Lemma 58.24.2 (0EKA)—The Stacks project

Loading web-font TeX/Main/Regular

Table of contents
Part 3: Topics in Scheme Theory
Chapter 58: Fundamental Groups of Schemes
Section 58.24: Finite étale covers of punctured spectra, IV
Lemma 58.24.2 (cite)

previous lemma

numberstags

history
statistics
1 tag refers to this tag

Lemma 58.24.2. Let $(A, \mathfrak m)$ be a Noetherian local ring. Let $f \in \mathfrak m$ . Assume

$A$ is $f$ -adically complete,
$f$ is a nonzerodivisor,
$H^1_\mathfrak m(A/fA)$ and $H^2_\mathfrak m(A/fA)$ are finite $A$ -modules,
for every maximal ideal $\mathfrak p \subset A_ f$ purity holds for $(A_ f)_\mathfrak p$ .

Then the restriction functor $\textit{FÉt}_ U \to \textit{FÉt}_{U_0}$ is essentially surjective.

Proof. The proof is identical to the proof of Lemma 58.24.1 using Lemma 58.23.2 in stead of Lemma 58.23.1. $\square$

Comments (0)

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