The Stacks project

Definition 42.10.2. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ be locally of finite type over $S$. Let $\mathcal{F}$ be a coherent $\mathcal{O}_ X$-module.

1. For any irreducible component $Z' \subset \text{Supp}(\mathcal{F})$ with generic point $\xi $ the integer $m_{Z', \mathcal{F}} = \text{length}_{\mathcal{O}_{X, \xi }} \mathcal{F}_\xi $ (Lemma 42.10.1) is called the multiplicity of $Z'$ in $\mathcal{F}$.

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 irreducible components of $\text{Supp}(\mathcal{F})$ of $\delta $-dimension $k$. (This is a $k$-cycle by Lemma 42.10.1.)