Lemma 85.19.4 (0D8M)—The Stacks project

Loading web-font TeX/Caligraphic/Regular

Table of contents
Part 5: Topics in Geometry
Chapter 85: Simplicial Spaces
Section 85.19: Cohomological descent for hypercoverings of an object
Lemma 85.19.4 (cite)

previous lemma

numberstags

history
statistics
1 tag refers to this tag

Lemma 85.19.4. Let $\mathcal{C}$ be a site with fibre products and $X \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ . Let $K$ be a hypercovering of $X$ . Let $\mathcal{A} \subset \textit{Ab}((\mathcal{C}/K)_{total})$ denote the weak Serre subcategory of cartesian abelian sheaves. Then the functor $a^{-1}$ defines an equivalence

$D^+(\mathcal{C}/X) \longrightarrow D_\mathcal {A}^+((\mathcal{C}/K)_{total})$

with quasi-inverse $Ra_*$ .

Proof. Via Remarks 85.15.5 and 85.16.4 this follows from Lemma 85.17.5. $\square$

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