## 81.18 Intersecting with an invertible sheaf

This section is the analogue of Chow Homology, Section 42.25. In this section we study the following construction.

Definition 81.18.1. In Situation 81.2.1 let $X/B$ be good. Let $\mathcal{L}$ be an invertible $\mathcal{O}_ X$-module. We define, for every integer $k$, an operation

$c_1(\mathcal{L}) \cap - : Z_{k + 1}(X) \to \mathop{\mathrm{CH}}\nolimits _ k(X)$

called intersection with the first Chern class of $\mathcal{L}$.

1. 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 81.17.1.

2. 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 $\mathop{\mathrm{CH}}\nolimits _ k(\coprod W_ i) = \prod \mathop{\mathrm{CH}}\nolimits _ k(W_ i)$. 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 81.18.2. In Situation 81.2.1 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 $\mathop{\mathrm{CH}}\nolimits _ k(X)$ 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 71.7.5 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. $\square$

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 70.7.6.

Lemma 81.18.3. In Situation 81.2.1 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

81.18.3.1
\begin{equation} \label{spaces-chow-equation-equal-as-cycles} [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 81.10.3. 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 42.25.3) 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. $\square$

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 31.14.8.

Lemma 81.18.4. In Situation 81.2.1 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 $\mathop{\mathrm{CH}}\nolimits _ k(X)$.

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 81.17.2 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 81.18.3. $\square$

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