# Cycle associated to a closed subscheme

Let $$X$$ be an algebraic variety (i.e. an integral $$k$$-scheme, such that $$X \to \mathrm{Spec \;}k$$ is separated and of finite type). Let $$Y$$ be a closed subscheme and $$Y_1, \dots Y_n$$ the irreducible components of $$Y_{\mathrm{red}}$$. Let $$\xi_i$$ be the generic point of $$Y_i$$. Since $$Y$$ is noetherian, then $$\mathcal{O}_{Y,\xi_i}$$ is a noetherian ring. We define the effective cycle associated to $$Y$$ as $$:=\sum_{i=1}^nl_iY_i$$, where $$l_i$$ is the length of the local ring $$\mathcal{O}_{Y,\xi_i}$$. In the book "3264 and all that" it is said that the ring $$\mathcal{O}_{Y,\xi_i}$$ has a finite composition series. I know that the length of $$\mathcal{O}_{Y,\xi_i}$$ is finite if and only if it is an artinian ring. But in this case i just see that it is noetherian. Why is it artinian too?

EDIT: I try to answer my own. Let $$\xi$$ be one of the $$\xi_i$$ and $$Z$$ the irreducible component associated to $$\xi$$. Choose $$U$$ to be an affine open subset of $$Y$$ such that $$\xi \in U$$. Then $$Z \cap U$$ is an irreducible component of $$U$$, with generic point $$\xi$$. If $$U= \mathrm{Spec \;}A$$, then $$\xi=\mathfrak{p}$$ is a minimal prime ideal and hence $$\mathcal{O}_{Y,\xi}=\mathcal{O}_{U,\mathfrak{p}}$$ is noetherian of dimension $$0$$, i.e. an artinian ring. Do you think it works?

• Yes, it works . – Georges Elencwajg Mar 28 at 9:23