Lemma 66.22.1 (04KF)—The Stacks project

Processing math: 100%

Table of contents
Part 4: Algebraic Spaces
Chapter 66: Properties of Algebraic Spaces
Section 66.22: Stalks of the structure sheaf
Lemma 66.22.1 (cite)

next definition

numberstags

Lemma 66.22.1. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$ . Let $\overline{x}$ be a geometric point of $X$ . Let $(U, \overline{u})$ be an étale neighbourhood of $\overline{x}$ where $U$ is a scheme. Then we have

$\mathcal{O}_{X, \overline{x}} = \mathcal{O}_{U, \overline{u}} = \mathcal{O}_{U, u}^{sh}$

where the left hand side is the stalk of the structure sheaf of $X$ , and the right hand side is the strict henselization of the local ring of $U$ at the point $u$ at which $\overline{u}$ is centered.

Proof. We know that the structure sheaf $\mathcal{O}_ U$ on $U_{\acute{e}tale}$ is the restriction of the structure sheaf of $X$ . Hence the first equality follows from Lemma 66.19.9 part (4). The second equality is explained in Étale Cohomology, Lemma 59.33.1. $\square$

Comments (0)

There are also:

2 comment(s) on Section 66.22: Stalks of the structure sheaf

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).

Comments (0)

Post a comment