Lemma 58.19.6 (0BM6)—The Stacks project

Loading [MathJax]/extensions/tex2jax.js

Table of contents
Part 3: Topics in Scheme Theory
Chapter 58: Fundamental Groups of Schemes
Section 58.19: Finite étale covers of punctured spectra, I
Lemma 58.19.6 (cite)

numberstags

[Corollary 1.11, Bhatt-local]

Lemma 58.19.6. In Situation 58.19.1. Assume

$H^1_\mathfrak m(A)$ and $H^2_\mathfrak m(A)$ are annihilated by a power of $f$, and
$A$ is henselian or more generally $(A, (f))$ is a henselian pair.

Then the restriction functor $\textit{FÉt}_ U \longrightarrow \textit{FÉt}_{U_0}$ is fully faithful.

Proof. By Lemma 58.19.4 we may assume that $A$ is a Noetherian complete local ring. (The assumptions carry over; use Dualizing Complexes, Lemma 47.9.3.) By Lemma 58.17.1 the result follows from Algebraic and Formal Geometry, Lemma 52.15.5. $\square$

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