗}f^{∗} = deg(f)Id_{Cycl(Y)}.

3 Relative cycles.

3.1 Relative cycles.

Definition 3.1.1 Let S be a Noetherian scheme, k be a field and x : Spec(k) → S be a k-point of S. A fat point over x is a triple (x_{0}, x_{1}, R), where R is a discrete valuation ring and x_{0} : Spec(k) → Spec(R), x_{1} : Spec(R) → S are morphisms such that

1. x = x_{1} ∘ x_{0}

2. The image of x_{0} is the closed point of Spec(R).

3. x_{1} takes the generic point of Spec(R) to a generic point of S.

Usually we will abbreviate the notation (x_{0}, x_{1}, R) to (x_{0}, x_{1}).

Lemma 3.1.2 Let S be a Noetherian scheme, X → S be a scheme over S and Z be a closed subscheme in X. Let further R be a discrete valuation ring and f : Spec(R) → S be a morphism. Then there exists a unique closed subscheme ϕ_{f}(Z) in Z ×_{S} Spec(R) such that:

1. The closed embedding ϕ_{f}(Z) → Z ×_{S} Spec(R) is an isomorphism over the generic point of Spec(R).

2. ϕ_{f}(Z) is flat over Spec(R).

Proof: See [8, 2.8.5].

Let S be a Noetheiran scheme, X → S be a scheme of finite type over S and Z be a closed subscheme of X. For any fat point (x_{0}, x_{1}) over a k-point x of S we denote by (x_{0}, x_{1})^{∗}(Z/S) the cycle on X ×_{Spec(k)} S associated with the closed subscheme ϕ_{x_{1}}(Z) ×_{Spec(R)} Spec(k).

If Z = ∑ m_{i}z_{i} is any cycle on X we denote by (x_{0}, x_{1})^{∗}(Z) the cycle ∑ m_{i}(x_{0}, x_{1})^{∗}(Z_{i}) where Z_{i} is the closure of the point z_{i} ( considered as a reduced closed subscheme of X).

Definition 3.1.3 Let S be a Noetherian scheme and X → S be a scheme of finite type over S. A relative cycle on X over S is a cycle Z = ∑ m_{i}z_{i} on X satisfying the following requirements:

1. The points z_{i} lie over generic points of S.

2. For any field k, k-point x of S and a pair of fat points (x_{0}, x_{1}), (y_{0}, y_{1}) of S over x one has:

(x_{0}, x_{1})^{∗}(Z) = (y_{0}, y_{1})^{∗}(Z).

We say that Z = ∑ n_{i}z_{i} is a relative cycle of dimension r if each point z_{i} has dimension r in its fiber over S. We denote the corresponding abelian groups by Cycl(X/S, r).

We say that Z is an equidimensional relative cycle of dimension r if supp(Z) is equidimensional of dimesnion r over S. We denote the corresponding abelian groups by Cycl_{equi}(X/S, r).

We say that Z is a proper relative cycle if supp(Z) is proper over S. We denote the corresponding abelian groups by P ropCycl(X/S, r) and P ropCycl_{equi}(X/S, r).

We will also use the notations Cycl^{eff}(X/S, r), P ropCycl^{eff}(X/S, r) etc. for the corresponding abelian monoids of effective relative cycles.

The following lemma gives us means to construct fat points.

Lemma 3.1.4 Let S be a Noetherian scheme, η be a generic point of S and s be a point in the closure of η. Let further L be an extension of finite type of the field of functions on S in η. Then there is a discrete valuation ring R and a morphism f : Spec(R) → S such that the following conditions hold:

1. f maps the generic point of Spec(R) to η and the field of functions of R is isomorphic to L over k_{η},

2. f maps the closed point of Spec(R) to s.

Proof: See [6, 7.1.7].

Let S be a Noetherian scheme, X → S be a scheme of finite type over S and Z = ∑ n_{i}z_{i} be a cycle on X such that the points z_{i} lie over generic points of S and are of dimension r in the corresponding fibers. Let Z_{i} denote the closure of z_{i} considered as a closed integral subscheme of X. It is clear from the above definition that Z ∈ Cycl(X/S, r) if and only if Z ∈ Cycl(X_{red}/S_{red}, r). Furthemore the schemes Z_{i} are flat over generic points of S_{red} and according to Theorem 2.2.2 one can find a blow-up S′ → S_{red} such that the proper transforms Z̃_{i} of Z_{i}’s are flat over S′. Now we can formulate the following usefull criterion.

Proposition 3.1.5 Under the above assumptions the following conditions are equivalent:

1. Z ∈ Cycl(X/S, r).

2. If x : Spec(k) → S is any geometric point of S and x′_{1}, x′_{2} : Spec(k) → S′ is a pair of its liftings to S′ then the cycles W_{1}, W_{2} on X_{x} = X ×_{S} Spec(k) given by the formulae

W_{1} = ∑_{i=1}^{k} n_{i}cycl_{X_{s}}(Z̃_{i} ×_{x′_{1}} Spec(k))

W_{2} = ∑_{i=1}^{k} n_{i}cycl_{X_{s}}(Z̃_{i} ×_{x′_{2}} Spec(k))

coincide.

Proof: (1=>2) the geometric points x, x′_{1}, x′_{2} give us set-theoretical points s ∈ S, s′_{1}, s′_{2} ∈ S′ such that s′_{1}, s′_{2} lie over s. We may assume that s (and hence also s′_{1}, s′_{2}) is not generic. Using Lemma 3.1.4 we construct discrete valuation rings R′_{i} and morphisms Spec(R′_{i}) → S′ which map the closed point of Spec(R′_{i}) to s′_{i} and the generic point of Spec(R′_{i}) to a generic point of S′.

Denote the residue fields of R′_{i} by k′_{i}. One checks easily that the scheme (Spec(k′_{1})×_{S} Spec(k′_{2}))×_{S′×_{S}S′} Spec(k) is not empty. Choosing any geometric L-point of this scheme for a field L we get a commutative diagram

Spec(k′_{1}) → Spec(R′_{1}) → S′
↗  ↗^{x′_{1}}  ↘
Spec(L)  →  Spec(k)  →^{x}  S
↘  ↘^{x′_{2}}  ↗
Spec(k′_{2}) → Spec(R′_{2}) → S′

Thus we get a geometric point Spec(L) → S and two fat points Spec(L) → Spec(R′_{i}) → S over it. The inverse images of the cycle Z with respect to these fat points are equal to ∑ n_{i}cycl[(Z̃_{i} ×_{x′_{1}} Spec(k)) ×_{Spec(k)} Spec(L)] and ∑ n_{i}cycl[(Z̃_{i} ×_{x′_{2}} Spec(k)) ×_{Spec(k)} Spec(L)] respectively. These cycles coincide according to the condition Z ∈ Cycl(X/S, r). Lemma 2.3.1 shows now that W_{1} = W_{2}. (2=>1) Let x : Spec(k) → S be a geometric point of S and let (x_{0}, x_{1}), (y_{0}, y_{1}) be a pair of fat points over x. According to the valuative criterion of properness (see [6, 7.3]) these fat points have canonical liftings to fat points (x′_{0}, x′_{1}), (y′_{0}, y′_{1}) of S′. This gives us two geometric points

x′ = x′_{1} ∘ x′_{0} : Spec(k) → S′
y′ = y′_{1} ∘ y′_{0} : Spec(k) → S′

of S′ over x. Our statement follows now from obvious equalities:

(x_{0}, x_{1})^{∗}(Z) = (x_{0}, x′_{1})^{∗}(∑ n_{i}Z̃_{i}) = ∑ n_{i}cycl(Z̃_{i} ×_{x′} Spec(k))
(y_{0}, y_{1})^{∗}(Z) = (y_{0}, y′_{1})^{∗}(∑ n_{i}Z̃_{i}) = ∑ n_{i}cycl(Z̃_{i} ×_{y′} Spec(k)).

Corollary 3.1.6 Let k be a field and X → Spec(k) be a scheme of finite type over k. Then the group Cycl(X/Spec(k), r) is the free abelian group generated by points of dimension r on X, i.e one has

Cycl(X/Spec(k), r) = Cycl_{equi}(X/Spec(k), r) = Cycl(X, r).

Proposition 3.1.7 Let S be a Noetherian scheme, X → S be a scheme of finite type over S and Z = ∑_{i=1}^{k} n_{i}z_{i} be an effective cycle on X which belongs to Cycl(X/S, r) for some r ≥ 0. Denote by Z_{i} the closure of the point z_{i} in X which we consider as an integral closed subscheme in X. Then Z_{i} is equidimensional of dimension r over S.

Proof: According to the Chevalley theorem 2.1.1 all components of all fibers of the projection Z_{i} → S are of dimension ≥ r. Assume that there exists a point s of S such that the fiber (Z_{0})_{s} of Z_{0} over s has a component of dimension > r. Let η be the generic point of this component.

By Theorem 2.2.2 there is a blow-up f : S′ → S_{red} of S_{red} such that the proper transforms Z̃_{i} of the subschemes Z_{i} with respect to f are flat (and hence equidimensional of dimension r over S′). The morphism Z̃_{0} → Z_{0} is proper and dominant and hence surjective. Let τ be any point of Z̃_{0} over η and let s′_{1} be its image in S′. On the other hand let s′_{2} be any closed point of the fiber (S′)_{s}. Choosing an algebraically closed field k which contains a composite of the fields k_{s′_{1}} and k_{s′_{2}} over k_{s} we get two geometric points x′_{1}, x′_{2} : Spec(k) → S′ over the same geometric point x : Spec(k) → S. Consider the following cycles on X_{k} = X ×_{S} Spec(k):

W_{1} = ∑ n_{i}cycl_{X_{k}}(Z̃_{i} ×_{s′_{1}} Spec(k))
W_{2} = ∑ n_{i}cycl_{X_{k}}(Z̃_{i} ×_{s′_{2}} Spec(k)).

In view of Proposition 3.1.5 it is sufficient to show that these cycles are different. We will do so by showing that the images of supp(W_{1}) and supp(W_{2}) in X_{s} are different. The image of supp(W_{2}) coincides with the image of ∪(Z̃_{i})_{s′_{2}} ⊂ X_{k_{s′_{2}}} in X_{s} and hence is of dimension ≤ r (since the morphism X_{s′_{2}} → X_{s} is finite). On the other hand the image of supp(W_{1}) contains η and hence is of dimension > r.

Corollary 3.1.8 Let S be a Noetherian scheme and X → S be a scheme of finite type over S. Then one has:

Cycl_{equi}^{eff}(X/S, r) = Cycl^{eff}(X/S, r)
P ropCycl_{equi}^{eff}(X/S, r) = P ropCycl^{eff}(X/S, r).

Example 3.1.9 Proposition 3.1.7 fails for cycles which are not effective. Consider the scheme X = P_{k}^{1} × A^{2} over S = A_{k}^{2} where k is a field. Consider the following two rational functions on S:

f(x, y) = y/x
g(x, y) = y/(x + y^{2}).

Let Γ_{f}, Γ_{g} ∈ X be the graphs of these functions and let Z be the cycle on X of the form Γ_{f} − Γ_{g}. Obviously supp(Z) is not equidimensional over S. We claim that Z nevertheless belongs to Cycl(X/S, 0).

Let S′ → S be the blow-up of the point (0, 0) in A^{2}. Denote by f′, g′ the rational functions on S′ which correspond to f and g. One can easily see that f′, g′ are in fact regular on S′ and moreover if S′_{0} ⊂ S′ denotes the exceptional divisor of S′ we have f′_{S′_{0}} = g′_{S′_{0}}. Since the proper transforms of the closed subschemes Γ_{f}, Γ_{g} are the graphs of f′ and g′ in S′ ×_{S} X = S′ ×P_{k}^{1} Proposition 3.1.5 shows that Z indeed belongs to Cycl(X/S, 0).

3.2 Cycles associated with flat subschemes.

Let p : X → S be a morphism of finite type of Noetherian schemes. We denote by Hilb(X/S, r) (resp. by P ropHilb(X/S, r)) the set of closed subschemes Z of X ×_{S} S which are flat (resp. flat and proper) and equidimensional of dimension r over S.

Let N(Hilb(X/S, r)), N(P ropHilb(X/S, r)) (resp. Z(Hilb(X/S, r)), Z(P ropHilb(X/S, r))) be the corresponding freely generated abelian monoids (resp. abelian groups).

The assigment S′/S → N(Hilb(X ×_{S} S′/S′, r)) etc. defines a presheaf of abelian monoids (groups) on the category of Noetherian schemes over S. If Z = ∑ n_{i}Z_{i} is an element of Z(Hilb(X/S, r)) and S′ is a Noetherian scheme over S we denote by Z ×_{S} S′ the corresponding element ∑ n_{i}(Z_{i} ×_{S} S′) of Z(Hilb(X ×_{S} S′/S′, r)).

Lemma 3.2.1 Let p : X → S be a finite flat morphism of Noetherian schemes of constant degree and S′/S be any Noetherian scheme over S. Let further τ′ be a generic point of S′ and η′_{1}, …, η′_{k} be all points of X′ = X ×_{S} S′ lying over τ′. Then one has

∑_{j=1}^{k} length(O_{X′,η′_{j}})[k_{η′_{j}} : k_{τ′}] = deg(p)length(O_{S′,τ′})

Proof: The module ∐ O_{X′,η′_{j}} = [(p′)_{∗}(O_{X′})]_{τ′} is a free O_{S′,τ′}-module of rank deg(p). Computing its length over O_{S′,τ′} in two different ways we get the desired equality.

Proposition 3.2.2 Let X → S be a scheme of finite type over a Noetherian scheme S and S′ → S be any Noetherian scheme over S. Let further Z = ∑ n_{i}Z_{i} be an element of Z(Hilb(X/S, r)). If cycl_{X}(Z) = 0 then cycl_{X×_{S}S′}(Z ×_{S} S′) = 0.

Proof: Replacing X by ∪Z_{i} we may assume that X is equidimensional of relative dimension r over S. Generic points of Z_{i} ×_{S} S′ lie over generic points of S′ and are generic in their fibers. Let η′ ∈ X′ = X ×_{S} S′ be any of these generic points. Computing the multiplicity of η′ in cycl(Z ×_{S} S′) we may replace X by any open neighborhood of the point η = pr_{1}(η′) ∈ X.

Lemma 3.2.3 Let p : X → S be a flat equidimensional morphism of dimension r and x be a generic point of a fiber of p. Then for any decomposition of p of the form X →^{p_{0}} A_{S}^{r} → S such that p_{0} is an equidimensional quasi-finite morphism there exists an open neighborhood U of x in X such that (p_{0})_{|U} is a flat quasi-finite morphism.

Proof: It follows immediately from the fact that p is flat and [5, Ex. IV, Cor. 5.9,p.99].

Using Lemma 3.2.3 and Proposition 2.1.3 and replacing S by A_{S}^{r} we see that it is sufficient to treat the case r = 0. Furthemore replacing S′ by Spec(O_{S′,τ′}) and S by Spec(O_{s,τ}) (where τ = p(η) and τ′ = p′(η′)) we may assume that S and S′ are local schemes, S′ is Artinian and f : S′ → S takes the closed point of S′ to the closed point of S.

Let S^{sh} (resp. (S′)^{sh}) denote the strict henselization of the local scheme S (resp. S′) in the closed point. Lemma 2.3.1 shows that cycl(Z ×_{S} S^{sh}) = 0 and moreover that it is sufficient to check that

cycl((Z ×_{S} S′) ×_{S′} (S′)^{sh}) = cycl(Z ×_{S} (S′)^{sh}) = 0.

Since there exists a morphism (S′)^{sh} → S^{sh} over S we see that we may replace S and S′ by S^{sh} and (S′)^{sh} and assume the schemes S and S′ to be strictly henselian.

Since η lies over the closed point of S and S is henselian we conclude that Spec(O_{X,η}) is an open neighborhood of η in X and is finite over S (see [11, I.4.2(c)]). Thus replacing X by Spec(O_{X,η}) we may additionally assume that X is local and finite over S. In these assumptions η is the only point over τ (and hence η′ is the only point over τ′) and the schemes Z_{i} are finite and flat over S of constant degree deg(Z_{i}/S). Lemma 3.2.1 shows now that the multiplicity of η′ in cycl_{X′}(Z ×_{S} S′) is equal to

∑_{i} n_{i}length(O_{Z′_{i},η′}) = (∑_{i} n_{i}deg(Z_{i}/S))length(O_{S′,τ′})/[k_{η′} : k_{τ′}].

We only have to show that ∑ n_{i}deg(Z_{i}/S) = 0. To do so let τ^{0} be a generic point of S and let η_{1}^{0}, …, η_{k}^{0} be all points of X over τ^{0}. The multiplicity of η_{j}^{0} in cycl(Z) is equal to ∑ n_{i}length(O_{Z_{i},η_{j}^{0}}). Using once again Lemma 3.2.1 we get

0 = ∑_{j}[k_{η_{j}^{0}} : k_{τ^{0}}] ∑_{i} n_{i}length(O_{Z_{i},η_{j}^{0}}) = (∑_{i} n_{i}deg(Z_{i}/S))length(O_{S,τ^{0}}).

Proposition is proven.

Corollary 3.2.4 Let X → S be a scheme of finite type over a Noetherian scheme S, Z be an element of Hilb(X/S, r) and (x_{0}, x_{1}, R) be a fat point over a k-point x : Spec(k) → S of S. Then

(x_{0}, x_{1})^{∗}(cycl_{X}(Z)) = cycl_{X×_{S}Spec(k)}(Z ×_{S} Spec(k)).

Proof: Let Z_{i} be the irreducible components of Z, z_{i} be their generic points and n_{i} be their multiplicities such that cycl_{X}(Z) = ∑ n_{i}z_{i}. One checks easily that

cycl_{X×_{S}Spec(R)}(Z ×_{S} Spec(R)) = ∑ n_{i}cycl_{X×_{S}Spec(R)}(ϕ_{x_{1}}(Z_{i})).

Proposition 3.2.2 shows now that

(x_{0}, x_{1})^{∗}(cycl_{X}(Z)) = ∑ n_{i}cycl_{X×_{S}Spec(k)}(ϕ_{x_{1}}(Z_{i}) ×_{Spec(R)} Spec(k)).

Corollary 3.2.5 Let X → S be a scheme of finite type over a Noetherian scheme S. Then the image of the cycle map Z(Hilb(X/S, r)) → Cycl(X) lies in Cycl_{equi}(X/S, r).

Corollary 3.2.6 Let R be a discrete valuation ring and X → Spec(R) be a scheme of finite type over Spec(R). Then a cycle Z = ∑ n_{i}Z_{i} belongs to Cycl(X/Spec(R), r) if and only if the points z_{i} belong to the generic fiber of X over Spec(R) and are of dimension r in this fiber.

3.3 Chow presheaves.

Theorem 3.3.1 Let X → S be a scheme of finite type over a Noetherian scheme S, Z be an element of the group Cycl(X/S, r) and f : T → S be a Noetherian scheme over S.

Then there is a unique element Z_{T} in Cycl(X ×_{S} T/T, r) ⊗_{Z} Q such that for any commutative diagram of the form

↗^{y_{0}} Spec(A) →^{y_{1}} T
Spec(k)  ↓ f
↘^{x_{0}} Spec(R) →^{x_{1}} S

where (x_{0}, x_{1}) and (y_{0}, y_{1}) are fat k-points of S and T respectively one has:

(y_{0}, y_{1})^{∗}(Z_{T}) = (x_{0}, x_{1})^{∗}(Z).

Proof: The uniqueness of Z_{T} follows easily form 3.1.4 and 2.3.1(2).

In the proof of existence we start with the special case T = Spec(k_{s}) where s is a set-theoretic point of S. In this case we have the following slightly more precise statement.

Lemma 3.3.2 Denote by p the exponential characteristic of the field k_{s}. Then there exists a unique cycle Z_{s} in Cycl(X_{s}, r)[1/p] such that for any field extension k/k_{s} and any fat point (x_{0}, x_{1}) over the k-point Spec(k) → Spec(k_{s}) → S one has (x_{0}, x_{1})^{∗}(Z) = Z_{s} ⊗_{k_{s}} k.

Proof: Denote by Z_{i} the closure of z_{i} which we consider as an integral closed subscheme in X and choose a blow-up S′ → S_{red} such that the proper transforms Z̃_{i} of Z_{i} are flat over S′. If k/k_{s} is a field extension such that the k-point Spec(k) → Spec(k_{s}) → S admits a lifting to S′ we get a cycle Z_{k} = ∑ n_{i}cycl(Z̃_{i}×_{S′} Spec(k)) ∈ Cycl(X ×_{S} Spec(k), r) which is independent of the choice of the lifting according to Proposition 3.1.5. Moreover if L/k is a field extension then the cycle Z_{L} is also defined and we have

Z_{L} = Z_{k} ⊗_{k} L

.

The morphism S′ → S being a surjective morphism of finite type we can find a finite normal extension k_{0}/k_{s} such that the point Spec(k_{0}) → Spec(k_{s}) → S admits a lifting to S′. The formula above shows that the cycle Z_{k_{0}} is Gal(k_{0}/k_{s})-invariant and hence descends to a cycle Z_{s} ∈ Cycl(X_{s}, r)[1/p] by Lemma 2.3.2.

Let now k be any extension of k_{s} such that the point Spec(k) → Spec(k_{s}) → S admits a lifting to S′ and let L be a composite of k and k_{0} over k_{s}. Then

Z_{k} ⊗_{k} L = Z_{L} = Z_{k_{0}} ⊗_{k_{0}} L = Z_{s} ⊗_{k_{s}} L = (Z_{s} ⊗_{k_{s}} k) ⊗_{k} L

and hence Z_{k} = Z_{s} ⊗_{k_{s}} k.

Finally let k/k_{s} be a field extension and (x_{0}, x_{1}, R) be a fat point over a k-point Spec(k) → Spec(k_{s}) → S. The morphism x_{1} : Spec(R) → S has a canonical lifting to S′ (according to the valuative criterion of properness).

This gives us a lifting to S′ of our k-point Spec(k) → S and it follows immediately from the construction of Z_{s} that one has (x_{0}, x_{1})^{∗}(Z) = Z_{k} = Z_{s} ⊗_{k_{s}} k. Lemma is proven.

In the course of the proof of Lemma 3.3.2 we have established that the cycle Z_{s} has the following somewhat more general property.

Lemma 3.3.3 Let S′ → S be a blow-up such that the proper transforms Z̃_{i} of Z_{i} are flat over S′ and let k/k_{s} be a field extension such that the kpoint Spec(k) → Spec(k_{s}) → S admits a lifting to S′. Then Z_{s} ⊗_{k_{s}} k = ∑ n_{i}cycl(Z̃_{i} ×_{S′} Spec(k)).

Let τ_{1}, …, τ_{n} be the generic points of T and σ_{1}, …, σ_{n} be their images in S. Consider the cycles

Z_{σ_{j}} ⊗_{k_{σ_{j}}} k_{τ_{j}} = ∑_{l} n_{jl}z_{jl} ∈ Cycl(X ×_{S} Spec(k_{τ_{j}}), r) ⊗_{Z} Q.

Here n_{jl} are rational numbers and z_{jl} are points of X ×_{S} T lying over τ_{j} and having dimension r in their fibers. Set Z_{T} = ∑_{j,l} n_{jl}z_{jl}. We are going to show that Z_{T} belongs to Cycl(X ×_{S} T/T, r) ⊗_{Z} Q and has the desired property. To do so we need the following lemma.

Lemma 3.3.4 Let f : S′ → S be a proper surjective morphism of finite type of Noetherian schemes, A be a discrete valuation ring and Spec(A) → S be a morphism of schemes. Then there exists a commutative diagram of the form:

Spec(A′) → S′
g ↓  ↓ f
Spec(A) → S

In which A′ is a discrete valuation ring and the morphism g is surjective.

Proof: Let K be the quotient field of A. Since f is a surjective morphism of finite type there exists a finite extension K′ of K such that the K′-point Spec(K′) → Spec(K) → S admits a lifting to S′. Let ν be the discrete valuation of K which corresponds to A. It can be extended to a discrete valuation ν′ of K′ and we take A′ to be the corresponding discrete valuation ring. The valuative criterion of properness ([6, 7.3]) shows that the morphism Spec(A′) → Spec(A) → S admits a lifting to S′.

Consider a commutative diagram of the form

↗^{y_{0}} Spec(A) →^{y_{1}} T
Spec(k)  ↓ f
↘^{x_{0}} Spec(R) →^{x_{1}} S

in which (x_{0}, x_{1}) (resp. (y_{0}, y_{1})) is a fat k-point of S (resp. of T). Let as before S′ → S_{red} denote a blow-up of S_{red} such that the proper transforms Z̃_{i} of Z_{i} are flat over S′. Lemma 3.3.4 shows that there is a discrete valuation ring A′ and a surjective morphism Spec(A′) → Spec(A) such that the morphism Spec(A′) → S admits a lifting to S′. Denote the residue field of A (resp. of A′) by k_{A} (resp. by k_{A′}) and let k′ be a composite of k and k_{A′} over k_{A} so that we have the following commutative diagram:

Spec(k′) → Spec(A′) → S′
↓ ↓ ↓
Spec(k) →^{y_{0}} Spec(A) →^{y_{1}} T → S

Assume that y_{1} maps the generic point of Spec(A) to τ_{1} and consider the following two elements in Q(Hilb(X ×_{S} Spec(A′)/Spec(A′), r)):

W = ∑_{i} n_{i}(Z̃_{i} ×_{S′} Spec(A′))

W_{1} = ∑_{j,l} n_{jl}(ϕ_{y_{1}}(Z_{jl}) ×_{Spec(A)} Spec(A′)) = ∑_{l} n_{1l}(ϕ_{y_{1}}(Z_{1l}) ×_{Spec(A)} Spec(A′))

where Z_{jl} denotes the closure of the points z_{jl} considered as a closed integral subscheme of X ×_{S} T.

Lemma 3.3.5

cycl(W) = cycl(W_{1})

Proof: Let K (resp. K′) denote the quotient field of A (resp. of A′). Since the map

Cycl(X×_{S}Spec(A′)/Spec(A′), r)⊗Q → Cycl(X×_{S}Spec(K′)/Spec(K′), r)⊗Q

is clearly injective we may replace A′ by K′ everywhere. Furthermore

cycl(W ×_{Spec(A′)} Spec(K)) = ∑ n_{i}cycl(Z̃_{i} ×_{S′} Spec(K′))

and according to Lemma 3.3.3 this cycle is equal to Z_{σ_{1}} ⊗_{k_{σ_{1}}} K′. On the other hand we have:

cycl(W_{1} ×_{Spec(A′)} Spec(K′)) =

= ∑_{l} n_{1l}cycl([ϕ_{y_{1}}(Z_{1l}) ×_{Spec(A)} Spec(K)] ×_{Spec(K)} Spec(K′)) =

= ∑_{l} n_{1l}cycl((Z_{1l} ×_{T} Spec(K)) ×_{Spec(K)} Spec(K′)) =

= [∑_{l} n_{1l}cycl(Z_{1l} ×_{T} Spec(k_{τ_{1}}))] ⊗_{k_{τ_{1}}} K′ =

= (∑_{l} n_{1l}z_{1l}) ⊗_{k_{τ_{1}}} K′ = (Z_{σ_{1}} ⊗_{k_{σ_{1}}} k_{τ_{1}}) ⊗_{k_{τ_{1}}} K′

= Z_{σ_{1}} ⊗_{k_{σ_{1}}} K′.

Lemma is proven.

Proposition 3.2.2 implies now that

cycl(W ×_{Spec(A′)} Spec(k′)) = cycl(W_{1} ×_{Spec(A′)} Spec(k′))

i.e.

∑ n_{i}cycl(Z̃_{i} ×_{S′} Spec(k′)) = ∑_{l} n_{1l}cycl(ϕ_{y_{1}}(Z_{1l}) ×_{Spec(A′)} Spec(k′)) =

= (y_{0}, y_{1})^{∗}(Z_{T}) ⊗_{k} k′.

On the other hand the cycle (x_{0}, x_{1})^{∗}(Z) ⊗_{k} k′ is equal to ∑ n_{i}cycl(Z̃_{i} ×_{S′} Spec(k′)) where this time the morphism Spec(k′) → S′ is a lifting of the same point Spec(k′) → Spec(k) → Spec(R) → S obtained using the canonical lifting of the morphism Spec(R) → S. Proposition 3.1.5 shows that (x_{0}, x_{1})^{∗}(Z)⊗_{k}k′ = (y_{0}, y_{1})^{∗}(Z_{T})⊗_{k}k′ and hence (x_{0}, x_{1})^{∗}(Z) = (y_{0}, y_{1})^{∗}(Z_{T}). Theorem 3.3.1 is proven.

Remark: In general cycles of the form cycl(f)(Z) do not have integral coefficients. See example 3.5.10(1) below.

Lemmas 3.3.6 and 3.3.8 below describe the behavior of supports of cycles with respect to the base change homomorphisms.

Lemma 3.3.6 In the notations and assumptions of Theorem 3.3.1 we have

supp(Z_{T}) ⊂ (supp(Z))_{T} = supp(Z) ×_{S} T.

Proof: Since supp(Z_{T}) = ∪supp(Z_{k_{τ_{j}}}) where τ_{j} are the generic points of T it is sufficient to consider the case T = Spec(k) where k is a field. According to Lemma 3.1.4 there exists an extension k′/k and a fat point (x_{0}, x_{1}, R) over the k′-point Spec(k′) → Spec(k) → S. The defining property of the cycle Z_{k} shows that

supp(Z_{k}) ×_{Spec(k)} Spec(k′) = supp(Z_{k′}) = supp((x_{0}, x_{1})^{∗}(Z)) ⊂

⊂ ∪_{i}ϕ_{x_{1}}(Z_{i}) ×_{Spec(R)} Spec(k′) ⊂ ∪_{i}(Z_{i} ×_{S} Spec(R)) ×_{Spec(R)} Spec(k′) =

= supp(Z) ×_{S} Spec(k′) = (supp(Z) ×_{S} Spec(k)) ×_{Spec(k)} Spec(k′).

Since the morphism X_{k′} → X_{k} is surjective the above inclusion implies the desired one supp(Z_{k}) ⊂ supp(Z) ×_{S} Spec(k).

Lemma 3.3.7 Consider a pull-back square of morphisms of finite type of Noetherian schemes of the form

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

and assume that the morphism f is universally open and any generic point of X lies over a generic point of S. Then any generic point of X′ lies over a generic point of S′.

Proof: Any generic point of X′ obviously lies over a generic point of S. Replacing S by this point we may assume that S = Spec(k) where k is a field. Then the morphism X → S is universally open (being flat) and hence the morphism X′ → S′ is open, which implies that it takes generic points to generic points.

Lemma 3.3.8 Let X → S be a scheme of finite type over a Noetherian scheme S, Z = ∑ n_{i}z_{i} be an element of Cycl(X/S, r) ⊗ Q, f : S′ → S be a Noetherian scheme over S and Z′ = cycl(f)(Z) be the corresponding element of Cycl(X ×_{S} S′/S′, r) ⊗ Q.

1. If f is a universally open morphism then supp(Z′) = (supp(Z) ×_{S} S′)_{red}.

2. If f is dominant then Supp(Z) is the closure of (X ×_{S} f)(Supp(Z′)).

Proof: 1. The inclusion

supp(Z′) ⊂ (supp(Z) ×_{S} S′)_{red}

follows from Lemma 3.3.6. Lemma 3.3.7 implies immediately that generic points of supp(Z)×_{S}S′ lie over generic points of S′. We may assume therefore that S′ = Spec(k) and the image of S′ in S is a generic point η of S. Then k is an extension of k_{η} and according to Lemma 2.3.1(2) we have

Supp(Z′) = Supp(Z_{η} ⊗_{k_{η}} k) = (SuppZ_{η} ×_{Spec(k_{η})} Spec(k))_{red}.

Now it suffices to note that

Z_{η} = ∑_{z_{i}/η} n_{i}z_{i}

(the sum being taken over those points z_{i} which lie over η) and hence Supp(Z_{η}) = Supp(Z) ×_{S} Spec(k_{η}). 3. It suffices to show that z_{i} ∈ (X ×_{S} f)(Supp(Z′)). Denote by η_{i} the image of z_{i} in S and let η′_{i} be a generic point of S′ over η_{i}. Using Lemma 3.3.6 and the part (1) of the present lemma we get

Supp(Z ×_{Spec(k_{η_{i}})} Spec(k_{η′_{i}}))_{red} = Supp(Z_{k_{η′_{i}}}) =
= Supp(Z′_{k_{η′_{i}}}) ⊂ Supp(Z′)

It suffices to note now that the morphism

Supp(Z) ×_{Spec(k_{η_{i}})} Spec(k_{η′_{i}}) → Supp(Z)

is surjective.

Let f : T → S be a morphism of Noetherian schemes and X → S be a scheme of finite type over S. We denote by

cycl(f) : Cycl(X/S, r) ⊗_{Z} Q → Cycl(X ×_{S} T/T, r) ⊗_{Z} Q

the homomorphism Z → Z_{T} constructed in Theorem 3.3.1. Lemma 3.3.6 shows that this homomorphism takes Cycl_{equi}(X/S, r)⊗_{Z}Q to Cycl_{equi}(X ×_{S} T/T, r) ⊗_{Z} Q, P ropCycl(X/S, r) ⊗_{Z} Q to P ropCycl(X ×_{S} T/T, r) ⊗_{Z} Q etc. We use the same notation cycl(f) for the homomorphisms induced on the corresponding groups and monoids.

It follows immediately from definitions and Lemma 3.1.4 that

cycl(g ∘ f) = cycl(f) ∘ cycl(g)

for any composable pair of morphisms of Noetherian schemes.

This shows that for any scheme of finite type X → S over a Noetherian scheme S there is a presheaf of Q-vector spaces Cycl(X/S, r)_{Q} on the category of Noetherian schemes over S such that for any Noetherian scheme S′ over S one has Cycl(X/S, r)_{Q}(S′) = Cycl(X ×_{S} S′/S′, r) ⊗ Q.

Similarly we have presheaves of Q-vector spaces P ropCycl(X/S, r)_{Q}, Cycl_{equi}(X/S, r)_{Q}, P ropCycl_{equi}(X/S, r)_{Q} and presheaves of uniquely divisible abelian monoids Cycl^{eff}(X/S, r)_{Q^{+}}, P ropCycl^{eff}(X/S, r)_{Q^{+}}.

For any Noetherian scheme S′ over S we have a canonical lattice Cycl(X ×_{S} S′, r) in the Q-vector space Cycl(X/S, r)_{Q}(S′). Unfortunately these lattices do not form in general a subpresheaf in Cycl(X/S, r)_{Q} (see example 3.5.10(1)).

Lemma 3.3.9 Let S be a Noetherian scheme, X → S be a scheme of finite type over S and Z be an element of Cycl(X/S, r). Then the following conditions are equivalent:

1. For any Noetherian scheme T over S the cycle Z_{T} belongs to Cycl(X×_{S} T/T, r).

2. For any point s ∈ S the cycle Z_{s} belongs to Cycl(X_{s}, r).

3. For any point s ∈ S there exists a separable field extension k/k_{s} such that the cycle Z_{k} = Z_{s} ⊗_{k_{s}} k belongs to Cycl(X ×_{S} Spec(k), r).

Proof: The implication (3=>2) follows from Lemma 2.3.2. The other implications are obvious.

We denote by z(X/S, r) (resp. c(X/S, r), z_{equi}(X/S, r), c_{equi}(X/S, r)) the subgroup of Cycl(X/S, r) (resp. of P ropCycl(X/S, r), Cycl_{equi}(X/S, r), P ropCycl_{equi}(X/S, r)) consisting of cycles satisfying the equivalent conditions of Lemma 3.3.9. It is clear that z(X/S, r) (resp. ...) is a subpresheaf in the presheaf Cycl(X/S, r)_{Q}. Moreover

c(X/S, r) = z(X/S, r) ∩ P ropCycl(X/S, r)_{Q}

etc.

Lemma 3.3.10 Let X → S be a scheme of finite type over a Noetherian scheme S, T → S be a Noetherian scheme over S and W be an element of Z(Hilb(X/S, r)). Then

[cycl_{X}(W)]_{T} = cycl_{X×_{S}T} (W_{T}).

Proof: It is sufficient to treat the case W = Z where Z is a closed subscheme of X flat and equidimensional of relative dimension r over S. Consider any commutative diagram of the form

Spec(A) →^{y_{1}} T

↗^{y_{0}}

Spec(k)  ↓ f

↘^{x_{0}}

Spec(R) →^{x_{1}} S

in which (x_{0}, x_{1}) (resp. (y_{0}, y_{1})) is a fat k-point of S (resp. of T). Corollary 3.2.4 shows that

(y_{0}, y_{1})^{∗}(cycl_{X×_{S}T} (Z ×_{S} T)) = cycl_{X×_{S}Spec(k)}(Z ×_{S} Spec(k)) =
= (x_{0}, x_{1})^{∗}(cycl_{X}(Z)).

Thus the cycle cycl_{X×_{S}T} (Z ×_{S} T) satisfies the requirements defining the cycle [cycl_{X}(Z)]_{T}.

Corollary 3.3.11 The homomorphisms cycl_{X} define a homomorphism of presheaves

cycl : Z(Hilb(X/S, r)) → z_{equi}(X/S, r).

Lemma 3.3.12 Let X → S be a scheme of finite type over a Noetherian scheme S and f : S′ → S be a flat morphism of Noetherian schemes. Assume further that the schemes S, S′ are reduced. Then for any element Z in Cycl(X/S, r) one has

cycl(f)(Z) = f_{X}^{∗}(Z)

where f_{X} = pr_{1} : X ×_{S} S′ → X and f_{X}^{∗} is the flat pull-back defined in section 2.3.

Proposition 3.3.13 Let S be a Noetherian scheme of exponential characteristic n and X → S be a scheme of finite type over S. Then the subgroups Cycl(X ×_{S} T/T, r)[1/n] in Cycl(X ×_{S} T/T, r)⊗Q for Noetherian schemes T over S form a subpresheaf Cycl(X/S, r)[1/n] in the presheaf Cycl(X/S, r)_{Q}.

Proof: It follows easily from Lemmas 3.3.2 and 3.3.9.

Proposition 3.3.14 Let X → S be a scheme of finite type over a Noetherian scheme S. Then the quotient presheaf Cycl(X/S, r)_{Q}/z(X/S, r) is a presheaf of torsion abelian groups.

Proof: We have to show that for any Z = ∑ n_{i}z_{i} ∈ Cycl(X/S, r) there exists an integer N > 0 such that NZ ∈ z(X/S, r). Using Noetherian induction we may assume that for any closed subscheme T ⊂ S such that T ≠ S there exists an integer N(T) such that N(T)Z_{T} ∈ z(X ×_{S} T/T, r). Denote by Z_{i} the closure of the point z_{i} considered as an integral closed subscheme of X. Let S′ → S_{red} be a blow-up such that the proper transforms Z̃_{i} of Z_{i} are flat over S′. Let U be a dense open subset of S over which the morphism S′ → S_{red} is an isomorphism and let T be the closed reduced subscheme S − U. We claim that N(T)Z ∈ z(X/S, r). By Lemma 3.3.9 it is sufficient to verify that N(T)Z_{s} ∈ Cycl(X_{s}, r) for any point s ∈ S. If s ∈ T it follows from our choice of N(T). If s ∈ U it follows from Lemma 3.3.3.

Proposition 3.3.15 Let S be a regular Noetherian scheme. Then for any scheme of finite type X over S and any r ≥ 0 one has:

Cycl(X/S, r) = z(X/S, r)
Cycl_{equi}(X/S, r) = z_{equi}(X/S, r)

etc.

Proof: Let Z be an element of Cycl(X/S, r) and let s ∈ S be a point of S. We have to show that Z_{s} ∈ Cycl(X_{s}, r). We proceed by induction on n = dim(O_{S,s}).

If n = 1 then O_{S,s} is a discrete valuation ring so that we have a canonical fat point Spec(k_{s}) →^{x_{0}} Spec(O_{S,s}) →^{x_{1}} S over the point Spec(k_{s}) → S. Thus Z_{s} = (x_{0}, x_{1})^{∗}(Z) ∈ Cycl(X_{s}, r).

If n > 1 choose a regular system of parameters t_{1}, …, t_{n} in O_{S,s}. Replacing S by an appropriate open neighborhood of s we may assume that S is affine , t_{1}, …, t_{n} ∈ O(S) and the closed subscheme T of S defined by the equation t_{1} = 0 is regular and irreducible. Let τ be the generic point of T. Since dim(O_{S,τ}) = 1 we conclude that Z_{τ} ∈ Cycl(X_{τ}, r) and hence Z_{T} ∈ Cycl(X ×_{S} T/T, r). The induction hypothesis shows now that Z_{s} ∈ Cycl(X_{s}, r).

3.4 Relative cycles over geometrically unibranch schemes.

Lemma 3.4.1 Let k be a field, X → Spec(k), S → Spec(k) be two schemes of finite type over k and Z be a closed subscheme in X ×_{Spec(k)} S defined by nilpotent sheaf of ideals which is flat over S. Let further E be an extension of k and s_{1}, s_{2} be two E-points of S over k. If S is geometrically connected then the cycles associated with the closed subschemes Z ×_{s_{1}} Spec(E) and Z ×_{s_{2}} Spec(E) in X ×_{Spec(k)} Spec(E) coincide.

Proof: We may replace E by its algebraic closure and thus assume that E is algebraically closed. Next we may replace X by X_{E} = X ×_{Spec(k)} Spec(E), S be S_{E} = S ×_{Spec(k)} Spec(E) and Z by Z_{E} = Z ×_{Spec(k)} Spec(E) and thus assume that E = k. The scheme S being connected we may find a chain of rational points of S starting with s_{1} and ending with s_{2} and such that any pair of consecutive points belongs to the same irreducible component of S. Thus we may assume that S is an integral scheme. Let X_{1}, …, X_{n} denote the irreducible components of X which are considered as closed integral subschemes of X. Since S is integral and the base field is algebraically closed we conclude that the schemes X_{i} × S are integral and coincide with the irreducible components of X × S.

This shows that the cycle cycl_{X×S}(Z) may be written (uniquely) in the form ∑ n_{i}cycl_{X×S}(X_{i} × S). Proposition 3.2.2 shows now that the cycles cycl_{X}(Z ×_{s_{j}} Spec(k)) both coincide with ∑ n_{i}cycl_{X}(X_{i}).

Theorem 3.4.2 Let S be a Noetherian geometrically unibranch scheme and X → S be a scheme of finite type over S. Let further Z ⊂ X be a closed subscheme which is equidimensional of relative dimension r over S. Then cycl_{X}(Z) ∈ Cycl_{equi}(X/S, r).

Proof: Replacing Z by its irreducible components (which are also equidimensional over S) we may assume that the scheme Z is integral. Choose a blowup S′ → S_{red} such that the proper transform Z̃ of Z is flat over S′. Let further k be a field, s : Spec(k) → S be a k-point of S and s_{1}, s_{2} : Spec(k) → S′ be two liftings of s to S′. According to Proposition 3.1.5 we have to show that the cycles cycl(Z̃ ×_{s_{1}} Spec(k)), cycl(Z̃ ×_{s_{2}} Spec(k)) coincide. Note that according to Proposition 2.1.7 and Lemma 2.1.11 the closed subscheme Z̃ in Z ×_{S} S′ is defined by a nilpotent sheaf of ideals and hence Z̃ ×_{S} Spec(k) is a closed subscheme of (Z ×_{S} S′) ×_{S} Spec(k) = (Z ×_{S} Spec(k)) ×_{Spec(k)} (S′ ×_{S} Spec(k)) defined by a nilpotent sheaf of ideals. The scheme S′ ×_{S} Spec(k) is geometrically connected according to Proposition 2.1.6. Thus our statement follows from Lemma 3.4.1.

Corollary 3.4.3 Let S be a Noetherian geometrically unibranch scheme and X → S be a scheme of finite type over S. Then the abelian group Cycl_{equi}(X/S, r) (resp. P ropCycl_{equi}(X/S, r)) is freely generated by cycles of integral closed subschemes Z in X which are equidimensional (resp. proper and equidimensional) of dimension r over S.

Corollary 3.4.4 Let S be a Noetherian geometrically unibranch scheme and X → S be a scheme of finite type over S. Then the abelian group Cycl_{equi}(X/S, r) (resp. the abelian group P ropCycl_{equi}(X/S, r)) is generated by the abelian monoid Cycl^{eff}(X/S, r) (resp. by the abelian monoid P ropCycl^{eff}(X/S, r)).

Corollary 3.4.5 Let S be a Noetherian regular scheme and X be a scheme of finite type over S. Then abelian group z_{equi}(X/S, r)(S) (resp. the abelian monoid z^{eff}(X/S, r)(S)) is the free abelian group (resp. the free abelian monoid) generated by closed integral subschemes of X which are equidimensional of dimension r over S.

Corollary 3.4.6 Let S be a Noetherian regular scheme and X be a scheme of finite type over S. Then the abelian group c_{equi}(X/S, r)(S) (resp. the abelian monoid c^{eff}(X/S, r)(S)) is the free abelian group (resp. the free abelian monoid) generated by closed integral subschemes of X which are proper and equidimensional of dimension r over S.

Example 3.4.7 The statement of Corollary 3.4.4 is false for schemes S which are not geometrically unibranch. Let us consider the following situation.

Let S = S_{1} ∪S_{2} be a union of two copies of affine line (i.e. S_{1} ≅ S_{2} ≅ A^{1}) such that the point {0} (resp. the point {1}) of S_{1} is identified with the point {0} (resp. the point {1}) of S_{2}.

We take X to be abstractly isomorphic to S, i.e. X = X_{1} ∪ X_{2} also is a union of two copies of affine line glued together in the same way. Consider the morphism X → S which maps X_{1}, X_{2} identically on S_{1}. Using Proposition 3.1.5 one can easily see that:

Cycl^{eff}(X/S, 0) = 0
Cycl(X/S, 0) = Z

and hence the abelian group of relative cycles is not generated by abelian monoid of effective relative cycles in this case.

Remarks:

1. The statement of Corollary 3.4.3 is false for the abelian groups Cycl(X/S, r) and P ropCycl(X/S, r) since if dim(S) > 1 there exist elements Z in these groups such that supp(Z) is not equidimensional over S (see exmple 3.1.9).

2. It is not true in general that for a geometrically unibranch scheme S the groups z_{equi}(X/S, r), c_{equi}(X/S, r) (or abelian monoids z^{eff}(X/S, r), c^{eff}(X/S, r) are generated by elements which correspond to integral closed subschemes of X (see example 3.5.10(2)) but in view of Proposition 3.3.15 it is true for regular schemes S.

Proposition 3.4.8 Let S be a normal Noetherian scheme and X → S be a smooth scheme of finite type of dimension r over S. Then one has:

z_{equi}(X/S, r − 1) = Cycl_{equi}(X/S, r − 1)

c_{equi}(X/S, r − 1) = P ropCycl_{equi}(X/S, r − 1)

and the first group is isomorphic to the group of relative Cartier divisors on X over S.

Proof: Note first that any normal scheme is geometrically unibranch by [8]. By Corollary 3.4.3 the group Cycl_{equi}(X/S, r − 1) is generated by integral closed subschemes Z in X which are equidimensional of relative dimension r − 1 over S. By [8, 21.14.3] any such Z is flat over S and our result follows from the definition of relative Cartier divisor (see [9]) and Corollary 3.3.11.

3.5 Multiplicities of components of inverse images of equidimensional cycles over geometrically unibranch schemes.

Let S be a Noetherian geometrically unibranch scheme and X → S be a scheme of finite type over S. Corollary 3.4.3 shows that the abelian group Cycl_{equi}(X/S, r) is the free abelian group generated by generic points of closed integral subschemes Z ⊂ X of X which are equidimensional of relative dimension r over S. Let now T → S be a morphism of Noetherian schemes. Lemma 3.3.6 shows that the cycle [Cycl_{X}(Z)]_{T} is a formal linear combination of generic points of Z ×_{S} T with certain multiplicities. The aim of this section is to give an explicit formula for these multiplicities. It is sufficient to consider the case T = Spec(k_{s}) where s is a certain point of S. Moreover since the groups Cycl_{equi}(X/S, r) and Cycl_{equi}(X_{red}/S_{red}, r) coincide we may assume that the scheme S is reduced and since irreducible components of S do not intersect we may further assume that S is integral. The formula we are going to provide involves multiplicities of certain ideals so we start by recalling briefly the necessary definitions and results from commutative algebra (see [10]).

Let O be a local Noetherian ring of dimension r and M be a finitely generated O module. An ideal I ⊂ O is called an ideal of definition if I contains a certain power of the maximal ideal. For any ideal of definition I of O one may consider the so called Samuel function:

χ_{M}^{I}(n) = length(M/I^{n+1}M).

It is known that for n big enough χ_{M}^{I}(n) is a polynomial in n of degree at most r. Furthermore it may be written in the form

χ_{M}^{I}(n) = (e/r!)n^{r} + (terms of lower degree)

where e is a nonnegative integer called the multiplicity of I with respect to M. We will denote it by e(I, M). The integer e(I, O) is called the multiplicity of I and is denoted e(I).

Proposition 3.5.1 Let μ_{1}, …, μ_{k} be all minimal prime ideals of O such that dim(O/μ) = r, then

e(I, M) = ∑_{i=1}^{k} e(I, O/μ_{i})length_{O_{μ_{i}}}(M_{μ_{i}}) = ∑_{i=1}^{k} e(I(O/μ_{i}))length_{O_{μ_{i}}}M_{μ_{i}}.

Proof: See [10] Theorem 14.7.

Lemma 3.5.2 Let O → O′ be a local homomorphism of Noetherian local rings. Let M_{O} be the maximal ideal of O and suppose that O′ is a flat Oalgebra and M_{O}O′ is an ideal of definition of O′. Then for any ideal of definition I of O the following formula holds

e(IO′) = e(I)length_{O′}(O′/M_{O}O′).

Proof: Flatness of O′ over O implies that for any n we have

χ_{O′}^{IO′}(n) = length_{O′}(O′ ⊗_{O} O/I^{n}) = length_{O}(O/I^{n})length_{O′}(O′/M_{O}O′) =

= χ_{O}^{I}(n)length_{O′}(O′/M_{O}O′).

The following property of multiplicities is obvious.

Lemma 3.5.3 Assume that O′ is a finite local O algebra such that dim(O′) = dim(O) and M is a finitely generated O′-module. Let k and k′ be the residue fields of O and O′. Then for any ideal of definition I of O the following formula holds:

e(IO′, M) = e(I, M)/[k′ : k].

Let Z → S be a scheme equidimensional of relative dimension r over an integral Noetherian geometrically unibranch scheme S. Let further s be a point of S, I be an ideal of definition of the local ring O_{S,s} and z be a generic point of the fiber Z_{s} = Z ×_{Spec(k_{s})} S. We set n_{I}(z) = e(IO_{Z,z})/e(I).

Lemma 3.5.4 1. n_{I}(z) is independent of the choice of I.

2. If Z is flat over S then n_{I}(z) coincides with the multiplicity of z in the cycle (cycl_{Z}(Z))_{s} = cycl_{Z_{s}}(Z_{s}).

Proof: The second statement follows immediately from Lemma 3.5.2. Replacing Z by an appropriate neighborhood of z we may assume that the morphism Z → S admits a decomposition of the form Z →^{p_{0}} A_{S}^{r} → S where p_{0} is an equidimensional quasi-finite morphism (see Proposition 2.1.3). Denote p_{0}(z) by x. Then x is the (unique) generic point of the fiber of A_{S}^{r} over s and according to the part (2) of our proposition we have e(IO_{A_{S}^{r},x}) = e(I). This shows that n_{I}(z) = e(IO_{Z,z})/e(IO_{A_{S}^{r},x}) so that we may replace S by A_{S}^{r} and assume that r = 0.

Denote by S′ the henselization of S at s and by s′ the closed point of S′. Set also Z′ = Z ×_{S} S′. The fiber of Z′ over s′ is isomorphic to the fiber of Z over s and thus there exists exactly one point z′ lying over z. Since O_{S′,s′} = O_{S,s}^{h} is ind-etale over O_{S,s} and O_{Z′,z′} is ind-etale over O_{Z,z} we conclude from Lemma 3.5.2 that e(I) = e(IO_{S′,s}), e(IO_{Z,z}) = e(IO_{Z′,z′}) and hence n_{I}(z) = n_{IO_{S′,s′}}(z′). Thus replacing S by S′ and Z by Z′ we may assume that S is a henselian local scheme (note that S′ is integral since S is geometrically unibranch). In this case O_{Z,z} is a finite O_{S,s}-algebra and according to Lemma 3.5.3 and Proposition 3.5.1 we have

n_{I}(z) = \frac{e(IO_{Z,z})}{e(I)} = \frac{e(I, O_{Z,z})}{[k_{z} : k_{s}]e(I)} = \frac{e(I)dim_{F (S)}(O_{X,x} ⊗_{O_{S,s}} F (S))}{e(I)[k_{z} : k_{s}]} =

= \frac{dim_{F (S)}(O_{X,x} ⊗_{O_{S,s}} F (S))}{[k_{z} : k_{s}]}

where F (S) is the quotient field of the integral domain O_{S}. Thus n_{I}(z) does not depend on the choice of I.

We will use the notation n(z) or n_{Z/S}(z) for the common value of multiplicities n_{I}(z). We will denote by [Z/S]_{s} the element of Cycl(Z_{s}, r) of the form ∑ n(z)z where the sum is taken over all generic points of Z_{s} = Z ×_{S} Spec(k_{s}). In the course of the proof of Lemma 3.5.4 we have established some useful properties of multiplicities n(z) which we would like to list now for future use.

Corollary 3.5.5 1. Assume that the morphism Z → S is factorized in the form Z → A_{S}^{r} → S where the first morphism is quasi-finite and equidimensional. Then for any point z ∈ Z generic in its fiber over S one has n_{Z/S}(z) = n_{Z/A_{S}^{r}}(z).

2. Let S′ be the henselization of S at s, s′ be the closed point of S′ and z′ be the only point of Z′ = Z ×_{S} S′ lying over z. Then n_{Z′/S′}(z′) = n_{Z/S}(z).

3. Let Z → S be a finite equidimensional morphism. Assume that the fiber Z_{s} consists of only one point z. Then

n(z) = \frac{dim_{F (S)}(O_{Z,z} ⊗_{O_{S,s}} F (S))}{[k_{z} : k_{s}]}

where F (S) is the function field of the integral scheme S.

Proposition 3.5.6 Let p : Z → S be an equidimensional scheme of relative dimension r over an integral Noetherian geometrically unibranch scheme S. Let S′ be another integral Noetherian geometrically unibranch scheme and let f : S′ → S be a dominant morphism. Set Z′ = Z ×_{S} S′. Let finally s′ be any point of S′ and s be its image in S. Then [Z′/S′]_{s′} = [Z/S]_{s} ⊗_{k_{s}} k_{s′}.

In other words if z′ is a generic point of the fiber Z′_{s′} and z is its image in Z then n(z′) = n(z)length((k_{s′} ⊗_{k_{s}} k_{z})_{μ_{z′}}) where μ_{z′} is the ideal ker(k_{s′} ⊗_{k_{s}} k_{z} → k_{z′}).

Proof: Replacing Z by an appropriate neighborhood of z we may assume by Proposition 2.1.3 that the morphism Z → S admits a decomposition of the form Z →^{p_{0}} A_{S}^{r} → S where p_{0} is an equidimensional quasi-finite morphism. Corollary 3.5.5(1) shows that n_{Z/S}(z) = n_{Z/A_{S}^{r}}(z) and n_{Z′/S′}(z′) = n_{Z′/A_{S′}^{r}}(z′). Furthermore denoting by x (resp. x′) the image of z (resp. z′) in A_{S}^{r} (resp. A_{S′}^{r}) one checks easily that the local rings (k_{s′} ⊗_{k_{s}} k_{z})_{μ_{z′}} and (k_{x′} ⊗_{k_{x}} k_{z})_{μ_{z′}} coincide. Thus we may replace S by A_{S}^{r}, S′ by A_{S′}^{r} and assume that r = 0. We certainly may also assume S, S′ to be local schemes and s, s′ to be their closed points.

Consider first the special case S′ = Spec(O_{S,s}^{sh}) where O_{S,s}^{sh} is the strict henselization of O_{S,s}. Since O_{S′,s′} (resp. O_{Z′,z′}) is ind-etale over O_{S,s} (resp. O_{Z,z}) we conclude from Lemma 3.5.2 that e(IO_{Z′,z′}) = e(IO_{Z,z}), e(IO_{S′,s′}) = e(IO_{S,s}) for any ideal of definition I of O_{S,s} and hence n(z) = n(z′). On the other hand k_{s′} is a separable algebraic extension of k_{s}. Thus (k_{s′} ⊗_{k_{s}} k_{z})_{μ_{z′}} is a reduced local Artinian ring and hence a field, i.e.

length((k_{s′} ⊗_{k_{s}} k_{z})_{μ_{z′}}) = 1.

In the general case there exists a morphism f′ : Spec(O_{S′,s′}^{sh}) → Spec(O_{S,s}^{sh}) such that the diagram

Spec(O_{S′,s′}^{sh}) →^{f′} Spec(O_{S,s}^{sh})
↓  ↓
S′  →^{f}  S

commutes. In view of the special case considered above we may replace f by f′ and assume that S and S′ are strictly henselian local schemes. Finally replacing Z by Spec(O_{Z,z}) we may assume that Z is finite over S and z is its only point over s. In this situation k_{z} is a purely inseparable extension of k_{s} and hence the Artinian ring k_{s′} ⊗_{k_{s}} k_{z} is local (i.e. z′ is the only point of Z′ over s′) and its residue field coincides with k_{z′}. Using Corollary 3.5.5(3) we conclude that

n(z) = \frac{dim_{F (S)}(O_{Z,z} ⊗_{O_{S,s}} F (S))}{[k_{z} : k_{s}]}

n(z′) = \frac{dim_{F (S′)}(O_{Z′,z′} ⊗_{O_{S′,s′}} F (S′))}{[k_{z′} : k_{s′}]}

length((k_{s′} ⊗_{k_{s}} k_{z})_{μ_{z′}}) = length(k_{s′} ⊗_{k_{s}} k_{z}) = \frac{[k_{z} : k_{s}]}{[k_{z′} : k_{s′}]}

Now it is sufficient to note that

O_{Z′,z′} ⊗_{O_{S′,s′}} F (S′) = (O_{Z,z} ⊗_{O_{S,s}} O_{S′,s′}) ⊗_{O_{S′,s′}} F (S′) =
= (O_{Z,z} ⊗_{O_{S,s}} F (S)) ⊗_{F (S)} F (S′)

and hence

dim_{F (S)}(O_{Z,z} ⊗_{O_{S,s}} F (S)) = dim_{F (S′)}(O_{Z′,z′} ⊗_{O_{S′,s′}} F (S′)).

Lemma 3.5.7 Under the assumptions of Proposition 3.5.6 let Z′ be a closed subscheme of Z defined by a nilpotent sheaf of ideals and such that the closed embedding Z′ → Z is an isomorphism over the generic point of S. Then for any point s of S the cycles [Z′/S]_{s} and [Z/S]_{s} coincide.

Proof: Let z be a generic point of the fiber Z_{s}. We have to show that n_{Z/S}(z) = n_{Z′/S}(z). Let I be an ideal of definition of the local ring O_{S,s}, let μ_{1}, …, μ_{n} be all minimal prime ideals of O_{Z,z} and let ν be the kernel of the natural surjective homomorphism O_{Z,z} → O_{Z′,z}. The ideal ν being nilpotent is contained in μ_{1} ∩ … ∩ μ_{n} and the ideals μ_{1}/ν, …, μ_{n}/ν are the minimal prime ideals of O_{Z′,z}. Furthermore our assumptions imply that (O_{Z,z})_{μ_{i}} = (O_{Z′,z})_{μ_{i}/ν}. Proposition 3.5.1 shows that

e(IO_{Z,z}) = ∑_{i} e(I(O_{Z,z}/μ_{i}))length((O_{Z,z})_{μ_{i}}) =

= ∑_{i} e(I(O_{Z′,z}/(μ_{i}/ν)))length((O_{Z′,z})_{μ_{i}/ν}) = e(IO_{Z′,z}).

Theorem 3.5.8 Let X → S be a scheme of finite type over an integral Noetherian geometrically unibranch scheme S. Let further Z be a closed integral subscheme of X equidimensional of relative dimension r over S and let Z be the corresponding element of Cycl(X/S, r). Then the cycles Z_{s} and [Z/S]_{s} coincide for any point s ∈ S of S.

Proof: Let k be an extension of k_{s} and let (x_{0}, x_{1}, R) be a fat point of S over the k-point Spec(k) → Spec(k_{s}) → S. To prove the theorem we have to show that (x_{0}, x_{1})^{∗}(Z) = [Z/S]_{s} ⊗_{k_{s}} k. Since Z ×_{S} Spec(R) is equidimensional over Spec(R) and the closed embedding ϕ_{x_{1}}(Z) → Z×_{S}Spec(R) is an isomorphism over the generic point of Spec(R) we conclude by Lemma 2.1.10 that ϕ_{x_{1}}(Z) is defined by a nilpotent sheaf of ideals. Let s′ be the closed point of Spec(R). Since ϕ_{x_{1}}(Z) is flat over Spec(R) we see from 3.5.4(2),3.5.7 and 3.5.6 that

(x_{0}, x_{1})^{∗}(Z) = cycl(ϕ_{x_{1}}(Z) ×_{Spec(R)} Spec(k)) = [ϕ_{x_{1}}(Z)/Spec(R)]_{s′} =

= [Z ×_{S} Spec(R)/Spec(R)]_{s′} = [Z/S]_{s} ⊗_{k_{s}} k.

For cycles equidimensional over regular Noetherian schemes the above formula for multiplicities of components of the inverse image reduces to the usual T or-formula as one sees from the following lemma.

Lemma 3.5.9 Let Z → S be an equidimensional scheme of relative dimension r over a regular scheme S and s ∈ S be a point of S. If z is a generic point of the fiber Z_{s} of Z over s then one has

n_{Z/S}(z) = ∑_{i=0}^{dim(O_{S,s})}(−1)^{i}length_{O_{Z,z}}(T or_{i}^{O_{S,s}}(O_{Z,z}, k_{s})).

Proof: Let t_{1}, …, t_{k} be a regular system of parameters of the regular local ring O_{S,s}. Take I = M = t_{1}O_{S,s} +…+t_{n}O_{S,s} to be the maximal ideal of this ring. Since O_{S,s} is regular the multiplicity e(I) equals one (see [10]). Thus n(z) = e(IO_{Z,z}). Theorem of Serre ([15]) shows that

e(IO_{Z,z}) = ∑_{i}(−1)^{i}length_{O_{Z,z}}(H_{i}(K(\underline{t}, O_{Z,z})))

where K(\underline{t}, O_{Z,z}) is the Koszul complex corresponding to the sequence \underline{t} = (t_{1}, …, t_{n}). On the other hand K(\underline{t}, O_{S,s}) is a projective resolution of k_{s} over O_{S,s} and hence

H_{i}(K(\underline{t}, O_{Z,z})) = H_{i}(K(\underline{t}, O_{S,s}) ⊗_{O_{S,s}} O_{Z,z}) = T or_{i}^{O_{S,s}}(O_{Z,z}, k_{s}).

Example 3.5.10 1. Let F be a field of characteristic p > 0 and let a, b ∈ F^{∗} be elements independent modulo (F^{∗})^{p}. Set

A = F [T_{0}, T_{1}, T_{2}]/(aT_{0}^{p} + bT_{1}^{p} − T_{2}^{p})

and let S = Spec(A). One verifies easily that A is an integrally closed domain and hence S is a normal integral scheme. Let X be the normalization of S in the field F (S)(γ) where γ^{p} = b/a. It is easy to check that X is isomorphic to Spec(F (α, β)[T_{1}, T_{2}]) where α^{p} = a, β^{p} = b and the homomorphism A = F [S] → F [X] = F (α, β)[T_{1}, T_{2}] maps T_{0} to α^{−1}T_{2} − γ^{−1}T_{1}.

Take s to be the only singular point of S (i.e. T_{0}(s) = T_{1}(s) = T_{2}(s) = 0) and x to be the only point of X over s. Take I to be the maximal ideal of O_{S,s}. Then IO_{X,x} is the maximal ideal of the regular local ring O_{X,x} and hence e(IO_{X,x}) = 1. On the other hand one checks easily that e(I) = p (see [10, 14.5]). Thus n(x) = 1/p. Therefore cycles of the form cycl(f)(Z) do not have in general integral coefficients^{1}.

1This example is due to A.S.Merkurjev.

2. In the notations of previous example let Y be the scheme obtained by gluing of p copies of X in the singular point. One can easily see that the fundamental cycle on Y is an element of c(Y/S, 0) which can not be represented as a sum of cycles which correspond to integral subschemes of Y.

3.6 Functoriality of Chow presheaves.

Let f : S_{1} → S_{2} be a morphism of Noetherian schemes. We say that a closed subscheme Z of S_{1} is proper with respect to f if the restriction of f to Z is a proper morphism. We say that a point s of S_{1} is proper with respect to f if the closure of s in S_{1} which we consider as a reduced closed subscheme is proper whith respect to f.

Let S be a Noetherian scheme and f : X → Y be a morphism of schemes of finite type over S. Let further Z = ∑ n_{i}z_{i} be a cycle on X which lies over generic points of S. We say that Z is proper with respect to f if all the points z_{i} are proper with respect to f. We define then a cycle f_{∗}(Z) on Y as the sum ∑ n_{i}m_{i}f(z_{i}) where m_{i} is the degree of the field extension k_{f(z_{i})}/k_{z_{i}} if this extension is finite and zero otherwise.

Theorem 3.6.1 Let S be a Noetherian scheme, p : X_{1} → X_{2} be a morphism of schemes of finite type over S, and f : S′ → S be a Noetherian scheme over S. Set X′_{i} = X_{i} ×_{S} S′ (i = 1, 2) and denote by p′ : X′_{1} → X′_{2} be the corresponding morphism over S′. Let further Z = ∑ n_{i}Z_{i} (resp. W = ∑ m_{j}W_{j}) be an element of Z(Hilb(X_{1}/S, r)) (resp. of Z(Hilb(X_{2}/S, r))). Assume that the closed subschemes Z_{i} are proper with respect to p and

p_{∗}(cycl_{X_{1}}(Z)) = cycl_{X_{2}}(W).

Then the cycle cycl_{X′_{1}}(Z ×_{S} S′) is proper with respect to p′ and we have

p′_{∗}(cycl_{X′_{1}}(Z ×_{S} S′)) = cycl_{X′_{2}}(W ×_{S} S′).

Proof: Replacing X_{1} by ∪Z_{i} and X_{2} by (∪W_{j}) ∪ (∪p(Z_{i})) we may assume that p is proper, X_{1} is equidimensional of relative dimension r over S and all fibers of X_{2} over S are of dimension ≤ r.

Both cycles in question are linear combinations of points of X′_{2} which lie over generic points of S′, are generic in their fibers over S′ and are of dimension r in these fibers. Let η′ ∈ X′_{2} be any of such points and let η be its image in X_{2}. Computing the multiplicities of η′ in our cycles we may replace X_{2} by any open neighborhood of η. Let V be an irreducible component of X_{2} which is not equidimensional of dimension r over S. Then V ⊂ p(Z_{i}) for a certain i and hence V is dominated by a component of Z_{i}. Proposition 2.1.9 shows that all fibers of V over S are of dimension < r and hence η does not belong to V. Thus throwing away bad components we may assume that X_{2} is equidimensional of relative dimension r over S. Proposition 2.1.9 shows also that we may assume that p is a finite morphism. According to Proposition 2.1.3 we may assume further that the morphism X_{2} → S has a factorization of the form X_{2} → A_{S}^{r} → S where the first arrow is an equidimensional quasi-finite morphism. Since the morphism p maps components of X_{1} onto components of X_{2} we conclude that the composition X_{1} → X_{2} → A_{S}^{r} is also equidimensional (and quasifinite). Let Z_{i}^{0} (resp. W_{j}^{0}) be the closed subset of Z_{i} (resp. of W_{j}) consisting of points where Z_{i} (resp. W_{j}) is not flat over A_{S}^{r}. Lemma 3.2.3 shows that Z_{i}^{0} and W_{j}^{0} contain no points generic in their fibers over S. Thus η does not belong to (∪W_{j}^{0}) ∩ (∪p(Z_{i}^{0})) and shrinking X_{2} further around η we may assume that Z_{i} and W_{j} are flat over A_{S}^{r}. This shows that we may replace S by A_{S}^{r} and assume that r = 0.

Let τ′ (resp. τ) be the image of η′ (resp. η) in S′ (resp. S). We may replace S′ by Spec(O_{S′,τ′}) and assume that S′ is a local Artinian scheme. Let O_{S′,τ′}^{sh} and O_{S,τ}^{sh} be the strict henselizations of the local rings O_{S′,τ′} and O_{S,τ} respectively. Find a morphism f_{0} : Spec(O_{S′,τ′}^{sh}) → Spec(O_{S,τ}^{sh}) making the following diagram commute

Spec(O_{S′,τ′}^{sh}) →^{f_{0}} Spec(O_{S,τ}^{sh})
↓  ↓
Spec(O_{S′,τ′}) = S′ →  S.

In view of Lemma 2.3.1 it is sufficient to check that the flat pull-backs of cycles in question to the scheme X′_{2} ×_{S′} Spec(O_{S′,τ′}^{sh}) coincide. Proposition 2.3.4 shows that these pull-backs are equal to p′_{∗}(cycl(Z ×_{S} Spec(O_{S′,τ′}^{sh}))) and cycl(W ×_{S} Spec(O_{S′,τ′}^{sh})) respectively. Thus we may replace S′ by Spec(O_{S′,τ′}^{sh}) in the same way we may replace S by Spec(O_{S,τ}^{sh}) and thus assume that S and S′ are strictly henselian local schemes and τ, τ′ are their closed points. Finally replacing X_{2} by Spec(O_{X_{2},η}) we may assume that X_{2} is a local scheme finite over S. Since S is strictly local it implies that X′_{2} is also local i.e. η′ is the only point of X′_{2}. Let α′_{1}, …, α′_{k} be all the points of X′_{1}. Using Lemma 3.2.1 we see that the multiplicity of η′ in the cycle p′_{∗}(cycl_{X′_{1}}(Z ×_{S} S′)) is equal to

∑_{l=1}^{k}[k_{α′_{l}} : k_{η′}] ∑_{i} lengthO_{Z′_{i},α′_{l}} =

= ∑_{i} \frac{n_{i}}{[k_{η′} : k_{τ′}]} ∑_{l}[k_{α′_{l}} : k_{τ′}]lengthO_{Z′_{i},α′_{l}} =

= \frac{lengthO_{S′,τ′}}{[k_{η′} : k_{τ′}]} ∑_{i} n_{i}deg(Z_{i}/S).

On the other hand the multiplicity of η′ in the cycle cycl_{X′_{2}}(W ×_{S} S′) is equal to

∑_{j} m_{j}lengthO_{W′_{j},η′} = \frac{lengthO_{S′,τ′}}{[k_{η′} : k_{τ′}]} ∑_{j} m_{j}deg(W_{j}/S).

Thus we have to show that ∑ n_{i}deg(Z_{i}/S) = ∑ m_{j}deg(W_{j}/S). To do so choose a generic point τ^{0} of S, let α_{1}^{0}, …, α_{n}^{0} be all points of X_{2} over τ^{0} and for each s = 1, …, n let α_{st}^{0}(t = 1, …, n_{s}) be the points of X_{1} over α_{s}^{0}. Our assumption imply that for every s = 1, …, n we have the following equality

∑_{t}[k_{α_{st}^{0}} : k_{α_{s}^{0}}] ∑_{i} lengthO_{Z_{i},α_{st}^{0}} = ∑_{j} m_{j}lengthO_{W_{j},α_{s}^{0}}.

Taking the sum of this equalities with coefficients [k_{α_{s}^{0}} : k_{τ^{0}}] and using once again Lemma 3.2.1 we get what we wanted.

Proposition 3.6.2 Let p : X → Y be a morphism of schemes of finite type over a Noetherian scheme S and Z = ∑ n_{i}z_{i} be an element of Cycl(X/S, r) such that the points z_{i} are proper with respect to p. Then the following statements hold:

1. The cycle p_{∗}(Z) on Y belongs to Cycl(Y/S, r).

2. For any morphism f : S′ → S of Noetherian schemes the cycle cycl(f)(Z) has the form ∑ m_{j}z′_{j} where the points z′_{j} are proper with respect to p′ = p ×_{S} S′ and moreover

p′_{∗}(cycl(f)(Z)) = cycl(f)(p_{∗}(Z)).

Proof: Denote by Z_{i} (resp. W_{i}) the closure of z_{i} (resp. p(z_{i})) considered as an integral closed subscheme of X (resp. of Y). Replacing X by ∪Z_{i} we may assume that the morphism p is proper. Let k be a field, x : Spec(k) → S be a k-point of S and (x_{0}, x_{1}, R) be a fat point of S over x. Set

Z_{0} = ∑ n_{i}ϕ_{x_{1}}(Z_{i}) ∈ Z(Hilb(X ×_{S} Spec(R)/Spec(R), r))
W_{0} = ∑ n_{i}m_{i}ϕ_{x_{1}}(W_{i}) ∈ Z(Hilb(Y ×_{S} Spec(R)/Spec(R), r))

where m_{i} = [k_{z_{i}} : k_{p(z_{i})}] if this field extension is finite and zero otherwise.

It is clear from the definitions that cycl(W_{0}) = (p×_{S} Spec(R))_{∗}(cycl(Z_{0})). Theorem 3.6.1 implies now that

(x_{0}, x_{1})^{∗}(p_{∗}(Z)) = cycl(W_{0} ×_{Spec(R)} Spec(k)) =

= (p ×_{S} Spec(k))_{∗}cycl(Z_{0} ×_{Spec(R)} Spec(k)) = (p ×_{S} Spec(k))_{∗}((x_{0}, x_{1})^{∗}(Z)).

Thus the cycle (x_{0}, x_{1})^{∗}(p_{∗}(Z)) is independent of the choice of fat point (x_{0}, x_{1}, R) over x. The same argument shows now that for any morphism f : S′ → S of Noetherian schemes the cycle p′_{∗}(cycl(f)(Z)) meets the property defining the cycle cycl(f)(p_{∗}(Z)) and hence is equal to this cycle.

Corollary 3.6.3 Let S be a Noetherian scheme and f : X → Y be a morphism (resp. a proper morphism) of schemes of finite type over S. Then there are homomorphisms:

f_{∗} : c(X/S, r) → c(Y/S, r)
f_{∗} : c_{equi}(X/S, r) → c_{equi}(Y/S, r)
f_{∗} : c^{eff}(X/S, r) → c^{eff}(Y/S, r)

(resp. homomorphisms

f_{∗} : z(X/S, r) → z(Y/S, r)
f_{∗} : z_{equi}(X/S, r) → z_{equi}(Y/S, r)
f_{∗} : z^{eff}(X/S, r) → z^{eff}(Y/S, r))

such that for any composable pair of morphisms X →^{f} Y →^{g} Z of schemes of finite type over S one has (gf)_{∗} = g_{∗}f_{∗}.

Let us consider now the contravariant functoriality of Chow presheaves.

Lemma 3.6.4 Let S be a Noetherian scheme and f : X → Y be a flat equidimensional morphism of dimension n of schemes of finite type over S. Then for any element Z in Cycl(Y/S, r)⊗Q one has f^{∗}(Z) ∈ Cycl(X/S, r+ n) and for any Noetherian scheme g : S′ → S we have

cycl(g)(f^{∗}(Z)) = (f ×_{S} S′)^{∗}(cycl(g)(Z)).

Proof: Easy.

Let S be a Noetherian scheme, f : X → Y be a flat (resp. flat and proper) equidimensional morphism of relative dimension n of schemes of finite type over S and F (−, −) be one of the presheaves z(−, −), z^{eff}(−, −), z_{equi}(−, −) (resp. c(−, −), c^{eff}(−, −), c_{equi}(−, −)). If Z is a cycle on Y which belongs to F (Y/S, r) then by Lemma 3.6.4 the cycle f^{∗}(Z) belongs to F (X/S, r + n) and this construction gives us homomorphisms of presheaves

f^{∗} : F (Y/S, r) → F (X/S, r + n).

For any composable pair X →^{f} Y →^{g} Z of flat (resp. flat and proper) equidimensional morphisms of schemes of finite type over S we obviously have (gf)^{∗} = f^{∗}g^{∗}.

Proposition 3.6.5 Let S be a Noetherian scheme. Consider a pull-back square of schemes of finite type over S of the form:

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

such that the morphism f is flat and equidimensional of dimension r. Assume further that f is proper and F (−, −) is one of the presheaves c(−, −), c^{eff}(−, −), c_{equi}(−, −) or that p is proper and F (−, −) is one of the presheaves z(−, −), z^{eff}(−, −), z_{equi}(−, −).

Then the following diagram of presheaves commutes:

F (Y/S, n) →^{g^{∗}} F (Y′/S, n + d)
p_{∗} ↓  ↓ p′_{∗}
F (X/S, n) →^{f^{∗}} F (X′/S, n + d)

Proof: It follows immediately from our definitions and 2.3.4.

As an application of our construction of push-forward homomorphisms we will show that for finite cycles over normal schemes our construction of base change homomorphisms gives the same answer as the one used in [16]. To start we will recall briefly the latter construction. For an integral scheme X we denote by F (X) its field of functions.

Definition 3.6.6 A finite surjective morphism of integral Noetherian schemes f : Y → S is called a pseudo-Galois covering if the field extension F (Y)/F (S) is normal and the canonical homomorphism

Aut_{S}(Y) → Aut_{F (S)}(F (Y)) = Gal(F (Y)/F (S))

is an isomorphism.

Let S be a normal integral Noetherian scheme, X be an integral scheme and p : X → S be a finite surjective morphism. Let g : S′ → S be any Noetherian integral scheme over S. Denote by X′_{i} the irreducible components of X′ = X ×_{S} S′ and by x_{i} (resp. x) denote the generic point of X′_{i} (resp. of X). Since any normal scheme is geometrically unibranch Theorem 3.4.2 implies that x ∈ Cycl_{equi}(X/S, 0). Consider the cycle cycl(g)(x) = ∑ n_{i}x′_{i}.

Assume that there exists^{2} a pseudo-Galois covering f : Y → S and an S-morphism q : Y → X. Let G be the Galois group Gal(F (Y)/F (S)) = Aut_{S}(Y). Denote by Y′_{j} the irreducible components of Y′ = Y ×_{S} S′. It is easy to check that the action of G permutes the components Y′_{j} transitively so that in particular the field extensions F (Y′_{j})/F (S′) are all isomorphic. Denote by l(i) the number of components Y′_{j} lying over X′_{i} and by l the total number of components Y′_{j}.

Proposition 3.6.7 In the above notations one has:

n_{i} = \frac{[F (X) : F (S)]l(i)}{[F (X′_{i}) : F (S′)]l}.

2Such a covering always exists. For an excellent scheme S it follows trivially from the finitness of normalizations of S in finite extensions of its field of functions. The proof in general case is a little more complicated.

Proof: Denote the generic point of Y (resp. of Y′_{j}) by y (resp. by y′_{j}). The cycle y is in Cycl(Y/S, 0) by Theorem 3.4.2 and has the following obvious properties:

1. f_{∗}(y) = [F (Y) : F (S)]s where s is the generic point of S.

2. q_{∗}(y) = [F (Y) : F (X)]x

3. σ_{∗}(y) = y for any σ ∈ G.

Consider the cycle cycl(g)(y) = ∑ m_{j}y′_{j}. Proposition 3.6.2 shows that (f ×_{S} S′)_{∗}(cycl(g)(y)) = [F (Y) : F (S)]s i.e. ∑ m_{j}[F (Y′_{j}) : F (S′)] = [F (Y) : F (S)]. Moreover for any σ ∈ G we have

(σ ×_{S} S′)_{∗}(cycl(g)(y)) = cycl(g)(σ_{∗}(y)) = cycl(g)(y).

Since the action of G on the set y′_{1}, …, y′_{l} is transitive we conclude that all multiplicities m_{j} are the same and equal to \frac{[F (Y):F (S)]}{l[F (Y′_{j}):F (S′)]}. Finally cycl(g)(x) =

\frac{1}{[F (Y):F (X)]}(q ×_{S} S′)_{∗}(cycl(g)(y)) and hence

n_{i} = \frac{1}{[F (Y) : F (X)]} ∑_{y′_{j}/x′_{i}} m_{j}[F (Y′_{j}) : F (X′_{i})] = \frac{[F (X) : F (S)]l(i)}{[F (X′_{i}) : F (S′)]l}.

3.7 Correspondence homomorphisms.

Let Y → X be a scheme of finite type over a Noetherian scheme X. For any cycle Y ∈ Cycl(Y/X, r) ⊗ Q define a homomorphism

Cor(Y, −) : Cycl(X) ⊗ Q → Cycl(Y) ⊗ Q

as follows. Let X = ∑ n_{i}x_{i} be an element of Cycl(X)⊗Q. Denote by X_{i} the closure of the point x_{i} which we consider as an integral closed subscheme of X. Let i_{X_{i}} : X_{i} → X be the corresponding embedding. We set

Cor(Y, X) = ∑ n_{i}(X_{i} ×_{X} Y → Y)_{∗}(cycl(i_{X_{i}})(Y)).

Lemma 3.7.1 Consider a pull-back square of Noetherian schemes of the form

\begin{matrix}Y′ & →^{q} & Y \\ ↓ & & ↓ \\ X′ & →^{p} & X. \end{matrix}

Assume that the morphism Y → X is of finite type and let Y be an element of Cycl(Y/X, r) ⊗ Q and X′ be a cycle on X′ which is proper with respect to p. Then one has:

Cor(Y, p_{∗}(X′)) = q_{∗}(Cor(cycl(p)(Y), X′)).

Proof: We may assume that X′ = cycl_{X′}(Z′) where Z′ is an integral closed subscheme of X′ and p_{Z′} : Z′ → X is a proper morphism. We have

q_{∗}(Cor(cycl(p)(Y), X′)) = (Z′ ×_{X} Y → Y)_{∗}cycl(Z′ → X)(Y)

and

Cor(Y, p_{∗}(X′)) = n(p(Z′) ×_{X} Y → Y)_{∗}cycl(p(Z′) → X)(Y)

where n = [F (Z′) : F (p(Z′))] if this extension is finite and zero otherwise. We have further

(Z′ ×_{X} Y → Y)_{∗}cycl(Z′ → X)(Y) =
= (p(Z′) ×_{X} Y → Y)_{∗}(q ×_{X} Y)_{∗}cycl(q)cycl(p(Z′) → X)(Y).

where q is the morphism Z′ → p(Z′). It is sufficient to show that

(Z′ ×_{X} Y → p(Z′) ×_{X} Y)_{∗}cycl(Z′ → p(Z′))(W) = n(W)

for any element W in Cycl(p(Z′) ×_{X} Y/p(Z′)). To prove the last statement we may replace p(Z′) by its generic point Spec(F (Z′)) and assume that W = cycl(W) where W is an integral closed subscheme of Y ×_{X} Spec(F (Z′)). For an infinite field extension F (Z′)/F (p(Z′)) we immediately conclude that the left hand side of our equality equals zero (as well as the right hand side). For finite field extension our equality follows from Lemmas 3.3.12 and 2.3.5.

The following lemma is straightforward.

\begin{matrix}Y′ & →^{q} & Y \\ ↓ & & ↓ \\ X′ & →^{p} & X. \end{matrix}

Assume that the morphism Y → X is of finite type, schemes X′ and X are reduced and the morphism p is flat. Then for any cycle X in Cycl(X) ⊗ Q and any cycle Y in Cycl(Y/X, r) ⊗ Q one has:

q^{∗}Cor(Y, X) = Cor(q^{∗}(Y), p^{∗}(X)).

Theorem 3.7.3 Let S be a Noetherain scheme and f : Y → X be a morphism of schemes of finite type over S. Let further Y = ∑ n_{i}y_{i} be an element of Cycl(Y/X, n) ⊗ Q and X be an element of Cycl(X/S, m) ⊗ Q. Then the element Cor(Y, X) belongs to the group Cycl(Y/S, n+m). Moreover for any Noetherian scheme g : S′ → S over S one has

cycl(g)(Cor(Y, X)) = Cor(cycl(g ×_{S} X)(Y), cycl(g)(X)).

Proof: We will first consider the following particular case of our theorem.

Lemma 3.7.4 The statement of the theorem holds if Y = cycl_{Y} (Y_{0}), X = cycl_{X}(X_{0}) where Y_{0} (resp. X_{0}) is a closed subscheme of Y (resp. of X) which is flat over X (resp. over S).

Proof: Set X′ = X ×_{S} S′, Y′ = Y ×_{S} S′. By Lemma 3.3.10 Cor(Y, X) coincides with the cycle associated with the closed subscheme Y_{0} ×_{X} X_{0} in Y which is clearly flat over S. Therefore Cor(Y, X) is a relative cycle over S by 3.2.5. We have further

cycl(g)(Cor(Y, X)) = cycl(g)(cycl_{Y} (Y_{0} ×_{X} X_{0})) = cycl_{Y′}((Y_{0} ×_{X} X_{0}) ×_{S} S′)

and

Cor(cycl(g×_{S}X)(Y), cycl(g)(X)) = Cor(cycl_{Y′}(Y_{0}×_{S}S′), cycl_{X′}(X_{0}×_{S}S′)) =

= cycl_{Y′}((Y_{0} ×_{S} S′) ×_{X′} (X_{0} ×_{S} S′))

Since

(Y_{0} ×_{S} S′) ×_{X′} (X_{0} ×_{S} S′) ≅ (Y_{0} ×_{X} X_{0}) ×_{S} S′

we conclude that our equality holds.

Let k be a field, x : Spec(k) → S be a k-valued point of S and (x_{0}, x_{1}, R) be a fat point of S over x. It is clearly sufficient to show that

(x_{0}, x_{1})^{∗}(Cor(Y, X)) = Cor(cycl(x ×_{S} X)(Y), cycl(x)(X)).

Let (x_{0}, Id) be the obvious fat point of Spec(R). By Lemma 3.7.2 (and Lemma 3.3.12) we have

(x_{0}, x_{1})^{∗}(Cor(Y, X)) = (x_{0}, Id)^{∗}(Cor(cycl(x_{1} ×_{S} X)(Y), cycl(x_{1})(X))).

We may assume now that S = Spec(R) where R is a discrete valuation ring. In this case all cycles which are formal linear combinations of points over the generic point of S are relative cycles over S (by Corollary 3.2.6). Thus we may assume that X = cycl_{X}(X_{0}) where X_{0} is a closed integral subscheme of X which is equidimensional of relative dimension m over S and we have to show that

cycl(x)(Cor(Y, X)) = (Cor(cycl(x ×_{S} X)(Y), cycl(x)(X))).

Denote by i the closed embedding X_{0} → X. We set

Y_{0} = Y ×_{Y} X_{0}
X_{x} = X ×_{S} Spec(k)  Y_{x} = Y ×_{S} Spec(k)
(X_{0})_{x} = X_{0} ×_{S} Spec(k) (Y_{0})_{x} = Y_{0} ×_{S} Spec(k)
u = pr_{1} : Y_{0} → Y  u_{x} = u ×_{S} Spec(k) : (Y_{0})_{x} → Y_{x}
v = pr_{1} : X_{x} → X  v_{0} = pr_{1} : (X_{0})_{x} → X_{0}

We have

x^{∗}(Cor(Y, X)) = x^{∗}(u_{∗}(cycl(i)(Y))) =

= ((u_{x})_{∗}(cycl(x)(cycl(i)(Y))) = (u_{x})_{∗}(cycl(x)(Cor(cycl(i)(Y), cycl_{X_{0}}(X_{0}))))

and

Cor(cycl(v)(Y), cycl(x)(X)) =
= (u_{x})_{∗}(Cor(cycl(v_{0})cycl(i)(Y), cycl(x)(cycl_{X_{0}}(X_{0})))).

We may replace now X by X_{0} and assume further that X is integral and equidimensional of relative dimension m over S and X is the fundamental cycle of X.

Denote by Y_{i} the closures of the points y_{i} which we consider as integral closed subschemes of Y. Let f_{X} : X̃ → X be a blow-up of X_{red} such that the proper transforms of Y_{i} with respect to f_{X} are flat and equidimensional of relative dimension n over X̃. Consider the pull-back square

\begin{matrix}Ỹ & →^{f_{Y}} & Y \\ ↓ & & ↓ \\ X̃ & →^{f_{X}} & X. \end{matrix}

Let X̃ be the fundamental cycle of X̃. Since S is the spectrum of a discrete valuation ring we have X̃ ∈ Cycl(X̃/S, m). Consider the following diagram of abelian groups (we write Cycl(−) instead of Cycl(−, −) ⊗ Q):

Cycl(Y/X)  →^{cycl(f_{X})} Cycl(Ỹ /X̃) →^{Cor(−,X̃)} Cycl(Ỹ /S) →^{(f_{Y} )_{∗}}  Cycl(Y/S)
↓ cycl(j)  ↓ cycl(j̃)  ↓ cycl(x)  ↓ cycl(x)
Cycl(Y_{x}/X_{x}) →^{cycl(f_{Xx})} Cycl(Ỹ_{x}/X̃_{x}) →^{Cor(−,X̃_{x})} Cycl(Ỹ_{x}/S_{x}) →^{(f_{Yx})_{∗}} Cycl(Y_{x}/S_{x})

where

S_{s} = Spec(k)
X_{x} = X ×_{S} Spec(k) Y_{x} = Y ×_{S} Spec(k)
X̃_{x} = X̃ ×_{S} Spec(k) Ỹ_{x} = Ỹ ×_{S} Spec(k)

and

j : X_{x} → X j̃ : X̃_{x} → X̃

are the obvious morphisms and X̃_{x} = cycl(x)(X̃).

The composition of the upper horizontal arrows equals Cor(−, X) by Lemma 3.7.1. By Proposition 3.6.2

(X̃_{x} → X̃)_{∗}(X̃_{x}) = cycl(x)(X)

and thus the composition of lower horizontal arrows equals Cor(−, cycl(x)(X)) by Lemma 3.7.1. We have only to show now that our diagram is commutative on Y ∈ Cycl(Y/X). The first square is commutative since f_{X} ∘ j̃ = j ∘ f_{X_{x}}. The last square is commutative by Proposition 3.6.2. Finally the middle square is commutative on cycl(f_{X})(Y) by our choice of the blow-up X̃ → X and Lemma 3.7.4.

Corollary 3.7.5 Let S be a Noetherian scheme and X_{1} → X_{2} →^{p} S be a morphism of schemes of finite type over S. Let F (−, −) be one of the presheaves z(−, −), z^{eff}(−, −), z_{equi}(−, −), c(−, −), c_{equi}(−, −), c^{eff}(−, −). Then for any n, m ≥ 0 there is a canonical morphism of presheaves of the form

Cor_{Y/X} : p_{∗}(F (X_{1}/X_{2}, n)) ⊗ F (X_{2}/S, m) → F (X_{1}/S, m + n)

Lemma 3.7.1 implies immediately the following proposition.

Proposition 3.7.6 Let S be a Noetherian scheme. Consider a pull-back square of schemes of finite type over S of the form:

\begin{matrix}Y′ & →^{g} & Y \\ ↓ & & ↓ \\ X′ & →^{f} & X \end{matrix}

Denote by p : X → S, p′ : X′ → S the structural morphisms of X and X′ respectively. Assume further that F (−) is one of the presheaves c(−, −), c^{eff}(−, −), c_{equi}(−, −) or that f is proper and F (−, −) is one of the presheaves z(−, −), z^{eff}(−, −), z_{equi}(−, −).

Then the following diagram of morphisms of presheaves commutes:

p_{∗}F (Y/X) ⊗ F (X′/S) →^{Id⊗f_{∗}} p_{∗}F (Y/X) ⊗ F (X/S) ↘
↓  F (Y/S)
p′_{∗}F (Y′/X′) ⊗ F (X′/S) →^{Cor_{Y′/X′}}  F (Y′/S)  ↗

Proposition 3.7.7 Let Z → Y → X be a composable pair of morphisms of finite type of Noetherian schemes. Let further X be an element of Cycl(X)⊗ Q and Z, Y be elements of Cycl(Z/Y, n) ⊗ Q and Cycl(Y/X, m) ⊗ Q respectively. Then one has

Cor(Cor(Z, Y), X) = Cor(Z, Cor(Y, X)).

Proof: It follows easily from definitions and Theorem 3.7.3.

Corollary 3.7.8 Let S be a Noetherian scheme and Z →^{f} Y →^{g} X be a composable pair of morphisms of schemes of finite type over S. Denote by h : Z → S the structural morphism and let F (−) be one of the presheaves z(−, −), z^{eff}(−, −), z_{equi}(−, −), c(−, −), c_{equi}(−, −), c^{eff}(−, −). Then the following diagram of morphisms commutes:

h_{∗}g_{∗}F (Z/Y) ⊗ h_{∗}F (Y/X) ⊗ F (X/S) → h_{∗}F (Z/X) ⊗ F (X/S)
↓  ↓
h_{∗}g_{∗}F (Z/Y, n) ⊗ F (Y/S)  →  F (Z/S).

Existence of correspondence homomorphisms for relative cycles allows us to construct the homomorphism of external product (see [4]). Let p_{X} : X → S, p_{Y} : Y → S be two schemes of finite type over a Noetherian scheme S. Let further F (−, −) be one of the groups z(−, −), z_{equi}(−, −), z^{eff}(−, −), c(−, −), c_{equi}(−, −), c^{eff}(−, −). We define the external product homomorphism

F (X/S, n) ⊗ F (Y/S, m) → F (X ×_{S} Y/S, n + m)

as the following composition

F (X/S,n)⊗F (Y/S,m) →^{cycl(p_{Y} )⊗Id} F (X×_{S}Y/Y,n)⊗F (Y/S,m) →^{Cor_{X×_{S} Y/Y}} F (X×_{S}Y/S,n+m).

One can verify easily using results of this section that it satisfies all the standard properties of external products of cycles and defines a homomorphism of the corresponding Chow presheaves.

4 Chow sheaves in the h-topologies.

4.1 The h-topologies.

In this section we will remind briefly the definitions and some basic properties of three Grothendieck topologies (the h-topology, the qfh-topology and the cdh-topology) on the categories of schemes. For more information on the hand the qfh-topologies see [18] or [16].

A morphism of schemes p : X → Y is called a topological epimorphism if the underlying Zariski topological space of Y is a quotient space of the underlying Zariski topological space of X (i.e. p is surjective and a subset A of Y is open if and only if p^{−1}(A) is open in X), p is called a universal topological epimorphism if for any Z/Y the morphism p_{Z} : X ×_{Y} Z → Z is a topological epimorphism.

An h-covering of a scheme X is a finite family of morphisms of finite type {p_{i} : X_{i} → X} such that ∐ p_{i} : ∐ X_{i} → X is a universal topological epimorphism.

A qfh-covering of a scheme X is an h-covering {p_{i}} such that all the morphisms p_{i} are quasi-finite.

h-coverings (resp. qfh-coverings) define a pretopology on the category of schemes, the h-topology (resp. the qfh-topology) is the associated topology.

The definition of the cdh-topology is a little less natural. Namely the cdhtopology on the category of schemes is the minimal Grothendieck topology such that the following two types of coverings are cdh-coverings.

1. Nisnevich coverings, i.e. etale coverings {U_{i} →^{p_{i}} X} such that for any point x of X there is a point x_{i} on one of the U_{i} such that p_{i}(x_{i}) = x and the morphism Spec(k_{x_{i}}) → Spec(k_{x}) is an isomorphism.

2. Coverings of the form X′ ∐ Z \underrightarrow{p_{X′} ∐ p_{Z}} X such that p_{X′} is a proper morphism, p_{Z} is a closed embedding and the morphism p_{X′}^{−1}(X − p_{Z}(Z)) → X − p_{Z}(Z) is an isomorphism.

Obviously the h-topology is stronger than both the qfh- and the cdhtopology, the cdh-topology is stronger than the Zariski topology and standard results on flat morphisms show that qfh-topology is stronger than the flat topology.

Lemma 4.1.1 Let {U_{i} →^{p_{i}} X} be an h-covering of a Noetherian scheme X. Denote by ∐ V_{j} the disjoint union of all irreducible components of ∐ U_{i} which dominate an irreducible component of X. Then the morphism q : ∐ V_{j} → X is surjective.

Proof: See [18].

Remark: In fact the property of h-coverings considered in Lemma 4.1.1 is characteristic. Namely one can show that a morphism of finite type X → S of Noetherian schemes is an h-covering if and only if for any Noetherian scheme T over S the union of irreducible components of X ×_{S} T which dominate irreducible components of T is surjective over T.

Proposition 4.1.2 Let X be a Noetherian scheme and U = {U_{i} → X} be an h-covering of X. Then there is a refinement {V_{j} → X} of U such that each

morphism q_{j} : V_{j} → X admits a decomposition of the form V_{j} →^{q_{j}^{f}} W_{j} →^{q_{j}^{p}} X such that q_{j}^{f} is a faithfully flat morphism, W_{j} is irreducible and q_{j}^{p} is an abstract blow-up (see Defenition 2.2.4) of an irreducible component of X.

Proof: To prove our proposition we may assume that X is integral and our covering is of the form U → X. By Theorem 2.2.2 there is a blow-up p : W → X such that the proper transform Ũ of U is flat over W. It is sufficient to show that the morphism Ũ → W is surjective. Let Z be a closed subscheme in X such that Z ≠ X and the morphism p : W → X is an isomorphism outside Z. Since W ×_{X} U → W is an h-covering and the closure of the complement W ×_{X} U − Ũ lies over p^{−1}(Z) and, therefore is not dominant over any irreducible component of W, the surjectivity of the morphism Ũ → W follows from Lemma 4.1.1.

Lemma 4.1.3 Let S be a Noetherian scheme and p : X → S be a scheme of finite type over S. Suppose that there is an h-covering f : S′ → S such that the scheme X′ = X ×_{S} S′ is proper over S′. Then X is proper over S.

Proof: Denote the projections X ×_{S} S′ → X, X ×_{S} S′ → S′ by f′ and p′ respectively. Let Z be a closed subset in X. It is sufficient to show that p(Z) is closed in S. Since f : S′ → S is an h-covering p(Z) is closed in S if and only if f^{−1}(p(Z)) is closed in S′. We obviously have f^{−1}(p(Z)) = p′((f′)^{−1}(Z)). Since p′ is proper we conclude that this set is closed.

Lemma 4.1.4 Let S be a Noetherian scheme and ϕ : F → G be a morphism of presheaves on the category of Noetherian schemes over S. Assume further that for any integral Noetherian scheme T over S and any section a ∈ G(T) of G over T there is an abstract blow-up f : T′ → T such that f^{∗}(a) belongs to the image of ϕ_{T′} : F (T′) → G(T′). Then ϕ_{cdh} : F_{cdh} → G_{cdh} is an epimorphism of the associated cdh-sheaves.

Proof: It is sufficient to show that for any section a ∈ G(S) of G over S there is a cdh-covering {X_{i} →^{p_{i}} S} such that p_{i}^{∗}(a) belongs to the image of ϕ_{X_{i}} : F (X_{i}) → G(X_{i}). Our condition implies that there is a closed subscheme S_{1} in S such that S_{1} ≠ S and a proper surjective morphism q : X_{1} → S such that X_{1} − q^{−1}(S_{1}) → S − S_{1} is an isomorphism and q^{∗}(a) belongs to Im(ϕ_{X_{1}}). Since X_{1} ∐ S_{1} → S is a cdh-covering of S we reduced our problem to S_{1}. Repeating this construction we get a sequence of closed subschemes … S_{i} ⊂ S_{i−1} ⊂ … ⊂ S_{1} ⊂ S such that S_{i} ≠ S_{i−1}. Since the scheme S is Noetherian this sequence must be finite, i.e. S_{i} = ∅ for i > n. The family of morphisms {X_{i} → S_{i} → S}_{i=1,...,n} is then a cdh-covering of S with the required property.

4.2 Sheaves in the h-topologies associated with Chow presheaves.

Lemma 4.2.1 Let X → S be a scheme of finite type over a Noetherian scheme S and Z be an element of Cycl(X/S, r). Then there is an abstract blow-up f : S′ → S such that

cycl(f)(Z) = ∑ n_{i}cycl(Z′_{i}) = ∑ n_{i}cycl((Z′_{i})_{red})

where Z′_{i} are irreducible closed subschemes of X ×_{S} S′ which are flat and equidimensional of relative dimension r over S′.

Proof: It follows immediately from Theorem 2.2.2.

Theorem 4.2.2 Let X → S be a scheme of finite type over a Noetherian scheme S. Then the presheaves Cycl(X/S, r)_{Q} and Cycl^{eff}(X/S, r)_{Q} are hsheaves and if S is a scheme of exponential characteristic p the same holds for the presheaves z(X/S, r) ⊗ Z[1/p] and z^{eff}(X/S, r) ⊗ Z[1/p].

Proof: We will only consider the case of Cycl(X/S, r)_{Q}. The proof for Cycl^{eff}(X/S, r)_{Q} is similar.

Note first that the presheaves Cycl(X/S, r)_{Q} are separated with respect to the h-topology, i.e. the canonical morphisms of presheaves Cycl(X/S, r)_{Q} → (Cycl(X/S, r)_{Q})_{h} are monomorphisms. Therefore according to the standard construction of the sheaf associated with a presheaf (see [11],[1]) it is sufficient to show that for any cofinial class of h-coverings {U_{i} → S}_{i=1,...,n} of S the following sequence of abelian groups is exact:

Cycl(X/S, r)_{Q}(S) → ⨁_{i} Cycl(X/S, r)_{Q}(U_{i}) → ⨁_{i,j} Cycl(X/S, r)_{Q}(U_{i} ×_{S} U_{j})

We may obviously replace the covering {U_{i} → S} by the covering

p : U = ∐ U_{i} → S

and hence assume that n = 1. By Lemma 4.1.1 we may also assume that any irreducible component of U dominates an irreducible component of S.

We will use the following simple lemma.

Lemma 4.2.3 Let p_{Y} : Y → Spec(k), p_{X} : X → Spec(k) be two schemes of finite type over a field k. Then the sequence of abelian groups

Cycl(X/Spec(k), r) ⊗ Q →^{cycl(p_{Y} )} Cycl(X ×_{Spec(k)} Y/Y, r) ⊗ Q →^{cycl(pr_{1})−cycl(pr_{2})}

→ Cycl(X ×_{Spec(k)} Y ×_{Spec(k)} Y/Y ×_{Spec(k)} Y, r) ⊗ Q

(where pr_{i} : Y ×_{Spec(k)} Y → Y are the projections) is exact.

Let Y be an element of Cycl(X/S, r)_{Q}(U) such that cycl(pr_{1})(Y) = cycl(pr_{2})(Y). Since any irreducible component of U dominates an irreducible component of S Lemma 4.2.3 implies that there exists a cycle Z on X such that for any generic point η : Spec(L) → U of U one has cycl(p ∘ η)(Z) = cycl(η)(Y). It is obviously sufficient to show that Z belongs to Cycl(X/S, r) ⊗ Q.

Let k be a field and (x_{0}, x_{1}, R), (y_{0}, y_{1}, Q) be two fat k points of S over a k-point s : Spec(k) → S of S. We have to show that (x_{0}, x_{1})^{∗}(Z) = (y_{0}, y_{1})^{∗}(Z).

Lemma 4.2.4 Let R be a discrete valuation ring and p : X → Spec(R) be an h-covering of Spec(R). Then there exist a discrete valuation ring Q, a dominant morphism p_{0} : Spec(Q) → Spec(R) which takes the closed point of Spec(Q) to the closed point of Spec(R) and a morphism s : spec(R) → X such that p ∘ s = p_{0}.

Proof: By Lemma 4.1.1 there is an irreducible component X_{0} of X which is dominant over S and whose image contains the closed point of Spec(R) i.e. X_{0} is irreducible and surjective over S. Then by [6] there is a discrete valuation ring Q and a dominant morphism Spec(Q) → X_{0} such that the image of the closed point of Spec(Q) lies in the closed fiber of X_{0} over Spec(R).

By Lemma 4.2.4 we may construct a commutative diagram of the form

Spec(R′)  →^{x′_{1}}  U

↘^{f}

↗^{x′_{0}}  Spec(R)  ↓ p
↗^{x_{0}}  ↘^{x_{1}}
Spec(L) →^{r} Spec(k)  S
↘^{y_{0}}  ↗^{y_{1}}
↘^{y′_{0}}  Spec(Q)  ↑ p

↗^{g}

Spec(Q′)  →^{y′_{1}}  U

such that (x′_{0}, x′_{1}, R′), (y′_{0}, y′_{1}, Q′) are fat points of U and f, g are dominant morphisms. It is obviously sufficient to show that (x_{0}, x_{1})^{∗}(Z) ⊗_{k} L = (y_{0}, y_{1})^{∗}(Z) ⊗_{k} L. We clearly have:

(x_{0}, x_{1})^{∗}(Z) ⊗_{k} L = (x′_{0}, x′_{1})^{∗}(Y)
(y_{0}, y_{1})^{∗}(Z) ⊗_{k} L = (y′_{0}, y′_{1})^{∗}(Y).

It is sufficient to show that the right hand sides of these two equalities coincide. Denote the L points x′_{1}∘x′_{0} and y′_{1}∘y′_{0} of U by x′ and y′ respectively. Since Y is an element of Cycl(X ×_{S} U/U, r) we have

(x′_{0}, x′_{1})^{∗}(Y) = cycl(x′)(Y) = cycl(x′ ×_{S} y′)(cycl(pr_{1})(Y))

(y′_{0}, y′_{1})^{∗}(Y) = cycl(y′)(Y) = cycl(x′ ×_{S} y′)(cycl(pr_{2})(Y))

(where pr_{i} : U ×_{S} U → U are the projections) and since

cycl(pr_{1})(Y) = cycl(pr_{2})(Y)

by our condition on Y we conclude that (x_{0}, x_{1})^{∗}(Z) = (y_{0}, y_{1})^{∗}(Z).

Proposition 4.2.5 Let X → S be a scheme of finite type over a scheme S. Then the presheaves Cycl_{equi}(X/S, r)_{Q} are qfh-sheaves and if S is a scheme of exponential characteristic p the same holds for the presheaves z_{equi}(X/S, r)⊗ Z[1/p].

Proof: It is clearly sufficient to consider the case of Cycl_{equi}(X/S, r)_{Q}. Let Z be an element of Cycl(X/S, r) ⊗ Q. In view of Theorem 4.2.2 it is sufficient to show that if there exists a qfh-covering f : S′ → S of S such that supp(cycl(f)(Z)) is equidimensional of dimension r over S′ then supp(Z) is equidimensional of dimension r over S. By Theorem 2.1.1 we have only to show that dim(supp(Z)/S) ≤ r. By Lemma 4.1.1 we may assume that any irreducible component of S′ dominates an irreducible component of S and dim(supp(cycl(f)(Z))/S′) ≤ r. Then supp(cycl(f)(Z)) is the closure in supp(Z)×_{S} S′ of the fibers of this scheme over generic points of S′. Since the projection supp(Z) ×_{S} S′ → supp(Z) is a qfh-covering using again Lemma 4.1.1 we conclude that the morphism g : supp(cycl(f)(Z)) → supp(Z) is surjective and hence dim(supp(Z)/S) ≤ r since the morphisms g and f are quasi-finite.

Proposition 4.2.6 Let X → S be a scheme of finite type over a Noetherian scheme S. Then the presheaves P ropCycl(X/S, r)_{Q} and P ropCycl^{eff}(X/S, r)_{Q} are h-sheaves and if S is a scheme of exponential characteristic p the same holds for the presheaves c(X/S, r) ⊗ Z[1/p] and c^{eff}(X/S, r) ⊗ Z[1/p].

Proof: In view of Theorem 4.2.2 it is sufficient to show that if Z is an element of Cycl_{equi}(X/S, r) ⊗ Q and there exists an h-covering f : S′ → S such that the support supp(cycl(f)(Z)) is proper over S′ then supp(Z) is proper over S.

We may obviously assume that S is reduced. By 4.1.2 we may assume further that f admits a decomposition of the form S′ →^{f_{0}} S′′ →^{f_{1}} S such that f_{1} is an abstract blow-up and f_{0} is faithfully flat. Let cycl(f_{1})(Z) = ∑ n_{i}cycl(Z′′_{i}) where Z′′_{i} are irreducible closed subschemes of X′′ = X ×_{S} S′′.

Since f_{0} is flat the closed subsets supp(cycl(f_{1}f_{0})(Z)) and supp(cycl(f_{1})(Z)) ×_{S′′} S′ coincide. Since f_{0} in particular is an h-covering we conclude by Lemma 4.1.3 that supp(cycl(f_{1}f_{0})(Z)) is proper over S′′.

Let Z = ∑ m_{j}cycl(Z_{j}) where Z_{j} are integral closed subschemes of X and m_{j} ≠ 0. We have to show that the morphism ∪Z_{j} → S is proper. Consider the commutative diagram:

∪Z′′_{i} → ∪Z_{i}
↓  ↓
S′′ → S

The upper horizontal arrow is proper since it is a composition of a closed embedding Z′′_{i} → Z_{i} ×_{S} S′′ with a proper morphism Z_{i} ×_{S} S′′ → Z_{i}. Being an isomorphism in generic points it is surjective. Therefore ∪Z_{i} → S is proper since f_{1} is proper and ∪Z′′_{i} → S′′ is proper.

The following proposition follows immediately from Propositions 4.2.5, 4.2.6.

Proposition 4.2.7 Let X → S be a scheme of finite type over a Noetherian scheme S. Then the presheaves P ropCycl_{equi}(X/S, r)_{Q} are qfh-sheaves and if S is a scheme of exponential characteristic p the same holds for the presheaves c_{equi}(X/S, r) ⊗ Z[1/p].

Proposition 4.2.8 Let X → S be a scheme of finite type over a Noetherian scheme S. Then for any r ≥ 0 the h-sheaves Cycl(X/S, r)_{Q}, P ropCycl(X/S, r)_{Q} are the h-sheaves of abelian groups associated in the obvious way with the h-sheaves of abelian monoids Cycl^{eff}(X/S, r)_{Q}, P ropCycl^{eff}(X/S, r)_{Q}. The same holds for the corresponding p-divisible sheaves if S is a scheme of exponential characteristic p.

Proof: By 4.2.2, 4.2.6 the presheaves Cycl(X/S, r)_{Q}, P ropCycl(X/S, r)_{Q} and their effective versions are h-sheaves. By Lemma 4.1.4 it is sufficient to show that for any element Z of Cycl(X/S, r)_{Q}(S) there is an abstract blow-up f : S′ → S such that cycl(f)(Z) is a linear combination of elements of Cycl^{eff}(X/S, r)(S′). It follows immediately from Lemma 4.2.1.

Theorem 4.2.9 Let X → S be a scheme of finite type over a Noetherian scheme S. Then one has:

1. For any r ≥ 0 the presheaves z(X/S, r), c(X/S, r), z^{eff}(X/S, r), c^{eff}(X/S, r) are sheaves in the cdh-topology.

2. The cdh-sheaves associated with the presheaves z_{equi}(X/S, r), c_{equi}(X/S, r) are isomorphic to z(X/S, r) and c(X/S, r) respectively.

Proof: Since we know already that Cycl(X/S, r)_{Q} etc. are h-sheaves to prove the first part of the theorem it is sufficient to show that for any element Z of Cycl(X/S, r)_{Q}(S) such that there exists a cdh-covering p : S′ → S such that cycl(p)(Z) ∈ z(X/S, r)(S′) one has Z ∈ z(X/S, r)(S).

It follows immediately from 3.3.9 and the fact that any cdh-covering of a spectrum of a field splits.

The second part follows trivially from the first part and Lemma 4.2.1.

Proposition 4.2.10 Let X → S be a scheme of finite type over a Noetherian scheme S. Then for any r ≥ 0 the cdh-sheaf z(X/S, r) (resp. c(X/S, r)) is the cdh-sheaf of abelian groups associated in the obvious way with the cdhsheaf of abelian monoids z^{eff}(X/S, r) (resp. c^{eff}(X/S, r)).

Proof: It follows from Lemma 4.1.4 and Lemma 4.2.1.

Let p : X → S be a morphism of finite type of Noetherian schemes. For any Noetherian scheme T over S consider an equivalence relation R_{T} ⊂ N(Hilb(X/S, r))(T) × N(Hilb(X/S, r))(T) such that (∑ n_{i}Z_{i}, ∑ m_{j}W_{j}) belongs to R_{T} if and only if cycl_{X_{T}} (∑ n_{i}Z_{i}) = cycl_{X_{T}} (∑ m_{j}W_{j}) where X_{T} = X ×_{S} T. Proposition 3.2.2 implies that it gives us equivalence relations R on the presheaves N(Hilb(X/S, r)), N(P ropHilb(X/S, r)), Z(Hilb(X/S, r)), Z(P ropHilb(X/S, r)) which are obviously consistent with the additive structure of these presheaves.

Theorem 4.2.11 Let X → S be a scheme of finite type over a Noetherian scheme S. Then one has canonical isomorphisms of cdh-sheaves:

z(X/S, r) = (Z(Hilb(X/S, r))/R)_{cdh}
z^{eff}(X/S, r) = (N(Hilb(X/S, r))/R)_{cdh}

c(X/S, r) = (Z(P ropHilb(X/S, r))/R)_{cdh}
c^{eff}(X/S, r) = (N(P ropHilb(X/S, r))/R)_{cdh}

Proof: We will only consider the first isomorphism. Proof in the other cases is similar. Note first that by 3.3.11 there is a morphism of presheaves Z(Hilb(X/S, r)) → z(X/S, r) and clearly the presheaf Z(Hilb(X/S, r))/R is the image of this morphism. The fact that the corresponding associated cdh-sheaf coincides with z(X/S, r) follows immediately from Lemma 4.1.4 and Lemma 4.2.1.

Theorem 4.2.12 Let X → S be a scheme of finite type over a Noetherian scheme S. Then one has:

1. The sheaf c_{equi}(X/S, 0)_{qfh} is canonically isomorphic to the qfh-sheaf Z_{qfh}(X/S) of abelian groups freely generated by the sheaf of sets representable by X.

2. The sheaf c^{eff}(X/S, 0)_{qfh} is canonically isomorphic to the qfh-sheaf N_{qfh}(X/S) of abelian monoids freely generated by the presheaf of sets representable by X.

Proof: We will only consider the first statement. The prove of the second one is similar.

Let δ ∈ c^{eff}(X/S, 0)(X) ⊂ Cycl(X ×_{S} X/X, 0) be the element which corresponds to the diagonal Δ : X → X ×_{S} X. By the universal property of the freely generated sheaves it gives us a morphism of sheaves Z_{qfh}(X) → c^{eff}(X/S, 0)_{qfh}. We have to show that it is an isomorphism. Note that it is a monomorphism since the functor of associated sheaf is exact and the corresponding morphism from the presheaf of abelian groups freely generated by X to the presheaf c(X/S, 0) is obviously a monomorphism.

To show that it is an epimorphism it is sufficient to verify that for any element Z of c^{eff}(X/S, 0)(S) there is a qfh-covering p : S′ → S such that cycl(p)(Z) is a formal linear combination of S′-points of X ×_{S} S′ over S′. We may assume that S is integral and Z = ∑ n_{i}cycl(Z_{i}) where Z_{i} ≠ Z_{j} are integral closed subscheme of X which are finite and surjective over S. We will use induction on deg(Z/S) = ∑ |n_{i}|deg(Z_{i}/S). If N = 0 then Z = 0 and there is nothing to prove.

Set Z = ∐ Z_{i}. Since p_{Z} : Z → S is finite and surjective it is a qfhcovering. It is sufficient (by the induction hypothesis) to show that there exists a cycle Z_{1} in c_{equi}(X ×_{S} Z_{1}/Z_{1}, 0) which is a linear combination of Z_{1}-points of X ×_{S} Z_{1} over Z_{1} and such that

deg(cycl(p_{Z_{1}})(Z) − Z_{1}) < deg(Z/S).

The cycle cycl(p_{Z_{1}})(Z) is of the form ∑ n_{i}(∑ m_{j}W_{ij}) where W_{ij} are the irreducible components of the schemes Z_{1} ×_{S} Z_{i}. We obviously have

∑ m_{j}deg(W_{ij}/Z_{1}) = deg(Z_{i}/S)

and hence

∑ n_{i}m_{j}deg(W_{ij}/Z_{1}) = deg(Z/S).

Let W_{11} be the irreducible component of Z_{1} ×_{S} Z_{1} which is the image of the diagonal embedding Z_{1} → Z_{1} ×_{S} Z_{1}. One can easily see that cycl(W_{11}) ∈ c_{equi}(X ×_{S} Z_{1}/Z_{1}, 0) and hence we may set Z_{1} = n_{1}m_{1}cycl(W_{11}).

Lemma 4.2.13 Let X → S be a scheme of finite type over a Noetherian scheme S. Then the h-sheaf z_{equi}(X/S, r)_{h} (resp. the sheaf c_{equi}(X/S, r)_{h}) is isomorphic to the h-sheaf z(X/S, r)_{h} (resp. to the sheaf c(X/S, r)_{h}).

Proof: It follows trivially from Lemma 4.2.1

Proposition 4.2.14 Let X → S be a scheme of finite type over a Noetherian scheme S. Then one has:

1. The sheaf c(X/S, 0)_{h} is canonically isomorphic to the h-sheaf Z_{h}(X/S) of abelian groups freely generated by the sheaf of sets representable by X.

2. The sheaf c^{eff}(X/S, 0)_{h} is canonically isomorphic to the h-sheaf N_{h}(X/S) of abelian monoids freely generated by the presheaf of sets representable by X.

Proof: It follows immediately from Theorem 4.2.12 and Lemma 4.2.13.

Proposition 4.2.15 Let X → S be a scheme of finite type over a Noetherian scheme S. Then the canonical morphisms of presheaves:

z(X/S, r)_{qfh} → z(X/S, r)_{h}
c(X/S, r)_{qfh} → c(X/S, r)_{h}

etc. are isomorphisms.

Proof: To prove this proposition we need the following two lemmas.

Lemma 4.2.16 Let S be a Noetherian scheme and {x_{1}, …, x_{n}} be a finite set of points of S. Let E_{1}, …, E_{n} be finite extensions of the fields k_{x_{1}}, …, k_{x_{n}}. Then there is a finite surjective morphism S′ → S such that for any i = 1, …, n and any point y over x_{i} the field extension k_{y}/k_{i} contains E_{i}.

Proof: Replacing S by the disjoint union of its irreducible components we may assume that S is integral. Using Zariski’s main theorem [11] one can reduce the problem to the case when S is a semi-local scheme and x_{i} are the closed points of S. We may obviously assume that E_{i} are normal extensions of k_{x_{i}} and since E_{i} are finitely generated over k_{x_{i}} using induction on the minimal number of generators we may further assume that E_{i} = k_{x_{i}}(z_{i}) for some elements z_{i} in E_{i}. Let

f_{i}(z) = z_{i}^{d} + a_{i1}z^{d_{i}−1} + … + a_{id_{i}}

be the minimal polinomial of z_{i} over k_{x_{i}} and

g_{i} = f_{i}^{d_{1}...d_{n}/d_{i}}.

Then g_{i} are of the same degree d = d_{1} … d_{n}. Let b_{ij}, i = 1, …, n, j = 1, …, d be the coefficients of g_{i}. We will find then elements B_{j} in O(S) such that the reduction of B_{j} in the point x_{i} equals b_{ij}. We set R = O(S)[z]/(Z^{d}+B_{1}z^{d−1}+ … + B_{d}). It is a finite algebra over O(S) and the morphism Spec(R) → S satisfies the required conditions.

Lemma 4.2.17 Let p : X → S be a scheme of finite type over a Noetherian scheme S. Assume further that p is surjective. Then there exists a finite surjective morphism S′ → S such that for any field k and any k-point s′ : Spec(k) → S′ there is a lifting of s′ to a k-point of X′ = X ×_{S} S′.

Proof: Let Z be a closed subscheme of S. We say that our lemma holds outside Z if there is a finite surjective morphism S_{Z} → S such that for any field k and any k-point s′ of S_{Z} ×_{S} (S − Z) there is a lifting of s′ to S_{Z} ×_{S} X.

Our lemma obviously holds for Z = X. By Noetherian induction it is sufficient to show that if our lemma holds for Z then it holds for a closed subset Z′ ⊂ Z such that Z′ ≠ Z.

Let η_{1}, …, η_{k} be all generic points of Z. Since the morphism p is of finite type there are finite extensions E_{1}, …, E_{k} of the fields k_{η_{1}}, …, k_{η_{k}} such that the points Spec(E_{i}) → S of S admit liftings to X. By Lemma 4.2.16 there is a finite surjective morphism f : S′ → S such that for any field k and any k-point s′ of S′ over one of the points η_{i} there is a lifting of s′ to X ×_{S} S′. Since p is of finite type this condition also holds for any point s′ of S′ which belongs to f^{−1}(U) for a dense open subset U of Z. Let Z′ = Z − U. Setting S_{Z′} = S_{Z} ×_{S} S′ we conclude that our lemma holds for Z′.

Let us now prove Proposition 4.2.15. We will only consider the case of z(X/S, r). Note first that by Proposition 4.2.2 both z(X/S, r)_{qfh} and z(X/S, r)_{h} are subpresheaves in Cycl(X/S, r)_{Q}. We have only to show that if Z is an element of Cycl(X/S, r) ⊗ Q and there is an h-covering f : S′ → S such that cycl(f)(Z) belongs to z(X/S, r)(S′) then there is a qfh-covering with the same property. It follows from 3.3.9 and Lemma 4.2.17.

Corollary 4.2.18 Let X → S be a scheme of finite type over a Noetherian scheme S. Then for any r ≥ 0 the qfh-sheaves Cycl_{equi}(X/S, r)_{Q}, P ropCycl_{equi}(X/S, r)_{Q} are isomorphic on the category of excellent Noetherian schemes over S to the qfh-sheaves of abelian groups associated in the obvious way with the qfh-sheaves of abelian monoids Cycl^{eff}(X/S, r)_{Q}, P ropCycl^{eff}(X/S, r)_{Q} and the same holds for the corresponding p-divisible sheaves if S is a scheme of exponential characteristic p.

Proof: Note first that one has by 3.1.7:

Cycl^{eff}(X/S, r)_{Q} ⊂ Cycl_{equi}(X/S, r)_{Q}
P ropCycl^{eff}(X/S, r)_{Q} ⊂ P ropCycl_{equi}(X/S, r)_{Q}

As in the proof of 4.2.8 it is sufficient to show that for any element Z = ∑ n_{i}z_{i} in Cycl_{equi}(X/S, r)_{Q} there is a qfh-covering f : S′ → S such that cycl(f)(Z) is a linear combination of elements of Cycl^{eff}(X/S, r)_{Q}(S′). Since S is an excellent scheme its normalization is a qfh-covering and our statement follows from Corollary 3.4.3.

Remark: Corollary 4.2.18 remains true without the excellency assumption.

4.3 Fundamental exact sequences for Chow sheaves.

Theorem 4.3.1 Let p : X → S be a scheme of finite type over a Noetherian scheme S, i : Z → X be a closed subscheme of X and j : U → X be the complement to Z in X. Then for any r ≥ 0 the following sequence of presheaves is left exact and it is also right exact as a sequence of cdh-sheaves:

0 → z(Z/S, r) →^{i_{∗}} z(X/S, r) →^{j^{∗}} z(U/S, r) → 0.

Proof: This sequence is obviously left exact as a sequence of presheaves. It is sufficient to show that the last arrow is a surjection in the cdh-topology. Let Z = ∑ n_{i}z_{i} be an element of z(U/S, r). By Lemma 4.1.4 it is sufficient to show that there is an abstract blow-up (2.2.4) of the form f : S′ → S such that cycl(f)(Z) belongs to the image of z(X/S, r)(S′) in z(U/S, r)(S′). Let Z_{i} be the closures of the points z_{i} in X wich we consider as closed integral subschemes. We may assume that S is reduced. Then by Theorem 2.2.2 there is a blow-up f : S′ → S such that the proper transforms Z̃_{i} of Z_{i} are flat over S′. We set Z′ = ∑ n_{i}cycl_{X×_{S}S′}(Z̃_{i}). Then by Corollary 3.3.11 one has Z′ ∈ z(X/S, r)(S′) and its restriction to U ×_{S} S′ obviously equals cycl(f)(Z).

Corollary 4.3.2 Let p : X → S be a scheme of finite type over a Noetherian scheme S and X = X_{1} ∪X_{2} be an open covering of X. Denote the inclusions X_{i} ⊂ X,X_{1} ∩ X_{2} ⊂ X_{i} by f_{i} and g_{i} respectively. Then for any r ≥ 0 the following sequence of presheaves is left exact and it is also right exact as a sequence of cdh-sheaves:

0 → z(X/S, r) →^{(f_{1})^{∗}+(f_{2})^{∗}} z(X_{1}/S, r) ⊕ z(X_{2}/S, r) →^{(g_{1})^{∗}−(g_{2})^{∗}}

→ z((X_{1} ∩ X_{2})/S, r) → 0.

Remark: Note that the sequence of Theorem 4.3.1 is in fact already exact in the topology where coverings are proper cdh-coverings.

Proposition 4.3.3 Let S be a Noetherian scheme, p : X → S be a scheme of finite type over S, i : Z → X be a closed subscheme of X and f : X′ → X be a proper morphism such that the morphism f^{−1}(X − Z) → X − Z is an isomorphism.

Consider the pull-back square

f^{−1}(Z) →^{i′} X′
f_{Z} ↓  ↓ f
Z  →^{i} X

Denote by F (−, r) one of the cdh-sheaves z(−, r), c(−, r). Then the following sequence of presheaves is left exact and it is also right exact as a sequence of cdh-sheaves

0 → F (f^{−1}(Z)/S, r) →^{i′_{∗}⊕(f_{Z})_{∗}} F (X′/S, r) ⊕ F (Z/S, r) →^{f_{∗}⊕(−i_{∗})}

→ F (X/S, r) → 0.

Proof: It is clearly sufficient to consider the case of the sheaves z(−, −).

Lemma 4.3.4 In the notations of Proposition 4.3.3 the following sequence of abelian groups is exact:

0→Cycl_{equi}(f^{−1}(Z)/S,r) →^{i′∗⊕(fZ)∗}Cycl_{equi}(X′/S,r)⊕Cycl_{equi}(Z/S,r) →^{f∗⊕(−i∗)}Cycl_{equi}(X/S,r).

Proof: The first arrow is a monomorphism since i′ : f^{−1}(Z) → X′ is a monomorphism.

Let us show that the sequence is exact in the middle term. Let Y = ∑ n_{i}y_{i} be an element of Cycl_{equi}(Z/S, r) and W = ∑ m_{j}w_{j} be an element of Cycl_{equi}(X′/S, r) such that f_{∗}(W) = i_{∗}(Y). The cycle W can be represented (uniquely) as a sum W = W_{0} + W_{1} such that supp(W_{0}) ⊂ f^{−1}(Z) and supp(W_{1}) ∩ f^{−1}(Z) = ∅. By our condition on W we have

f_{∗}(W_{0}) = i_{∗}(Y)
f_{∗}(W_{1}) = 0

and since f is an isomorphism outside f^{−1}(Z) we conclude that W_{1} = 0 and hence W ⊕ Y belongs to the image of the homomorphism i′_{∗} ⊕ (f_{Z})_{∗}.

One can easily see that Lemma 4.3.4 implies that the sequence of abelian groups

0 → z_{equi}(f^{−1}(Z)/S, r) →^{i′_{∗}⊕(f_{Z})_{∗}} z_{equi}(X′/S, r) ⊕ z_{equi}(Z/S, r) →^{f_{∗}⊕(−i_{∗})}

→ z_{equi}(X/S, r).

is also exact. Hence, our sequence of sheaves is left exact and it is sufficient to show that the homomorphism

z_{equi}(X′/S, r) ⊕ z_{equi}(Z/S, r) →^{f_{∗}⊕(−i_{∗})} z_{equi}(X/S, r)

is surjective as a homomorphism of cdh-sheaves. By Lemma 4.1.4 we may assume that S is integral and it is sufficient to show that for any element W of the group z_{equi}(X/S, r) there is a blow-up g : S′ → S such that cycl(g)(W) belongs to the image of z_{equi}(X′×_{S} S′/S′, r)⊕z_{equi}(Z ×_{S} S′/S′, r). By Lemma 4.2.1 we may assume that W = cycl_{X}(W) where W is an integral closed subscheme of X which is equidimensional of relative dimension r over S. If W ⊂ Z our statement is obvious. Let w be the generic point of W and W′ be the closure of f^{−1}(w) in X′. By Theorem 2.2.2 there is a blowup g : S′ → S such that the proper transform W̃′ of W′ is flat over S′. Denote by f′ the morphism f ×_{S} S′ : X′ ×_{S} S′ → X ×_{S} S′. Since g is an isomorphism in generic points and f is an isomorphism ouside Z we clearly have f′_{∗}(cycl(W̃′)) = cycl(g)(W). To finish the proof it is sufficient to note that W̃′ belongs to z(X′ ×_{S} S′/S′, r) by Corollary 3.3.11.

Corollary 4.3.5 Let S be a Noetherian scheme, p : X → S be a scheme of finite type over S and X = Z_{1} ∪ Z_{2} be a covering of X by two closed subschemes. Denote the inclusions Z_{1} ∩ Z_{2} ⊂ Z_{i}, Z_{i} ⊂ X by f_{i} and g_{i} respectively and let F (−, r) be one of the cdh-sheaves z(−, r), c(−, r).

Then the following sequence of cdh-sheaves is exact

0 → F (Z_{1} ∩ Z_{2}, r) →^{(f_{1})_{∗}+(f_{2})_{∗}} F (Z_{1}, r) ⊕ F (Z_{2}, r)) →^{(g_{1})_{∗}−(g_{2})_{∗}} F (X, r) → 0.

Proof: It is sufficient to apply Proposition 4.3.3 in the case X′ = Z_{1} ∐ Z_{2}, Z = Z_{1} ∩ Z_{2}.

Proposition 4.3.6 In the notations of Proposition 4.3.3 assume in addition that f is a finite morphism and S is a geometrically unibranch scheme. Then the following sequence of abelian groups is exact:

0→Cycl_{equi}(f^{−1}(Z)/S,r) →^{i′∗⊕(fZ)∗}Cycl_{equi}(X′/S,r)⊕Cycl_{equi}(Z/S,r) →^{f∗⊕(−i∗)}Cycl_{equi}(X/S,r)→0.

The same statement holds for the groups P ropCycl_{equi}(−, −).

Proof: It is clearly sufficient to consider the case of Cycl_{equi}(−, −). By Lemma 4.3.4 our sequence is left exact.

Let W be an element of Cycl_{equi}(X/S, r). By Corollary 3.4.3 we may assume that W = cycl_{X}(W) where W is an integral closed subscheme of X which is equidimensional of relative dimension r over S. If W ⊂ Z then W belongs to the image of the homomorphism i_{∗}. Otherwise let w be the generic point of W and let W′ be the closure of w′ = f^{−1}(w) in X′. Since w belongs to X − Z we have f_{∗}(cycl_{X′}(W′)) = W. To finish the proof of our proposition it is sufficient to show that cycl_{X′}(W′) ∈ Cycl_{equi}(X′/S, r). It follows from the fact that f is finite and Theorem 3.4.2.

Remark: Note that in Proposition 4.3.6 one can not replace in general the groups Cycl_{equi}(−, −) by the groups z_{equi}(−, −). An example of the situation when the corresponding sequence is not right exact for the groups z_{equi}(−, −) can be easily deduced from example 3.5.10(2).

Proposition 4.3.7 Let S be a Noetherian scheme, p : X → S be a scheme of finite type over S and X = U_{1} ∪ U_{2} be an open covering of X. Denote the inclusions U_{1} ∩ U_{2} ⊂ U_{i}, U_{i} ⊂ X by f_{i} and g_{i} respectively.

Then the sequence of presheaves

0 → c_{equi}(U_{1} ∩ U_{2}/S, 0) →^{(f_{1})_{∗}+(f_{2})_{∗}} c_{equi}(U_{1}/S, 0) ⊕ c_{equi}(U_{2}/S, 0) →^{(g_{1})_{∗}−(g_{2})_{∗}}

→ c_{equi}(X/S, 0) → 0.

is exact in the Nisnevich topology.

Proof: Note that our sequence is left exact as a sequence of presheaves by obvious reason. To prove that the last arrow is a surjection in the Nisnevich topology it is sufficient to show that the map

c_{equi}(U_{1}/S, 0) ⊕ c_{equi}(U_{2}/S, 0) →^{(g_{1})_{∗}−(g_{2})_{∗}} c_{equi}(X/S, 0)

is surjective if S is a local henselian scheme (see [12]). It follows trivially from the fact that for any element Z of c_{equi}(X/S, 0) its support is finite over S and the existence of decomposition of schemes finite over local henselian schemes into a disjoint union of local schemes (see [11, I.4.2.9(c)]).

Corollary 4.3.8 In the notations of Proposition 4.3.7 the sequence of cdhsheaves

0 → c(U_{1} ∩ U_{2}, 0) →^{(f_{1})_{∗}+(f_{2})_{∗}} c(U_{1}, 0) ⊕ c(U_{2}, 0) →^{(g_{1})_{∗}−(g_{2})_{∗}} c(X, 0) → 0.

is exact.

Proof: It follows immediately from Proposition 4.3.7, Theorem 4.2.9 and exactness of the functor of associated sheaf.

Remark: Note that the exact sequence of Proposition 4.3.7 is quite different from the exact sequence of Corollary 4.3.2. In particular while the sequence for finite cycles requires only Nisnevich coverings to be exact the sequence for general cycles requires only abstract blow-ups, i.e. proper cdh-coverings to be exact.

Proposition 4.3.9 Let p : X → S be a scheme of finite type over a Noetherian scheme S, Z → X be a closed subscheme of X and f : X′ → X be an etale morphism such that the morphism f^{−1}(Z) → Z is an isomorphism. Let further U be the complement to Z in X and U′ be the complement to f^{−1}(Z) in X′. Then the canonical morphism of quotient sheaves in the Nisnevich topology

c_{equi}(X′/S, 0)/c_{equi}(U′/S, 0) → c_{equi}(X/S, 0)/c_{equi}(U/S, 0)

is an isomorphism.

Proof: It is sufficient to show that for a local henselian scheme S the morphism of abelian groups

c_{equi}(X′/S, 0)/c_{equi}(U′/S, 0) →^{f_{∗}} c_{equi}(X/S, 0)/c_{equi}(U/S, 0).

is an isomorphism. The following lemma is an easy corollary of the standard properties of henselian schemes (see [11]).

Lemma 4.3.10 Let q : X → S be an etale morphism such that the scheme S is henselian and let x be a closed point of X over the closed point s of S such that the morphism Spec(k_{x}) → Spec(k_{s}) is an isomorphism. Then q is an isomorphism in a neighborhood of x.

(Injectivity.) Let W′ be an element of c_{equi}(X′/S, 0) and assume that f_{∗}(W′) belongs to c_{equi}(U/S, 0). Since S is henselian the support W′ = supp(W′) of W′ is a disjoint union of local henselian schemes and we may assume that the closed points of W belong to f^{−1}(Z) and f_{∗}(W) = 0.

Since f is an isomorphism on f^{−1}(Z) we may further assume that W′ is local. Then by Lemma 4.3.10 the morphism supp(W′) → supp(f_{∗}(W′)) is an isomorphism and hence W′ = 0. (Surjectivity.) Let W be an element of c_{equi}(X/S, 0). As above we may assume that W = supp(W) is local and its closed point belongs to Z. Let W′ be the local scheme of the closed point of f^{−1}(W) over the closed point of W. Then by Lemma 4.3.10 the morphism W′ → W is an isomorphism. Denote by W′ the cycle on W′ (and hence on X′) which corresponds to W. We obviously have f_{∗}(W′) = W and W′ ∈ c_{equi}(X′/S, 0).

Corollary 4.3.11 In notations of Proposition 4.3.9 there is a canonical isomorphism of quotient sheaves in the cdh-topology of the form:

c(X′/S, 0)/c(U′/S, 0) → c(X/S, 0)/c(U/S, 0).

Proof: It follows from Proposition 4.3.9 and Theorem 4.2.9.

4.4 Representability of Chow sheaves.

In this section we consider representability of Chow sheaves of effective proper cycles on quasi-projective schemes over a Noetherian scheme S. Let us begin with the following definition.

Definition 4.4.1 Let S be a Noetherian scheme and F be a presheaf of sets on the category of Noetherian schemes over S. We say that F is hrepresentable by a scheme X over S if there is an isomorphism F_{h} → L_{h}(X) of the h-sheaf associated with F with the h-sheaf associated with the presheaf represented by X.

Note that the scheme X which h-represents a presheaf F is not uniquely defined up to isomorphism since the h-topology is not subcanonical. Nevertheless as was shown in [18] X is well defined up to a universal homeomorphism. In particular if for some X which h-represents F all generic points of X are of characteristic zero then there exists a unique semi-normal scheme which h-represents F.

Let S be a Noetherian scheme and F be a presheaf on the category of Noetherian schemes over S. A k-point of F is a pair of the form (Spec(k) → S, ϕ) where k is a field and ϕ ∈ F (Spec(k)/S) is a section of F over Spec(k). We say that a k-point ϕ of F is equivalent to a k′-point ϕ′ of F if there is a field k′′ a morphism Spec(k′′) → S and morphisms u : Spec(k′′) → Spec(k), u′ : Spec(k′′) → Spec(k′) over S such that the sections u^{∗}(ϕ) and (u′)^{∗}(ϕ′) of F over Spec(k′′) coincide. A point of F is by definition an equivalence class of k-points of F. Denote the set of points of F by T op(F). One can easily see that for any morphism of presheaves f : F → G one has a map of sets T op(f) : T op(F) → T op(G) which is a monomorphism (resp. an epimorphism) if f is a monomorphism (resp. an epimorphism). In particular for any subpresheaf F_{0} in F we get a subset T op(F_{0}) in T op(F).

The following lemma is trivial.

Lemma 4.4.2 Let t be a Grothendieck topology on the category of Noetherian schemes over S such that for any algebraically closed field k any morphism Spec(k) → S and any t-covering p : U → Spec(k) there exists a section s : Spec(k) → U of p. For a presheaf F on the category of Noetherian schemes over S denote by F_{t} the associated t-sheaf. Then the map T op(F) → T op(F_{t}) is a bijection.

Note that all topologies we use in this paper satisfty the condition of Lemma 4.4.2.

For any presheaf of sets F on the category of Noetherian schemes over S, any Noetherian scheme S′ over S and any section ϕ of F over S′ denote by T op(ϕ) the obvious map of sets S′ → T op(F).

Let now A ⊂ T op(F) be a subset in T op(F). Denote by F_{A} the subpresheaf in F such that for any Noetherian scheme S′ over S the subset F_{A}(S′) ⊂ F (S′) consists of sections ϕ such that T op(ϕ)(S′) ⊂ A. If a subpresheaf F_{0} in F is of the form F_{A} for a subset A in T op(F) we say that F_{0} is defined by a pointwise condition. One can easily see that F_{0} is defined by a pointwise condition if and only if for any Noetherian scheme S′ over S and a section ϕ of F over S′ such that T op(ϕ)(S′) ⊂ T op(F_{0}) one has ϕ ∈ F_{0}(S′).

Let A be a subset in T op(F). We say that A is open (resp. closed, constructible) if for any Noetherian scheme S′ over S and any section ϕ of T on S′ the set T op(ϕ)^{−1}(A) is open (resp. closed, constructible ) in S′. We will say further that a subpresheaf F_{0} of F is open (resp. closed, constructible) if F_{0} is of the form F_{A} for an open (resp. closed, constructible) subset in T op(F).

One can easily see that open subsets of T op(F) form a topology on this set and that a subset is closed if and only if its complement is open. Note also that for any morphism of presheaves f : F → G the corresponding map T op(f) : T op(F) → T op(G) is a continuous map with respect to this topology. The following lemma is straightforward.

Lemma 4.4.3 Let X → S be a Noetherian scheme over a Noetherian scheme S. Denote by L(X/S) the presheaf of sets represented by X on the category of Noetherian schemes over S.

1. The map T op(ϕ) : X → T op(L(X/S)) defined by the tautological section of L(X/S) over X is a homeomorphism of the corresponding topological spaces.

2. Let t be a topology on the category of Noetherian schemes over S satisfying the condition of Lemma 4.4.2 and such that for any Noetherian scheme S′ over S the morphism S′_{red} → S′ is a t-covering. Then for any open (resp. closed) subset A ⊂ T op(L_{t}(X/S)) the subpresheaf L_{t}(X/S)_{A} is t-representable by the corresponding open (resp. closed) subscheme in X.

Note that the h-topologies (h-,cdh-,qfh-) satisfy the conditions of Lemma 4.4.3(2).

We say that a presheaf F on the category of Noetherian schemes over a Noetherian scheme S is topologically separated if for any Noetherian scheme T over S and any dominant morphism T′ → T the map F (T) → F (T′) is injective. Note that all the Chow presheaves considered in this paper are topologically separated.

Lemma 4.4.4 Let S be a Noetherian scheme, F be a topologically separated presheaf on the category of Noetherian schemes over S, X → S be a scheme of finite type over S and f : z(X/S, r) → F be a monomorphism of presheaves. Consider a closed subset A in T op(z(X/S, r)). Then the subsheaf f(z(X/S, r)_{A})_{h} of F_{h} is a closed subpresheaf of F if and only if T op(f)(A) is a constructible subset in T op(F) = T op(F_{h}).

Proof: We obviously have T op(f)(A) = T op(f(z(X/S, r)_{A})) ⊂ T op(F) which proves the “only if” part.

Assume that T op(f)(A) is a constructible subset in T op(F). Let us show first that f(z(X/S, r)_{A})_{h} is a subpresheaf in F_{h} given by a pointwise condition. Let T be a Noetherian scheme over S and ϕ ∈ F_{h}(T) be a section of F_{h} such that for any geometrical point x : Spec(k) → T of T we have x^{∗}(ϕ) ∈ z(X/S, r)_{A}. We have to show that ϕ ∈ f(z(X/S, r)_{A})_{h}(T). Our problem is h-local with respect to T. Replacing T by the union of its irreducible components we may assume that T is an integral scheme. Let η be the generic point of T, k̄_{η} be an algebraic closure of the function field of T and η̄ : Spec(k̄_{η}) → T be the corresponding geometrical point of T. Then η̄^{∗}(ϕ) corresponds to an element Z in z(X ×_{T} Spec(k̄_{η})/Spec(k̄_{η}), r). Since X is of finite type over S and A is a closed subset of T op(Z(X/S, r)) there is a quasi-finite dominant morphism p : U → T and an elememt Z_{U} in z(X/S, r)_{A}(U) such that the restriction of Z_{U} to Spec(k̄_{η}) equals Z. Theorem 2.2.2 implies easily now that there is an h-covering p̄: Ū → T which is finite over the generic point of T and a section ZŪ of z(X/S, r)_{A} over Ū such that its restriction to Spec(k̄_{η}) equals Z. Since F is topologically separated we conclude that p̄^{∗}(ϕ) = ZŪ and hence f(z(X/S, A))_{h} is indeed defined by a pointwise condition. To prove that T op(f(z(X/S, A))_{h}) is a closed subset in T op(F) = T op(F_{h}) is trivial.

Proposition 4.4.5 Let X → S be a scheme of finite type over a Noetherian scheme S and X_{0} be a closed subscheme in X. Then the sheaf z(X_{0}/S, r) is a closed subpresheaf in the sheaf z(X/S, r) .

Proof: Note first that the sheaf z(X_{0}/S, r) is a subpresheaf in z(X/S, r) given by a pointwise condition. Moreover a section of z(X/S, r) belongs to z(X_{0}/S, r) if and only if it belongs to z(X_{0}/S, r) over the generic points of S. It implies easily that the only thing we have to show is that for any cycle Z in z(X/S, r) which does not belong to z(X_{0}/S, r) in a generic point η of S there is a neighborhood U of η in S such that Z_{s} does not belong to z((X_{0})_{s}/Spec(k_{s}), r) for any point s in U. It is obvious.

Proposition 4.4.6 Let X → S be a scheme of finite type over a Noetherian scheme S and X_{0} be an open subscheme in X. Then the sheaf c^{eff}(X_{0}/S, r) is an open subpresheaf in the sheaf c^{eff}(X/S, r) .

Proof: Let us show first that the subpresheaf c^{eff}(X_{0}/S, r) is given by a pointwise condition. Let Z be an element of c^{eff}(X/S, r). Assume that for any geometrical point x : Spec(k) → S of S we have cycl(x)(Z) ∈ c^{eff}(X_{0} ×_{S} Spec(k)/Spec(k), r). We have to show that Z ∈ c^{eff}(X_{0}/S, r). Since c^{eff}(X_{0}/S, r) is a cdh-subsheaf in c^{eff}(X/S, r) it is sufficient to show that there is a blow-up f : S′ → S such that cycl(f)(Z) ∈ c^{eff}(X_{0} ×_{S} S′/S′, r). We may assume therefore that Z = ∑ n_{i}cycl(Z_{i}) where Z_{i} are irreducible closed subschemes of X which are flat over S. It is sufficient to show that our condition on Z implies that supp(Z) ⊂ X_{0}. It follows immediately from the fact that Z is effective since supp(cycl(x)(cycl(Z_{i}))) = Z_{i} ×_{S} Spec(k) for any geometrical point Spec(k) → S of S.

Let now Z be an arbitrary element of c^{eff}(X/S, r). We have to show that the set of points s of S such that Zk̄_{s} ∈ c^{eff}(X_{0} ×_{S} Spec(k̄_{s})/Spec(k̄_{s}), r) is open in S. Again we may replace S by its blow-up and assume that Z = ∑ n_{i}cycl(Z_{i}) where Z_{i} are irreducible closed subschemes of X which are flat over S. Then clearly our subset is the intersection of subsets U_{i} where s ∈ U_{i} if and only if Z_{i}×_{S}Spec(k_{s}) ⊂ X_{0}×_{S}Spec(k_{s}). Then U_{i} = S−p(Z_{i}∩(X−X_{0})) and since p : Z_{i} → S is proper we conclude that U_{i} are open subsets of S.

Example 4.4.7 The analog of Proposition 4.4.6 for the sheaves c(X/S, r) is false. Let us consider the scheme X = P_{k}^{2} × A^{1} → A^{1} over the affine line. Let further Z_{1} (resp. Z_{2}) be the closed subscheme given by the equation x_{0} +x_{1} +tx_{2} = 0 (resp. x_{0} +x_{1} −tx_{2} = 0) where x_{i} are the coordinates on P^{2} and t is the coordinate on A^{1}. Let Z = Z_{1}−Z_{2} and X_{0} = A^{2}×A^{1} ⊂ P_{k}^{2}×A^{1}. Then the set of points t ∈ A^{1} where Z_{t} ∈ (X_{0})_{t} consists of one point t = 0.

Let S be a Noetherian scheme. For any multi-index I = (i_{1}, …, i_{k}) denote by P_{S}^{I} the product of projective spaces P_{S}^{i_{1}} ×_{S} … ×_{S} P^{i_{k}} over S. Let Z be an element of Cycl^{eff}(P_{S}^{I}/S, r). For any point s of S denote by deg_{s}(Z) the (multi-)degree of the cycle Z_{s} on P_{k_{s}}^{I} (note that apriory deg_{s}(Z) is sequence of rational numbers).

Proposition 4.4.8 Let S be a Noetherian scheme and Z be an element of Cycl^{eff}(P_{S}^{I}/S, r). Then the function s → deg_{s}(Z) is locally constant on S.

Proof: It is sufficient to show that if η is a generic point of S and s is a point in the closure of η then deg_{η}(Z) = deg_{s}(Z). Since for any cycle W on P_{k}^{I} and any field extension k′/k we have deg(Z) = deg(Z ⊗_{k} k′) it is sufficient to show that for some field extensions L, E of k_{η} and k_{s} respectively the cycles Z_{Spec(L)} and Z_{Spec(E)} have the same degree. Let (x_{0}, x_{1}, R) be a fat point on S such that the image of x_{1} is {η, s}. Replacing S by Spec(R) we may assume that S is the spectrum of a discrete valuation ring. In this case Z = ∑ n_{i}cycl(Z_{i}) where Z_{i} are closed subschemes of P_{S}^{I} which are flat and equidimensional over S and our statement follows from the invariance of (multi-)degree in flat families.

Corollary 4.4.9 Let S be a connected Noetherian scheme. Then for any cycle Z in Cycl(P_{S}^{I}/S, r) and any point s of S the degree deg_{s}(Z) is a sequence of integers which does not depend on s.

For a Noetherian scheme S, a multi-index I = (i_{1}, …, i_{k}) and a sequence of nonnegative integers D = (d_{1}, …, d_{n}) denote by z_{D}^{eff}(P_{S}^{I}/S, r) the subset in z(P_{S}^{I}/S, r) which consists of cycles Z such that for any point s of S one has deg_{s}(Z) = D. One can easily see that z_{d}^{eff}(P_{S}^{I}/S, r) is in fact a cdh-subsheaf in z^{eff}(P_{S}^{I}/S, r). Proposition 4.4.8 implies further that for a connected Noetherian scheme one has

z^{eff}(P_{S}^{I}/S, r) = ⋃_{D} z_{D}^{eff}(P_{S}^{I}/S, r).

The proof of the following lemma is standard.

Lemma 4.4.10 Let S be a Noetherian scheme. Then for any multi-index I = (i_{1}, …, i_{k}) and any sequence of nonnegative integers D = (d_{1}, …, d_{k}) the sheaf z_{D}^{eff}(P_{S}^{I}/S, (∑ i_{j}) − 1) is h-representable by the projective space P_{S}^{N} for some N = N(I, D).

Denote by G the product (P^{n})^{∗} ×_{Spec(Z)} …_{Spec(Z)} (P^{n})^{∗} of r + 1-copies of the projective space dual to the standard projective space (i.e (P^{n})^{∗} is the scheme which parameterizes hyperplanes in P^{n}). Let further L ∈ P^{n} ×_{Spec(Z)} G be the closed subscheme of points (x, H_{1}, …, H_{r+1}) such that x ∈ H_{i} for all i = 1, …, r +1. It is smooth over P^{n} and fibers of the projection f : L → P^{n} are isomorphic to (P^{n−1})^{r+1}.

Denote further by Div_{d}(G) the projective space which parametrizes cycles of codimension 1 and degree (d, …, d) on ((P^{n})^{∗})^{r+1} (see Lemma 4.4.10). Let Div_{d}^{irr}(G) be the open subspace in Div_{d}(G) which parametrizes irreducible divisors on G and let Γ ∈ G × Div_{d}^{irr}(G) be the support of the corresponding relative cycle on G × Div_{d}^{irr}(G) over Div_{d}^{irr}(G). Set

U = L × Div_{d}^{irr}(G) − (L × Div_{d}^{irr}(G) ∩ P^{n} × Γ)

Let finally Φ be the subset (pr : L × Div_{d}^{irr}(G) → P^{n} × Div_{d}^{irr}(G))(U). Since L is smooth (and hence universally open) over P^{n} this subset is open in P^{n} × Div_{d}^{irr}(G). We define C_{r,d}^{irr} to be the subset in Div_{d}^{irr}(G) which consists of points s such that dim((P^{n}×Div_{d}^{irr}(G)−U)×_{Div_{d}^{irr}(G)}Spec(k_{s})) ≥ r. Since the projection P^{n} × Div_{d}^{irr}(G) → Div_{d}^{irr}(G) is proper Chevalley theorem (2.1.1) implies that C_{r,d}^{irr} is a closed subset in Div_{d}^{irr}(G).

Theorem 4.4.11 For any Noetherian scheme S and any r, d ≥ 0 the sheaf of sets z_{d}^{eff}(P_{S}^{n}/S, r) is h-representable by a projective scheme C_{r,d} over S.

Proof: Note that it is obviously sufficient to prove our theorem for S = Spec(Z). We will use the notations which we introduced in the construction of C_{r,d}^{irr} above. Let Z be an element of z^{eff}(P_{S}^{n}/S, r). Let

f : L → P^{n}
p : L → G

be the obvious morphisms. Since f is smooth and p is proper we get a homomorphism of presheaves

Chow = p_{∗}f^{∗} : z^{eff}(P^{n}/Spec(Z), r) → z^{eff}(G/Spec(Z), (n − 1)(r + 1) + r)

The following lemma is straighforward.

Lemma 4.4.12 1. The homomorphism Chow is a monomorphism.

2. The homomorphism Chow takes the subsheaf z_{d}^{eff}(P_{S}^{n}/S, r) to the subsheaf z_{D}^{eff}(((P_{S}^{n})^{∗})^{r+1}/S, (n − 1)(r + 1) + r) where D = (d, …, d).

In view of Lemma 4.4.12 and Lemma 4.4.10 the homomorphism Chow gives us an embedding of the sheaf z_{d}^{eff}(P_{S}^{n}/S, r) to the cdh-sheaf representable by the projective space Div_{d}(G). Since z_{d}^{eff}(P^{n}/Spec(Z), r) is clearly a closed subpresheaf in the sheaf z(P^{n}/Spec(Z), r) it is sufficient by 4.4.3 and 4.4.4 to show that the subset C_{r,d} = Im(T op(Chow)) in Div^{d}(G) is constructible.

Denote by F_{d} the subpresheaf in z_{d}^{eff}(P^{n}/Spec(Z), r) such that for a Noetherian scheme S the subset F_{d}(S) in z_{d}^{eff}(P_{S}^{n}/S, r) consists of cycles Z such that for any algebraically closed field k and a k-point x : Spec(k) → S the cycle cycl(x)(Z) on P_{k}^{n} is of the form cycl(Z) for a closed integral subscheme Z in P_{k}^{n} of dimension r and degree d. Let further

F_{d} = ∐_{k=1}^{d} ∐_{d_{1}+...+d_{k}=d} ∏_{i=1,...,k} F_{d_{i}}.

We have the following diagram of morphisms of presheaves

F_{d} →^{Chow} ∐_{k=1}^{d} ∐_{d_{1}+...+d_{k}=d} ∏_{i=1,...,k} L(Div_{d_{i}}^{irr}(G))
↓  ↓
z_{d}^{eff}(P^{n}/Spec(Z), r) → L(Div_{d}(G))

The first vertical arrow is a surjection on the corresponding topological spaces and the second vertical arrow being induced by a morphism of schemes of finite type takes constructible sets to constructible sets. It is sufficient therefore to show that the image of T op(Chow^{irr}) where Chow^{irr} is the morphism

F_{d} → L(Div_{d}^{irr}(G))

is constructible. Let us show that in fact Im(T op(Chow^{irr})) = C_{r,d}^{irr}.

(“ Im(Top(Chow^{irr})) ⊂ C_{r,d}^{irr}”) Let k be an algebraically closed field and Z ∈ F_{d}(P_{k}^{n}). We have to show that the point x on Div_{d}^{irr}(G)_{k} which corresponds to Chow^{irr}(Z) belongs to C_{r,d}^{irr}. It follows immediately from our definition of C_{r,d}^{irr} since in this case the fiber of the projection P^{n}×Div_{d}^{irr}(G)− U → Div_{d}^{irr}(G) over x contains the support of Z.

(“C_{r,d}^{irr} ⊂ Im(Top(Chow^{irr})) ”) Let k be an algebraically closed field and x : Spec(k) → Div_{d}^{irr}(G) be a point of Div_{d}^{irr}(G) which belongs to C_{r,d}^{irr}, i.e. such that dim((P^{n} × Div_{d}^{irr}(G) − U) ×_{Div_{d}^{irr}(G)} Spec(k)) ≥ r. Let D_{x} be the divisor on G_{k} which corresponds to x. One can verify easily that fibers of P^{n} × Div_{d}(G) − U over Div_{d}(G) are of dimension ≤ r. Denote by Z_{i} the irreducible components of the fiber of P^{n} × Div_{d}(G) − U over x which have dimension r. Then p(f^{−1}(Z_{i})) is an irreducible divisor in G which is obviously contained in supp(D). Since p(f^{−1}(Z_{i})) ≠ p(f^{−1}(Z_{j})) for i ≠ j (Lemma 4.4.12(1)) and supp(D) is irreducible we conclude that there is only one component Z of dimension r and p(f^{−1}(Z)) = supp(D). It implies easily that D = Chow^{irr}(cycl(Z)). Theorem is proven.

Let S be a Noetherian scheme and i : X → P_{S}^{n} be a projective scheme over S. Denote by z_{d}^{eff}((X, i)/S, r) the subpresheaf in the Chow presheaf z^{eff}(X/S, r) such that for a Noetherian scheme T over S the subset z_{d}^{eff}((X, i)/S, r)(T) in z^{eff}(X/S, r)(T) consists of relative cycles of degree d with respect to the embedding i ×_{S} Id_{T}. If U ⊂ X is an open subset of X we denote by c_{d}^{eff}((U, i)/S, r) the preimage of z_{d}^{eff}((X, i)/S, r) with respect to the obvious morphism c^{eff}(U/S, r) → z^{eff}(X/S, r).

Corollary 4.4.13 For any Noetherian scheme S and a projective scheme i : X → P_{S}^{n} over S the presheaf z_{d}^{eff}((X, i)/S, r) is h-representable by a projective scheme C_{r,d}(X, i) over S and

z^{eff}(X/S, r) = ∐_{d≥0} z_{d}^{eff}((X, i)/S, r).

If U is an open subscheme in X then c_{d}^{eff}((U, i)/S, r) is representable by an open subscheme C_{r,d}(U, i) in C_{r,d}(X, i) and

c^{eff}(U/S, r) = ∐_{d≥0} c_{d}^{eff}((U, i)/S, r).

Proof: It follows immediately from Theorem 4.4.11, Propositions 4.4.5, 4.4.6 and Lemma 4.4.3.

Let X → S be a scheme of finite type over a Noetherian scheme S. A (closed) equivalence relation on X is a closed subscheme R ⊂ X ×_{S} X such that for any Noetherian scheme T over S the subset Hom_{S}(T, R) of the set Hom_{S}(T, X ×_{S} X) = Hom_{S}(T, X) × Hom_{S}(T, X) is an equivalence relation on Hom_{S}(T, X). An equivalence relation R is called a proper equivalence relation if the projections R → X are proper morphisms.

Let R be an equivalence relation on X and Y be a Noetherian scheme over S. Let Γ → Y ×_{S} X be a closed subset of Y ×_{S} X. Denote by p_{Y} : Γ → Y, p_{X} : Γ → X the obvious morphisms. We call Γ a graph-like closed subset (with respect to R) if the following conditions hold:

1. The morphism p_{Y} : Γ → Y is a universal topological epimorphism (i.e. an h-covering).

2. For any algebraically closed field k̄ and a k̄-valued point ȳ: Spec(k̄) → Y all the elements of the subset p_{X}(p_{Y}^{−1}(ȳ)) in X(k̄) are equivalent with respect to R(k̄).

Any such Γ defines for any algebraically closed field k̄ a map of sets f_{Γ} : Y (k̄) → X(k̄)/R(k̄). We say that two graph-like subsets Γ_{1}, Γ_{2} are equivalent if for any algebraically closed field k̄ the corresponding maps f_{Γ_{1}} and f_{Γ_{2}} coincide. A continuous algebraic map from Y to X/R (over S) is an equivalence class of graph-like closed subschemes of Y ×_{S} X with respect to this equivalence relation^{3}. We denote the set of continuous algebraic maps from Y to X/R over S by Hom_{S}^{a.c.}(Y, X/R).

Let now R be a proper equivalence relation. Denote the projections R → X by pr_{1} and pr_{2} respectively. For any closed subset Γ in Y ×_{S} X which is of finite type over Y consider the subset Γ_{R} ⊂ Y ×_{S} X of the form (Id_{Y} ×_{S} pr_{2})(Id_{Y} ×_{S} pr_{1})^{−1}(Γ). Since R is proper Γ_{R} is a closed subset. One can verify easily that if Γ is a graph-like subset with respect to R then Γ_{R} is graph-like and Γ is equivalent to Γ_{R}. Moreover two graph-like closed subsets Γ_{1} and Γ_{2} are equivalent if and only if (Γ_{1})_{R} = (Γ_{2})_{R}.

Lemma 4.4.14 Let X → S be a scheme of finite type over a Noetherian scheme S and R ⊂ X ×_{S} X be a proper equivalence relation on X. Let L_{h}(X) be the h-sheaf represented by X on the category of Noetherian schemes over S and let L_{h}(X)/R be the quotient sheaf (in the h-topology) of L_{h}(X) with respect to the equivalence relation defined by R. Then for any Noetherian scheme Y over S there is a canonical bijection:

Hom_{S}^{a.c.}(Y, X/R) = (L_{h}(X)/R)(Y).

Proof: Clearly Hom_{S}^{a.c.}(Y, X/R) is a presheaf on the category of Noetherian schemes over S with respect to Y. Let us show that it is in fact an h-sheaf. Consider an h-covering p : U → Y of Y. Note first that if Γ_{1}, Γ_{2} are two graph-like closed subsets in Y ×_{S} X such that (p ×_{S} Id_{X})^{−1}(Γ_{1}) is equivalent to (p ×_{S} Id_{X})^{−1}(Γ_{2}) then Γ_{1} is equivalent to Γ_{2}.

3Our definition of continuous algebraic maps is a natural generalization of the definition given in [3].

Let now Γ ⊂ U ×_{S} X be a graph-like closed subset such that pr_{1}^{−1}(Γ) is equivalent to pr_{2}^{−1}(Γ) where pr_{i} : U ×_{Y} U ×_{S} X → U ×_{S} X are the projections. We may assume that Γ = Γ_{R}. Then the same obviously holds for pr_{i}^{−1}(Γ) and we conclude that pr_{1}^{−1}(Γ) = pr_{2}^{−1}(Γ). Since p is a universal topological epimorphism it implies trivially that Γ = (p×_{S} Id_{X})^{−1}(Γ_{0}) for a closed subset Γ_{0} in Y ×_{S} X. Since Γ_{0} is obvioulsy graph-like with respect to R it proves that Hom_{S}^{a.c.}(−, X/R) is indeed an h-sheaf.

Let us construct a morphism of sheaves

ϕ : Hom_{S}^{a.c.}(−, X/R) → L_{h}(X)/R.

Let Γ be a graph-like closed subset in Y ×_{S} X. We have a morphism Γ → X. Since Γ → Y is an h-covering our definition of a graph-like subset implies trivially that the corresponding section of L_{h}(X)/R on Γ can be descended to a section of L_{h}(X)/R on Y which does not depend on the choice of Γ in its equivalence class.

The proof of the fact that ϕ is an isomorphism is trivial.

Let X → S be a quasi-projective scheme over a Noetherian scheme S and i : X̄ → P_{S}^{n} be a projective scheme over S such that there is an open embedding X ⊂ X̄ over S. Consider the scheme C_{r,d}(X̄, i) which h-represents the presheaf z_{d}^{eff}(X̄/S, r) by Corollary 4.4.13. Set

C_{r,≤d}(X̄, i) = ∐_{i≤d} C_{r,i}(X̄, r).

Consider the canonical section Z of z(X̄/S, r)_{h} over C_{r,≤d}(X̄, i) and let Z̃ be the section (pr_{1}^{∗}(Z) − pr_{2}^{∗}(Z)) − (pr_{3}^{∗}(Z) − pr_{4}^{∗}(Z)) of z(X̄/S, r)_{h} over the product (C_{r,≤d}(X̄, i))_{S}^{4} . Denote by R_{d}^{X} the closed subset of points of (C_{r,≤d}(X̄, i))_{S}^{4} where Z̃ belongs to z((X̄ − X)/S, r) ⊂ z(X̄/S, r) (see 4.4.5). One can easily see that R_{d}^{X} is an equivalence relation on (C_{r,≤d}(X̄, i))_{S}^{2} and since C_{r,≤d}(X̄, i) is proper over S it is a proper equivalence relation. Note further that the obviuos embeddings C_{r,≤d}(X̄, i) → C_{r,≤(d+1)}(X̄, i) take R_{d}^{X} to R_{d+1}^{X} and hence for any Noetherian scheme T over S there is a family of maps

Hom_{S}^{a.c.}(T, (C_{r,≤d}(X̄, i))_{S}^{2}/R_{d}^{X}) → Hom_{S}^{a.c.}(T, (C_{r,≤(d+1)}(X̄, i))_{S}^{2}/R_{d+1}^{X}).

Proposition 4.4.15 For any Noetherian scheme T over S there is a canonical bijection

colim_{d}Hom_{S}^{a.c.}(T, (C_{r,≤d}(X̄, i) ×_{S} C_{r,≤d}(X̄, i))/R_{d}^{X}) = z(X/S, r)_{h}(T).

Proof: It follows immediately from Lemma 4.4.14 and our definition of the equivalence relations R_{d}^{X}.

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