Lemma 20.34.1 (0A3B)—The Stacks project

Table of contents
Part 1: Preliminaries
Chapter 20: Cohomology of Sheaves
Section 20.34: Cohomology with support in a closed subset, II
Lemma 20.34.1 (cite)

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Lemma 20.34.1. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $i : Z \to X$ be the inclusion of a closed subset.

$R\mathcal{H}_ Z : D(\mathcal{O}_ X) \to D(\mathcal{O}_ X|_ Z)$ is right adjoint to $i_* : D(\mathcal{O}_ X|_ Z) \to D(\mathcal{O}_ X)$.
For $K$ in $D(\mathcal{O}_ X|_ Z)$ we have $R\mathcal{H}_ Z(i_*K) = K$.
Let $\mathcal{G}$ be a sheaf of $\mathcal{O}_ X|_ Z$-modules on $Z$. Then $\mathcal{H}^ p_ Z(i_*\mathcal{G}) = 0$ for $p > 0$.

Proof. The functor $i_*$ is exact, so $i_* = Ri_* = Li_*$. Hence part (1) of the lemma follows from Modules, Lemma 17.13.6 and Derived Categories, Lemma 13.30.3. Let $K$ be as in (2). We can represent $K$ by a K-injective complex $\mathcal{I}^\bullet $ of $\mathcal{O}_ X|_ Z$-modules. By Lemma 20.32.9 the complex $i_*\mathcal{I}^\bullet $, which represents $i_*K$, is a K-injective complex of $\mathcal{O}_ X$-modules. Thus $R\mathcal{H}_ Z(i_*K)$ is computed by $\mathcal{H}_ Z(i_*\mathcal{I}^\bullet ) = \mathcal{I}^\bullet $ which proves (2). Part (3) is a special case of (2). $\square$

Comments (2)

Comment #1801 by Keenan Kidwell on January 20, 2016 at 17:07

Currently the proof cites 20.11.11, which says that for $f:X\to Y$ a flat map of ringed spaces, $f_*$ preserves injectives (the reason being that $f^*$ is exact in this case). This seems awkward to me since the current lemma is in the context of abelian sheaves on a topological space. Wouldn't it make more sense to cite the fact that $i^{-1}$ is (always) exact? Although upon doing some searching, this is only stated for maps of ringed spaces as 17.3.3 (3)...but this seems a bit awkward for the same reason. Should there be a separate tag, perhaps in the section 6.22, deducing exactness of $i^{-1}$ for all maps of spaces from the description on stalks?

Comment #1826 by Johan on February 04, 2016 at 19:35

OK, yes we could and should state separate lemmas for abelian sheaves. The general idea is that once sheaves of modules have been defined, then most results should be stated for cohomology and derived categories of sheaves of modules with results for abelian sheaves as special cases. However, in this section we only do the material for abelian sheaves, because a good notion of closed immersions for ringed spaces is a bit tricky (for a bunch of reasons).

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