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 10.52.12.

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 $\dim _\delta (Z) \leq k$.


Comments (0)


Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 02QS. Beware of the difference between the letter 'O' and the digit '0'.