Chow Groups of Spaces

Introduction

In this chapter we first discuss Chow groups of algebraic spaces. Having defined these, we define Chern classes of vector bundles as operators on these chow groups. The strategy will be entirely the same as the strategy in the case of schemes. Therefore we urge the reader to take a look at the introduction (Chow Homology, Section 02P4) of the corresponding chapter for schemes.

Some related papers: [edidin-graham] and [kresch_cycle].

Setup

We first fix the category of algebraic spaces we will be working with. Please keep in mind throughout this chapter that “decent \(+\) locally Noetherian” is the same as “quasi-separated \(+\) locally Noetherian” according to Decent Spaces, Lemma 0BB6.

Situation

Here \(S\) is a scheme and \(B\) is an algebraic space over \(S\). We assume \(B\) is quasi-separated, locally Noetherian, and universally catenary (Decent Spaces, Definition 0ED7). Moreover, we assume given a dimension function \(\delta : |B| \longrightarrow \mathbf{Z}\). We say \(X/B\) is good if \(X\) is an algebraic space over \(B\) whose structure morphism \(f : X \to B\) is quasi-separated and locally of finite type. In this case we define \[\delta = \delta_{X/B} : |X| \longrightarrow \mathbf{Z}\] as the map sending \(x\) to \(\delta(f(x))\) plus the transcendence degree of \(x/f(x)\) (Morphisms of Spaces, Definition 04NM). This is a dimension function by More on Morphisms of Spaces, Lemma 0EDN.

A special case is when \(S = B\) is a scheme and \((S, \delta)\) is as in Chow Homology, Situation 02QL. Thus \(B\) might be the spectrum of a field (Chow Homology, Example 02QM) or \(B = \Spec(\mathbf{Z})\) (Chow Homology, Example 02QN).

Many lemma, proposition, theorems, definitions on algebraic spaces are easier in the setting of Situation 0EDT because the algebraic spaces we are working with are quasi-separated (and thus a fortiori decent) and locally Noetherian. We will sprinkle this chapter with remarks such as the following to point this out.

Remark

In Situation 0EDT if \(X/B\) is good, then \(|X|\) is a sober topological space. See Properties of Spaces, Lemma 06NJ or Decent Spaces, Proposition 03K6. We will use this without further mention to choose generic points of irreducible closed subsets of \(|X|\).

Remark

In Situation 0EDT if \(X/B\) is good, then \(X\) is integral (Spaces over Fields, Definition 0AD4) if and only if \(X\) is reduced and \(|X|\) is irreducible. Moreover, for any point \(\xi \in |X|\) there is a unique integral closed subspace \(Z \subset X\) such that \(\xi\) is the generic point of the closed subset \(|Z| \subset |X|\), see Spaces over Fields, Lemma 0ENF.

If \(B\) is Jacobson and \(\delta\) sends closed points to zero, then \(\delta\) is the function sending a point to the dimension of its closure.

Lemma

In Situation 0EDT assume \(B\) is Jacobson and that \(\delta(b) = 0\) for every closed point \(b\) of \(|B|\). Let \(X/B\) be good. If \(Z \subset X\) is an integral closed subspace with generic point \(\xi \in |Z|\), then the following integers are the same:

\(\delta(\xi) = \delta_{X/B}(\xi)\),

\(\dim(|Z|)\),

\(\text{codim}(\{z\}, |Z|)\) for \(z \in |Z|\) closed,

the dimension of the local ring of \(Z\) at \(z\) for \(z \in |Z|\) closed, and

\(\dim(\mathcal{O}_{Z, \overline{z}})\) for \(z \in |Z|\) closed.

Proof

Let \(X\), \(Z\), \(\xi\) be as in the lemma. Since \(X\) is locally of finite type over \(B\) we see that \(X\) is Jacobson, see Decent Spaces, Lemma 0BA3. Hence \(X_{\text{ft-pts}} \subset |X|\) is the set of closed points by Decent Spaces, Lemma 0BA5. Given a chain \(T_0 \supset \ldots \supset T_e\) of irreducible closed subsets of \(|Z|\) we have \(T_e \cap X_{\text{ft-pts}}\) nonempty by Morphisms of Spaces, Lemma 06EK. Thus we can always assume such a chain ends with \(T_e = \{z\}\) for some \(z \in |Z|\) closed. It follows that \(\dim(Z) = \sup_z \text{codim}(\{z\}, |Z|)\) where \(z\) runs over the closed points of \(|Z|\). We have \(\text{codim}(\{z\}, Z) = \delta(\xi) - \delta(z)\) by Topology, Lemma 02IA. By Morphisms of Spaces, Lemma 06EI the image of \(z\) is a finite type point of \(B\), i.e., a closed point of \(|B|\). By Morphisms of Spaces, Lemma 0ECY the transcendence degree of \(z/b\) is \(0\). We conclude that \(\delta(z) = \delta(b) = 0\) by assumption. Thus we obtain equality \[\dim(|Z|) = \text{codim}(\{z\}, Z) = \delta(\xi)\] for all \(z \in |Z|\) closed. Finally, we have that \(\text{codim}(\{z\}, Z)\) is equal to the dimension of the local ring of \(Z\) at \(z\) by Decent Spaces, Lemma 0ED1 which in turn is equal to \(\dim(\mathcal{O}_{Z, \overline{z}})\) by Properties of Spaces, Lemma 04N9.

In the situation of the lemma above the value of \(\delta\) at the generic point of a closed irreducible subset is the dimension of the irreducible closed subset. This motivates the following definition.

Definition

In Situation 0EDT for any good \(X/B\) and any irreducible closed subset \(T \subset |X|\) we define \[\dim_\delta(T) = \delta(\xi)\] where \(\xi \in T\) is the generic point of \(T\). We will call this the \(\delta\)-dimension of \(T\). If \(T \subset |X|\) is any closed subset, then we define \(\dim_\delta(T)\) as the supremum of the \(\delta\)-dimensions of the irreducible components of \(T\). If \(Z\) is a closed subspace of \(X\), then we set \(\dim_\delta(Z) = \dim_\delta(|Z|)\).

Of course this just means that \(\dim_\delta(T) = \sup \{\delta(t) \mid t \in T\}\).

Cycles

This is the analogue of Chow Homology, Section 02QQ

Since we are not assuming our spaces are quasi-compact we have to be a little careful when defining cycles. We have to allow infinite sums because a rational function may have infinitely many poles for example. In any case, if \(X\) is quasi-compact then a cycle is a finite sum as usual.

Definition

In Situation 0EDT let \(X/B\) be good. Let \(k \in \mathbf{Z}\).

A cycle on \(X\) is a formal sum \[\alpha = \sum n_Z [Z]\] where the sum is over integral closed subspaces \(Z \subset X\), each \(n_Z \in \mathbf{Z}\), and \(\{|Z|; n_Z \not = 0\}\) is a locally finite collection of subsets of \(|X|\) (Topology, Definition 0BDS).

A \(k\)-cycle on \(X\) is a cycle \[\alpha = \sum n_Z [Z]\] where \(n_Z \not = 0 \Rightarrow \dim_\delta(Z) = k\).

The abelian group of all \(k\)-cycles on \(X\) is denoted \(Z_k(X)\).

In other words, a \(k\)-cycle on \(X\) is a locally finite formal \(\mathbf{Z}\)-linear combination of integral closed subspaces (Remark 0EDV) of \(\delta\)-dimension \(k\). Addition of \(k\)-cycles \(\alpha = \sum n_Z[Z]\) and \(\beta = \sum m_Z[Z]\) is given by \[\alpha + \beta = \sum (n_Z + m_Z)[Z],\] i.e., by adding the coefficients.

Multiplicities

A section with a few simple results on lengths and multiplicities.

Lemma

Let \(S\) be a scheme and let \(X\) be an algebraic space over \(S\). Let \(\mathcal{F}\) be a quasi-coherent \(\mathcal{O}_X\)-module. Let \(x \in |X|\). Let \(d \in \{0, 1, 2, \ldots, \infty\}\). The following are equivalent

\(\text{length}_{\mathcal{O}_{X, \overline{x}}} \mathcal{F}_{\overline{x}} = d\)

for some étale morphism \(U \to X\) with \(U\) a scheme and \(u \in U\) mapping to \(x\) we have \(\text{length}_{\mathcal{O}_{U, u}} (\mathcal{F}|_U)_u = d\)

for any étale morphism \(U \to X\) with \(U\) a scheme and \(u \in U\) mapping to \(x\) we have \(\text{length}_{\mathcal{O}_{U, u}} (\mathcal{F}|_U)_u = d\)

Proof

Let \(U \to X\) and \(u \in U\) be as in (2) or (3). Then we know that \(\mathcal{O}_{X, \overline{x}}\) is the strict henselization of \(\mathcal{O}_{U, u}\) and that \[\mathcal{F}_{\overline{x}} = (\mathcal{F}|_U)_u \otimes_{\mathcal{O}_{U, u}} \mathcal{O}_{X, \overline{x}}\] See Properties of Spaces, Lemmas 04KF and 05VP. Thus the equality of the lengths follows from Algebra, Lemma 02M1 the fact that \(\mathcal{O}_{U, u} \to \mathcal{O}_{X, \overline{x}}\) is flat and the fact that \(\mathcal{O}_{X, \overline{x}}/\mathfrak m_u\mathcal{O}_{X, \overline{x}}\) is equal to the residue field of \(\mathcal{O}_{X, \overline{x}}\). These facts about strict henselizations can be found in More on Algebra, Lemma 07QM.

Definition

Let \(S\) be a scheme and let \(X\) be an algebraic space over \(S\). Let \(\mathcal{F}\) be a quasi-coherent \(\mathcal{O}_X\)-module. Let \(x \in |X|\). Let \(d \in \{0, 1, 2, \ldots, \infty\}\). We say \(\mathcal{F}\) has length \(d\) at \(x\) if the equivalent conditions of Lemma 0EE1 are satisfied.

Lemma

Let \(S\) be a scheme. Let \(i : Y \to X\) be a closed immersion of algebraic spaces over \(S\). Let \(\mathcal{G}\) be a quasi-coherent \(\mathcal{O}_Y\)-module. Let \(y \in |Y|\) with image \(x \in |X|\). Let \(d \in \{0, 1, 2, \ldots, \infty\}\). The following are equivalent

\(\mathcal{G}\) has length \(d\) at \(y\), and

\(i_*\mathcal{G}\) has length \(d\) at \(x\).

Proof

Choose an étale morphism \(f : U \to X\) with \(U\) a scheme and \(u \in U\) mapping to \(x\). Set \(V = Y \times_X U\). Denote \(g : V \to Y\) and \(j : V \to U\) the projections. Then \(j : V \to U\) is a closed immersion and there is a unique point \(v \in V\) mapping to \(y \in |Y|\) and \(u \in U\) (use Properties of Spaces, Lemma 03H4 and Spaces, Lemma 02YW). We have \(j_*(\mathcal{G}|_V) = (i_*\mathcal{G})|_U\) as modules on the scheme \(V\) and \(j_*\) the “usual” pushforward of modules for the morphism of schemes \(j\), see discussion surrounding Cohomology of Spaces, Equation (071Z). In this way we reduce to the case of schemes: if \(i : Y \to X\) is a closed immersion of schemes, then \[(i_*\mathcal{G})_x = \mathcal{G}_y\] as modules over \(\mathcal{O}_{X, x}\) where the module structure on the right hand side is given by the surjection \(i_y^\sharp : \mathcal{O}_{X, x} \to \mathcal{O}_{Y, y}\). Thus equality by Algebra, Lemma 00IX.

Lemma

Let \(S\) be a scheme and let \(X\) be a locally Noetherian algebraic space over \(S\). Let \(\mathcal{F}\) be a coherent \(\mathcal{O}_X\)-module. Let \(x \in |X|\). The following are equivalent

for some étale morphism \(U \to X\) with \(U\) a scheme and \(u \in U\) mapping to \(x\) we have \(u\) is a generic point of an irreducible component of \(\text{Supp}(\mathcal{F}|_U)\),

for any étale morphism \(U \to X\) with \(U\) a scheme and \(u \in U\) mapping to \(x\) we have \(u\) is a generic point of an irreducible component of \(\text{Supp}(\mathcal{F}|_U)\),

the length of \(\mathcal{F}\) at \(x\) is finite and nonzero.

If \(X\) is decent (equivalently quasi-separated) then these are also equivalent to

\(x\) is a generic point of an irreducible component of \(\text{Supp}(\mathcal{F})\).

Proof

Assume \(f : U \to X\) is an étale morphism with \(U\) a scheme and \(u \in U\) maps to \(x\). Then \(\mathcal{F}|_U = f^*\mathcal{F}\) is a coherent \(\mathcal{O}_U\)-module on the locally Noetherian scheme \(U\) and in particular \((\mathcal{F}|_U)_u\) is a finite \(\mathcal{O}_{U, u}\)-module, see Cohomology of Spaces, Lemma 07UB and Cohomology of Schemes, Lemma 01XZ. Recall that the support of \(\mathcal{F}|_U\) is a closed subset of \(U\) (Morphisms, Lemma 056J) and that the support of \((\mathcal{F}|_U)_u\) is the pullback of the support of \(\mathcal{F}|_U\) by the morphism \(\Spec(\mathcal{O}_{U, u}) \to U\). Thus \(u\) is a generic point of an irreducible component of \(\text{Supp}(\mathcal{F}|_U)\) if and only if the support of \((\mathcal{F}|_U)_u\) is equal to the maximal ideal of \(\mathcal{O}_{U, u}\). Now the equivalence of (1), (2), (3) follows from by Algebra, Lemma 00L5.

If \(X\) is decent we choose an étale morphism \(f : U \to X\) and a point \(u \in U\) mapping to \(x\). The support of \(\mathcal{F}\) pulls back to the support of \(\mathcal{F}|_U\), see Morphisms of Spaces, Lemma 07TZ. Also, specializations \(x' \leadsto x\) in \(|X|\) lift to specializations \(u' \leadsto u\) in \(U\) and any nontrivial specialization \(u' \leadsto u\) in \(U\) maps to a nontrivial specialization \(f(u') \leadsto f(u)\) in \(|X|\), see Decent Spaces, Lemmas 03IL and 03K5. Using that \(|X|\) and \(U\) are sober topological spaces (Decent Spaces, Proposition 03K6 and Schemes, Lemma 01IS) we conclude \(x\) is a generic point of the support of \(\mathcal{F}\) if and only if \(u\) is a generic point of the support of \(\mathcal{F}|_U\). We conclude (4) is equivalent to (1).

The parenthetical statement follows from Decent Spaces, Lemma 0BB6.

Lemma

In Situation 0EDT let \(X/B\) be good. Let \(T \subset |X|\) be a closed subset and \(t \in T\). If \(\dim_\delta(T) \leq k\) and \(\delta(t) = k\), then \(t\) is a generic point of an irreducible component of \(T\).

Proof

We know \(t\) is contained in an irreducible component \(T' \subset T\). Let \(t' \in T'\) be the generic point. Then \(k \geq \delta(t') \geq \delta(t)\). Since \(\delta\) is a dimension function we see that \(t = t'\).

Cycle associated to a closed subspace

This section is the analogue of Chow Homology, Section 02QS.

Remark

In Situation 0EDT let \(X/B\) be good. Let \(Y \subset X\) be a closed subspace. By Remarks 0EDU and 0EDV there are \(1\)-to-\(1\) correspondences between

irreducible components \(T\) of \(|Y|\),

generic points of irreducible components of \(|Y|\), and

integral closed subspaces \(Z \subset Y\) with the property that \(|Z|\) is an irreducible component of \(|Y|\).

In this chapter we will call \(Z\) as in (3) an irreducible component of \(Y\) and we will call \(\xi \in |Z|\) its generic point.

Definition

In Situation 0EDT let \(X/B\) be good. Let \(Y \subset X\) be a closed subspace.

For an irreducible component \(Z \subset Y\) with generic point \(\xi\) the length of \(\mathcal{O}_Y\) at \(\xi\) (Definition 0EE2) is called the multiplicity of \(Z\) in \(Y\). By Lemma 0EE4 applied to \(\mathcal{O}_Y\) on \(Y\) this is a positive integer.

Assume \(\dim_\delta(Y) \leq k\). The \(k\)-cycle associated to \(Y\) is \[[Y]_k = \sum m_{Z, Y}[Z]\] where the sum is over the irreducible components \(Z\) of \(Y\) of \(\delta\)-dimension \(k\) and \(m_{Z, Y}\) is the multiplicity of \(Z\) in \(Y\). This is a \(k\)-cycle by Spaces over Fields, Lemma 0EE5.

It is important to note that we only define \([Y]_k\) if the \(\delta\)-dimension of \(Y\) does not exceed \(k\). In other words, by convention, if we write \([Y]_k\) then this implies that \(\dim_\delta(Y) \leq k\).

Cycle associated to a coherent sheaf

This is the analogue of Chow Homology, Section 02QV.

Definition

In Situation 0EDT let \(X/B\) be good. Let \(\mathcal{F}\) be a coherent \(\mathcal{O}_X\)-module.

For an integral closed subspace \(Z \subset X\) with generic point \(\xi\) such that \(|Z|\) is an irreducible component of \(\text{Supp}(\mathcal{F})\) the length of \(\mathcal{F}\) at \(\xi\) (Definition 0EE2) is called the multiplicity of \(Z\) in \(\mathcal{F}\). By Lemma 0EE4 this is a positive integer.

Assume \(\dim_\delta(\text{Supp}(\mathcal{F})) \leq k\). The \(k\)-cycle associated to \(\mathcal{F}\) is \[[\mathcal{F}]_k = \sum m_{Z, \mathcal{F}}[Z]\] where the sum is over the integral closed subspaces \(Z \subset X\) corresponding to irreducible components of \(\text{Supp}(\mathcal{F})\) of \(\delta\)-dimension \(k\) and \(m_{Z, \mathcal{F}}\) is the multiplicity of \(Z\) in \(\mathcal{F}\). This is a \(k\)-cycle by Spaces over Fields, Lemma 0EE5.

It is important to note that we only define \([\mathcal{F}]_k\) if \(\mathcal{F}\) is coherent and the \(\delta\)-dimension of \(\text{Supp}(\mathcal{F})\) does not exceed \(k\). In other words, by convention, if we write \([\mathcal{F}]_k\) then this implies that \(\mathcal{F}\) is coherent on \(X\) and \(\dim_\delta(\text{Supp}(\mathcal{F})) \leq k\).

Lemma

In Situation 0EDT let \(X/B\) be good. Let \(\mathcal{F}\) be a coherent \(\mathcal{O}_X\)-module with \(\dim_\delta(\text{Supp}(\mathcal{F})) \leq k\). Let \(Z\) be an integral closed subspace of \(X\) with \(\dim_\delta(Z) = k\). Let \(\xi \in |Z|\) be the generic point. Then the coefficient of \(Z\) in \([\mathcal{F}]_k\) is the length of \(\mathcal{F}\) at \(\xi\).

Proof

Observe that \(|Z|\) is an irreducible component of \(\text{Supp}(\mathcal{F})\) if and only if \(\xi \in \text{Supp}(\mathcal{F})\), see Lemma 0EE6. Moreover, the length of \(\mathcal{F}\) at \(\xi\) is zero if \(\xi \not \in \text{Supp}(\mathcal{F})\). Combining this with Definition 0EEB we conclude.

Lemma

In Situation 0EDT let \(X/B\) be good. Let \(Y \subset X\) be a closed subspace. If \(\dim_\delta(Y) \leq k\), then \([Y]_k = [i_*\mathcal{O}_Y]_k\) where \(i : Y \to X\) is the inclusion morphism.

Proof

Let \(Z\) be an integral closed subspace of \(X\) with \(\dim_\delta(Z) = k\). If \(Z \not \subset Y\) the \(Z\) has coefficient zero in both \([Y]_k\) and \([i_*\mathcal{O}_Y]_k\). If \(Z \subset Y\), then the generic point of \(Z\) may be viewed as a point \(y \in |Y|\) whose image \(x \in |X|\). Then the coefficient of \(Z\) in \([Y]_k\) is the length of \(\mathcal{O}_Y\) at \(y\) and the coefficient of \(Z\) in \([i_*\mathcal{O}_Y]_k\) is the length of \(i_*\mathcal{O}_Y\) at \(x\). Thus the equality of the coefficients follows from Lemma 0EE3.

Lemma

In Situation 0EDT let \(X/B\) be good. Let \(0 \to \mathcal{F} \to \mathcal{G} \to \mathcal{H} \to 0\) be a short exact sequence of coherent \(\mathcal{O}_X\)-modules. Assume that the \(\delta\)-dimension of the supports of \(\mathcal{F}\), \(\mathcal{G}\), and \(\mathcal{H}\) are \(\leq k\). Then \([\mathcal{G}]_k = [\mathcal{F}]_k + [\mathcal{H}]_k\).

Proof

Let \(Z\) be an integral closed subspace of \(X\) with \(\dim_\delta(Z) = k\). It suffices to show that the coefficients of \(Z\) in \([\mathcal{G}]_k\), \([\mathcal{F}]_k\), and \([\mathcal{H}]_k\) satisfy the corresponding additivity. By Lemma 0EEC it suffices to show \[\text{the length of }\mathcal{G}\text{ at }x = \text{the length of }\mathcal{F}\text{ at }x + \text{the length of }\mathcal{H}\text{ at }x\] for any \(x \in |X|\). Looking at Definition 0EE2 this follows immediately from additivity of lengths, see Algebra, Lemma 00IV.

Preparation for proper pushforward

This section is the analogue of Chow Homology, Section 02R0.

Lemma

In Situation 0EDT let \(X,Y/B\) be good and let \(f : X \to Y\) be a morphism over \(B\). If \(Z \subset X\) is an integral closed subspace, then there exists a unique integral closed subspace \(Z' \subset Y\) such that there is a commutative diagram \[\xymatrix{ Z \ar[r] \ar[d] & X \ar[d]^f \\ Z' \ar[r] & Y }\] with \(Z \to Z'\) dominant. If \(f\) is proper, then \(Z \to Z'\) is proper and surjective.

Proof

Let \(\xi \in |Z|\) be the generic point. Let \(Z' \subset Y\) be the integral closed subspace whose generic point is \(\xi' = f(\xi)\), see Remark 0EDV. Since \(\xi \in |f^{-1}(Z')| = |f|^{-1}(|Z'|)\) by Properties of Spaces, Lemma 03H4 and since \(Z\) is the reduced with \(|Z| = \overline{\{\xi\}}\) we see that \(Z \subset f^{-1}(Z')\) as closed subspaces of \(X\) (see Properties of Spaces, Lemma 03JJ). Thus we obtain our morphism \(Z \to Z'\). This morphism is dominant as the generic point of \(Z\) maps to the generic point of \(Z'\). Uniqueness of \(Z'\) is clear. If \(f\) is proper, then \(Z \to Y\) is proper as a composition of proper morphisms (Morphisms of Spaces, Lemmas 04WP and 04XZ). Then we conclude that \(Z \to Z'\) is proper by Morphisms of Spaces, Lemma 04NX. Surjectivity then follows as the image of a proper morphism is closed.

Remark

In Situation 0EDT let \(X/B\) be good. Every \(x \in |X|\) can be represented by a (unique) monomorphism \(\Spec(k) \to X\) where \(k\) is a field, see Decent Spaces, Lemma 03K4. Then \(k\) is the residue field of \(x\) and is denoted \(\kappa(x)\). Recall that \(X\) has a dense open subscheme \(U \subset X\) (Properties of Spaces, Proposition 06NH). If \(x \in U\), then \(\kappa(x)\) agrees with the residue field of \(x\) on \(U\) as a scheme. See Decent Spaces, Section 0EMV.

Remark

In Situation 0EDT let \(X/B\) be good. Assume \(X\) is integral. In this case the function field \(R(X)\) of \(X\) is defined and is equal to the residue field of \(X\) at its generic point. See Spaces over Fields, Definition 0ENE. Combining this with Remark 0EDV we find that for any \(x \in X\) the residue field \(\kappa(x)\) is the function field of the unique integral closed subspace \(Z \subset X\) whose generic point is \(x\).

Lemma

In Situation 0EDT let \(X, Y/B\) be good and let \(f : X \to Y\) be a morphism over \(B\). Assume \(X\), \(Y\) integral and \(\dim_\delta(X) = \dim_\delta(Y)\). Then either \(f\) factors through a proper closed subspace of \(Y\), or \(f\) is dominant and the extension of function fields \(R(X) / R(Y)\) is finite.

Proof

By Lemma 0EEG there is a unique integral closed subspace \(Z \subset Y\) such that \(f\) factors through a dominant morphism \(X \to Z\). Then \(Z = Y\) if and only if \(\dim_\delta(Z) = \dim_\delta(Y)\). On the other hand, by our construction of dimension functions (see Situation 0EDT) we have \(\dim_\delta(X) = \dim_\delta(Z) + r\) where \(r\) the transcendence degree of the extension \(R(X)/R(Z)\). Combining this with Spaces over Fields, Lemma 0AD5 the lemma follows.

Lemma

In Situation 0EDT let \(X, Y/B\) be good. Let \(f : X \to Y\) be a morphism over \(B\). Assume \(f\) is quasi-compact, and \(\{T_i\}_{i \in I}\) is a locally finite collection of closed subsets of \(|X|\). Then \(\{\overline{|f|(T_i)}\}_{i \in I}\) is a locally finite collection of closed subsets of \(|Y|\).

Proof

Let \(V \subset |Y|\) be a quasi-compact open subset. Then \(|f|^{-1}(V) \subset |X|\) is quasi-compact by Morphisms of Spaces, Lemma 0EMK. Hence the set \(\{i \in I : T_i \cap |f|^{-1}(V) \not = \emptyset \}\) is finite by a simple topological argument which we omit. Since this is the same as the set \[\{i \in I : |f|(T_i) \cap V \not = \emptyset \} = \{i \in I : \overline{|f|(T_i)} \cap V \not = \emptyset \}\] the lemma is proved.

Proper pushforward

This section is the analogue of Chow Homology, Section 02R3.

Definition

In Situation 0EDT let \(X, Y/B\) be good. Let \(f : X \to Y\) be a morphism over \(B\). Assume \(f\) is proper.

Let \(Z \subset X\) be an integral closed subspace with \(\dim_\delta(Z) = k\). Let \(Z' \subset Y\) be the image of \(Z\) as in Lemma 0EEG. We define \[f_*[Z] = \left\{ \begin{matrix} 0 & \text{if} & \dim_\delta(Z')< k, \\ \deg(Z/Z') [Z'] & \text{if} & \dim_\delta(Z') = k. \end{matrix} \right.\] The degree of \(Z\) over \(Z'\) is defined and finite if \(\dim_\delta(Z') = \dim_\delta(Z)\) by Lemma 0ENY and Spaces over Fields, Definition 0AD6.

Let \(\alpha = \sum n_Z [Z]\) be a \(k\)-cycle on \(X\). The pushforward of \(\alpha\) as the sum \[f_* \alpha = \sum n_Z f_*[Z]\] where each \(f_*[Z]\) is defined as above. The sum is locally finite by Lemma 0ENZ above.

By definition the proper pushforward of cycles \[f_* : Z_k(X) \longrightarrow Z_k(Y)\] is a homomorphism of abelian groups. It turns \(X \mapsto Z_k(X)\) into a covariant functor on the category whose object are good algebraic spaces over \(B\) and whose morphisms are proper morphisms over \(B\).

Lemma

In Situation 0EDT let \(X, Y, Z/B\) be good. Let \(f : X \to Y\) and \(g : Y \to Z\) be proper morphisms over \(B\). Then \(g_* \circ f_* = (g \circ f)_*\) as maps \(Z_k(X) \to Z_k(Z)\).

Proof

Let \(W \subset X\) be an integral closed subspace of dimension \(k\). Consider the integral closed subspaces \(W' \subset Y\) and \(W'' \subset Z\) we get by applying Lemma 0EEG to \(f\) and \(W\) and then to \(g\) and \(W'\). Then \(W \to W'\) and \(W' \to W''\) are surjective and proper. We have to show that \(g_*(f_*[W]) = (f \circ g)_*[W]\). If \(\dim_\delta(W'') < k\), then both sides are zero. If \(\dim_\delta(W'') = k\), then we see \(W \to W'\) and \(W' \to W''\) both satisfy the hypotheses of Lemma 0ENY. Hence \[g_*(f_*[W]) = \deg(W/W')\deg(W'/W'')[W''], \quad (f \circ g)_*[W] = \deg(W/W'')[W''].\] Then we can apply Spaces over Fields, Lemma 0ENH to conclude.

Lemma

In Situation 0EDT let \(f : X \to Y\) be a proper morphism of good algebraic spaces over \(B\).

Let \(Z \subset X\) be a closed subspace with \(\dim_\delta(Z) \leq k\). Then \[f_*[Z]_k = [f_*{\mathcal O}_Z]_k.\]

Let \(\mathcal{F}\) be a coherent sheaf on \(X\) such that \(\dim_\delta(\text{Supp}(\mathcal{F})) \leq k\). Then \[f_*[\mathcal{F}]_k = [f_*{\mathcal F}]_k.\]

Note that the statement makes sense since \(f_*\mathcal{F}\) and \(f_*\mathcal{O}_Z\) are coherent \(\mathcal{O}_Y\)-modules by Cohomology of Spaces, Lemma 08AR.

Proof

Part (1) follows from (2) and Lemma 0EED. Let \(\mathcal{F}\) be a coherent sheaf on \(X\). Assume that \(\dim_\delta(\text{Supp}(\mathcal{F})) \leq k\). By Cohomology of Spaces, Lemma 07UG there exists a closed immersion \(i : Z \to X\) and a coherent \(\mathcal{O}_Z\)-module \(\mathcal{G}\) such that \(i_*\mathcal{G} \cong \mathcal{F}\) and such that the support of \(\mathcal{F}\) is \(Z\). Let \(Z' \subset Y\) be the scheme theoretic image of \(f|_Z : Z \to Y\), see Morphisms of Spaces, Definition 082Y. Consider the commutative diagram \[\xymatrix{ Z \ar[r]_i \ar[d]_{f|_Z} & X \ar[d]^f \\ Z' \ar[r]^{i'} & Y }\] of algebraic spaces over \(B\). Observe that \(f|_Z\) is surjective (follows from Morphisms of Spaces, Lemma 082Z and the fact that \(|f|\) is closed) and proper (follows from Morphisms of Spaces, Lemmas 04WP, 04XZ, and 04NX). We have \(f_*\mathcal{F} = f_*i_*\mathcal{G} = i'_*(f|_Z)_*\mathcal{G}\) by going around the diagram in two ways. Suppose we know the result holds for closed immersions and for \(f|_Z\). Then we see that \[f_*[\mathcal{F}]_k = f_*i_*[\mathcal{G}]_k = (i')_*(f|_Z)_*[\mathcal{G}]_k = (i')_*[(f|_Z)_*\mathcal{G}]_k = [(i')_*(f|_Z)_*\mathcal{G}]_k = [f_*\mathcal{F}]_k\] as desired. The case of a closed immersion follows from Lemma 0EE3 and the definitions. Thus we have reduced to the case where \(\dim_\delta(X) \leq k\) and \(f : X \to Y\) is proper and surjective.

Assume \(\dim_\delta(X) \leq k\) and \(f : X \to Y\) is proper and surjective. For every irreducible component \(Z \subset Y\) with generic point \(\eta\) there exists a point \(\xi \in X\) such that \(f(\xi) = \eta\). Hence \(\delta(\eta) \leq \delta(\xi) \leq k\). Thus we see that in the expressions \[f_*[\mathcal{F}]_k = \sum n_Z[Z], \quad \text{and} \quad [f_*\mathcal{F}]_k = \sum m_Z[Z].\] whenever \(n_Z \not = 0\), or \(m_Z \not = 0\) the integral closed subspace \(Z\) is actually an irreducible component of \(Y\) of \(\delta\)-dimension \(k\) (see Lemma 0EE6). Pick such an integral closed subspace \(Z \subset Y\) and denote \(\eta\) its generic point. Note that for any \(\xi \in X\) with \(f(\xi) = \eta\) we have \(\delta(\xi) \geq k\) and hence \(\xi\) is a generic point of an irreducible component of \(X\) of \(\delta\)-dimension \(k\) as well (see Lemma 0EE6). By Spaces over Fields, Lemma 0AD2 there exists an open subspace \(\eta \in V \subset Y\) such that \(f^{-1}(V) \to V\) is finite. Since \(\eta\) is a generic point of an irreducible component of \(|Y|\) we may assume \(V\) is an affine scheme, see Properties of Spaces, Proposition 06NH. Replacing \(Y\) by \(V\) and \(X\) by \(f^{-1}(V)\) we reduce to the case where \(Y\) is affine, and \(f\) is finite. In particular \(X\) and \(Y\) are schemes and we reduce to the corresponding result for schemes, see Chow Homology, Lemma 02R6 (applied with \(S = Y\)).

Preparation for flat pullback

This section is the analogue of Chow Homology, Section 02R7.

Recall that a morphism of algebraic spaces is said to have relative dimension \(r\) if étale locally on the source and the target we get a morphism of schemes which has relative dimension \(r\). The precise definition is equivalent, but in fact slightly different, see Morphisms of Spaces, Definition 06LR.

Lemma

In Situation 0EDT let \(X, Y/B\) be good. Let \(f : X \to Y\) be a morphism over \(B\). Assume \(f\) is flat of relative dimension \(r\). For any closed subset \(T \subset |Y|\) we have \[\dim_\delta(|f|^{-1}(T)) = \dim_\delta(T) + r.\] provided \(|f|^{-1}(T)\) is nonempty. If \(Z \subset Y\) is an integral closed subscheme and \(Z' \subset f^{-1}(Z)\) is an irreducible component, then \(Z'\) dominates \(Z\) and \(\dim_\delta(Z') = \dim_\delta(Z) + r\).

Proof

Since the \(\delta\)-dimension of a closed subset is the supremum of the \(\delta\)-dimensions of the irreducible components, it suffices to prove the final statement. We may replace \(Y\) by the integral closed subscheme \(Z\) and \(X\) by \(f^{-1}(Z) = Z \times_Y X\). Hence we may assume \(Z = Y\) is integral and \(f\) is a flat morphism of relative dimension \(r\). Since \(Y\) is locally Noetherian the morphism \(f\) which is locally of finite type, is actually locally of finite presentation. Hence Morphisms of Spaces, Lemma 042S applies and we see that \(f\) is open. Let \(\xi \in X\) be a generic point of an irreducible component of \(X\). By the openness of \(f\) we see that \(f(\xi)\) is the generic point \(\eta\) of \(Z = Y\). Thus \(Z'\) dominates \(Z = Y\). Finally, we see that \(\xi\) and \(\eta\) are in the schematic locus of \(X\) and \(Y\) by Properties of Spaces, Proposition 06NH. Since \(\xi\) is a generic point of \(X\) we see that \(\mathcal{O}_{X, \xi} = \mathcal{O}_{X_\eta, \xi}\) has only one prime ideal and hence has dimension \(0\) (we may use usual local rings as \(\xi\) and \(\eta\) are in the schematic loci of \(X\) and \(Y\)). Thus by Morphisms of Spaces, Lemma 04NQ (and the definition of morphisms of given relative dimension) we conclude that the transcendence degree of \(\kappa(\xi)\) over \(\kappa(\eta)\) is \(r\). In other words, \(\delta(\xi) = \delta(\eta) + r\) as desired.

Here is the lemma that we will use to prove that the flat pullback of a locally finite collection of closed subschemes is locally finite.

Lemma

In Situation 0EDT let \(X, Y/B\) be good. Let \(f : X \to Y\) be a morphism over \(B\). Assume \(\{T_i\}_{i \in I}\) is a locally finite collection of closed subsets of \(|Y|\). Then \(\{|f|^{-1}(T_i)\}_{i \in I}\) is a locally finite collection of closed subsets of \(X\).

Proof

Let \(U \subset |X|\) be a quasi-compact open subset. Since the image \(|f|(U) \subset |Y|\) is a quasi-compact subset there exists a quasi-compact open \(V \subset |Y|\) such that \(|f|(U) \subset V\). Note that \[\{i \in I : |f|^{-1}(T_i) \cap U \not = \emptyset \} \subset \{i \in I : T_i \cap V \not = \emptyset \}.\] Since the right hand side is finite by assumption we win.

Flat pullback

This section is the analogue of Chow Homology, Section 02RA.

Let \(S\) be a scheme and let \(f : X \to Y\) be a morphism of algebraic spaces over \(S\). Let \(Z \subset Y\) be a closed subspace. In this chapter we will sometimes use the terminology scheme theoretic inverse image for the inverse image \(f^{-1}(Z)\) of \(Z\) constructed in Morphisms of Spaces, Definition 083Q. The scheme theoretic inverse image is the fibre product \[\xymatrix{ f^{-1}(Z) \ar[r] \ar[d] & X \ar[d] \\ Z \ar[r] & Y }\] If \(\mathcal{I} \subset \mathcal{O}_Y\) is the quasi-coherent sheaf of ideals corresponding to \(Z\) in \(Y\), then \(f^{-1}(\mathcal{I})\mathcal{O}_X\) is the quasi-coherent sheaf of ideals corresponding to \(f^{-1}(Z)\) in \(X\).

Definition

In Situation 0EDT let \(X, Y/B\) be good. Let \(f : X \to Y\) be a morphism over \(B\). Assume \(f\) is flat of relative dimension \(r\).

Let \(Z \subset Y\) be an integral closed subspace of \(\delta\)-dimension \(k\). We define \(f^*[Z]\) to be the \((k+r)\)-cycle on \(X\) associated to the scheme theoretic inverse image \[f^*[Z] = [f^{-1}(Z)]_{k+r}.\] This makes sense since \(\dim_\delta(f^{-1}(Z)) = k + r\) by Lemma 0EP5.

Let \(\alpha = \sum n_i [Z_i]\) be a \(k\)-cycle on \(Y\). The flat pullback of \(\alpha\) by \(f\) is the sum \[f^* \alpha = \sum n_i f^*[Z_i]\] where each \(f^*[Z_i]\) is defined as above. The sum is locally finite by Lemma 0EP6.

We denote \(f^* : Z_k(Y) \to Z_{k + r}(X)\) the map of abelian groups so obtained.

An open immersion is flat. This is an important though trivial special case of a flat morphism. If \(U \subset X\) is open then sometimes the pullback by \(j : U \to X\) of a cycle is called the restriction of the cycle to \(U\). Note that in this case the maps \[j^* : Z_k(X) \longrightarrow Z_k(U)\] are all surjective. The reason is that given any integral closed subspace \(Z' \subset U\), we can take the closure of \(Z\) of \(Z'\) in \(X\) and think of it as a reduced closed subspace of \(X\) (see Properties of Spaces, Definition 047X). And clearly \(Z \cap U = Z'\), in other words \(j^*[Z] = [Z']\) whence the surjectivity. In fact a little bit more is true.

Lemma

In Situation 0EDT let \(X/B\) be good. Let \(U \subset X\) be an open subspace. Let \(Y\) be the reduced closed subspace of \(X\) with \(|Y| = |X| \setminus |U|\) and denote \(i : Y \to X\) the inclusion morphism. For every \(k \in \mathbf{Z}\) the sequence \[\xymatrix{ Z_k(Y) \ar[r]^{i_*} & Z_k(X) \ar[r]^{j^*} & Z_k(U) \ar[r] & 0 }\] is an exact complex of abelian groups.

Proof

Surjectivity of \(j^*\) we saw above. First assume \(X\) is quasi-compact. Then \(Z_k(X)\) is a free \(\mathbf{Z}\)-module with basis given by the elements \([Z]\) where \(Z \subset X\) is integral closed of \(\delta\)-dimension \(k\). Such a basis element maps either to the basis element \([Z \cap U]\) of \(Z_k(U)\) or to zero if \(Z \subset Y\). Hence the lemma is clear in this case. The general case is similar and the proof is omitted.

Lemma

In Situation 0EDT let \(f : X \to Y\) be an étale morphism of good algebraic spaces over \(B\). If \(Z \subset Y\) is an integral closed subspace, then \(f^*[Z] = \sum [Z']\) where the sum is over the irreducible components (Remark 0EE8) of \(f^{-1}(Z)\).

Proof

The meaning of the lemma is that the coefficient of \([Z']\) is \(1\). This follows from the fact that \(f^{-1}(Z)\) is a reduced algebraic space because it is étale over the integral algebraic space \(Z\).

Lemma

In Situation 0EDT let \(X, Y, Z/B\) be good. Let \(f : X \to Y\) and \(g : Y \to Z\) be flat morphisms of relative dimensions \(r\) and \(s\) over \(B\). Then \(g \circ f\) is flat of relative dimension \(r + s\) and \[f^* \circ g^* = (g \circ f)^*\] as maps \(Z_k(Z) \to Z_{k + r + s}(X)\).

Proof

The composition is flat of relative dimension \(r + s\) by Morphisms of Spaces, Lemmas 04NR and 03MN. Suppose that

\(A \subset Z\) is a closed integral subspace of \(\delta\)-dimension \(k\),

\(A' \subset Y\) is a closed integral subspace of \(\delta\)-dimension \(k + s\) with \(A' \subset g^{-1}(A)\), and

\(A'' \subset Y\) is a closed integral subspace of \(\delta\)-dimension \(k + s + r\) with \(A'' \subset f^{-1}(W')\).

We have to show that the coefficient \(n\) of \([A'']\) in \((g \circ f)^*[A]\) agrees with the coefficient \(m\) of \([A'']\) in \(f^*(g^*[A])\). We may choose a commutative diagram \[\xymatrix{ U \ar[d] \ar[r] & V \ar[d] \ar[r] & W \ar[d] \\ X \ar[r] & Y \ar[r] & Z }\] where \(U, V, W\) are schemes, the vertical arrows are étale, and there exist points \(u \in U\), \(v \in V\), \(w \in W\) such that \(u \mapsto v \mapsto w\) and such that \(u, v, w\) map to the generic points of \(A'', A', A\). (Details omitted.) Then we have flat local ring homorphisms \(\mathcal{O}_{W, w} \to \mathcal{O}_{V, v}\), \(\mathcal{O}_{V, v} \to \mathcal{O}_{U, u}\), and repeatedly using Lemma 0EE1 we find \[n = \text{length}_{\mathcal{O}_{U, u}}( \mathcal{O}_{U, u}/\mathfrak m_w\mathcal{O}_{U, u})\] and \[m = \text{length}_{\mathcal{O}_{V, v}}( \mathcal{O}_{V, v}/\mathfrak m_w\mathcal{O}_{V, v}) \text{length}_{\mathcal{O}_{U, u}}( \mathcal{O}_{U, u}/\mathfrak m_v\mathcal{O}_{U, u})\] Hence the equality follows from Algebra, Lemma 02M2.

Lemma

In Situation 0EDT let \(X, Y/B\) be good. Let \(f : X \to Y\) be a flat morphism of relative dimension \(r\).

Let \(Z \subset Y\) be a closed subspace with \(\dim_\delta(Z) \leq k\). Then we have \(\dim_\delta(f^{-1}(Z)) \leq k + r\) and \([f^{-1}(Z)]_{k + r} = f^*[Z]_k\) in \(Z_{k + r}(X)\).

Let \(\mathcal{F}\) be a coherent sheaf on \(Y\) with \(\dim_\delta(\text{Supp}(\mathcal{F})) \leq k\). Then we have \(\dim_\delta(\text{Supp}(f^*\mathcal{F})) \leq k + r\) and \[f^*[{\mathcal F}]_k = [f^*{\mathcal F}]_{k+r}\] in \(Z_{k + r}(X)\).

Proof

Part (1) follows from part (2) by Lemma 0EED and the fact that \(f^*\mathcal{O}_Z = \mathcal{O}_{f^{-1}(Z)}\).

Proof of (2). As \(X\), \(Y\) are locally Noetherian we may apply Cohomology of Spaces, Lemma 07UB to see that \(\mathcal{F}\) is of finite type, hence \(f^*\mathcal{F}\) is of finite type (Modules on Sites, Lemma 03DO), hence \(f^*\mathcal{F}\) is coherent (Cohomology of Spaces, Lemma 07UB again). Thus the lemma makes sense. Let \(W \subset Y\) be an integral closed subspace of \(\delta\)-dimension \(k\), and let \(W' \subset X\) be an integral closed subspace of dimension \(k + r\) mapping into \(W\) under \(f\). We have to show that the coefficient \(n\) of \([W']\) in \(f^*[{\mathcal F}]_k\) agrees with the coefficient \(m\) of \([W']\) in \([f^*{\mathcal F}]_{k+r}\). We may choose a commutative diagram \[\xymatrix{ U \ar[d] \ar[r] & V \ar[d] \\ X \ar[r] & Y }\] where \(U, V\) are schemes, the vertical arrows are étale, and there exist points \(u \in U\), \(v \in V\) such that \(u \mapsto v\) and such that \(u, v\) map to the generic points of \(W', W\). (Details omitted.) Consider the stalk \(M = (\mathcal{F}|_V)_v\) as an \(\mathcal{O}_{V, v}\)-module. (Note that \(M\) has finite length by our dimension assumptions, but we actually do not need to verify this. See Lemma 0EE4.) We have \((f^*\mathcal{F}|_U)_u = \mathcal{O}_{U, u} \otimes_{\mathcal{O}_{V, v}} M\). Thus we see that \[n = \text{length}_{\mathcal{O}_{U, u}} (\mathcal{O}_{U, u} \otimes_{\mathcal{O}_{V, v}} M) \quad \text{and} \quad m = \text{length}_{\mathcal{O}_{V, v}}(M) \text{length}_{\mathcal{O}_{V, v}}( \mathcal{O}_{U, u}/\mathfrak m_v \mathcal{O}_{U, u})\] Thus the equality follows from Algebra, Lemma 02M1.

Push and pull

This section is the analogue of Chow Homology, Section 02RF.

In this section we verify that proper pushforward and flat pullback are compatible when this makes sense. By the work we did above this is a consequence of cohomology and base change.

Lemma

In Situation 0EDT let \[\xymatrix{ X' \ar[r]_{g'} \ar[d]_{f'} & X \ar[d]^f \\ Y' \ar[r]^g & Y }\] be a fibre product diagram of good algebraic spaces over \(B\). Assume \(f : X \to Y\) proper and \(g : Y' \to Y\) flat of relative dimension \(r\). Then also \(f'\) is proper and \(g'\) is flat of relative dimension \(r\). For any \(k\)-cycle \(\alpha\) on \(X\) we have \[g^*f_*\alpha = f'_*(g')^*\alpha\] in \(Z_{k + r}(Y')\).

Proof

The assertion that \(f'\) is proper follows from Morphisms of Spaces, Lemma 04WP. The assertion that \(g'\) is flat of relative dimension \(r\) follows from Morphisms of Spaces, Lemmas 04NS and 03MO. It suffices to prove the equality of cycles when \(\alpha = [W]\) for some integral closed subspace \(W \subset X\) of \(\delta\)-dimension \(k\). Note that in this case we have \(\alpha = [\mathcal{O}_W]_k\), see Lemma 0EED. By Lemmas 0EP3 and 0EPB it therefore suffices to show that \(f'_*(g')^*\mathcal{O}_W\) is isomorphic to \(g^*f_*\mathcal{O}_W\). This follows from cohomology and base change, see Cohomology of Spaces, Lemma 073K.

Lemma

In Situation 0EDT let \(X, Y/B\) be good. Let \(f : X \to Y\) be a finite locally free morphism of degree \(d\) (see Morphisms of Spaces, Definition 03ZV). Then \(f\) is both proper and flat of relative dimension \(0\), and \[f_*f^*\alpha = d\alpha\] for every \(\alpha \in Z_k(Y)\).

Proof

A finite locally free morphism is flat and finite by Morphisms of Spaces, Lemma 0416, and a finite morphism is proper by Morphisms of Spaces, Lemma 04NZ. We omit showing that a finite morphism has relative dimension \(0\). Thus the formula makes sense. To prove it, let \(Z \subset Y\) be an integral closed subscheme of \(\delta\)-dimension \(k\). It suffices to prove the formula for \(\alpha = [Z]\). Since the base change of a finite locally free morphism is finite locally free (Morphisms of Spaces, Lemma 03ZY) we see that \(f_*f^*\mathcal{O}_Z\) is a finite locally free sheaf of rank \(d\) on \(Z\). Thus clearly \(f_*f^*\mathcal{O}_Z\) has length \(d\) at the generic point of \(Z\). Hence \[f_*f^*[Z] = f_*f^*[\mathcal{O}_Z]_k = [f_*f^*\mathcal{O}_Z]_k = d[Z]\] where we have used Lemmas 0EPB and 0EP3.

Preparation for principal divisors

This section is the analogue of Chow Homology, Section 02RI. Some of the material in this section partially overlaps with the discussion in Spaces over Fields, Section 0ENI.

Lemma

In Situation 0EDT let \(X/B\) be good. Assume \(X\) is integral.

If \(Z \subset X\) is an integral closed subspace, then the following are equivalent:

\(Z\) is a prime divisor,

\(|Z|\) has codimension \(1\) in \(|X|\), and

\(\dim_\delta(Z) = \dim_\delta(X) - 1\).

If \(Z\) is an irreducible component of an effective Cartier divisor on \(X\), then \(\dim_\delta(Z) = \dim_\delta(X) - 1\).

Proof

Part (1) follows from the definition of a prime divisor (Spaces over Fields, Definition 0ENJ), Decent Spaces, Lemma 0ED1, and the definition of a dimension function (Topology, Definition 02I9).

Let \(D \subset X\) be an effective Cartier divisor. Let \(Z \subset D\) be an irreducible component and let \(\xi \in |Z|\) be the generic point. Choose an étale neighbourhood \((U, u) \to (X, \xi)\) where \(U = \Spec(A)\) and \(D \times_X U\) is cut out by a nonzerodivisor \(f \in A\), see Divisors on Spaces, Lemma 083C. Then \(u\) is a generic point of \(V(f)\) by Decent Spaces, Lemma 0ABV. Hence \(\mathcal{O}_{U, u}\) has dimension \(1\) by Krull’s Hauptidealsatz (Algebra, Lemma 00KV). Thus \(\xi\) is a codimension \(1\) point on \(X\) and \(Z\) is a prime divisor as desired.

Principal divisors

This section is the analogue of Chow Homology, Section 02RN. The following definition is the analogue of Spaces over Fields, Definition 0ENP in our current setup.

Definition

In Situation 0EDT let \(X/B\) be good. Assume \(X\) is integral with \(\dim_\delta(X) = n\). Let \(f \in R(X)^*\). The principal divisor associated to \(f\) is the \((n - 1)\)-cycle \[\text{div}(f) = \text{div}_X(f) = \sum \text{ord}_Z(f) [Z]\] defined in Spaces over Fields, Definition 0ENP. This makes sense because prime divisors have \(\delta\)-dimension \(n - 1\) by Lemma 0EPZ.

In the situation of the definition for \(f, g \in R(X)^*\) we have \[\text{div}_X(fg) = \text{div}_X(f) + \text{div}_X(g)\] in \(Z_{n - 1}(X)\). See Spaces over Fields, Lemma 0ENQ. The following lemma will allow us to reduce statements about principal divisors to the case of schemes.

Lemma

In Situation 0EDT let \(f : X \to Y\) be an étale morphism of good algebraic spaces over \(B\). Assume \(Y\) is integral. Let \(g \in R(Y)^*\). As cycles on \(X\) we have \[f^*(\text{div}_Y(g)) = \sum\nolimits_{X'} (X' \to X)_*\text{div}_{X'}(g \circ f|_{X'})\] where the sum is over the irreducible components of \(X\) (Remark 0EE8).

Proof

The map \(|X| \to |Y|\) is open. The set of irreducible components of \(|X|\) is locally finite in \(|X|\). We conclude that \(f|_{X'} : X' \to Y\) is dominant for every irreducible component \(X' \subset X\). Thus \(g \circ f|_{X'}\) is defined (Morphisms of Spaces, Section 0EML), hence \(\text{div}_{X'}(g \circ f|_{X'})\) is defined. Moreover, the sum is locally finite and we find that the right hand side indeed is a cycle on \(X\). The left hand side is defined by Definition 0EP8 and the fact that an étale morphism is flat of relative dimension \(0\).

Since \(f\) is étale we see that \(\delta_X(x) = \delta_y(f(x))\) for all \(x \in |X|\). Thus if \(\dim_\delta(Y) = n\), then \(\dim_\delta(X') = n\) for every irreducible component \(X'\) of \(X\) (since generic points of \(X\) map to the generic point of \(Y\), see above). Thus both left and right hand side are \((n - 1)\)-cycles.

Let \(Z \subset X\) be an integral closed subspace with \(\dim_\delta(Z) = n - 1\). To prove the equality, we need to show that the coefficients of \(Z\) are the same. Let \(Z' \subset Y\) be the integral closed subspace constructed in Lemma 0EEG. Then \(\dim_\delta(Z') = n - 1\) too. Let \(\xi \in |Z|\) be the generic point. Then \(\xi' = f(\xi) \in |Z'|\) is the generic point. Consider the commutative diagram \[\xymatrix{ \Spec(\mathcal{O}_{X, \xi}^h) \ar[r] \ar[d] & X \ar[d] \\ \Spec(\mathcal{O}_{Y, \xi'}^h) \ar[r] & Y }\] of Decent Spaces, Remark 0EPL. We have to be slightly careful as the reduced Noetherian local rings \(\mathcal{O}_{X, \xi}^h\) and \(\mathcal{O}_{Y, \xi'}^h\) need not be domains. Thus we work with total rings of fractions \(Q(-)\) rather than fraction fields. By definition, to get the coefficient of \(Z'\) in \(\text{div}_Y(g)\) we write the image of \(g\) in \(Q(\mathcal{O}_{Y, \xi'}^h)\) as \(a/b\) with \(a, b \in \mathcal{O}_{Y, \xi'}^h\) nonzerodivisors and we take \[\text{ord}_{Z'}(g) = \text{length}_{\mathcal{O}_{Y, \xi'}^h} (\mathcal{O}_{Y, \xi'}^h/a \mathcal{O}_{Y, \xi'}^h) - \text{length}_{\mathcal{O}_{Y, \xi'}^h} (\mathcal{O}_{Y, \xi'}^h/b \mathcal{O}_{Y, \xi'}^h)\] Observe that the coefficient of \(Z\) in \(f^*\text{div}_Y(G)\) is the same integer, see Lemma 0EPY. Suppose that \(\xi \in X'\). Then we can consider the maps \[\mathcal{O}_{Y, \xi'}^h \to \mathcal{O}_{X, \xi}^h \to \mathcal{O}_{X', \xi}^h\] The first arrow is flat and the second arrow is a surjective map of reduced local Noetherian rings of dimension \(1\). Therefore both these maps send nonzerodivisors to nonzerodivisors and we conclude the coefficient of \(Z'\) in \(\text{div}_{X'}(g \circ f|_{X'})\) is \[\text{ord}_Z(g \circ f|_{X'}) = \text{length}_{\mathcal{O}_{X', \xi}^h} (\mathcal{O}_{X', \xi}^h/a \mathcal{O}_{X', \xi}^h) - \text{length}_{\mathcal{O}_{Y, \xi'}^h} (\mathcal{O}_{X', \xi}^h/b \mathcal{O}_{X', \xi}^h)\] by the same prescription as above. Thus it suffices to show \[\text{length}_{\mathcal{O}_{Y, \xi'}^h} (\mathcal{O}_{Y, \xi'}^h/a \mathcal{O}_{Y, \xi'}^h) = \sum\nolimits_{\xi \in |X'|} \text{length}_{\mathcal{O}_{X', \xi}^h} (\mathcal{O}_{X', \xi}^h/a \mathcal{O}_{X', \xi}^h)\] First, since the ring map \(\mathcal{O}_{Y, \xi'}^h \to \mathcal{O}_{X, \xi}^h\) is flat and unramified, we have \[\text{length}_{\mathcal{O}_{Y, \xi'}^h} (\mathcal{O}_{Y, \xi'}^h/a \mathcal{O}_{Y, \xi'}^h) = \text{length}_{\mathcal{O}_{X, \xi}^h} (\mathcal{O}_{X, \xi}^h/a \mathcal{O}_{X, \xi}^h)\] by Algebra, Lemma 02M1. Let \(\mathfrak q_1, \ldots, \mathfrak q_t\) be the nonmaximal primes of \(\mathcal{O}_{X, \xi}^h\) and set \(R_j = \mathcal{O}_{X, \xi}^h/\mathfrak q_j\). For \(X'\) as above, denote \(J(X') \subset \{1, \ldots, t\}\) the set of indices such that \(\mathfrak q_j\) corresponds to a point of \(X'\), i.e., such that under the surjection \(\mathcal{O}_{X, \xi}^h \to \mathcal{O}_{X', \xi}\) the prime \(\mathfrak q_j\) corresponds to a prime of \(\mathcal{O}_{X', \xi}\). By Chow Homology, Lemma 02QG we get \[\text{length}_{\mathcal{O}_{X, \xi}^h} (\mathcal{O}_{X, \xi}^h/a \mathcal{O}_{X, \xi}^h) = \sum\nolimits_j \text{length}_{R_j}(R_j/a R_j)\] and \[\text{length}_{\mathcal{O}_{X', \xi}^h} (\mathcal{O}_{X', \xi}^h/a \mathcal{O}_{X', \xi}^h) = \sum\nolimits_{j \in J(X')} \text{length}_{R_j}(R_j/a R_j)\] Thus the result of the lemma holds because \(\{1, \ldots, t\}\) is the disjoint union of the sets \(J(X')\): each point of codimension \(0\) on \(X\) lies on a unique \(X'\).

Principal divisors and pushforward

This section is the analogue of Chow Homology, Section 02RS.

Lemma

In Situation 0EDT let \(X, Y/B\) be good. Assume \(X\), \(Y\) are integral and \(n = \dim_\delta(X) = \dim_\delta(Y)\). Let \(p : X \to Y\) be a dominant proper morphism. Let \(f \in R(X)^*\). Set \[g = \text{Nm}_{R(X)/R(Y)}(f).\] Then we have \(p_*\text{div}(f) = \text{div}(g)\).

Proof

We are going to deduce this from the case of schemes by étale localization. Let \(Z \subset Y\) be an integral closed subspace of \(\delta\)-dimension \(n - 1\). We want to show that the coefficient of \([Z]\) in \(p_*\text{div}(f)\) and \(\text{div}(g)\) are equal. Apply Spaces over Fields, Lemma 0AD2 to the morphism \(p : X \to Y\) and the generic point \(\xi \in |Z|\). We find that we may replace \(Y\) by an open subspace containing \(\xi\) and assume that \(p : X \to Y\) is finite. Pick an étale neighbourhood \((V, v) \to (Y, \xi)\) where \(V\) is an affine scheme. By Lemma 0EPY it suffices to prove the equality of cycles after pulling back to \(V\). Set \(U = V \times_Y X\) and consider the commutative diagram \[\xymatrix{ U \ar[r]_a \ar[d]_{p'} & X \ar[d]^p \\ V \ar[r]^b & Y }\] Let \(V_j \subset V\), \(j = 1, \ldots, m\) be the irreducible components of \(V\). For each \(i\), let \(U_{j, i}\), \(i = 1, \ldots, n_j\) be the irreducible components of \(U\) dominating \(V_j\). Denote \(p'_{j, i} : U_{j, i} \to V_j\) the restriction of \(p' : U \to V\). By the case of schemes (Chow Homology, Lemma 02RT) we see that \[p'_{j, i, *}\text{div}_{U_{j, i}}(f_{j, i}) = \text{div}_{V_j}(g_{j, i})\] where \(f_{j, i}\) is the restriction of \(f\) to \(U_{j, i}\) and \(g_{j, i}\) is the norm of \(f_{j, i}\) along the finite extension \(R(U_{j, i})/R(V_j)\). We have \[\begin{align*} b^* p_*\text{div}_X(f) & = p'_* a^* \text{div}_X(f) \\ & = p'_*\left(\sum\nolimits_{j, i} (U_{j, i} \to U)_*\text{div}_{U_{j, i}}(f_{j, i})\right) \\ & = \sum\nolimits_{j, i} (V_j \to V)_*p'_{j, i, *}\text{div}_{U_{j, i}}(f_{j, i}) \\ & = \sum\nolimits_j (V_j \to V)_*\left(\sum\nolimits_i \text{div}_{V_j}(g_{j, i})\right) \\ & = \sum\nolimits_j (V_j \to V)_*\text{div}_{V_j}(\prod\nolimits_i g_{j, i}) \end{align*}\] by Lemmas 0EPD, 0EQ2, and 0EP2. To finish the proof, using Lemma 0EQ2 again, it suffices to show that \[g \circ b|_{V_j} = \prod\nolimits_i g_{j, i}\] as elements of the function field of \(V_j\). In terms of fields this is the following statement: let \(L/K\) be a finite extension. Let \(M/K\) be a finite separable extension. Write \(M \otimes_K L = \prod M_i\). Then for \(t \in L\) with images \(t_i \in M_i\) the image of \(\text{Norm}_{L/K}(t)\) in \(M\) is \(\prod \text{Norm}_{M_i/M}(t_i)\). We omit the proof.

Rational equivalence

This section is the analogue of Chow Homology, Section 02RV. In this section we define rational equivalence on \(k\)-cycles. We will allow locally finite sums of images of principal divisors (under closed immersions). This leads to some pretty strange phenomena (see examples in the chapter on schemes). However, if we do not allow these then we do not know how to prove that capping with Chern classes of line bundles factors through rational equivalence.

Definition

In Situation 0EDT let \(X/B\) be good. Let \(k \in \mathbf{Z}\).

Given any locally finite collection \(\{W_j \subset X\}\) of integral closed subspaces with \(\dim_\delta(W_j) = k + 1\), and any \(f_j \in R(W_j)^*\) we may consider \[\sum (i_j)_*\text{div}(f_j) \in Z_k(X)\] where \(i_j : W_j \to X\) is the inclusion morphism. This makes sense as the morphism \(\coprod i_j : \coprod W_j \to X\) is proper.

We say that \(\alpha \in Z_k(X)\) is rationally equivalent to zero if \(\alpha\) is a cycle of the form displayed above.

We say \(\alpha, \beta \in Z_k(X)\) are rationally equivalent and we write \(\alpha \sim_{rat} \beta\) if \(\alpha - \beta\) is rationally equivalent to zero.

We define \[\CH_k(X) = Z_k(X) / \sim_{rat}\] to be the Chow group of \(k\)-cycles on \(X\). This is sometimes called the Chow group of \(k\)-cycles modulo rational equivalence on \(X\).

There are many other interesting equivalence relations. Rational equivalence is the coarsest of them all. A very simple but important lemma is the following.

Lemma

In Situation 0EDT let \(X/B\) be good. Let \(U \subset X\) be an open subspace. Let \(Y\) be the reduced closed subspace of \(X\) with \(|Y| = |X| \setminus |U|\) and denote \(i : Y \to X\) the inclusion morphism. Let \(k \in \mathbf{Z}\). Suppose \(\alpha, \beta \in Z_k(X)\). If \(\alpha|_U \sim_{rat} \beta|_U\) then there exist a cycle \(\gamma \in Z_k(Y)\) such that \[\alpha \sim_{rat} \beta + i_*\gamma.\] In other words, the sequence \[\xymatrix{ \CH_k(Y) \ar[r]^{i_*} & \CH_k(X) \ar[r]^{j^*} & \CH_k(U) \ar[r] & 0 }\] is an exact complex of abelian groups.

Proof

Let \(\{W_j\}_{j \in J}\) be a locally finite collection of integral closed subspaces of \(U\) of \(\delta\)-dimension \(k + 1\), and let \(f_j \in R(W_j)^*\) be elements such that \((\alpha - \beta)|_U = \sum (i_j)_*\text{div}(f_j)\) as in the definition. Let \(W_j' \subset X\) be the corresponding integral closed subspace of \(X\), i.e., having the same generic point as \(W_j\). Suppose that \(V \subset X\) is a quasi-compact open. Then also \(V \cap U\) is quasi-compact open in \(U\) as \(V\) is Noetherian. Hence the set \(\{j \in J \mid W_j \cap V \not = \emptyset\} = \{j \in J \mid W'_j \cap V \not = \emptyset\}\) is finite since \(\{W_j\}\) is locally finite. In other words we see that \(\{W'_j\}\) is also locally finite. Since \(R(W_j) = R(W'_j)\) we see that \[\alpha - \beta - \sum (i'_j)_*\text{div}(f_j)\] is a cycle on \(X\) whose restriction to \(U\) is zero. The lemma follows by applying Lemma 0EP9.

Remark

In Situation 0EDT let \(X/B\) be good. Suppose we have infinite collections \(\alpha_i, \beta_i \in Z_k(X)\), \(i \in I\) of \(k\)-cycles on \(X\). Suppose that the supports of \(\alpha_i\) and \(\beta_i\) form locally finite collections of closed subsets of \(X\) so that \(\sum \alpha_i\) and \(\sum \beta_i\) are defined as cycles. Moreover, assume that \(\alpha_i \sim_{rat} \beta_i\) for each \(i\). Then it is not clear that \(\sum \alpha_i \sim_{rat} \sum \beta_i\). Namely, the problem is that the rational equivalences may be given by locally finite families \(\{W_{i, j}, f_{i, j} \in R(W_{i, j})^*\}_{j \in J_i}\) but the union \(\{W_{i, j}\}_{i \in I, j\in J_i}\) may not be locally finite.

In many cases in practice, one has a locally finite family of closed subsets \(\{T_i\}_{i \in I}\) of \(|X|\) such that \(\alpha_i, \beta_i\) are supported on \(T_i\) and such that \(\alpha_i \sim_{rat} \beta_i\) “on” \(T_i\). More precisely, the families \(\{W_{i, j}, f_{i, j} \in R(W_{i, j})^*\}_{j \in J_i}\) consist of integral closed subspaces \(W_{i, j}\) with \(|W_{i, j}| \subset T_i\). In this case it is true that \(\sum \alpha_i \sim_{rat} \sum \beta_i\) on \(X\), simply because the family \(\{W_{i, j}\}_{i \in I, j\in J_i}\) is automatically locally finite in this case.

Rational equivalence and push and pull

This section is the analogue of Chow Homology, Section 02S0. In this section we show that flat pullback and proper pushforward commute with rational equivalence.

Lemma

In Situation 0EDT let \(X, Y/B\) be good. Assume \(Y\) integral with \(\dim_\delta(Y) = k\). Let \(f : X \to Y\) be a flat morphism of relative dimension \(r\). Then for \(g \in R(Y)^*\) we have \[f^*\text{div}_Y(g) = \sum m_{X', X} (X' \to X)_*\text{div}_{X'}(g \circ f|_{X'})\] as \((k + r - 1)\)-cycles on \(X\) where the sum is over the irreducible components \(X'\) of \(X\) and \(m_{X', X}\) is the multiplicity of \(X'\) in \(X\).

Proof

Observe that any irreducible component of \(X\) dominates \(Y\) (Lemma 0EP5) and hence the composition \(g \circ f|_{X'}\) is defined (Morphisms of Spaces, Section 0EML). We will reduce this to the case of schemes. Choose a scheme \(V\) and a surjective étale morphism \(V \to Y\). Choose a scheme \(U\) and a surjective étale morphism \(U \to V \times_Y X\). Picture \[\xymatrix{ U \ar[r]_a \ar[d]_h & X \ar[d]^f \\ V \ar[r]^b & Y }\] Since \(a\) is surjective and étale it follows from Lemma 0EPY that it suffices to prove the equality of cycles after pulling back by \(a\). We can use Lemma 0EQ2 to write \[b^*\text{div}_Y(g) = \sum (V' \to V)_*\text{div}_{V'}(g \circ b|_{V'})\] where the sum is over the irreducible components \(V'\) of \(V\). Using Lemma 0EPD we find \[h^*b^*\text{div}_Y(g) = \sum (V' \times_V U \to U)_*(h')^*\text{div}_{V'}(g \circ b|_{V'})\] where \(h' : V' \times_V U \to V'\) is the projection. We may apply the lemma in the case of schemes (Chow Homology, Lemma 0EPH) to the morphism \(h' : V' \times_V U \to V'\) to see that we have \[(h')^*\text{div}_{V'}(g \circ b|_{V'}) = \sum m_{U', V' \times_V U} (U' \to V' \times_V U)_*\text{div}_{U'}(g \circ b|_{V'} \circ h'|_{U'})\] where the sum is over the irreducible components \(U'\) of \(V' \times_V U\). Each \(U'\) occurring in this sum is an irreducible component of \(U\) and conversely every irreducible component \(U'\) of \(U\) is an irreducible component of \(V' \times_V U\) for a unique irreducible component \(V' \subset V\). Given an irreducible component \(U' \subset U\), denote \(\overline{a(U')} \subset X\) the “image” in \(X\) (Lemma 0EEG); this is an irreducible component of \(X\) for example by Lemma 0EP5. The muplticity \(m_{U', V' \times_V U}\) is equal to the multiplicity \(m_{\overline{a(U')}, X}\). This follows from the equality \(h^*a^*[Y] = b^*f^*[Y]\) (Lemma 0EPA), the definitions, and Lemma 0EPY. Combining all of what we just said we obtain \[a^*f^*\text{div}_Y(g) = h^*b^*\text{div}_Y(g) = \sum m_{\overline{a(U')}, X} (U' \to U)_*\text{div}_{U'}(g \circ (f \circ a)|_{U'})\] Next, we analyze what happens with the right hand side of the formula in the statement of the lemma if we pullback by \(a\). First, we use Lemma 0EPD to get \[a^*\sum m_{X', X} (X' \to X)_*\text{div}_{X'}(g \circ f|_{X'}) = \sum m_{X', X} (X' \times_X U \to U)_*(a')^*\text{div}_{X'}(g \circ f|_{X'})\] where \(a' : X' \times_X U \to X'\) is the projection. By Lemma 0EQ2 we get \[(a')^*\text{div}_{X'}(g \circ f|_{X'}) = \sum (U' \to X' \times_X U)_*\text{div}_{U'}(g \circ (f \circ a)|_{U'})\] where the sum is over the irreducible components \(U'\) of \(X' \times_X U\). These \(U'\) are irreducible components of \(U\) and in fact are exactly the irreducible components of \(U\) such that \(\overline{a(U')} = X'\). Comparing with what we obtained above we conclude.

Lemma

In Situation 0EDT let \(X, Y/B\) be good. Let \(f : X \to Y\) be a flat morphism of relative dimension \(r\). Let \(\alpha \sim_{rat} \beta\) be rationally equivalent \(k\)-cycles on \(Y\). Then \(f^*\alpha \sim_{rat} f^*\beta\) as \((k + r)\)-cycles on \(X\).

Proof

What do we have to show? Well, suppose we are given a collection \[i_j : W_j \longrightarrow Y\] of closed immersions, with each \(W_j\) integral of \(\delta\)-dimension \(k + 1\) and rational functions \(g_j \in R(W_j)^*\). Moreover, assume that the collection \(\{|i_j|(|W_j|)\}_{j \in J}\) is locally finite in \(|Y|\). Then we have to show that \[f^*(\sum i_{j, *}\text{div}(g_j)) = \sum f^*i_{j, *}\text{div}(g_j)\] is rationally equivalent to zero on \(X\). The sum on the right makes sense by Lemma 0EP6.

Consider the fibre products \[i'_j : W'_j = W_j \times_Y X \longrightarrow X.\] and denote \(f_j : W'_j \to W_j\) the first projection. By Lemma 0EPD we can write the sum above as \[\sum i'_{j, *}(f_j^*\text{div}(g_j))\] By Lemma 0EQA we see that each \(f_j^*\text{div}(g_j)\) is rationally equivalent to zero on \(W'_j\). Hence each \(i'_{j, *}(f_j^*\text{div}(g_j))\) is rationally equivalent to zero. Then the same is true for the displayed sum by the discussion in Remark 0EQ8.

Lemma

In Situation 0EDT let \(X, Y/B\) be good. Let \(p : X \to Y\) be a proper morphism. Suppose \(\alpha, \beta \in Z_k(X)\) are rationally equivalent. Then \(p_*\alpha\) is rationally equivalent to \(p_*\beta\).

Proof

What do we have to show? Well, suppose we are given a collection \[i_j : W_j \longrightarrow X\] of closed immersions, with each \(W_j\) integral of \(\delta\)-dimension \(k + 1\) and rational functions \(f_j \in R(W_j)^*\). Moreover, assume that the collection \(\{i_j(W_j)\}_{j \in J}\) is locally finite on \(X\). Then we have to show that \[p_*\left(\sum i_{j, *}\text{div}(f_j)\right)\] is rationally equivalent to zero on \(X\).

Note that the sum is equal to \[\sum p_*i_{j, *}\text{div}(f_j).\] Let \(W'_j \subset Y\) be the integral closed subspace which is the image of \(p \circ i_j\), see Lemma 0EEG. The collection \(\{W'_j\}\) is locally finite in \(Y\) by Lemma 0ENZ. Hence it suffices to show, for a given \(j\), that either \(p_*i_{j, *}\text{div}(f_j) = 0\) or that it is equal to \(i'_{j, *}\text{div}(g_j)\) for some \(g_j \in R(W'_j)^*\).

The arguments above therefore reduce us to the case of a since integral closed subspace \(W \subset X\) of \(\delta\)-dimension \(k + 1\). Let \(f \in R(W)^*\). Let \(W' = p(W)\) as above. We get a commutative diagram of morphisms \[\xymatrix{ W \ar[r]_i \ar[d]_{p'} & X \ar[d]^p \\ W' \ar[r]^{i'} & Y }\] Note that \(p_*i_*\text{div}(f) = i'_*(p')_*\text{div}(f)\) by Lemma 0EP2. As explained above we have to show that \((p')_*\text{div}(f)\) is the divisor of a rational function on \(W'\) or zero. There are three cases to distinguish.

The case \(\dim_\delta(W') < k\). In this case automatically \((p')_*\text{div}(f) = 0\) and there is nothing to prove.

The case \(\dim_\delta(W') = k\). Let us show that \((p')_*\text{div}(f) = 0\) in this case. Since \((p')_*\text{div}(f)\) is a \(k\)-cycle, we see that \((p')_*\text{div}(f) = n[W']\) for some \(n \in \mathbf{Z}\). In order to prove that \(n = 0\) we may replace \(W'\) by a nonempty open subspace. In particular, we may and do assume that \(W'\) is a scheme. Let \(\eta \in W'\) be the generic point. Let \(K = \kappa(\eta) = R(W')\) be the function field. Consider the base change diagram \[\xymatrix{ W_\eta \ar[r] \ar[d]_c & W \ar[d]^{p'} \\ \Spec(K) \ar[r]^\eta & W' }\] Observe that \(c\) is proper. Also \(|W_\eta|\) has dimension \(1\): use Decent Spaces, Lemma 0AC8 to identify \(|W_\eta|\) as the subspace of \(|W|\) of points mapping to \(\eta\) and note that since \(\dim_\delta(W) = k + 1\) and \(\delta(\eta) = k\) points of \(W_\eta\) must have \(\delta\)-value either \(k\) or \(k + 1\). Thus the local rings have dimension \(\leq 1\) by Decent Spaces, Lemma 0ED1. By Spaces over Fields, Lemma 0ADD we find that \(W_\eta\) is a scheme. Since \(\Spec(K)\) is the limit of the nonempty affine open subschemes of \(W'\) we conclude that we may assume that \(W\) is a scheme by Limits of Spaces, Lemma 07SR. Then finally by the case of schemes (Chow Homology, Lemma 02S2) we find that \(n = 0\).

The case \(\dim_\delta(W') = k + 1\). In this case Lemma 0EQ4 applies, and we see that indeed \(p'_*\text{div}(f) = \text{div}(g)\) for some \(g \in R(W')^*\) as desired.

The divisor associated to an invertible sheaf

This section is the analogue of Chow Homology, Section 02SI. The following definition is the analogue of Spaces over Fields, Definition 0EPU in our current setup.

Definition

In Situation 0EDT let \(X/B\) be good. Assume \(X\) is integral and \(n = \dim_\delta(X)\). Let \(\mathcal{L}\) be an invertible \(\mathcal{O}_X\)-module.

For any nonzero meromorphic section \(s\) of \(\mathcal{L}\) we define the Weil divisor associated to \(s\) is the \((n - 1)\)-cycle \[\text{div}_\mathcal{L}(s) = \sum \text{ord}_{Z, \mathcal{L}}(s) [Z]\] defined in Spaces over Fields, Definition 0EPU. This makes sense because Weil divisors have \(\delta\)-dimension \(n - 1\) by Lemma 0EPZ.

We define Weil divisor associated to \(\mathcal{L}\) as \[c_1(\mathcal{L}) \cap [X] = \text{class of }\text{div}_\mathcal{L}(s) \in \CH_{n - 1}(X)\] where \(s\) is any nonzero meromorphic section of \(\mathcal{L}\) over \(X\). This is well defined by Spaces over Fields, Lemma 0EPT.

The zero scheme of a nonzero section is an effective Cartier divisor whose Weil divisor class computes the Weil divisor associated to the invertible module.

Lemma

In Situation 0EDT let \(X/B\) be good. Assume \(X\) is integral and \(n = \dim_\delta(X)\). Let \(\mathcal{L}\) be an invertible \(\mathcal{O}_X\)-module. Let \(s \in \Gamma(X, \mathcal{L})\) be a nonzero global section. Then \[\text{div}_\mathcal{L}(s) = [Z(s)]_{n - 1}\] in \(Z_{n - 1}(X)\) and \[c_1(\mathcal{L}) \cap [X] = [Z(s)]_{n - 1}\] in \(\CH_{n - 1}(X)\).

Proof

Let \(Z \subset X\) be an integral closed subspace of \(\delta\)-dimension \(n - 1\). Let \(\xi \in |Z|\) be its generic point. To prove the first equality we compare the coefficients of \(Z\) on both sides. Choose an elementary étale neighbourhood \((U, u) \to (X, \xi)\), see Decent Spaces, Section 0EMV and recall that \(\mathcal{O}_{X, \xi}^h = \mathcal{O}_{U, u}^h\) in this case. After replacing \(U\) by an open neighbourhood of \(u\) we may assume there is a trivializing section \(s_U\) of \(\mathcal{L}|_U\). Write \(s|_U = f s_U\) for some \(f \in \Gamma(U, \mathcal{O}_U)\). Then \(Z \times_X U\) is equal to \(V(f)\) as a closed subscheme of \(U\), see Divisors on Spaces, Definition 0845. As in Spaces over Fields, Section 0ENV denote \(\mathcal{L}_\xi\) the pullback of \(\mathcal{L}\) under the canonical morphism \(c_\xi : \Spec(\mathcal{O}_{X, \xi}^h) \to X\). Denote \(s_\xi\) the pullback of \(s_U\); it is a trivialization of \(\mathcal{L}_\xi\). Then we see that \(c_\xi^*(s) = fs_\xi\). The coefficient of \(Z\) in \([Z(s)]_{n - 1}\) is by definition \[\text{length}_{\mathcal{O}_{U, u}}(\mathcal{O}_{U, u}/f\mathcal{O}_{U, u})\] Since \(\mathcal{O}_{U, u} \to \mathcal{O}_{X, \xi}^h\) is flat and identifies residue fields this is equal to \[\text{length}_{\mathcal{O}_{X, \xi}^h} (\mathcal{O}_{X, \xi}^h/f\mathcal{O}_{X, \xi}^h)\] by Algebra, Lemma 02M1. This final quantity is equal to \(\text{ord}_{Z, \mathcal{L}}(s)\) by Spaces over Fields, Definition 0EPR, i.e., to the coefficient of \(Z\) in \(\text{div}_\mathcal{L}(s)\) as desired.

Lemma

In Situation 0EDT let \(X/B\) be good. Let \(\mathcal{L}\) be an invertible \(\mathcal{O}_X\)-module. The morphism \[q : T = \underline{\Spec}\left( \bigoplus\nolimits_{n \in \mathbf{Z}} \mathcal{L}^{\otimes n}\right) \longrightarrow X\] has the following properties:

\(q\) is surjective, smooth, affine, of relative dimension \(1\),

there is an isomorphism \(\alpha : q^*\mathcal{L} \cong \mathcal{O}_T\),

formation of \((q : T \to X, \alpha)\) commutes with base change,

\(q^* : Z_k(X) \to Z_{k + 1}(T)\) is injective,

if \(Z \subset X\) is an integral closed subspace, then \(q^{-1}(Z) \subset T\) is an integral closed subspace,

if \(Z \subset X\) is a closed subspace of \(X\) of \(\delta\)-dimension \(\leq k\), then \(q^{-1}(Z)\) is a closed subspace of \(T\) of \(\delta\)-dimension \(\leq k + 1\) and \(q^*[Z]_k = [q^{-1}(Z)]_{k + 1}\),

if \(\xi' \in |T|\) is the generic point of the fibre of \(|T| \to |X|\) over \(\xi\), then the ring map \(\mathcal{O}_{X, \xi}^h \to \mathcal{O}_{T, \xi'}^h\) is flat, we have \(\mathfrak m_{\xi'}^h = \mathfrak m_\xi^h \mathcal{O}_{T, \xi'}^h\), and the residue field extension is purely transcendental of transcendence degree \(1\), and

add more here as needed.

Proof

Let \(U \to X\) be an étale morphism such that \(\mathcal{L}|_U\) is trivial. Then \(T \times_X U \to U\) is isomorphic to the projection morphism \(\mathbf{G}_m \times U \to U\), where \(\mathbf{G}_m\) is the multipliciative group scheme, see Groupoids, Example 022U. Thus (1) is clear.

To see (2) observe that \(q_*q^*\mathcal{L} = \bigoplus_{n \in \mathbf{Z}} \mathcal{L}^{\otimes n + 1}\). Thus there is an obvious isomorphism \(q_*q^*\mathcal{L} \to q_*\mathcal{O}_T\) of \(q_*\mathcal{O}_T\)-modules. By Morphisms of Spaces, Lemma 08AI this determines an isomorphism \(q^*\mathcal{L} \to \mathcal{O}_T\).

Part (3) holds because forming the relative spectrum commutes with arbitrary base change and the same thing is clearly true for the isomorphism \(\alpha\).

Part (4) follows immediately from (1) and the definitions.

Part (5) follows from the fact that if \(Z\) is an integral algebraic space, then \(\mathbf{G}_m \times Z\) is an integral algebraic space.

Part (6) follows from the fact that lengths are preserved: if \((A, \mathfrak m)\) is a local ring and \(B = A[x]_{\mathfrak m A[x]}\) and if \(M\) is an \(A\)-module, then \(\text{length}_A(M) = \text{length}_B(M \otimes_A B)\). This implies that if \(\mathcal{F}\) is a coherent \(\mathcal{O}_X\)-module and \(\xi \in |X|\) with \(\xi' \in |T|\) the generic point of the fibre over \(\xi\), then the length of \(\mathcal{F}\) at \(\xi\) is the same as the length of \(q^*\mathcal{F}\) at \(\xi'\). Tracing through the definitions this gives (6) and more.

The map in part (7) comes from Decent Spaces, Remark 0EPL. However, in our case we have \[\Spec(\mathcal{O}_{X, \xi}^h) \times_X T = \mathbf{G}_m \times \Spec(\mathcal{O}_{X, \xi}^h) = \Spec(\mathcal{O}_{X, \xi}^h[t, t^{-1}])\] and \(\xi'\) corresponds to the generic point of the special fibre of this over \(\Spec(\mathcal{O}_{X, \xi}^h)\). Thus \(\mathcal{O}_{T, \xi'}^h\) is the henselization of the localization of \(\mathcal{O}_{X, \xi}^h[t, t^{-1}]\) at the corresponding prime. Part (7) follows from this and some commutative algebra; details omitted.

Lemma

In Situation 0EDT let \(X/B\) be good. Let \(\mathcal{L}\) be an invertible \(\mathcal{O}_X\)-module. Assume \(X\) is integral. Let \(s\) be a nonzero meromorphic section of \(\mathcal{L}\). Let \(q : T \to X\) be the morphism of Lemma 0EQG. Then \[q^*\text{div}_\mathcal{L}(s) = \text{div}_T(q^*(s))\] where we view the pullback \(q^*(s)\) as a nonzero meromorphic function on \(T\) using the isomorphism \(q^*\mathcal{L} \to \mathcal{O}_T\)

Proof

Observe that \(\text{div}_T(q^*(s)) = \text{div}_{\mathcal{O}_T}(q^*(s))\) by the compatibility between the constructions given in Spaces over Fields, Sections 0ENI and 0ENV. We will show the agreement with \(\text{div}_{\mathcal{O}_T}(q^*(s))\) in this proof. We will use all the properties of \(q : T \to X\) stated in Lemma 0EQG without further mention. Let \(Z \subset T\) be a prime divisor. Then either \(Z \to X\) is dominant or \(Z = q^{-1}(Z')\) for some prime divisor \(Z' \subset X\). If \(Z \to X\) is dominant, then the coefficient of \(Z\) in either side of the equality of the lemma is zero. Thus we may assume \(Z = q^{-1}(Z')\) where \(Z' \subset X\) is a prime divisor. Let \(\xi' \in |Z'|\) and \(\xi \in |Z|\) be the generic points. Then we obtain a commutative diagram \[\xymatrix{ \Spec(\mathcal{O}_{T, \xi}^h) \ar[r]_-{c_\xi} \ar[d]_h & T \ar[d]^q \\ \Spec(\mathcal{O}_{X, \xi'}^h) \ar[r]^-{c_{\xi'}} & X }\] see Decent Spaces, Remark 0EPL. Choose a trivialization \(s_{\xi'}\) of \(\mathcal{L}_{\xi'} = c_{\xi'}^*\mathcal{L}\). Then we can use the pullback \(s_\xi\) of \(s_{\xi'}\) via \(h\) as our trivialization of \(\mathcal{L}_\xi = c_\xi^* q^*\mathcal{L}\). Write \(s/s_{\xi'} = a/b\) for \(a, b \in \mathcal{O}_{X, \xi'}\) nonzerodivisors. By definition the coefficient of \(Z'\) in \(\text{div}_\mathcal{L}(s)\) is \[\text{length}_{\mathcal{O}_{X, \xi'}^h}( \mathcal{O}_{X, \xi'}^h/a \mathcal{O}_{X, \xi'}^h) - \text{length}_{\mathcal{O}_{X, \xi'}^h}( \mathcal{O}_{X, \xi'}^h/b \mathcal{O}_{X, \xi'}^h)\] Since \(Z = q^{-1}(Z')\), this is also the coefficient of \(Z\) in \(q^*\text{div}_\mathcal{L}(s)\). Since \(\mathcal{O}_{X, \xi'}^h \to \mathcal{O}_{T, \xi}^h\) is flat the elements \(a, b\) map to nonzerodivisors in \(\mathcal{O}_{T, \xi}^h\). Thus \(q^*(s)/s_\xi = a/b\) in the total quotient ring of \(\mathcal{O}_{T, \xi}^h\). By definition the coefficient of \(Z\) in \(\text{div}_T(q^*(s))\) is \[\text{length}_{\mathcal{O}_{T, \xi}^h}( \mathcal{O}_{T, \xi}^h/a \mathcal{O}_{T, \xi}^h) - \text{length}_{\mathcal{O}_{T, \xi}^h}( \mathcal{O}_{T, \xi}^h/b \mathcal{O}_{T, \xi}^h)\] The proof is finished because these lengths are the same as before by Algebra, Lemma 02M1 and the fact that \(\mathfrak m_\xi^h = \mathfrak m_{\xi'}^h\mathcal{O}_{T, \xi}^h\) shown in Lemma 0EQG.

Intersecting with an invertible sheaf

This section is the analogue of Chow Homology, Section 02SN. In this section we study the following construction.

Definition

In Situation 0EDT let \(X/B\) be good. Let \(\mathcal{L}\) be an invertible \(\mathcal{O}_X\)-module. We define, for every integer \(k\), an operation \[c_1(\mathcal{L}) \cap - : Z_{k + 1}(X) \to \CH_k(X)\] called intersection with the first Chern class of \(\mathcal{L}\).

Given an integral closed subspace \(i : W \to X\) with \(\dim_\delta(W) = k + 1\) we define \[c_1(\mathcal{L}) \cap [W] = i_*(c_1({i^*\mathcal{L}}) \cap [W])\] where the right hand side is defined in Definition 0EQE.

For a general \((k + 1)\)-cycle \(\alpha = \sum n_i [W_i]\) we set \[c_1(\mathcal{L}) \cap \alpha = \sum n_i c_1(\mathcal{L}) \cap [W_i]\]

Write each \(c_1(\mathcal{L}) \cap W_i = \sum_j n_{i, j} [Z_{i, j}]\) with \(\{Z_{i, j}\}_j\) a locally finite sum of integral closed subspaces of \(W_i\). Since \(\{W_i\}\) is a locally finite collection of integral closed subspaces on \(X\), it follows easily that \(\{Z_{i, j}\}_{i, j}\) is a locally finite collection of closed subspaces of \(X\). Hence \(c_1(\mathcal{L}) \cap \alpha = \sum n_in_{i, j}[Z_{i, j}]\) is a cycle. Another, often more convenient, way to think about this is to observe that the morphism \(\coprod W_i \to X\) is proper. Hence \(c_1(\mathcal{L}) \cap \alpha\) can be viewed as the pushforward of a class in \(\CH_k(\coprod W_i) = \prod \CH_k(W_i)\). This also explains why the result is well defined up to rational equivalence on \(X\).

The main goal for the next few sections is to show that intersecting with \(c_1(\mathcal{L})\) factors through rational equivalence. This is not a triviality.

Lemma

In Situation 0EDT let \(X/B\) be good. Let \(\mathcal{L}\), \(\mathcal{N}\) be an invertible sheaves on \(X\). Then \[c_1(\mathcal{L}) \cap \alpha + c_1(\mathcal{N}) \cap \alpha = c_1(\mathcal{L} \otimes_{\mathcal{O}_X} \mathcal{N}) \cap \alpha\] in \(\CH_k(X)\) for every \(\alpha \in Z_{k - 1}(X)\). Moreover, \(c_1(\mathcal{O}_X) \cap \alpha = 0\) for all \(\alpha\).

Proof

The additivity follows directly from Spaces over Fields, Lemma 0EPV and the definitions. To see that \(c_1(\mathcal{O}_X) \cap \alpha = 0\) consider the section \(1 \in \Gamma(X, \mathcal{O}_X)\). This restricts to an everywhere nonzero section on any integral closed subspace \(W \subset X\). Hence \(c_1(\mathcal{O}_X) \cap [W] = 0\) as desired.

Recall that \(Z(s) \subset X\) denotes the zero scheme of a global section \(s\) of an invertible sheaf on an algebraic space \(X\), see Divisors on Spaces, Definition 0845.

Lemma

In Situation 0EDT let \(Y/B\) be good. Let \(\mathcal{L}\) be an invertible \(\mathcal{O}_Y\)-module. Let \(s \in \Gamma(Y, \mathcal{L})\) be a regular section and assume \(\dim_\delta(Y) \leq k + 1\). Write \([Y]_{k + 1} = \sum n_i[Y_i]\) where \(Y_i \subset Y\) are the irreducible components of \(Y\) of \(\delta\)-dimension \(k + 1\). Set \(s_i = s|_{Y_i} \in \Gamma(Y_i, \mathcal{L}|_{Y_i})\). Then [0EQM]\[\begin{equation} [Z(s)]_k = \sum n_i[Z(s_i)]_k \end{equation}\] as \(k\)-cycles on \(Y\).

Proof

Let \(\varphi : V \to Y\) be a surjective étale morphism where \(V\) is a scheme. It suffices to prove the equality after pulling back by \(\varphi\), see Lemma 0EPY. That same lemma tells us that \(\varphi^*[Y_i] = [\varphi^{-1}(Y_i)] = \sum [V_{i, j}]\) where \(V_{i, j}\) are the irreducible components of \(V\) lying over \(Y_i\). Hence if we first apply the case of schemes (Chow Homology, Lemma 0EPI) to \(\varphi^*s_i\) on \(Y_i \times_Y V\) we find that \(\varphi^*[Z(s_i)]_k = [Z(\varphi^*s_i)] = \sum [Z(s_{i, j})]_k\) where \(s_{i, j}\) is the pullback of \(s\) to \(V_{i, j}\). Applying the case of schemes to \(\varphi^*s\) we get \[\varphi^*[Z(s)]_k = [Z(\varphi^*s)]_k = \sum n_i[Z(s_{i, j})]_k\] by our remark on multiplicities above. Combining all of the above the proof is complete.

The following lemma is a useful result in order to compute the intersection product of the \(c_1\) of an invertible sheaf and the cycle associated to a closed subscheme. Recall that \(Z(s) \subset X\) denotes the zero scheme of a global section \(s\) of an invertible sheaf on a scheme \(X\), see Divisors, Definition 02OQ.

Lemma

In Situation 0EDT let \(X/B\) be good. Let \(\mathcal{L}\) be an invertible \(\mathcal{O}_X\)-module. Let \(Y \subset X\) be a closed subscheme with \(\dim_\delta(Y) \leq k + 1\) and let \(s \in \Gamma(Y, \mathcal{L}|_Y)\) be a regular section. Then \[c_1(\mathcal{L}) \cap [Y]_{k + 1} = [Z(s)]_k\] in \(\CH_k(X)\).

Proof

Write \[[Y]_{k + 1} = \sum n_i[Y_i]\] where \(Y_i \subset Y\) are the irreducible components of \(Y\) of \(\delta\)-dimension \(k + 1\) and \(n_i > 0\). By assumption the restriction \(s_i = s|_{Y_i} \in \Gamma(Y_i, \mathcal{L}|_{Y_i})\) is not zero, and hence is a regular section. By Lemma 0EQF we see that \([Z(s_i)]_k\) represents \(c_1(\mathcal{L}|_{Y_i})\). Hence by definition \[c_1(\mathcal{L}) \cap [Y]_{k + 1} = \sum n_i[Z(s_i)]_k\] Thus the result follows from Lemma 0EQL.

Intersecting with an invertible sheaf and push and pull

This section is the analogue of Chow Homology, Section 0AYA. In this section we prove that the operation \(c_1(\mathcal{L}) \cap -\) commutes with flat pullback and proper pushforward.

Lemma

In Situation 0EDT let \(X, Y/B\) be good. Let \(f : X \to Y\) be a flat morphism of relative dimension \(r\). Let \(\mathcal{L}\) be an invertible sheaf on \(Y\). Assume \(Y\) is integral and \(n = \dim_\delta(Y)\). Let \(s\) be a nonzero meromorphic section of \(\mathcal{L}\). Then we have \[f^*\text{div}_\mathcal{L}(s) = \sum n_i\text{div}_{f^*\mathcal{L}|_{X_i}}(s_i)\] in \(Z_{n + r - 1}(X)\). Here the sum is over the irreducible components \(X_i \subset X\) of \(\delta\)-dimension \(n + r\), the section \(s_i = f|_{X_i}^*(s)\) is the pullback of \(s\), and \(n_i = m_{X_i, X}\) is the multiplicity of \(X_i\) in \(X\).

Proof

Using sleight of hand we will deduce this from Lemma 0EQA. (An alternative is to redo the proof of that lemma in the setting of meromorphic sections of invertible modules.) Namely, let \(q : T \to Y\) be the morphism of Lemma 0EQG constructed using \(\mathcal{L}\) on \(Y\). We will use all the properties of \(T\) stated in this lemma. Consider the fibre product diagram \[\xymatrix{ T' \ar[r]_{q'} \ar[d]_h & X \ar[d]^f \\ T \ar[r]^q & Y }\] Then \(q' : T' \to X\) is the morphism constructed using \(f^*\mathcal{L}\) on \(X\). Then it suffices to prove \[(q')^*f^*\text{div}_\mathcal{L}(s) = \sum n_i (q')^*\text{div}_{f^*\mathcal{L}|_{X_i}}(s_i)\] Observe that \(T'_i = q^{-1}(X_i)\) are the irreducible components of \(T'\) and that \(n_i\) is the multiplicity of \(T'_i\) in \(T'\). The left hand side is equal to \[h^*q^*\text{div}_\mathcal{L}(s) = h^*\text{div}_T(q^*(s))\] by Lemma 0EQH (and Lemma 0EPA). On the other hand, denoting \(q'_i : T'_i \to X_i\) the restriction of \(q'\) we find that Lemma 0EQH also tells us the right hand side is equal to \[\sum n_i \text{div}_{T_i}((q'_i)^*(s_i))\] In these two formulas the expressions \(q^*(s)\) and \((q'_i)^*(s_i)\) represent the rational functions corresponding to the pulled back meromorphic sections of \(q^*\mathcal{L}\) and \((q'_i)^*f^*\mathcal{L}|_{X_i}\) via the isomorphism \(\alpha : q^*\mathcal{L} \to \mathcal{O}_T\) and its pullbacks to spaces over \(T\). With this convention it is clear that \((q'_i)^*(s_i)\) is the composition of the rational function \(q^*(s)\) on \(T\) and the morphism \(h|_{T'_i} : T'_i \to T\). Thus Lemma 0EQA exactly says that \[h^*\text{div}_T(q^*(s)) = \sum n_i \text{div}_{T_i}((q'_i)^*(s_i))\] as desired.

Lemma

In Situation 0EDT let \(X, Y/B\) be good. Let \(f : X \to Y\) be a flat morphism of relative dimension \(r\). Let \(\mathcal{L}\) be an invertible sheaf on \(Y\). Let \(\alpha\) be a \(k\)-cycle on \(Y\). Then \[f^*(c_1(\mathcal{L}) \cap \alpha) = c_1(f^*\mathcal{L}) \cap f^*\alpha\] in \(\CH_{k + r - 1}(X)\).

Proof

Write \(\alpha = \sum n_i[W_i]\). We will show that \[f^*(c_1(\mathcal{L}) \cap [W_i]) = c_1(f^*\mathcal{L}) \cap f^*[W_i]\] in \(\CH_{k + r - 1}(X)\) by producing a rational equivalence on the closed subspace \(f^{-1}(W_i)\) of \(X\). By the discussion in Remark 0EQ8 this will prove the equality of the lemma is true.

Let \(W \subset Y\) be an integral closed subspace of \(\delta\)-dimension \(k\). Consider the closed subspace \(W' = f^{-1}(W) = W \times_Y X\) so that we have the fibre product diagram \[\xymatrix{ W' \ar[r] \ar[d]_h & X \ar[d]^f \\ W \ar[r] & Y }\] We have to show that \(f^*(c_1(\mathcal{L}) \cap [W]) = c_1(f^*\mathcal{L}) \cap f^*[W]\). Choose a nonzero meromorphic section \(s\) of \(\mathcal{L}|_W\). Let \(W'_i \subset W'\) be the irreducible components of \(\delta\)-dimension \(k + r\). Write \([W']_{k + r} = \sum n_i[W'_i]\) with \(n_i\) the multiplicity of \(W'_i\) in \(W'\) as per definition. So \(f^*[W] = \sum n_i[W'_i]\) in \(Z_{k + r}(X)\). Since each \(W'_i \to W\) is dominant we see that \(s_i = s|_{W'_i}\) is a nonzero meromorphic section for each \(i\). By Lemma 0EQQ we have the following equality of cycles \[h^*\text{div}_{\mathcal{L}|_W}(s) = \sum n_i\text{div}_{f^*\mathcal{L}|_{W'_i}}(s_i)\] in \(Z_{k + r - 1}(W')\). This finishes the proof since the left hand side is a cycle on \(W'\) which pushes to \(f^*(c_1(\mathcal{L}) \cap [W])\) in \(\CH_{k + r - 1}(X)\) and the right hand side is a cycle on \(W'\) which pushes to \(c_1(f^*\mathcal{L}) \cap f^*[W]\) in \(\CH_{k + r - 1}(X)\).

Lemma

In Situation 0EDT let \(X, Y/B\) be good. Let \(f : X \to Y\) be a proper morphism. Let \(\mathcal{L}\) be an invertible sheaf on \(Y\). Assume \(X\), \(Y\) integral, \(f\) dominant, and \(\dim_\delta(X) = \dim_\delta(Y)\). Let \(s\) be a nonzero meromorphic section \(s\) of \(\mathcal{L}\) on \(Y\). Then \[f_*\left(\text{div}_{f^*\mathcal{L}}(f^*s)\right) = [R(X) : R(Y)]\text{div}_\mathcal{L}(s).\] as cycles on \(Y\). In particular \[f_*(c_1(f^*\mathcal{L}) \cap [X]) = c_1(\mathcal{L}) \cap f_*[Y].\]

Proof

The last equation follows from the first since \(f_*[X] = [R(X) : R(Y)][Y]\) by definition. Proof of the first equaltion. Let \(q : T \to Y\) be the morphism of Lemma 0EQG constructed using \(\mathcal{L}\) on \(Y\). We will use all the properties of \(T\) stated in this lemma. Consider the fibre product diagram \[\xymatrix{ T' \ar[r]_{q'} \ar[d]_h & X \ar[d]^f \\ T \ar[r]^q & Y }\] Then \(q' : T' \to X\) is the morphism constructed using \(f^*\mathcal{L}\) on \(X\). It suffices to prove the equality after pulling back to \(T'\). The left hand side pulls back to \[\begin{align*} q^*f_*\left(\text{div}_{f^*\mathcal{L}}(f^*s)\right) & = h_*(q')^*\text{div}_{f^*\mathcal{L}}(f^*s) \\ & = h_*\text{div}_{(q')^*f^*\mathcal{L}}((q')^*f^*s) \\ & = h_*\text{div}_{h^*q^*\mathcal{L}}(h^*q^*s) \end{align*}\] The first equality by Lemma 0EPD. The second by Lemma 0EQQ using that \(T'\) is integral. The third because the displayed diagram commutes. The right hand side pulls back to \[[R(X) : R(Y)]q^*\text{div}_\mathcal{L}(s) = [R(T') : R(T)]\text{div}_{q^*\mathcal{L}}(q^*s)\] This follows from Lemma 0EQQ, the fact that \(T\) is integral, and the equality \([R(T') : R(T)] = [R(X) : R(Y)]\) whose proof we omit (it follows from Lemma 0EPD but that would be a silly way to prove the equality). Thus it suffices to prove the lemma for \(h : T' \to T\), the invertible module \(q^\mathcal{L}\) and the section \(q^*s\). Since \(q^*\mathcal{L} = \mathcal{O}_T\) we reduce to the case where \(\mathcal{L} \cong \mathcal{O}\) discussed in the next paragraph.

Assume that \(\mathcal{L} = \mathcal{O}_Y\). In this case \(s\) corresponds to a rational function \(g \in R(Y)\). Using the embedding \(R(Y) \subset R(X)\) we may think of \(g\) as a rational on \(X\) and we are simply trying to prove \[f_*\left(\text{div}_X(g)\right) = [R(X) : R(Y)]\text{div}_Y(g).\] Comparing with the result of Lemma 0EQ4 we see this true since \(\text{Nm}_{R(X)/R(Y)}(g) = g^{[R(X) : R(Y)]}\) as \(g \in R(Y)^*\).

Lemma

In Situation 0EDT let \(X, Y/B\) be good. Let \(p : X \to Y\) be a proper morphism. Let \(\alpha \in Z_{k + 1}(X)\). Let \(\mathcal{L}\) be an invertible sheaf on \(Y\). Then \[p_*(c_1(p^*\mathcal{L}) \cap \alpha) = c_1(\mathcal{L}) \cap p_*\alpha\] in \(\CH_k(Y)\).

Proof

Suppose that \(p\) has the property that for every integral closed subspace \(W \subset X\) the map \(p|_W : W \to Y\) is a closed immersion. Then, by definition of capping with \(c_1(\mathcal{L})\) the lemma holds.

We will use this remark to reduce to a special case. Namely, write \(\alpha = \sum n_i[W_i]\) with \(n_i \not = 0\) and \(W_i\) pairwise distinct. Let \(W'_i \subset Y\) be the “image” of \(W_i\) as in Lemma 0EEG. Consider the diagram \[\xymatrix{ X' = \coprod W_i \ar[r]_-q \ar[d]_{p'} & X \ar[d]^p \\ Y' = \coprod W'_i \ar[r]^-{q'} & Y. }\] Since \(\{W_i\}\) is locally finite on \(X\), and \(p\) is proper we see that \(\{W'_i\}\) is locally finite on \(Y\) and that \(q, q', p'\) are also proper morphisms. We may think of \(\sum n_i[W_i]\) also as a \(k\)-cycle \(\alpha' \in Z_k(X')\). Clearly \(q_*\alpha' = \alpha\). We have \(q_*(c_1(q^*p^*\mathcal{L}) \cap \alpha') = c_1(p^*\mathcal{L}) \cap q_*\alpha'\) and \((q')_*(c_1((q')^*\mathcal{L}) \cap p'_*\alpha') = c_1(\mathcal{L}) \cap q'_*p'_*\alpha'\) by the initial remark of the proof. Hence it suffices to prove the lemma for the morphism \(p'\) and the cycle \(\sum n_i[W_i]\). Clearly, this means we may assume \(X\), \(Y\) integral, \(f : X \to Y\) dominant and \(\alpha = [X]\). In this case the result follows from Lemma 0EQS.

The key formula

This section is the analogue of Chow Homology, Section 0AYB. We strongly urge the reader to read the proof in that case first.

In Situation 0EDT let \(X/B\) be good. Assume \(X\) is integral and \(\dim_\delta(X) = n\). Let \(\mathcal{L}\) and \(\mathcal{N}\) be invertible \(\mathcal{O}_X\)-modules. Let \(s\) be a nonzero meromorphic section of \(\mathcal{L}\) and let \(t\) be a nonzero meromorphic section of \(\mathcal{N}\). Let \(Z \subset X\) be a prime divisor with generic point \(\xi \in |Z|\). Consider the morphism \[c_\xi : \Spec(\mathcal{O}_{X, \xi}^h) \longrightarrow X\] used in Spaces over Fields, Section 0ENV. We denote \(\mathcal{L}_\xi\) and \(\mathcal{N}_\xi\) the pullbacks of \(\mathcal{L}\) and \(\mathcal{N}\) by \(c_\xi\); we often think of \(\mathcal{L}_\xi\) and \(\mathcal{N}_\xi\) as the rank \(1\) free \(\mathcal{O}_{X, \xi}^h\)-modules they give rise to. Note that the pullback of \(s\), resp. \(t\) is a regular meromorphic section of \(\mathcal{L}_\xi\), resp. \(\mathcal{N}_\xi\).

Let \(Z_i \subset X\), \(i \in I\) be a locally finite set of prime divisors with the following property: If \(Z \not \in \{Z_i\}\), then \(s\) is a generator for \(\mathcal{L}_\xi\) and \(t\) is a generator for \(\mathcal{N}_\xi\). Such a set exists by Spaces over Fields, Lemma 0EPS. Then \[\text{div}_\mathcal{L}(s) = \sum \text{ord}_{Z_i, \mathcal{L}}(s) [Z_i]\] and similarly \[\text{div}_\mathcal{N}(t) = \sum \text{ord}_{Z_i, \mathcal{N}}(t) [Z_i]\] Unwinding the definitions more, we pick for each \(i\) generators \(s_i \in \mathcal{L}_{\xi_i}\) and \(t_i \in \mathcal{N}_{\xi_i}\) where \(\xi_i\) is the generic point of \(Z_i\). Then we can write \[s = f_i s_i \quad\text{and}\quad t = g_i t_i\] with \(f_i, g_i\) invertible elements of the total ring of fractions \(Q(\mathcal{O}_{X, \xi_i}^h)\). We abbreviate \(B_i = \mathcal{O}_{X, \xi_i}^h\). Let us denote \[\text{ord}_{B_i} : Q(B_i)^* \longrightarrow \mathbf{Z},\quad a/b \longmapsto \text{length}_{B_i}(B_i/aB_i) - \text{length}_{B_i}(B_i/bB_i)\] In other words, we temporarily extend Algebra, Definition 02MD to these reduced Noetherian local rings of dimension \(1\). Then by definition \[\text{ord}_{Z_i, \mathcal{L}}(s) = \text{ord}_{B_i}(f_i) \quad\text{and}\quad \text{ord}_{Z_i, \mathcal{N}}(t) = \text{ord}_{B_i}(g_i)\] Since \(\xi_i\) is the generic point of \(Z_i\) we see that the residue field \(\kappa(\xi_i)\) is the function field of \(Z_i\). Moreover \(\kappa(\xi_i)\) is the residue field of \(B_i\), see Decent Spaces, Lemma 0EN0. Since \(t_i\) is a generator of \(\mathcal{N}_{\xi_i}\) we see that its image in the fibre \(\mathcal{N}_{\xi_i} \otimes_{B_i} \kappa(\xi_i)\) is a nonzero meromorphic section of \(\mathcal{N}|_{Z_i}\). We will denote this image \(t_i|_{Z_i}\). From our definitions it follows that \[c_1(\mathcal{N}) \cap \text{div}_\mathcal{L}(s) = \sum \text{ord}_{B_i}(f_i) (Z_i \to X)_*\text{div}_{\mathcal{N}|_{Z_i}}(t_i|_{Z_i})\] and similarly \[c_1(\mathcal{L}) \cap \text{div}_\mathcal{N}(t) = \sum \text{ord}_{B_i}(g_i) (Z_i \to X)_*\text{div}_{\mathcal{L}|_{Z_i}}(s_i|_{Z_i})\] in \(\CH_{n - 2}(X)\). We are going to find a rational equivalence between these two cycles. To do this we consider the tame symbol \[\partial_{B_i}(f_i, g_i) \in \kappa(\xi_i)^* = R(Z_i)^*\] see Chow Homology, Section 0EAH.

Lemma

In the situation above the cycle \[\sum (Z_i \to X)_*\left( \text{ord}_{B_i}(f_i) \text{div}_{\mathcal{N}|_{Z_i}}(t_i|_{Z_i}) - \text{ord}_{B_i}(g_i) \text{div}_{\mathcal{L}|_{Z_i}}(s_i|_{Z_i}) \right)\] is equal to the cycle \[\sum (Z_i \to X)_*\text{div}(\partial_{B_i}(f_i, g_i))\]

Proof

The strategy of the proof will be: first reduce to the case where \(\mathcal{L}\) and \(\mathcal{N}\) are trivial invertible modules, then change our choices of local trivializations, and then finally use étale localization to reduce to the case of schemes¹.

First step. Let \(q : T \to X\) be the morphism constructed in Lemma 0EQG. We will use all properties stated in that lemma without further mention. In particular, it suffices to show that the cycles are equal after pulling back by \(q\). Denote \(s'\) and \(t'\) the pullbacks of \(s\) and \(t\) to meromorphic sections of \(q^*\mathcal{L}\) and \(q^*\mathcal{N}\). Denote \(Z'_i = q^{-1}(Z_i)\), denote \(\xi'_i \in |Z'_i|\) the generic point, denote \(B'_i = \mathcal{O}_{T, \xi'_i}^h\), denote \(\mathcal{L}_{\xi'_i}\) and \(\mathcal{N}_{\xi'_i}\) the pullbacks of \(\mathcal{L}\) and \(\mathcal{N}\) to \(\Spec(B'_i)\). Recall that we have commutative diagrams \[\xymatrix{ \Spec(B'_i) \ar[r]_-{c_{\xi'_i}} \ar[d] & T \ar[d]^q \\ \Spec(B_i) \ar[r]^-{c_{\xi_i}} & X }\] see Decent Spaces, Remark 0EPL. Denote \(s'_i\) and \(t'_i\) the pullbacks of \(s_i\) and \(t_i\) which are generators of \(\mathcal{L}_{\xi'_i}\) and \(\mathcal{N}_{\xi'_i}\). Then we have \[s' = f'_i s'_i \quad\text{and}\quad t' = g'_i t'_i\] where \(f'_i\) and \(g'_i\) are the images of \(f_i, g_i\) under the map \(Q(B_i) \to Q(B'_i)\) induced by \(B_i \to B'_i\). By Algebra, Lemma 02M1 we have \[\text{ord}_{B_i}(f_i) = \text{ord}_{B'_i}(f'_i) \quad\text{and}\quad \text{ord}_{B_i}(g_i) = \text{ord}_{B'_i}(g'_i)\] By Lemma 0EQQ applied to \(q : Z'_i \to Z_i\) we have \[q^*\text{div}_{\mathcal{N}|_{Z_i}}(t_i|_{Z_i}) = \text{div}_{q^*\mathcal{N}|_{Z'_i}}(t'_i|_{Z'_i}) \quad\text{and}\quad q^*\text{div}_{\mathcal{L}|_{Z_i}}(s_i|_{Z_i}) = \text{div}_{q^*\mathcal{L}|_{Z'_i}}(s'_i|_{Z'_i})\] This already shows that the first cycle in the statement of the lemma pulls back to the corresponding cycle for \(s', t', Z'_i, s'_i, t'_i\). To see the same is true for the second, note that by Chow Homology, Lemma 0EPG we have \[\partial_{B_i}(f_i, g_i) \mapsto \partial_{B'_i}(f'_i, g'_i) \quad\text{via}\quad \kappa(\xi_i) \to \kappa(\xi'_i)\] Hence the same lemma as before shows that \[q^*\text{div}(\partial_{B_i}(f_i, g_i)) = \text{div}(\partial_{B'_i}(f'_i, g'_i))\] Since \(q^*\mathcal{L} \cong \mathcal{O}_T\) we find that it suffices to prove the equality in case \(\mathcal{L}\) is trivial. Exchanging the roles of \(\mathcal{L}\) and \(\mathcal{N}\) we see that we may similarly assume \(\mathcal{N}\) is trivial. This finishes the proof of the first step.

Second step. Assume \(\mathcal{L} = \mathcal{O}_X\) and \(\mathcal{N} = \mathcal{O}_X\). Denote \(1\) the trivializing section of \(\mathcal{L}\). Then \(s_i = u \cdot 1\) for some unit \(u \in B_i\). Let us examine what happens if we replace \(s_i\) by \(1\). Then \(f_i\) gets replaced by \(u f_i\). Thus the first part of the first expression of the lemma is unchanged and in the second part we add \[\text{ord}_{B_i}(g_i)\text{div}(u|_{Z_i})\] where \(u|_{Z_i}\) is the image of \(u\) in the residue field by Spaces over Fields, Lemma 0EPT and in the second expression we add \[\text{div}(\partial_{B_i}(u, g_i))\] by bi-linearity of the tame symbol. These terms agree by the property of the tame symbol given in Chow Homology, Equation (0EAN).

Let \(Y \subset X\) be an integral closed subspace with \(\dim_\delta(Y) = n - 2\). To show that the coefficients of \(Y\) of the two cycles of the lemma is the same, we may do a replacement of \(s_i\) by \(1\) as in the previous paragraph. In exactly the same way one shows that we may do a replacement of \(t_i\) by \(1\). Since there are only a finite number of \(Z_i\) such that \(Y \subset Z_i\) we may assume \(s_i = 1\) and \(t_i = 1\) for all these \(Z_i\).

Third step. Here we prove the coefficients of \(Y\) in the cycles of the lemma agree for an integral closed subspace \(Y\) with \(\dim_\delta(Y) = n - 2\) such that moreover \(\mathcal{L} = \mathcal{O}_X\) and \(\mathcal{N} = \mathcal{O}_X\) and \(s_i = 1\) and \(t_i = 1\) for all \(Z_i\) such that \(Y \subset Z_i\). After replacing \(X\) by a smaller open subspace we may in fact assume that \(s_i\) and \(t_i\) are equal to \(1\) for all \(i\). In this case the first cycle is zero. Our task is to show that the coefficient of \(Y\) in the second cycle is zero as well.

First, since \(\mathcal{L} = \mathcal{O}_X\) and \(\mathcal{N} = \mathcal{O}_X\) we may and do think of \(s, t\) as rational functions \(f, g\) on \(X\). Since \(s_i\) and \(t_i\) are equal to \(1\) we find that \(f_i\), resp. \(g_i\) is the image of \(f\), resp. \(g\) in \(Q(B_i)\) for all \(i\). Let \(\zeta \in |Y|\) be the generic point. Choose an étale neighbourhood \[(U, u) \longrightarrow (X, \zeta)\] and denote \(Y' = \overline{\{u\}} \subset U\). Since an étale morphism is flat, we can pullback \(f\) and \(g\) to regular meromorphic functions on \(U\) which we will also denote \(f\) and \(g\). For every prime divisor \(Y \subset Z \subset X\) the scheme \(Z \times_X U\) is a union of prime divisors of \(U\). Conversely, given a prime divisor \(Y' \subset Z' \subset U\), there is a prime divisor \(Y \subset Z \subset X\) such that \(Z'\) is a component of \(Z \times_X U\). Given such a pair \((Z, Z')\) the ring map \[\mathcal{O}_{X, \xi}^h \to \mathcal{O}_{U, \xi'}^h\] is étale (in fact it is finite étale). Hence we find that \[\partial_{\mathcal{O}_{X, \xi}^h}(f, g) \mapsto \partial_{\mathcal{O}_{U, \xi'}^h}(f, g) \quad\text{via}\quad \kappa(\xi) \to \kappa(\xi')\] by Chow Homology, Lemma 0EPG. Thus Lemma 0EQ2 applies to show \[(Z \times_X U \to Z)^*\text{div}_Z(\partial_{\mathcal{O}_{X, \xi}^h}(f, g)) = \sum\nolimits_{Z' \subset Z \times_X U} \text{div}_{Z'}(\partial_{\mathcal{O}_{U, \xi'}^h}(f, g))\] Since flat pullback commutes with pushforward along closed immersions (Lemma 0EPD) we see that it suffices to prove that the coefficient of \(Y'\) in \[\sum\nolimits_{Z' \subset U} (Z' \to U)_*\text{div}_{Z'}(\partial_{\mathcal{O}_{U, \xi'}^h}(f, g))\] is zero.

Let \(A = \mathcal{O}_{U, u}\). Then \(f, g \in Q(A)^*\). Thus we can write \(f = a/b\) and \(g = c/d\) with \(a, b, c, d \in A\) nonzerodivisors. The coefficient of \(Y'\) in the expression above is \[\sum\nolimits_{\mathfrak q \subset A\text{ height }1} \text{ord}_{A/\mathfrak q}(\partial_{A_\mathfrak q}(f, g))\] By bilinearity of \(\partial_A\) it suffices to prove \[\sum\nolimits_{\mathfrak q \subset A\text{ height }1} \text{ord}_{A/\mathfrak q}(\partial_{A_\mathfrak q}(a, c))\] is zero and similarly for the other pairs \((a, d)\), \((b, c)\), and \((b, d)\). This is true by Chow Homology, Lemma 0EAW.

Intersecting with an invertible sheaf and rational equivalence

This section is the analogue of Chow Homology, Section 02TG. Applying the key lemma we obtain the fundamental properties of intersecting with invertible sheaves. In particular, we will see that \(c_1(\mathcal{L}) \cap -\) factors through rational equivalence and that these operations for different invertible sheaves commute.

Lemma

In Situation 0EDT let \(X/B\) be good. Assume \(X\) integral and \(\dim_\delta(X) = n\). Let \(\mathcal{L}\), \(\mathcal{N}\) be invertible on \(X\). Choose a nonzero meromorphic section \(s\) of \(\mathcal{L}\) and a nonzero meromorphic section \(t\) of \(\mathcal{N}\). Set \(\alpha = \text{div}_\mathcal{L}(s)\) and \(\beta = \text{div}_\mathcal{N}(t)\). Then \[c_1(\mathcal{N}) \cap \alpha = c_1(\mathcal{L}) \cap \beta\] in \(\CH_{n - 2}(X)\).

Proof

Immediate from the key Lemma 0EQV and the discussion preceding it.

Lemma

In Situation 0EDT let \(X/B\) be good. Let \(\mathcal{L}\) be invertible on \(X\). The operation \(\alpha \mapsto c_1(\mathcal{L}) \cap \alpha\) factors through rational equivalence to give an operation \[c_1(\mathcal{L}) \cap - : \CH_{k + 1}(X) \to \CH_k(X)\]

Proof

Let \(\alpha \in Z_{k + 1}(X)\), and \(\alpha \sim_{rat} 0\). We have to show that \(c_1(\mathcal{L}) \cap \alpha\) as defined in Definition 0EQJ is zero. By Definition 0EQ6 there exists a locally finite family \(\{W_j\}\) of integral closed subspaces with \(\dim_\delta(W_j) = k + 2\) and rational functions \(f_j \in R(W_j)^*\) such that \[\alpha = \sum (i_j)_*\text{div}_{W_j}(f_j)\] Note that \(p : \coprod W_j \to X\) is a proper morphism, and hence \(\alpha = p_*\alpha'\) where \(\alpha' \in Z_{k + 1}(\coprod W_j)\) is the sum of the principal divisors \(\text{div}_{W_j}(f_j)\). By Lemma 0EQT we have \(c_1(\mathcal{L}) \cap \alpha = p_*(c_1(p^*\mathcal{L}) \cap \alpha')\). Hence it suffices to show that each \(c_1(\mathcal{L}|_{W_j}) \cap \text{div}_{W_j}(f_j)\) is zero. In other words we may assume that \(X\) is integral and \(\alpha = \text{div}_X(f)\) for some \(f \in R(X)^*\).

Assume \(X\) is integral and \(\alpha = \text{div}_X(f)\) for some \(f \in R(X)^*\). We can think of \(f\) as a regular meromorphic section of the invertible sheaf \(\mathcal{N} = \mathcal{O}_X\). Choose a meromorphic section \(s\) of \(\mathcal{L}\) and denote \(\beta = \text{div}_\mathcal{L}(s)\). By Lemma 0EQX we conclude that \[c_1(\mathcal{L}) \cap \alpha = c_1(\mathcal{O}_X) \cap \beta.\] However, by Lemma 0EQK we see that the right hand side is zero in \(\CH_k(X)\) as desired.

In Situation 0EDT let \(X/B\) be good. Let \(\mathcal{L}\) be invertible on \(X\). We will denote \[c_1(\mathcal{L})^s \cap - : \CH_{k + s}(X) \to \CH_k(X)\] the operation \(c_1(\mathcal{L}) \cap -\). This makes sense by Lemma 0EQY. We will denote \(c_1(\mathcal{L}^s \cap -\) the \(s\)-fold iterate of this operation for all \(s \geq 0\).

Lemma

In Situation 0EDT let \(X/B\) be good. Let \(\mathcal{L}\), \(\mathcal{N}\) be invertible on \(X\). For any \(\alpha \in \CH_{k + 2}(X)\) we have \[c_1(\mathcal{L}) \cap c_1(\mathcal{N}) \cap \alpha = c_1(\mathcal{N}) \cap c_1(\mathcal{L}) \cap \alpha\] as elements of \(\CH_k(X)\).

Proof

Write \(\alpha = \sum m_j[Z_j]\) for some locally finite collection of integral closed subspaces \(Z_j \subset X\) with \(\dim_\delta(Z_j) = k + 2\). Consider the proper morphism \(p : \coprod Z_j \to X\). Set \(\alpha' = \sum m_j[Z_j]\) as a \((k + 2)\)-cycle on \(\coprod Z_j\). By several applications of Lemma 0EQT we see that \(c_1(\mathcal{L}) \cap c_1(\mathcal{N}) \cap \alpha = p_*(c_1(p^*\mathcal{L}) \cap c_1(p^*\mathcal{N}) \cap \alpha')\) and \(c_1(\mathcal{N}) \cap c_1(\mathcal{L}) \cap \alpha = p_*(c_1(p^*\mathcal{N}) \cap c_1(p^*\mathcal{L}) \cap \alpha')\). Hence it suffices to prove the formula in case \(X\) is integral and \(\alpha = [X]\). In this case the result follows from Lemma 0EQX and the definitions.

Intersecting with effective Cartier divisors

This section is the analogue of Chow Homology, Section 02T7. Please read the introduction of that section we motivation.

Recall that effective Cartier divisors correspond \(1\)-to-\(1\) to isomorphism classes of pairs \((\mathcal{L}, s)\) where \(\mathcal{L}\) is an invertible sheaf and \(s\) is a global section, see Divisors on Spaces, Lemma 0847. If \(D\) corresponds to \((\mathcal{L}, s)\), then \(\mathcal{L} = \mathcal{O}_X(D)\). Please keep this in mind while reading this section.

Definition

In Situation 0EDT let \(X/B\) be good. Let \((\mathcal{L}, s)\) be a pair consisting of an invertible sheaf and a global section \(s \in \Gamma(X, \mathcal{L})\). Let \(D = Z(s)\) be the vanishing locus of \(s\), and denote \(i : D \to X\) the closed immersion. We define, for every integer \(k\), a (refined) Gysin homomorphism \[i^* : Z_{k + 1}(X) \to \CH_k(D).\] by the following rules:

Given an integral closed subspace \(W \subset X\) with \(\dim_\delta(W) = k + 1\) we define

if \(W \not \subset D\), then \(i^*[W] = [D \cap W]_k\) as a \(k\)-cycle on \(D\), and

if \(W \subset D\), then \(i^*[W] = i'_*(c_1(\mathcal{L}|_W) \cap [W])\), where \(i' : W \to D\) is the induced closed immersion.

For a general \((k + 1)\)-cycle \(\alpha = \sum n_j[W_j]\) we set \[i^*\alpha = \sum n_j i^*[W_j]\]

If \(D\) is an effective Cartier divisor, then we denote \(D \cdot \alpha = i_*i^*\alpha\) the pushforward of the class to a class on \(X\).

In fact, as we will see later, this Gysin homomorphism \(i^*\) can be viewed as an example of a non-flat pullback. Thus we will sometimes informally call the class \(i^*\alpha\) the pullback of the class \(\alpha\).

Remark

Let \(S\), \(B\), \(X\), \(\mathcal{L}\), \(s\), \(i : D \to X\) be as in Definition 0ER1 and assume that \(\mathcal{L}|_D \cong \mathcal{O}_D\). In this case we can define a canonical map \(i^* : Z_{k + 1}(X) \to Z_k(D)\) on cycles, by requiring that \(i^*[W] = 0\) whenever \(W \subset D\). The possibility to do this will be useful later on.

Remark

Let \(f : X' \to X\) be a morphism of good algebraic spaces over \(B\) as in Situation 0EDT. Let \((\mathcal{L}, s, i : D \to X)\) be a triple as in Definition 0ER1. Then we can set \(\mathcal{L}' = f^*\mathcal{L}\), \(s' = f^*s\), and \(D' = X' \times_X D = Z(s')\). This gives a commutative diagram \[\xymatrix{ D' \ar[d]_g \ar[r]_{i'} & X' \ar[d]^f \\ D \ar[r]^i & X }\] and we can ask for various compatibilities between \(i^*\) and \((i')^*\).

Lemma

In Situation 0EDT let \(X/B\) be good. Let \((\mathcal{L}, s, i : D \to X)\) be as in Definition 0ER1. Let \(\alpha\) be a \((k + 1)\)-cycle on \(X\). Then \(i_*i^*\alpha = c_1(\mathcal{L}) \cap \alpha\) in \(\CH_k(X)\). In particular, if \(D\) is an effective Cartier divisor, then \(D \cdot \alpha = c_1(\mathcal{O}_X(D)) \cap \alpha\).

Proof

Write \(\alpha = \sum n_j[W_j]\) where \(i_j : W_j \to X\) are integral closed subspaces with \(\dim_\delta(W_j) = k\). Since \(D\) is the vanishing locus of \(s\) we see that \(D \cap W_j\) is the vanishing locus of the restriction \(s|_{W_j}\). Hence for each \(j\) such that \(W_j \not \subset D\) we have \(c_1(\mathcal{L}) \cap [W_j] = [D \cap W_j]_k\) by Lemma 0EQN. So we have \[c_1(\mathcal{L}) \cap \alpha = \sum\nolimits_{W_j \not \subset D} n_j[D \cap W_j]_k + \sum\nolimits_{W_j \subset D} n_j i_{j, *}(c_1(\mathcal{L})|_{W_j}) \cap [W_j])\] in \(\CH_k(X)\) by Definition 0EQJ. The right hand side matches (termwise) the pushforward of the class \(i^*\alpha\) on \(D\) from Definition 0ER1. Hence we win.

Lemma

In Situation 0EDT. Let \(f : X' \to X\) be a proper morphism of good algebraic spaces over \(B\). Let \((\mathcal{L}, s, i : D \to X)\) be as in Definition 0ER1. Form the diagram \[\xymatrix{ D' \ar[d]_g \ar[r]_{i'} & X' \ar[d]^f \\ D \ar[r]^i & X }\] as in Remark 0ER3. For any \((k + 1)\)-cycle \(\alpha'\) on \(X'\) we have \(i^*f_*\alpha' = g_*(i')^*\alpha'\) in \(\CH_k(D)\) (this makes sense as \(f_*\) is defined on the level of cycles).

Proof

Suppose \(\alpha = [W']\) for some integral closed subspace \(W' \subset X'\). Let \(W \subset X\) be the “image” of \(W'\) as in Lemma 0EEG. In case \(W' \not \subset D'\), then \(W \not \subset D\) and we see that \[[W' \cap D']_k = \text{div}_{\mathcal{L}'|_{W'}}({s'|_{W'}}) \quad\text{and}\quad [W \cap D]_k = \text{div}_{\mathcal{L}|_W}(s|_W)\] and hence \(f_*\) of the first cycle equals the second cycle by Lemma 0EQS. Hence the equality holds as cycles. In case \(W' \subset D'\), then \(W \subset D\) and \(f_*(c_1(\mathcal{L}|_{W'}) \cap [W'])\) is equal to \(c_1(\mathcal{L}|_W) \cap [W]\) in \(\CH_k(W)\) by the second assertion of Lemma 0EQS. By Remark 0EQ8 the result follows for general \(\alpha'\).

Lemma

In Situation 0EDT. Let \(f : X' \to X\) be a flat morphism of relative dimension \(r\) of good algebraic spaces over \(B\). Let \((\mathcal{L}, s, i : D \to X)\) be as in Definition 0ER1. Form the diagram \[\xymatrix{ D' \ar[d]_g \ar[r]_{i'} & X' \ar[d]^f \\ D \ar[r]^i & X }\] as in Remark 0ER3. For any \((k + 1)\)-cycle \(\alpha\) on \(X\) we have \((i')^*f^*\alpha = g^*i^*\alpha'\) in \(\CH_{k + r}(D)\) (this makes sense as \(f^*\) is defined on the level of cycles).

Proof

Suppose \(\alpha = [W]\) for some integral closed subspace \(W \subset X\). Let \(W' = f^{-1}(W) \subset X'\). In case \(W \not \subset D\), then \(W' \not \subset D'\) and we see that \[W' \cap D' = g^{-1}(W \cap D)\] as closed subspaces of \(D'\). Hence the equality holds as cycles, see Lemma 0EPB. In case \(W \subset D\), then \(W' \subset D'\) and \(W' = g^{-1}(W)\) with \([W']_{k + 1 + r} = g^*[W]\) and equality holds in \(\CH_{k + r}(D')\) by Lemma 0EQR. By Remark 0EQ8 the result follows for general \(\alpha'\).

Lemma

In Situation 0EDT let \(X/B\) be good. Let \((\mathcal{L}, s, i : D \to X)\) be as in Definition 0ER1. Let \(Z \subset X\) be a closed subscheme such that \(\dim_\delta(Z) \leq k + 1\) and such that \(D \cap Z\) is an effective Cartier divisor on \(Z\). Then \(i^*([Z]_{k + 1}) = [D \cap Z]_k\).

Proof

The assumption means that \(s|_Z\) is a regular section of \(\mathcal{L}|_Z\). Thus \(D \cap Z = Z(s)\) and we get \[[D \cap Z]_k = \sum n_i [Z(s_i)]_k\] as cycles where \(s_i = s|_{Z_i}\), the \(Z_i\) are the irreducible components of \(\delta\)-dimension \(k + 1\), and \([Z]_{k + 1} = \sum n_i[Z_i]\). See Lemma 0EQL. We have \(D \cap Z_i = Z(s_i)\). Comparing with the definition of the Gysin map we conclude.

Gysin homomorphisms

This section is the analogue of Chow Homology, Section 02TK. In this section we use the key formula to show the Gysin homomorphism factor through rational equivalence.

Lemma

In Situation 0EDT let \(X/B\) be good. Assume \(X\) integral and \(n = \dim_\delta(X)\). Let \(i : D \to X\) be an effective Cartier divisor. Let \(\mathcal{N}\) be an invertible \(\mathcal{O}_X\)-module and let \(t\) be a nonzero meromorphic section of \(\mathcal{N}\). Then \(i^*\text{div}_\mathcal{N}(t) = c_1(\mathcal{N}) \cap [D]_{n - 1}\) in \(\CH_{n - 2}(D)\).

Proof

Write \(\text{div}_\mathcal{N}(t) = \sum \text{ord}_{Z_i, \mathcal{N}}(t)[Z_i]\) for some integral closed subspaces \(Z_i \subset X\) of \(\delta\)-dimension \(n - 1\). We may assume that the family \(\{Z_i\}\) is locally finite, that \(t \in \Gamma(U, \mathcal{N}|_U)\) is a generator where \(U = X \setminus \bigcup Z_i\), and that every irreducible component of \(D\) is one of the \(Z_i\), see Spaces over Fields, Lemmas 0EE5, 0ENN, and 0EPS.

Set \(\mathcal{L} = \mathcal{O}_X(D)\). Denote \(s \in \Gamma(X, \mathcal{O}_X(D)) = \Gamma(X, \mathcal{L})\) the canonical section. We will apply the discussion of Section 0EQU to our current situation. For each \(i\) let \(\xi_i \in |Z_i|\) be its generic point. Let \(B_i = \mathcal{O}_{X, \xi_i}^h\). For each \(i\) we pick generators \(s_i\) of \(\mathcal{L}_{\xi_i}\) and \(t_i\) of \(\mathcal{N}_{\xi_i}\) over \(B_i\) but we insist that we pick \(s_i = s\) if \(Z_i \not \subset D\). Write \(s = f_i s_i\) and \(t = g_i t_i\) with \(f_i, g_i \in B_i\). Then \(\text{ord}_{Z_i, \mathcal{N}}(t) = \text{ord}_{B_i}(g_i)\). On the other hand, we have \(f_i \in B_i\) and \[[D]_{n - 1} = \sum \text{ord}_{B_i}(f_i)[Z_i]\] because of our choices of \(s_i\). We claim that \[i^*\text{div}_\mathcal{N}(t) = \sum \text{ord}_{B_i}(g_i) \text{div}_{\mathcal{L}|_{Z_i}}(s_i|_{Z_i})\] as cycles. More precisely, the right hand side is a cycle representing the left hand side. Namely, this is clear by our formula for \(\text{div}_\mathcal{N}(t)\) and the fact that \(\text{div}_{\mathcal{L}|_{Z_i}}(s_i|_{Z_i}) = [Z(s_i|_{Z_i})]_{n - 2} = [Z_i \cap D]_{n - 2}\) when \(Z_i \not \subset D\) because in that case \(s_i|_{Z_i} = s|_{Z_i}\) is a regular section, see Lemma 0EQF. Similarly, \[c_1(\mathcal{N}) \cap [D]_{n - 1} = \sum \text{ord}_{B_i}(f_i) \text{div}_{\mathcal{N}|_{Z_i}}(t_i|_{Z_i})\] The key formula (Lemma 0EQV) gives the equality \[\sum \left( \text{ord}_{B_i}(f_i) \text{div}_{\mathcal{N}|_{Z_i}}(t_i|_{Z_i}) - \text{ord}_{B_i}(g_i) \text{div}_{\mathcal{L}|_{Z_i}}(s_i|_{Z_i}) \right) = \sum \text{div}_{Z_i}(\partial_{B_i}(f_i, g_i))\] of cycles. If \(Z_i \not \subset D\), then \(f_i = 1\) and hence \(\text{div}_{Z_i}(\partial_{B_i}(f_i, g_i)) = 0\). Thus we get a rational equivalence between our specific cycles representing \(i^*\text{div}_\mathcal{N}(t)\) and \(c_1(\mathcal{N}) \cap [D]_{n - 1}\) on \(D\). This finishes the proof.

Lemma

In Situation 0EDT let \(X/B\) be good. Let \((\mathcal{L}, s, i : D \to X)\) be as in Definition 0ER1. The Gysin homomorphism factors through rational equivalence to give a map \(i^* : \CH_{k + 1}(X) \to \CH_k(D)\).

Proof

Let \(\alpha \in Z_{k + 1}(X)\) and assume that \(\alpha \sim_{rat} 0\). This means there exists a locally finite collection of integral closed subspaces \(W_j \subset X\) of \(\delta\)-dimension \(k + 2\) and \(f_j \in R(W_j)^*\) such that \(\alpha = \sum i_{j, *}\text{div}_{W_j}(f_j)\). Set \(X' = \coprod W_i\) and consider the diagram \[\xymatrix{ D' \ar[d]_q \ar[r]_{i'} & X' \ar[d]^p \\ D \ar[r]^i & X }\] of Remark 0ER3. Since \(X' \to X\) is proper we see that \(i^*p_* = q_*(i')^*\) by Lemma 0ER5. As we know that \(q_*\) factors through rational equivalence (Lemma 0EQC), it suffices to prove the result for \(\alpha' = \sum \text{div}_{W_j}(f_j)\) on \(X'\). Clearly this reduces us to the case where \(X\) is integral and \(\alpha = \text{div}(f)\) for some \(f \in R(X)^*\).

Assume \(X\) is integral and \(\alpha = \text{div}(f)\) for some \(f \in R(X)^*\). If \(X = D\), then we see that \(i^*\alpha\) is equal to \(c_1(\mathcal{L}) \cap \alpha\). This is rationally equivalent to zero by Lemma 0EQY. If \(D \not = X\), then we see that \(i^*\text{div}_X(f)\) is equal to \(c_1(\mathcal{O}_D) \cap [D]_{n - 1}\) in \(\CH_k(D)\) by Lemma 0ER9. Of course capping with \(c_1(\mathcal{O}_D)\) is the zero map.

Lemma

In Situation 0EDT let \(X/B\) be good. Let \((\mathcal{L}, s, i : D \to X)\) be a triple as in Definition 0ER1. Let \(\mathcal{N}\) be an invertible \(\mathcal{O}_X\)-module. Then \(i^*(c_1(\mathcal{N}) \cap \alpha) = c_1(i^*\mathcal{N}) \cap i^*\alpha\) in \(\CH_{k - 2}(D)\) for all \(\alpha \in \CH_k(Z)\).

Proof

With exactly the same proof as in Lemma 0ERA this follows from Lemmas 0EQT, 0EQZ, and 0ER9.

Lemma

In Situation 0EDT let \(X/B\) be good. Let \((\mathcal{L}, s, i : D \to X)\) and \((\mathcal{L}', s', i' : D' \to X)\) be two triples as in Definition 0ER1. Then the diagram \[\xymatrix{ \CH_k(X) \ar[r]_{i^*} \ar[d]_{(i')^*} & \CH_{k - 1}(D) \ar[d] \\ \CH_{k - 1}(D') \ar[r] & \CH_{k - 2}(D \cap D') }\] commutes where each of the maps is a Gysin map.

Proof

Denote \(j : D \cap D' \to D\) and \(j' : D \cap D' \to D'\) the closed immersions corresponding to \((\mathcal{L}|_{D'}, s|_{D'}\) and \((\mathcal{L}'_D, s|_D)\). We have to show that \((j')^*i^*\alpha = j^* (i')^*\alpha\) for all \(\alpha \in \CH_k(X)\). Let \(W \subset X\) be an integral closed subscheme of dimension \(k\). We will prove the equality in case \(\alpha = [W]\). The general case will then follow from the observation in Remark 0EQ8 (and the specific shape of our rational equivalence produced below). We will deduce the equality for \(\alpha = [W]\) from the key formula.

We let \(\sigma\) be a nonzero meromorphic section of \(\mathcal{L}|_W\) which we require to be equal to \(s|_W\) if \(W \not \subset D\). We let \(\sigma'\) be a nonzero meromorphic section of \(\mathcal{L}'|_W\) which we require to be equal to \(s'|_W\) if \(W \not \subset D'\). Write \[\text{div}_{\mathcal{L}|_W}(\sigma) = \sum \text{ord}_{Z_i, \mathcal{L}|_W}(\sigma)[Z_i] = \sum n_i[Z_i]\] and similarly \[\text{div}_{\mathcal{L}'|_W}(\sigma') = \sum \text{ord}_{Z_i, \mathcal{L}'|_W}(\sigma')[Z_i] = \sum n'_i[Z_i]\] as in the discussion in Section 0EQU. Then we see that \(Z_i \subset D\) if \(n_i \not = 0\) and \(Z'_i \subset D'\) if \(n'_i \not = 0\). For each \(i\), let \(\xi_i \in |Z_i|\) be the generic point. As in Section 0EQU we choose for each \(i\) an element \(\sigma_i \in \mathcal{L}_{\xi_i}\), resp. \(\sigma'_i \in \mathcal{L}'_{\xi_i}\) which generates over \(B_i = \mathcal{O}_{W, \xi_i}^h\) and which is equal to the image of \(s\), resp. \(s'\) if \(Z_i \not \subset D\), resp. \(Z_i \not \subset D'\). Write \(\sigma = f_i \sigma_i\) and \(\sigma' = f'_i\sigma'_i\) so that \(n_i = \text{ord}_{B_i}(f_i)\) and \(n'_i = \text{ord}_{B_i}(f'_i)\). From our definitions it follows that \[(j')^*i^*[W] = \sum \text{ord}_{B_i}(f_i) \text{div}_{\mathcal{L}'|_{Z_i}}(\sigma'_i|_{Z_i})\] as cycles and \[j^*(i')^*[W] = \sum \text{ord}_{B_i}(f'_i) \text{div}_{\mathcal{L}|_{Z_i}}(\sigma_i|_{Z_i})\] The key formula (Lemma 0EQV) now gives the equality \[\sum \left( \text{ord}_{B_i}(f_i) \text{div}_{\mathcal{L}'|_{Z_i}}(\sigma'_i|_{Z_i}) - \text{ord}_{B_i}(f'_i) \text{div}_{\mathcal{L}|_{Z_i}}(\sigma_i|_{Z_i}) \right) = \sum \text{div}_{Z_i}(\partial_{B_i}(f_i, f'_i))\] of cycles. Note that \(\text{div}_{Z_i}(\partial_{B_i}(f_i, f'_i)) = 0\) if \(Z_i \not \subset D \cap D'\) because in this case either \(f_i = 1\) or \(f'_i = 1\). Thus we get a rational equivalence between our specific cycles representing \((j')^*i^*[W]\) and \(j^*(i')^*[W]\) on \(D \cap D' \cap W\).

Relative effective Cartier divisors

This section is the analogue of Chow Homology, Section 02TP. Relative effective Cartier divisors are defined in Divisors on Spaces, Section 0EPM. To develop the basic results on Chern classes of vector bundles we only need the case where both the ambient scheme and the effective Cartier divisor are flat over the base.

Lemma

In Situation 0EDT. Let \(X, Y/B\) be good. Let \(p : X \to Y\) be a flat morphism of relative dimension \(r\). Let \(i : D \to X\) be a relative effective Cartier divisor (Divisors on Spaces, Definition 0EPP). Let \(\mathcal{L} = \mathcal{O}_X(D)\). For any \(\alpha \in \CH_{k + 1}(Y)\) we have \[i^*p^*\alpha = (p|_D)^*\alpha\] in \(\CH_{k + r}(D)\) and \[c_1(\mathcal{L}) \cap p^*\alpha = i_* ((p|_D)^*\alpha)\] in \(\CH_{k + r}(X)\).

Proof

Let \(W \subset Y\) be an integral closed subspace of \(\delta\)-dimension \(k + 1\). By Divisors on Spaces, Lemma 0EPN we see that \(D \cap p^{-1}W\) is an effective Cartier divisor on \(p^{-1}W\). By Lemma 0ER7 we get the first equality in \[i^*[p^{-1}W]_{k + r + 1} = [D \cap p^{-1}W]_{k + r} = [(p|_D)^{-1}(W)]_{k + r}.\] and the second because \(D \cap p^{-1}(W) = (p|_D)^{-1}(W)\) as algebraic spaces. Since by definition \(p^*[W] = [p^{-1}W]_{k + r + 1}\) we see that \(i^*p^*[W] = (p|_D)^*[W]\) as cycles. If \(\alpha = \sum m_j[W_j]\) is a general \(k + 1\) cycle, then we get \(i^*\alpha = \sum m_j i^*p^*[W_j] = \sum m_j(p|_D)^*[W_j]\) as cycles. This proves then first equality. To deduce the second from the first apply Lemma 0ER4.

Affine bundles

This section is the analogue of Chow Homology, Section 02TS. For an affine bundle the pullback map is surjective on Chow groups.

Lemma

In Situation 0EDT let \(X, Y/B\) be good. Let \(f : X \to Y\) be a quasi-compact flat morphism over \(B\) of relative dimension \(r\). Assume that for every \(y \in Y\) we have \(X_y \cong \mathbf{A}^r_{\kappa(y)}\). Then \(f^* : \CH_k(Y) \to \CH_{k + r}(X)\) is surjective for all \(k \in \mathbf{Z}\).

Proof

Let \(\alpha \in \CH_{k + r}(X)\). Write \(\alpha = \sum m_j[W_j]\) with \(m_j \not = 0\) and \(W_j\) pairwise distinct integral closed subspaces of \(\delta\)-dimension \(k + r\). Then the family \(\{W_j\}\) is locally finite in \(X\). Let \(Z_j \subset Y\) be the integral closed subspace such that we obtain a dominant morphism \(W_j \to Z_j\) as in Lemma 0EEG. For any quasi-compact open \(V \subset Y\) we see that \(f^{-1}(V) \cap W_j\) is nonempty only for finitely many \(j\). Hence the collection \(Z_j\) of closures of images is a locally finite collection of integral closed subspaces of \(Y\).

Consider the fibre product diagrams \[\xymatrix{ f^{-1}(Z_j) \ar[r] \ar[d]_{f_j} & X \ar[d]^f \\ Z_j \ar[r] & Y }\] Suppose that \([W_j] \in Z_{k + r}(f^{-1}(Z_j))\) is rationally equivalent to \(f_j^*\beta_j\) for some \(k\)-cycle \(\beta_j \in \CH_k(Z_j)\). Then \(\beta = \sum m_j \beta_j\) will be a \(k\)-cycle on \(Y\) and \(f^*\beta = \sum m_j f_j^*\beta_j\) will be rationally equivalent to \(\alpha\) (see Remark 0EQ8). This reduces us to the case \(Y\) integral, and \(\alpha = [W]\) for some integral closed subscheme of \(X\) dominating \(Y\). In particular we may assume that \(d = \dim_\delta(Y) < \infty\).

Hence we can use induction on \(d = \dim_\delta(Y)\). If \(d < k\), then \(\CH_{k + r}(X) = 0\) and the lemma holds; this is the base case of the induction. Consider a nonempty open \(V \subset Y\). Suppose that we can show that \(\alpha|_{f^{-1}(V)} = f^*\beta\) for some \(\beta \in Z_k(V)\). By Lemma 0EP9 we see that \(\beta = \beta'|_V\) for some \(\beta' \in Z_k(Y)\). By the exact sequence \(\CH_k(f^{-1}(Y \setminus V)) \to \CH_k(X) \to \CH_k(f^{-1}(V))\) of Lemma 0EQ7 we see that \(\alpha - f^*\beta'\) comes from a cycle \(\alpha' \in \CH_{k + r}(f^{-1}(Y \setminus V))\). Since \(\dim_\delta(Y \setminus V) < d\) we win by induction on \(d\).

In particular, by replacing \(Y\) by a suitable open we may assume \(Y\) is a scheme with generic point \(\eta\). The isomorphism \(Y_\eta \cong \mathbf{A}^r_\eta\) extends to an isomorphism over a nonempty open \(V \subset Y\), see Limits of Spaces, Lemma 07SK. This reduces us to the case of schemes which is Chow Homology, Lemma 02TT.

Lemma

In Situation 0EDT let \(X/B\) be good. Let \(\mathcal{L}\) be an invertible \(\mathcal{O}_X\)-module. Let \[p : L = \underline{\Spec}(\text{Sym}^*(\mathcal{L})) \longrightarrow X\] be the associated vector bundle over \(X\). Then \(p^* : \CH_k(X) \to \CH_{k + 1}(L)\) is an isomorphism for all \(k\).

Proof

For surjectivity see Lemma 0ERG. Let \(o : X \to L\) be the zero section of \(L \to X\), i.e., the morphism corresponding to the surjection \(\text{Sym}^*(\mathcal{L}) \to \mathcal{O}_X\) which maps \(\mathcal{L}^{\otimes n}\) to zero for all \(n > 0\). Then \(p \circ o = \text{id}_X\) and \(o(X)\) is an effective Cartier divisor on \(L\). Hence by Lemma 0ERE we see that \(o^* \circ p^* = \text{id}\) and we conclude that \(p^*\) is injective too.

Bivariant intersection theory

This section is the analogue of Chow Homology, Section 0B75. In order to intelligently talk about higher Chern classes of vector bundles we introduce the following notion, following [FM]. It follows from [F, Theorem 17.1] that our definition agrees with that of [F] modulo the caveat that we are working in different settings.

Definition

In Situation 0EDT let \(f : X \to Y\) be a morphism of good algebraic spaces over \(B\). Let \(p \in \mathbf{Z}\). A bivariant class \(c\) of degree \(p\) for \(f\) is given by a rule which assigns to every morphism \(Y' \to Y\) of good algebraic spaces over \(B\) and every \(k\) a map \[c \cap - : \CH_k(Y') \longrightarrow \CH_{k - p}(X')\] where \(X' = Y' \times_Y X\), satisfying the following conditions

if \(Y'' \to Y'\) is a proper morphism, then \(c \cap (Y'' \to Y')_*\alpha'' = (X'' \to X')_*(c \cap \alpha'')\) for all \(\alpha''\) on \(Y''\),

if \(Y'' \to Y'\) a morphism of good algebraic spaces over \(B\) which is flat of relative dimension \(r\), then \(c \cap (Y'' \to Y')^*\alpha' = (X'' \to X')^*(c \cap \alpha')\) for all \(\alpha'\) on \(Y'\),

if \((\mathcal{L}', s', i' : D' \to Y')\) is as in Definition 0ER1 with pullback \((\mathcal{N}', t', j' : E' \to X')\) to \(X'\), then we have \(c \cap (i')^*\alpha' = (j')^*(c \cap \alpha')\) for all \(\alpha'\) on \(Y'\).

The collection of all bivariant classes of degree \(p\) for \(f\) is denoted \(A^p(X \to Y)\).

In Situation 0EDT let \(X \to Y\) and \(Y \to Z\) be morphisms of good algebraic spaces over \(B\). Let \(p \in \mathbf{Z}\). It is clear that \(A^p(X \to Y)\) is an abelian group. Moreover, it is clear that we have a bilinear composition \[A^p(X \to Y) \times A^q(Y \to Z) \to A^{p + q}(X \to Z)\] which is associative. We will be most interested in \(A^p(X) = A^p(X \to X)\), which will always mean the bivariant cohomology classes for \(\text{id}_X\). Namely, that is where Chern classes will live.

Definition

In Situation 0EDT let \(X/B\) be good. The Chow cohomology of \(X\) is the graded \(\mathbf{Z}\)-algebra \(A^*(X)\) whose degree \(p\) component is \(A^p(X \to X)\).

Warning: It is not clear that the \(\mathbf{Z}\)-algebra structure on \(A^*(X)\) is commutative, but we will see that Chern classes live in its center.

Remark

In Situation 0EDT let \(f : X \to Y\) be a morphism of good algebraic spaces over \(B\). Then there is a canonical \(\mathbf{Z}\)-algebra map \(A^*(Y) \to A^*(X)\). Namely, given \(c \in A^p(Y)\) and \(X' \to X\), then we can let \(f^*c\) be defined by the map \(c \cap - : \CH_k(X') \to \CH_{k - p}(X')\) which is given by thinking of \(X'\) as an algebraic space over \(Y\).

Lemma

In Situation 0EDT let \(X/B\) be good. Let \(\mathcal{L}\) be an invertible \(\mathcal{O}_X\)-module. Then the rule that to \(f : X' \to X\) assigns \(c_1(f^*\mathcal{L}) \cap - : \CH_k(X') \to \CH_{k - 1}(X')\) is a bivariant class of degree \(1\).

Proof

This follows from Lemmas 0EQY, 0EQT, 0EQR, and 0ERB.

Lemma

In Situation 0EDT let \(f : X \to Y\) be a morphism of good algebraic spaces over \(B\) which is flat of relative dimension \(r\). Then the rule that to \(Y' \to Y\) assigns \((f')^* : \CH_k(Y') \to \CH_{k + r}(X')\) where \(X' = X \times_Y Y'\) is a bivariant class of degree \(-r\).

Proof

This follows from Lemmas 0EQB, 0EPA, 0EPD, and 0ER6.

Lemma

In Situation 0EDT let \(X/B\) be good. Let \((\mathcal{L}, s, i : D \to X)\) be a triple as in Definition 0ER1. Then the rule that to \(f : X' \to X\) assigns \((i')^* : \CH_k(X') \to \CH_{k - 1}(D')\) where \(D' = D \times_X X'\) is a bivariant class of degree \(1\).

Proof

This follows from Lemmas 0ERA, 0ER5, 0ER6, and 0ERC.

Lemma

In Situation 0EDT let \(f : X \to Y\) and \(g : Y \to Z\) be morphisms of good algebraic spaces over \(B\). Let \(c \in A^p(X \to Z)\) and assume \(f\) is proper. Then the rule that to \(X' \to X\) assigns \(\alpha \longmapsto f_*(c \cap \alpha)\) is a bivariant class of degree \(p\).

Proof

This follows from Lemmas 0EP2, 0EPD, and 0ER5.

Here we see that \(c_1(\mathcal{L})\) is in the center of \(A^*(X)\).

Lemma

In Situation 0EDT let \(X/B\) be good. Let \(\mathcal{L}\) be an invertible \(\mathcal{O}_X\)-module. Then \(c_1(\mathcal{L}) \in A^1(X)\) commutes with every element \(c \in A^p(X)\).

Proof

Let \(p : L \to X\) be as in Lemma 0ERH and let \(o : X \to L\) be the zero section. Observe that \(p^*\mathcal{L}^{\otimes -1}\) has a canonical section whose vanishing locus is exactly the effective Cartier divisor \(o(X)\). Let \(\alpha \in \CH_k(X)\). Then we see that \[p^*(c_1(\mathcal{L}^{\otimes -1}) \cap \alpha) = c_1(p^*\mathcal{L}^{\otimes -1}) \cap p^*\alpha = o_* o^* p^*\alpha\] by Lemmas 0EQR and 0ERE. Since \(c\) is a bivariant class we have \[\begin{align*} p^*(c \cap c_1(\mathcal{L}^{\otimes -1}) \cap \alpha) & = c \cap p^*(c_1(\mathcal{L}^{\otimes -1}) \cap \alpha) \\ & = c \cap o_* o^* p^*\alpha \\ & = o_* o^* p^*(c \cap \alpha) \\ & = p^*(c_1(\mathcal{L}^{\otimes -1}) \cap c \cap \alpha) \end{align*}\] (last equality by the above applied to \(c \cap \alpha\)). Since \(p^*\) is injective by a lemma cited above we get that \(c_1(\mathcal{L}^{\otimes -1})\) is in the center of \(A^*(X)\). This proves the lemma.

Here a criterion for when a bivariant class is zero.

Lemma

In Situation 0EDT let \(X/B\) be good. Let \(c \in A^p(X)\). Then \(c\) is zero if and only if \(c \cap [Y] = 0\) in \(\CH_*(Y)\) for every integral algebraic space \(Y\) locally of finite type over \(X\).

Proof

The if direction is clear. For the converse, assume that \(c \cap [Y] = 0\) in \(\CH_*(Y)\) for every integral algebraic space \(Y\) locally of finite type over \(X\). Let \(X' \to X\) be locally of finite type. Let \(\alpha \in \CH_k(X')\). Write \(\alpha = \sum n_i [Y_i]\) with \(Y_i \subset X'\) a locally finite collection of integral closed subschemes of \(\delta\)-dimension \(k\). Then we see that \(\alpha\) is pushforward of the cycle \(\alpha' = \sum n_i[Y_i]\) on \(X'' = \coprod Y_i\) under the proper morphism \(X'' \to X'\). By the properties of bivariant classes it suffices to prove that \(c \cap \alpha' = 0\) in \(\CH_{k - p}(X'')\). We have \(\CH_{k - p}(X'') = \prod \CH_{k - p}(Y_i)\) as follows immediately from the definitions. The projection maps \(\CH_{k - p}(X'') \to \CH_{k - p}(Y_i)\) are given by flat pullback. Since capping with \(c\) commutes with flat pullback, we see that it suffices to show that \(c \cap [Y_i]\) is zero in \(\CH_{k - p}(Y_i)\) which is true by assumption.

Projective space bundle formula

In Situation 0EDT let \(X/B\) be good. Consider a finite locally free \(\mathcal{O}_X\)-module \(\mathcal{E}\) of rank \(r\). Our convention is that the projective bundle associated to \(\mathcal{E}\) is the morphism \[\xymatrix{ \mathbf{P}(\mathcal{E}) = \underline{\text{Proj}}_X(\text{Sym}^*(\mathcal{E})) \ar[r]^-\pi & X }\] over \(X\) with \(\mathcal{O}_{\mathbf{P}(\mathcal{E})}(1)\) normalized so that \(\pi_*(\mathcal{O}_{\mathbf{P}(\mathcal{E})}(1)) = \mathcal{E}\). In particular there is a surjection \(\pi^*\mathcal{E} \to \mathcal{O}_{\mathbf{P}(\mathcal{E})}(1)\). We will say informally “let \((\pi : P \to X, \mathcal{O}_P(1))\) be the projective bundle associated to \(\mathcal{E}\)” to denote the situation where \(P = \mathbf{P}(\mathcal{E})\) and \(\mathcal{O}_P(1) = \mathcal{O}_{\mathbf{P}(\mathcal{E})}(1)\).

Lemma

In Situation 0EDT let \(X/B\) be good. Let \(\mathcal{E}\) be a finite locally free \(\mathcal{O}_X\)-module \(\mathcal{E}\) of rank \(r\). Let \((\pi : P \to X, \mathcal{O}_P(1))\) be the projective bundle associated to \(\mathcal{E}\). For any \(\alpha \in \CH_k(X)\) the element \[\pi_*\left( c_1(\mathcal{O}_P(1))^s \cap \pi^*\alpha \right) \in \CH_{k + r - 1 - s}(X)\] is \(0\) if \(s < r - 1\) and is equal to \(\alpha\) when \(s = r - 1\).

Proof

Let \(Z \subset X\) be an integral closed subspace of \(\delta\)-dimension \(k\). We will prove the lemma for \(\alpha = [Z]\). We omit the argument deducing the general case from this special case; hint: argue as in Remark 0EQ8.

Let \(P_Z = P \times_X Z\) be the base change; of course \(\pi_Z : P_Z \to Z\) is the projective bundle associated to \(\mathcal{E}|_Z\) and \(\mathcal{O}_P(1)\) pulls back to the corresponding invertible module on \(P_Z\). Since \(c_1(\mathcal{O}_P(1) \cap -\), and \(\pi^*\) are bivariant classes by Lemmas 0ERM and 0ERN we see that \[\pi_*\left( c_1(\mathcal{O}_P(1))^s \cap \pi^*[Z] \right) = (Z \to X)_*\pi_{Z, *}\left( c_1(\mathcal{O}_{P_Z}(1))^s \cap \pi_Z^*[Z] \right)\] Hence it suffices to prove the lemma in case \(X\) is integral and \(\alpha = [X]\).

Assume \(X\) is integral, \(\dim_\delta(X) = k\), and \(\alpha = [X]\). Note that \(\pi^*[X] = [P]\) as \(P\) is integral of \(\delta\)-dimension \(r - 1\). If \(s < r - 1\), then by construction \(c_1(\mathcal{O}_P(1))^s \cap [P]\) a \((k + r - 1 - s)\)-cycle. Hence the pushforward of this cycle is zero for dimension reasons.

Let \(s = r - 1\). By the argument given above we see that \(\pi_*(c_1(\mathcal{O}_P(1))^s \cap [P]) = n [X]\) for some \(n \in \mathbf{Z}\). We want to show that \(n = 1\). For the same dimension reasons as above it suffices to prove this result after replacing \(X\) by a dense open. Thus we may assume \(X\) is a scheme and the result follows from Chow Homology, Lemma 02TW.

Lemma

Let \((S, \delta)\) be as in Situation 0EDT. Let \(X\) be locally of finite type over \(S\). Let \(\mathcal{E}\) be a finite locally free \(\mathcal{O}_X\)-module \(\mathcal{E}\) of rank \(r\). Let \((\pi : P \to X, \mathcal{O}_P(1))\) be the projective bundle associated to \(\mathcal{E}\). The map \[\bigoplus\nolimits_{i = 0}^{r - 1} \CH_{k + i}(X) \longrightarrow \CH_{k + r - 1}(P),\] \[(\alpha_0, \ldots, \alpha_{r-1}) \longmapsto \pi^*\alpha_0 + c_1(\mathcal{O}_P(1)) \cap \pi^*\alpha_1 + \ldots + c_1(\mathcal{O}_P(1))^{r - 1} \cap \pi^*\alpha_{r-1}\] is an isomorphism.

Proof

Fix \(k \in \mathbf{Z}\). We first show the map is injective. Suppose that \((\alpha_0, \ldots, \alpha_{r - 1})\) is an element of the left hand side that maps to zero. By Lemma 0ERU we see that \[0 = \pi_*(\pi^*\alpha_0 + c_1(\mathcal{O}_P(1)) \cap \pi^*\alpha_1 + \ldots + c_1(\mathcal{O}_P(1))^{r - 1} \cap \pi^*\alpha_{r-1}) = \alpha_{r - 1}\] Next, we see that \[0 = \pi_*(c_1(\mathcal{O}_P(1)) \cap (\pi^*\alpha_0 + c_1(\mathcal{O}_P(1)) \cap \pi^*\alpha_1 + \ldots + c_1(\mathcal{O}_P(1))^{r - 2} \cap \pi^*\alpha_{r - 2})) = \alpha_{r - 2}\] and so on. Hence the map is injective.

To prove the map is surjective, we will argue exactly as in the proof of Lemma 0ERG to reduce to the case of schemes. We urge the reader to skip the proof.

Let \(\beta \in \CH_{k + r - 1}(P)\). Write \(\beta = \sum m_j[W_j]\) with \(m_j \not = 0\) and \(W_j\) pairwise distinct integral closed subspaces of \(\delta\)-dimension \(k + r\). Then the family \(\{W_j\}\) is locally finite in \(P\). Let \(Z_j \subset X\) be the “image” of \(W_j\) as in Lemma 0EEG. For any quasi-compact open \(U \subset X\) we see that \(\pi^{-1}(U) \cap W_j\) is nonempty only for finitely many \(j\). Hence the collection \(Z_j\) of images is a locally finite collection of integral closed subspaces of \(X\).

Consider the fibre product diagrams \[\xymatrix{ P_j \ar[r] \ar[d]_{\pi_j} & P \ar[d]^\pi \\ Z_j \ar[r] & X }\] Suppose that \([W_j] \in Z_{k + r - 1}(P_j)\) is rationally equivalent to \[\pi_j^*\alpha_{j, 0} + c_1(\mathcal{O}(1)) \cap \pi_j^*\alpha_{j, 1} + \ldots + c_1(\mathcal{O}(1))^{r - 1} \cap \pi_j^*\alpha_{j, r - 1}\] for some \((k + i)\)-cycle \(\alpha_{j, i} \in \CH_{k + i}(Z_j)\). Then \(\alpha_i = \sum m_j \beta_{j, i}\) will be a \((k + i)\)-cycle on \(X\) and \[\pi^*\alpha_0 + c_1(\mathcal{O}(1)) \cap \pi^*\alpha_1 + \ldots + c_1(\mathcal{O}(1))^{r - 1} \cap \pi^*\alpha_{r - 1}\] will be rationally equivalent to \(\beta\) (see Remark 0EQ8). This reduces us to the case \(X\) integral, and \(\alpha = [W]\) for some integral closed subscheme of \(P\) dominating \(X\). In particular we may assume that \(d = \dim_\delta(X) < \infty\).

Hence we can use induction on \(d = \dim_\delta(X)\). If \(d < k\), then \(\CH_{k + r - 1}(X) = 0\) and the lemma holds; this is the base case of the induction. Consider a nonempty open \(U \subset X\). Suppose that we can show that \[\beta|_{\pi^{-1}(U)} = \pi^*\alpha_0 + c_1(\mathcal{O}(1)) \cap \pi^*\alpha_1 + \ldots + c_1(\mathcal{O}(1))^{r - 1} \cap \pi^*\alpha_{r - 1}\] for some \(\alpha_i \in Z_{k + i}(U)\). By Lemma 0EP9 we see that \(\alpha_i = \alpha'_i|_U\) for some \(\alpha'_i \in Z_{k + i}(X)\). By the exact sequences \(\CH_{k + i}(\pi^{-1}(X \setminus U)) \to \CH_{k + i}(P) \to \CH_{k + i}(\pi^{-1}(U))\) of Lemma 0EQ7 we see that \[\beta - \left(\pi^*\alpha'_0 + c_1(\mathcal{O}(1)) \cap \pi^*\alpha'_1 + \ldots + c_1(\mathcal{O}(1))^{r - 1} \cap \pi^*\alpha'_{r - 1}\right)\] comes from a cycle \(\beta' \in \CH_{k + r}(\pi^{-1}(X \setminus U))\). Since \(\dim_\delta(X \setminus U) < d\) we win by induction on \(d\).

In particular, by replacing \(X\) by a suitable open we may assume \(X\) is a scheme and we have reduced our problem to Chow Homology, Lemma 02TX.

Lemma

In Situation 0EDT let \(X/B\) be good. Let \(\mathcal{E}\) be a finite locally free sheaf of rank \(r\) on \(X\). Let \[p : E = \underline{\Spec}(\text{Sym}^*(\mathcal{E})) \longrightarrow X\] be the associated vector bundle over \(X\). Then \(p^* : \CH_k(X) \to \CH_{k + r}(E)\) is an isomorphism for all \(k\).

Proof

(For the case of linebundles, see Lemma 0ERH.) For surjectivity see Lemma 0ERG. Let \((\pi : P \to X, \mathcal{O}_P(1))\) be the projective space bundle associated to the finite locally free sheaf \(\mathcal{E} \oplus \mathcal{O}_X\). Let \(s \in \Gamma(P, \mathcal{O}_P(1))\) correspond to the global section \((0, 1) \in \Gamma(X, \mathcal{E} \oplus \mathcal{O}_X)\). Let \(D = Z(s) \subset P\). Note that \((\pi|_D : D \to X , \mathcal{O}_P(1)|_D)\) is the projective space bundle associated to \(\mathcal{E}\). We denote \(\pi_D = \pi|_D\) and \(\mathcal{O}_D(1) = \mathcal{O}_P(1)|_D\). Moreover, \(D\) is an effective Cartier divisor on \(P\). Hence \(\mathcal{O}_P(D) = \mathcal{O}_P(1)\) (see Divisors on Spaces, Lemma 0847). Also there is an isomorphism \(E \cong P \setminus D\). Denote \(j : E \to P\) the corresponding open immersion. For injectivity we use that the kernel of \[j^* : \CH_{k + r}(P) \longrightarrow \CH_{k + r}(E)\] are the cycles supported in the effective Cartier divisor \(D\), see Lemma 0EQ7. So if \(p^*\alpha = 0\), then \(\pi^*\alpha = i_*\beta\) for some \(\beta \in \CH_{k + r}(D)\). By Lemma 0ERV we may write \[\beta = \pi_D^*\beta_0 + \ldots + c_1(\mathcal{O}_D(1))^{r - 1} \cap \pi_D^* \beta_{r - 1}.\] for some \(\beta_i \in \CH_{k + i}(X)\). By Lemmas 0ERE and 0EQT this implies \[\pi^*\alpha = i_*\beta = c_1(\mathcal{O}_P(1)) \cap \pi^*\beta_0 + \ldots + c_1(\mathcal{O}_D(1))^r \cap \pi^*\beta_{r - 1}.\] Since the rank of \(\mathcal{E} \oplus \mathcal{O}_X\) is \(r + 1\) this contradicts Lemma 0EQT unless all \(\alpha\) and all \(\beta_i\) are zero.

The Chern classes of a vector bundle

This section is the analogue of Chow Homology, Sections 02TZ and 02U4. However, contrary to what is done there, we directly define the Chern classes of a vector bundle as bivariant classes. This saves a considerable amount of work.

Lemma

In Situation 0EDT let \(X/B\) be good. Let \(\mathcal{E}\) be a finite locally free sheaf of rank \(r\) on \(X\). Let \((\pi : P \to X, \mathcal{O}_P(1))\) be the projective space bundle associated to \(\mathcal{E}\). For every morphism \(X' \to X\) of good algebraic spaces over \(B\) there are unique maps \[c_i(\mathcal{E}) \cap - : \CH_k(X') \longrightarrow \CH_{k - i}(X'),\quad i = 0, \ldots, r\] such that for \(\alpha \in \CH_k(X')\) we have \(c_0(\mathcal{E}) \cap \alpha = \alpha\) and \[\sum\nolimits_{i = 0, \ldots, r} (-1)^i c_1(\mathcal{O}_{P'}(1))^i \cap (\pi')^*\left(c_{r - i}(\mathcal{E}) \cap \alpha\right) = 0\] where \(\pi' : P' \to X'\) is the base change of \(\pi\). Moreover, these maps define a bivariant class \(c_i(\mathcal{E})\) of degree \(i\) on \(X\).

Proof

Uniqueness and existence of the maps \(c_i(\mathcal{E}) \cap -\) follows immediately from Lemma 0ERV and the given description of \(c_0(\mathcal{E})\). For every \(i \in \mathbf{Z}\) the rule which to every morphism \(X' \to X\) of good algebraic spaces over \(B\) assigns the map \[t_i(\mathcal{E}) \cap - : \CH_k(X') \longrightarrow \CH_{k - i}(X'),\quad \alpha \longmapsto \pi'_*(c_1(\mathcal{O}_{P'}(1))^{r - 1 + i} \cap (\pi')^*\alpha)\] is a bivariant class² by Lemmas 0ERM, 0ERN, and 0ERQ. By Lemma 0ERU we have \(t_i(\mathcal{E}) = 0\) for \(i < 0\) and \(t_0(\mathcal{E}) = 1\). Applying pushforward to the equation in the statement of the lemma we find from Lemma 0ERU that \[(-1)^r t_1(\mathcal{E}) + (-1)^{r - 1}c_1(\mathcal{E}) = 0\] In particular we find that \(c_1(\mathcal{E})\) is a bivariant class. If we multiply the equation in the statement of the lemma by \(c_1(\mathcal{O}_{P'}(1))\) and push the result forward to \(X'\) we find \[(-1)^r t_2(\mathcal{E}) + (-1)^{r - 1} t_1(\mathcal{E}) \cap c_1(\mathcal{E}) + (-1)^{r - 2} c_2(\mathcal{E}) = 0\] As before we conclude that \(c_2(\mathcal{E})\) is a bivariant class. And so on.

Definition

In Situation 0EDT let \(X/B\) be good. Let \(\mathcal{E}\) be a finite locally free sheaf of rank \(r\) on \(X\). For \(i = 0, \ldots, r\) the \(i\)th Chern class of \(\mathcal{E}\) is the bivariant class \(c_i(\mathcal{E}) \in A^i(X)\) of degree \(i\) constructed in Lemma 0ERY. The total Chern class of \(\mathcal{E}\) is the formal sum \[c(\mathcal{E}) = c_0(\mathcal{E}) + c_1(\mathcal{E}) + \ldots + c_r(\mathcal{E})\] which is viewed as a nonhomogeneous bivariant class on \(X\).

For convenience we often set \(c_i(\mathcal{E}) = 0\) for \(i > r\) and \(i < 0\). By definition we have \(c_0(\mathcal{E}) = 1 \in A^0(X)\). Here is a sanity check.

Lemma

In Situation 0EDT let \(X/B\) be good. Let \(\mathcal{L}\) be an invertible \(\mathcal{O}_X\)-module. The first Chern class of \(\mathcal{L}\) on \(X\) of Definition 0ERZ is equal to the bivariant class of Lemma 0ERM.

Proof

Namely, in this case \(P = \mathbf{P}(\mathcal{L}) = X\) with \(\mathcal{O}_P(1) = \mathcal{L}\) by our normalization of the projective bundle, see Section 0ERT. Hence the equation in Lemma 0ERY reads \[(-1)^0 c_1(\mathcal{L})^0 \cap c^{new}_1(\mathcal{L}) \cap \alpha + (-1)^1 c_1(\mathcal{L})^1 \cap c^{new}_0(\mathcal{L}) \cap \alpha = 0\] where \(c_i^{new}(\mathcal{L})\) is as in Definition 0ERZ. Since \(c_0^{new}(\mathcal{L}) = 1\) and \(c_1(\mathcal{L})^0 = 1\) we conclude.

Next we see that Chern classes are in the center of the bivariant Chow cohomology ring \(A^*(X)\).

Lemma

In Situation 0EDT let \(X/B\) be good. Let \(\mathcal{E}\) be a locally free \(\mathcal{O}_X\)-module of rank \(r\). Then \(c_j(\mathcal{L}) \in A^j(X)\) commutes with every element \(c \in A^p(X)\). In particular, if \(\mathcal{F}\) is a second locally free \(\mathcal{O}_X\)-module on \(X\) of rank \(s\), then \[c_i(\mathcal{E}) \cap c_j(\mathcal{F}) \cap \alpha = c_j(\mathcal{F}) \cap c_i(\mathcal{E}) \cap \alpha\] as elements of \(\CH_{k - i - j}(X)\) for all \(\alpha \in \CH_k(X)\).

Proof

Let \(X' \to X\) be a morphism of good algebraic spaces over \(B\). Let \(\alpha \in \CH_k(X')\). Write \(\alpha_j = c_j(\mathcal{E}) \cap \alpha\), so \(\alpha_0 = \alpha\). By Lemma 0ERY we have \[\sum\nolimits_{i = 0}^r (-1)^i c_1(\mathcal{O}_{P'}(1))^i \cap (\pi')^*(\alpha_{r - i}) = 0\] in the chow group of the projective bundle \((\pi' : P' \to X', \mathcal{O}_{P'}(1))\) associated to \((X' \to X)^*\mathcal{E}\). Applying \(c \cap -\) and using Lemma 0ERR and the properties of bivariant classes we obtain \[\sum\nolimits_{i = 0}^r (-1)^i c_1(\mathcal{O}_{P'}(1))^i \cap \pi^*(c \cap \alpha_{r - i}) = 0\] in the Chow group of \(P'\). Hence we see that \(c \cap \alpha_j\) is equal to \(c_j(\mathcal{E}) \cap (c \cap \alpha)\) by the uniqueness in Lemma 0ERY. This proves the lemma.

Remark

In Situation 0EDT let \(X/B\) be good. Let \(\mathcal{E}\) be a finite locally free \(\mathcal{O}_X\)-module. If the rank of \(\mathcal{E}\) is not constant then we can still define the Chern classes of \(\mathcal{E}\). Namely, in this case we can write \[X = X_0 \amalg X_1 \amalg X_2 \amalg \ldots\] where \(X_r \subset X\) is the open and closed subspace where the rank of \(\mathcal{E}\) is \(r\). If \(X' \to X\) is a morphism of good algebraic spaces over \(B\), then we obtain by pullback a corresponding decomposition of \(X'\) and we find that \[\CH_*(X') = \prod\nolimits_{r \geq 0} \CH_*(X'_r)\] by our definitions. Then we simply define \(c_i(\mathcal{E})\) to be the bivariant class which preserves these direct product decompositions and acts by the already defined operations \(c_i(\mathcal{E}|_{X_r}) \cap -\) on the factors. Observe that in this setting it may happen that \(c_i(\mathcal{E})\) is nonzero for infinitely many \(i\).

Polynomial relations among Chern classes

In Situation 0EDT let \(X/B\) be good. Let \(\mathcal{E}_i\) be a finite collection of finite locally free \(\mathcal{O}_X\)-modules. By Lemma 0ES1 we see that the Chern classes \[c_j(\mathcal{E}_i) \in A^*(X)\] generate a commutative (and even central) \(\mathbf{Z}\)-subalgebra of the Chow cohomology \(A^*(X)\). Thus we can say what it means for a polynomial in these Chern classes to be zero, or for two polynomials to be the same. As an example, saying that \(c_1(\mathcal{E}_1)^5 + c_2(\mathcal{E}_2)c_3(\mathcal{E}_3) = 0\) means that the operations \[\CH_k(Y) \longrightarrow \CH_{k - 5}(Y), \quad \alpha \longmapsto c_1(\mathcal{E}_1)^5 \cap \alpha + c_2(\mathcal{E}_2) \cap c_3(\mathcal{E}_3) \cap \alpha\] are zero for all morphisms \(f : Y \to X\) of good algebraic spaces over \(B\). By Lemma 0ERS this is equivalent to the requirement that given any morphism \(f : Y \to X\) where \(Y\) is an integral algebraic space locally of finite type over \(X\) the cycle \[c_1(\mathcal{E}_1)^5 \cap [Y] + c_2(\mathcal{E}_2) \cap c_3(\mathcal{E}_3) \cap [Y]\] is zero in \(\CH_{\dim(Y) - 5}(Y)\).

A specific example is the relation \[c_1(\mathcal{L} \otimes_{\mathcal{O}_X} \mathcal{N}) = c_1(\mathcal{L}) + c_1(\mathcal{N})\] proved in Lemma 0EQK. More generally, here is what happens when we tensor an arbitrary locally free sheaf by an invertible sheaf.

Lemma

In Situation 0EDT let \(X/B\) be good. Let \(\mathcal{E}\) be a finite locally free sheaf of rank \(r\) on \(X\). Let \(\mathcal{L}\) be an invertible sheaf on \(X\). Then we have [0ES5]\[\begin{equation} c_i({\mathcal E} \otimes {\mathcal L}) = \sum\nolimits_{j = 0}^i \binom{r - i + j}{j} c_{i - j}({\mathcal E}) c_1({\mathcal L})^j \end{equation}\] in \(A^*(X)\).

Proof

The proof is identical to the proof of Chow Homology, Lemma 02UD replacing the lemmas used there by Lemmas 0ERS and 0ERY.

Additivity of Chern classes

This section is the analogue of Chow Homology, Section 02UF.

Lemma

In Situation 0EDT let \(X/B\) be good. Let \(\mathcal{E}\), \(\mathcal{F}\) be finite locally free sheaves on \(X\) of ranks \(r\), \(r - 1\) which fit into a short exact sequence \[0 \to \mathcal{O}_X \to \mathcal{E} \to \mathcal{F} \to 0\] Then we have \[c_r(\mathcal{E}) = 0, \quad c_j(\mathcal{E}) = c_j(\mathcal{F}), \quad j = 0, \ldots, r - 1\] in \(A^*(X)\).

Proof

The proof is identical to the proof of Chow Homology, Lemma 02UG replacing the lemmas used there by Lemmas 0ERS, 0ERE, 0EQT, and 0ERY.

Lemma

In Situation 0EDT let \(X/B\) be good. Let \(\mathcal{E}\), \(\mathcal{F}\) be finite locally free sheaves on \(X\) of ranks \(r\), \(r - 1\) which fit into a short exact sequence \[0 \to \mathcal{L} \to \mathcal{E} \to \mathcal{F} \to 0\] where \(\mathcal{L}\) is an invertible sheaf. Then \[c(\mathcal{E}) = c(\mathcal{L}) c(\mathcal{F})\] in \(A^*(X)\).

Proof

The proof is identical to the proof of Chow Homology, Lemma 02UH replacing the lemmas used there by Lemmas 0ES7 and 0ES4.

Lemma

In Situation 0EDT let \(X/B\) be good. Suppose that \(\mathcal{E}\) sits in an exact sequence \[0 \to \mathcal{E}_1 \to \mathcal{E} \to \mathcal{E}_2 \to 0\] of finite locally free sheaves \(\mathcal{E}_i\) of rank \(r_i\). The total Chern classes satisfy \[c(\mathcal{E}) = c(\mathcal{E}_1) c(\mathcal{E}_2)\] in \(A^*(X)\).

Proof

The proof is identical to the proof of Chow Homology, Lemma 02UI replacing the lemmas used there by Lemmas 0ERS, 0ES8, and 0ERY.

Lemma

In Situation 0EDT let \(X/B\) be good. Let \({\mathcal L}_i\), \(i = 1, \ldots, r\) be invertible \(\mathcal{O}_X\)-modules. Let \(\mathcal{E}\) be a locally free rank \(\mathcal{O}_X\)-module endowed with a filtration \[0 = \mathcal{E}_0 \subset \mathcal{E}_1 \subset \mathcal{E}_2 \subset \ldots \subset \mathcal{E}_r = \mathcal{E}\] such that \(\mathcal{E}_i/\mathcal{E}_{i - 1} \cong \mathcal{L}_i\). Set \(c_1({\mathcal L}_i) = x_i\). Then \[c(\mathcal{E}) = \prod\nolimits_{i = 1}^r (1 + x_i)\] in \(A^*(X)\).

Proof

Apply Lemma 0ES8 and induction.

The splitting principle

This section is the analogue of Chow Homology, Section 02UF.

Lemma

In Situation 0EDT let \(X/B\) be good. Let \(\mathcal{E}_i\) be a finite collection of locally free \(\mathcal{O}_X\)-modules of rank \(r_i\). There exists a projective flat morphism \(\pi : P \to X\) of relative dimension \(d\) such that

for any morphism \(f : Y \to X\) of good algebraic spaces over \(B\) the map \(\pi_Y^* : \CH_*(Y) \to \CH_{* + d}(Y \times_X P)\) is injective, and

each \(\pi^*\mathcal{E}_i\) has a filtration whose successive quotients \(\mathcal{L}_{i, 1}, \ldots, \mathcal{L}_{i, r_i}\) are invertible \({\mathcal O}_P\)-modules.

Proof

We prove this by induction on the integer \(r = \sum r_i\). If \(r = 0\) we can take \(\pi = \text{id}_X\). If \(r_i = 1\) for all \(i\), then we can also take \(\pi = \text{id}_X\). Assume that \(r_{i_0} > 1\) for some \(i_0\). Let \((\pi : P \to X, \mathcal{O}_P(1))\) be the projective bundle associated to \(\mathcal{E}_{i_0}\). The canonical map \(\pi^*\mathcal{E}_{i_0} \to \mathcal{O}_P(1)\) is surjective and hence its kernel \(\mathcal{E}'_{i_0}\) is finite locally free of rank \(r_{i_0} - 1\). Observe that \(\pi_Y^*\) is injective for any morphism \(f : Y \to X\) of good algebraic spaces over \(B\), see Lemma 0ERV. Thus it suffices to prove the lemma for \(P\) and the locally free sheaves \(\pi^*\mathcal{E}_i\). However, because we have the subbundle \(\mathcal{E}_{i_0} \subset \pi^*\mathcal{E}_{i_0}\) with invertible quotient, it now suffices to prove the lemma for the collection \(\{\mathcal{E}_i\}_{i \not = i_0} \cup \{\mathcal{E}'_{i_0}\}\). This decreases \(r\) by \(1\) and we win by induction hypothesis.

Rather than explaining what the splitting principle says, let us use it in the proof of some lemmas.

Lemma

In Situation 0EDT let \(X/B\) be good. Let \(\mathcal{E}\) be a finite locally free \(\mathcal{O}_X\)-module with dual \(\mathcal{E}^\vee\). Then \[c_i(\mathcal{E}^\vee) = (-1)^i c_i(\mathcal{E})\] in \(A^i(X)\).

Proof

Choose a morphism \(\pi : P \to X\) as in Lemma 0ESC. By the injectivity of \(\pi^*\) (after any base change) it suffices to prove the relation between the Chern classes of \(\mathcal{E}\) and \(\mathcal{E}^\vee\) after pulling back to \(P\). Thus we may assume there exist invertible \(\mathcal{O}_X\)-modules \({\mathcal L}_i\), \(i = 1, \ldots, r\) and a filtration \[0 = \mathcal{E}_0 \subset \mathcal{E}_1 \subset \mathcal{E}_2 \subset \ldots \subset \mathcal{E}_r = \mathcal{E}\] such that \(\mathcal{E}_i/\mathcal{E}_{i - 1} \cong \mathcal{L}_i\). Then we obtain the dual filtration \[0 = \mathcal{E}_r^\perp \subset \mathcal{E}_1^\perp \subset \mathcal{E}_2^\perp \subset \ldots \subset \mathcal{E}_0^\perp = \mathcal{E}^\vee\] such that \(\mathcal{E}_{i - 1}^\perp/\mathcal{E}_i^\perp \cong \mathcal{L}_i^{\otimes -1}\). Set \(x_i = c_1(\mathcal{L}_i)\). Then \(c_1(\mathcal{L}_i^{\otimes -1}) = - x_i\) by Lemma 0EQK. By Lemma 0ESA we have \[c(\mathcal{E}) = \prod\nolimits_{i = 1}^r (1 + x_i) \quad\text{and}\quad c(\mathcal{E}^\vee) = \prod\nolimits_{i = 1}^r (1 - x_i)\] in \(A^*(X)\). The result follows from a formal computation which we omit.

Lemma

In Situation 0EDT let \(X/B\) be good. Let \(\mathcal{E}\) and \(\mathcal{F}\) be a finite locally free \(\mathcal{O}_X\)-modules of ranks \(r\) and \(s\). Then we have \[c_1(\mathcal{E} \otimes \mathcal{F}) = r c_1(\mathcal{F}) + s c_1(\mathcal{E})\] \[c_2(\mathcal{E} \otimes \mathcal{F}) = r^2 c_2(\mathcal{F}) + rs c_1(\mathcal{F})c_1(\mathcal{E}) + s^2 c_2(\mathcal{E})\] and so on (see proof).

Proof

Arguing exactly as in the proof of Lemma 0ESD we may assume we have invertible \(\mathcal{O}_X\)-modules \({\mathcal L}_i\), \(i = 1, \ldots, r\) \({\mathcal N}_i\), \(i = 1, \ldots, s\) filtrations \[0 = \mathcal{E}_0 \subset \mathcal{E}_1 \subset \mathcal{E}_2 \subset \ldots \subset \mathcal{E}_r = \mathcal{E} \quad\text{and}\quad 0 = \mathcal{F}_0 \subset \mathcal{F}_1 \subset \mathcal{F}_2 \subset \ldots \subset \mathcal{F}_s = \mathcal{F}\] such that \(\mathcal{E}_i/\mathcal{E}_{i - 1} \cong \mathcal{L}_i\) and such that \(\mathcal{F}_j/\mathcal{F}_{j - 1} \cong \mathcal{N}_j\). Ordering pairs \((i, j)\) lexicographically we obtain a filtration \[0 \subset \ldots \subset \mathcal{E}_i \otimes \mathcal{F}_j + \mathcal{E}_{i - 1} \otimes \mathcal{F} \subset \ldots \subset \mathcal{E} \otimes \mathcal{F}\] with successive quotients \[\mathcal{L}_1 \otimes \mathcal{N}_1, \mathcal{L}_1 \otimes \mathcal{N}_2, \ldots, \mathcal{L}_1 \otimes \mathcal{N}_s, \mathcal{L}_2 \otimes \mathcal{N}_1, \ldots, \mathcal{L}_r \otimes \mathcal{N}_s\] By Lemma 0ESA we have \[c(\mathcal{E}) = \prod (1 + x_i), \quad c(\mathcal{F}) = \prod (1 + y_j), \quad\text{and}\quad c(\mathcal{F}) = \prod (1 + x_i + y_j),\] in \(A^*(X)\). The result follows from a formal computation which we omit.

Degrees of zero cycles

This section is the analogue of Chow Homology, Section 0AZ0. We start with defining the degree of a zero cycle on a proper algebraic space over a field.

Definition

Let \(k\) be a field. Let \(p : X \to \Spec(k)\) be a proper morphism of algebraic spaces. The degree of a zero cycle on \(X\) is given by proper pushforward \[p_* : \CH_0(X) \longrightarrow \CH_0(\Spec(k)) \longrightarrow \mathbf{Z}\] (Lemma 0EQC) composed with the natural isomorphism \(\CH_0(\Spec(k)) \to \mathbf{Z}\) which maps \([\Spec(k)]\) to \(1\). Notation: \(\deg(\alpha)\).

Let us spell this out further.

Lemma

Let \(k\) be a field. Let \(X\) be a proper algebraic space over \(k\). Let \(\alpha = \sum n_i[Z_i]\) be in \(Z_0(X)\). Then \[\deg(\alpha) = \sum n_i\deg(Z_i)\] where \(\deg(Z_i)\) is the degree of \(Z_i \to \Spec(k)\), i.e., \(\deg(Z_i) = \dim_k \Gamma(Z_i, \mathcal{O}_{Z_i})\).

Proof

This is the definition of proper pushforward (Definition 0EP1).

Lemma

Let \(k\) be a field. Let \(X\) be a proper algebraic space over \(k\). Let \(Z \subset X\) be a closed subspace of dimension \(d\). Let \(\mathcal{L}_1, \ldots, \mathcal{L}_d\) be invertible \(\mathcal{O}_X\)-modules. Then \[(\mathcal{L}_1 \cdots \mathcal{L}_d \cdot Z) = \deg( c_1(\mathcal{L}_1) \cap \ldots \cap c_1(\mathcal{L}_1) \cap [Z]_d)\] where the left hand side is defined in Spaces over Fields, Definition 0EDF.

Proof

Let \(Z_i \subset Z\), \(i = 1, \ldots, t\) be the irreducible components of dimension \(d\). Let \(m_i\) be the multiplicity of \(Z_i\) in \(Z\). Then \([Z]_d = \sum m_i[Z_i]\) and \(c_1(\mathcal{L}_1) \cap \ldots \cap c_1(\mathcal{L}_d) \cap [Z]_d\) is the sum of the cycles \(m_i c_1(\mathcal{L}_1) \cap \ldots \cap c_1(\mathcal{L}_d) \cap [Z_i]\). Since we have a similar decomposition for \((\mathcal{L}_1 \cdots \mathcal{L}_d \cdot Z)\) by Spaces over Fields, Lemma 0EDE it suffices to prove the lemma in case \(Z = X\) is a proper integral algebraic space over \(k\).

By Chow’s lemma there exists a proper morphism \(f : X' \to X\) which is an isomorphism over a dense open \(U \subset X\) such that \(X'\) is a scheme. See More on Morphisms of Spaces, Lemma 088U. Then \(X'\) is a proper scheme over \(k\). After replacing \(X'\) by the scheme theoretic closure of \(f^{-1}(U)\) we may assume that \(X'\) is integral. Then \[(f^*\mathcal{L}_1 \cdots f^*\mathcal{L}_d \cdot X') = (\mathcal{L}_1 \cdots \mathcal{L}_d \cdot X)\] by Spaces over Fields, Lemma 0EDJ and we have \[f_*(c_1(f^*\mathcal{L}_1) \cap \ldots \cap c_1(f^*\mathcal{L}_d) \cap [Y]) = c_1(\mathcal{L}_1) \cap \ldots \cap c_1(\mathcal{L}_d) \cap [X]\] by Lemma 0EQT. Thus we may replace \(X\) by \(X'\) and assume that \(X\) is a proper scheme over \(k\). This case was proven in Chow Homology, Lemma 0BFI.

It is possible that a shorter proof can be given by immediately applying étale localization.↩︎
Up to signs these are the Segre classes of \(\mathcal{E}\).↩︎