The Stacks project

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 ├ętale 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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