Lemma 42.25.4 (02SQ)—The Stacks project

Loading [MathJax]/extensions/tex2jax.js

Table of contents
Part 3: Topics in Scheme Theory
Chapter 42: Chow Homology and Chern Classes
Section 42.25: Intersecting with an invertible sheaf
Lemma 42.25.4 (cite)

previous lemma

numberstags

Lemma 42.25.4. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ be locally of finite type over $S$. Let $\mathcal{L}$ be an invertible $\mathcal{O}_ X$-module. Let $Y \subset X$ be a closed subscheme. Let $s \in \Gamma (Y, \mathcal{L}|_ Y)$. Assume

$\dim _\delta (Y) \leq k + 1$,
$\dim _\delta (Z(s)) \leq k$, and
for every generic point $\xi $ of an irreducible component of $Z(s)$ of $\delta $-dimension $k$ the multiplication by $s$ induces an injection $\mathcal{O}_{Y, \xi } \to (\mathcal{L}|_ Y)_\xi $¹.

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 42.24.2 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 42.25.3. $\square$

[1] For example, this holds if $s$ is a regular section of $\mathcal{L}|_ Y$.

Comments (0)

There are also:

2 comment(s) on Section 42.25: Intersecting with an invertible sheaf

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$ ). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Comments (0)

Post a comment