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.

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 $\mathcal{F}$*.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.)

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