## 80.5 Cycle associated to a closed subspace

This section is the analogue of Chow Homology, Section 42.9.

Definition 80.5.2. In Situation 80.2.1 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 80.4.2) is called the *multiplicity of $Z$ in $Y$*. By Lemma 80.4.4 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 70.6.1.

It is important to note that we only define $[Y]_ k$ if the $\delta $-dimension of $Y$ does not exceed $k$. In other words, by convention, if we write $[Y]_ k$ then this implies that $\dim _\delta (Y) \leq k$.

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