Lemma 21.31.10 (0DCY)—The Stacks project

Table of contents
Part 1: Preliminaries
Chapter 21: Cohomology on Sites
Section 21.31: Cohomology on Hausdorff and locally quasi-compact spaces
Lemma 21.31.10 (cite)

Lemma 21.31.10. With notation as above.

For $X \in \mathop{\mathrm{Ob}}\nolimits (\textit{LC}_{qc})$ and an abelian sheaf $\mathcal{F}$ on $X$ we have $\epsilon _{X, *}a_ X^{-1}\mathcal{F} = \pi _ X^{-1}\mathcal{F}$ and $R^ i\epsilon _{X, *}(a_ X^{-1}\mathcal{F}) = 0$ for $i > 0$.
For a proper morphism $f : X \to Y$ in $\textit{LC}_{qc}$ and abelian sheaf $\mathcal{F}$ on $X$ we have $a_ Y^{-1}(R^ if_*\mathcal{F}) = R^ if_{qc, *}(a_ X^{-1}\mathcal{F})$ for all $i$.
For $X \in \mathop{\mathrm{Ob}}\nolimits (\textit{LC}_{qc})$ and $K$ in $D^+(X)$ the map $\pi _ X^{-1}K \to R\epsilon _{X, *}(a_ X^{-1}K)$ is an isomorphism.
For a proper morphism $f : X \to Y$ in $\textit{LC}_{qc}$ and $K$ in $D^+(X)$ we have $a_ Y^{-1}(Rf_*K) = Rf_{qc, *}(a_ X^{-1}K)$.

Proof. By Lemma 21.31.9 the lemmas in Section 21.30 all apply to our current setting. To translate the results observe that the category $\mathcal{A}_ X$ of Lemma 21.30.2 is the essential image of $a_ X^{-1} : \textit{Ab}(X) \to \textit{Ab}(\textit{LC}_{qc}/X)$.

Part (1) is equivalent to $(V_ n)$ for all $n$ which holds by Lemma 21.30.8.

Part (2) follows by applying $\epsilon _ Y^{-1}$ to the conclusion of Lemma 21.30.3.

Part (3) follows from Lemma 21.30.8 part (1) because $\pi _ X^{-1}K$ is in $D^+_{\mathcal{A}'_ X}(\textit{LC}_{Zar}/X)$ and $a_ X^{-1} = \epsilon _ X^{-1} \circ a_ X^{-1}$.

Part (4) follows from Lemma 21.30.8 part (2) for the same reason. $\square$

The Stacks project

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