Lemma 61.27.4 (09AL)—The Stacks project

Processing math: 100%

Table of contents
Part 3: Topics in Scheme Theory
Chapter 61: Pro-étale Cohomology
Section 61.27: Constructible sheaves on the pro-étale site
Lemma 61.27.4 (cite)

numberstags

Lemma 61.27.4. Let $X$ be a scheme. Let $\Lambda$ be a Noetherian ring. Let $D_ c(X_{\acute{e}tale}, \Lambda )$ , resp. $D_ c(X_{pro\text{-}\acute{e}tale}, \Lambda )$ be the full subcategory of $D(X_{\acute{e}tale}, \Lambda )$ , resp. $D(X_{pro\text{-}\acute{e}tale}, \Lambda )$ consisting of those complexes whose cohomology sheaves are constructible sheaves of $\Lambda$ -modules. Then

$\epsilon ^{-1} : D_ c^+(X_{\acute{e}tale}, \Lambda ) \longrightarrow D_ c^+(X_{pro\text{-}\acute{e}tale}, \Lambda )$

is an equivalence of categories.

Proof. The categories $D_ c(X_{\acute{e}tale}, \Lambda )$ and $D_ c(X_{pro\text{-}\acute{e}tale}, \Lambda )$ are strictly full, saturated, triangulated subcategories of $D(X_{\acute{e}tale}, \Lambda )$ and $D(X_{pro\text{-}\acute{e}tale}, \Lambda )$ by Étale Cohomology, Lemma 59.71.6 and Lemma 61.27.3 and Derived Categories, Section 13.17. The statement of the lemma follows by combining Lemmas 61.19.8 and 61.27.2. $\square$

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