The Stacks project

Definition 82.6.1. In Situation 82.2.1 let $X/B$ be good. Let $\mathcal{F}$ be a coherent $\mathcal{O}_ X$-module.

  1. 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 82.4.2) is called the multiplicity of $Z$ in $\mathcal{F}$. By Lemma 82.4.4 this is a positive integer.

  2. 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 72.6.1.

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