This section is the analogue of Chow Homology, Section 42.9.
Remark 82.5.1. In Situation 82.2.1 let $X/B$ be good. Let $Y \subset X$ be a closed subspace. By Remarks 82.2.2 and 82.2.3 there are $1$-to-$1$ correspondences between
irreducible components $T$ of $|Y|$,
generic points of irreducible components of $|Y|$, and
integral closed subspaces $Z \subset Y$ with the property that $|Z|$ is an irreducible component of $|Y|$.
In this chapter we will call $Z$ as in (3) an irreducible component of $Y$ and we will call $\xi \in |Z|$ its generic point.
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
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.
\[ [Y]_ k = \sum m_{Z, Y}[Z] \]
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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