Lemma 85.25.5 (0DAH)—The Stacks project

Processing math: 100%

Table of contents
Part 5: Topics in Geometry
Chapter 85: Simplicial Spaces
Section 85.25: Proper hypercoverings in topology
Lemma 85.25.5 (cite)

numberstags

Lemma 85.25.5. Let $U$ be a simplicial object of $\textit{LC}$ and let $a : U \to X$ be an augmentation. Let $\mathcal{A} \subset \textit{Ab}(U_{Zar})$ denote the weak Serre subcategory of cartesian abelian sheaves. If $U$ is a proper hypercovering of $X$ , then the functor $a^{-1}$ defines an equivalence

$D^+(X) \longrightarrow D_\mathcal {A}^+(U_{Zar})$

with quasi-inverse $Ra_*$ where $a : \mathop{\mathit{Sh}}\nolimits (U_{Zar}) \to \mathop{\mathit{Sh}}\nolimits (X)$ is as in Lemma 85.2.8.

Proof. Observe that $\mathcal{A}$ is a weak Serre subcategory by Lemma 85.12.6. The equivalence is a formal consequence of the results obtained so far. Use Lemmas 85.25.2 and 85.25.3 and Cohomology on Sites, Lemma 21.28.5. $\square$

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