The Stacks project

42.9 Cycle associated to a closed subscheme

Lemma 42.9.1. 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. Let $Z' \subset Z$ be an irreducible component and let $\xi \in Z'$ be its generic point. Then

    \[ \text{length}_{\mathcal{O}_{X, \xi }} \mathcal{O}_{Z, \xi } < \infty \]
  2. If $\dim _\delta (Z) \leq k$ and $\xi \in Z$ with $\delta (\xi ) = k$, then $\xi $ is a generic point of an irreducible component of $Z$.

Proof. Let $Z' \subset Z$, $\xi \in Z'$ be as in (1). Then $\dim (\mathcal{O}_{Z, \xi }) = 0$ (for example by Properties, Lemma 28.10.3). Hence $\mathcal{O}_{Z, \xi }$ is Noetherian local ring of dimension zero, and hence has finite length over itself (see Algebra, Proposition 10.60.7). Hence, it also has finite length over $\mathcal{O}_{X, \xi }$, see Algebra, Lemma 10.52.5.

Assume $\xi \in Z$ and $\delta (\xi ) = k$. Consider the closure $Z' = \overline{\{ \xi \} }$. It is an irreducible closed subscheme with $\dim _\delta (Z') = k$ by definition. Since $\dim _\delta (Z) = k$ it must be an irreducible component of $Z$. Hence we see (2) holds. $\square$

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.)

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

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