Lemma 42.9.1 (02QT)—The Stacks project

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.

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

$\text{length}_{\mathcal{O}_{X, \xi }} \mathcal{O}_{Z, \xi } < \infty$
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. $\square$

Comments (4)

Comment #7442 by old friend on June 21, 2022 at 18:48

"Hence, it also has finite length over OX,ξ, see Algebra, Lemma 10.52.12." In this case, the two lengths be would even be equal right? We have OX,ξ ---> OZ,ξ is surjective. If this is the case, citing tag 00IX would be helpful too.

Comment #7443 by David Holmes on June 22, 2022 at 06:09

Dear old friend, What you propose seems to me to work, and 00IX seems simpler to apply here than 02M0.

Comment #7446 by old friend on June 22, 2022 at 15:33

Thanks David Holmes for confirming. I was confused by the definition given in e.g. Fulton or 3264. I think stacks insists on using O_X,ξ rather than O_Z,ξ because they later define cycle associated to a coherent sheaf F on X. There the length of F_ξ only makes sense over O_X,ξ but when F is O_Z, we could take length over O_Z,ξ.

Comment #7599 by Stacks Project on August 13, 2022 at 18:34

Thanks, see here.

Comments (4)

Post a comment