The Stacks project

81.5 Cycle associated to a closed subspace

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

Remark 81.5.1. In Situation 81.2.1 let $X/B$ be good. Let $Y \subset X$ be a closed subspace. By Remarks 81.2.2 and 81.2.3 there are $1$-to-$1$ correspondences between

  1. irreducible components $T$ of $|Y|$,

  2. generic points of irreducible components of $|Y|$, and

  3. 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 81.5.2. In Situation 81.2.1 let $X/B$ be good. Let $Y \subset X$ be a closed subspace.

  1. For an irreducible component $Z \subset Y$ with generic point $\xi $ the length of $\mathcal{O}_ Y$ at $\xi $ (Definition 81.4.2) is called the multiplicity of $Z$ in $Y$. By Lemma 81.4.4 applied to $\mathcal{O}_ Y$ on $Y$ this is a positive integer.

  2. 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 71.6.1.

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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