Definition 82.5.2. In Situation 82.2.1 let $X/B$ be good. Let $Y \subset X$ be a closed subspace.

For an irreducible component $Z \subset Y$ with generic point $\xi $ the length of $\mathcal{O}_ Y$ at $\xi $ (Definition 82.4.2) is called the

*multiplicity of $Z$ in $Y$*. By Lemma 82.4.4 applied to $\mathcal{O}_ Y$ on $Y$ this is a positive integer.Assume $\dim _\delta (Y) \leq k$. The

*$k$-cycle associated to $Y$*is\[ [Y]_ k = \sum m_{Z, Y}[Z] \]where the sum is over the irreducible components $Z$ of $Y$ of $\delta $-dimension $k$ and $m_{Z, Y}$ is the multiplicity of $Z$ in $Y$. This is a $k$-cycle by Spaces over Fields, Lemma 72.6.1.

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