The Stacks project

Definition 42.9.2. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ be locally of finite type over $S$. Let $Z \subset X$ be a closed subscheme.

1. For any irreducible component $Z' \subset Z$ with generic point $\xi $ the integer $m_{Z', Z} = \text{length}_{\mathcal{O}_{X, \xi }} \mathcal{O}_{Z, \xi }$ (Lemma 42.9.1) is called the multiplicity of $Z'$ in $Z$.

2. Assume $\dim _\delta (Z) \leq k$. The $k$-cycle associated to $Z$ is

   \[ [Z]_ k = \sum m_{Z', Z}[Z'] \]

   where the sum is over the irreducible components of $Z$ of $\delta $-dimension $k$. (This is a $k$-cycle by Divisors, Lemma 31.26.1.)