21.21 Localization and cohomology

Let $\mathcal{C}$ be a site. Let $f : X \to Y$ be a morphism of $\mathcal{C}$. Then we obtain a morphism of topoi

$j_{X/Y} : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}/X) \longrightarrow \mathop{\mathit{Sh}}\nolimits (\mathcal{C}/Y)$

See Sites, Sections 7.25 and 7.27. Some questions about cohomology are easier for this type of morphisms of topoi. Here is an example where we get a trivial type of base change theorem.

Lemma 21.21.1. Let $\mathcal{C}$ be a site. Let

$\xymatrix{ X' \ar[d] \ar[r] & X \ar[d] \\ Y' \ar[r] & Y }$

be a cartesian diagram of $\mathcal{C}$. Then we have $j_{Y'/Y}^{-1} \circ Rj_{X/Y, *} = Rj_{X'/Y', *} \circ j_{X'/X}^{-1}$ as functors $D(\mathcal{C}/X) \to D(\mathcal{C}/Y')$.

Proof. Let $E \in D(\mathcal{C}/X)$. Choose a K-injective complex $\mathcal{I}^\bullet$ of abelian sheaves on $\mathcal{C}/X$ representing $E$. By Lemma 21.20.1 we see that $j_{X'/X}^{-1}\mathcal{I}^\bullet$ is K-injective too. Hence we may compute $Rj_{X'/Y'}(j_{X'/X}^{-1}E)$ by $j_{X'/Y', *}j_{X'/X}^{-1}\mathcal{I}^\bullet$. Thus we see that the equality holds by Sites, Lemma 7.27.5. $\square$

If we have a ringed site $(\mathcal{C}, \mathcal{O})$ and a morphism $f : X \to Y$ of $\mathcal{C}$, then $j_{X/Y}$ becomes a morphism of ringed topoi

$j_{X/Y} : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/X), \mathcal{O}_ X) \longrightarrow (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/Y), \mathcal{O}_ Y)$

See Modules on Sites, Lemma 18.19.5.

Lemma 21.21.2. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let

$\xymatrix{ X' \ar[d] \ar[r] & X \ar[d] \\ Y' \ar[r] & Y }$

be a cartesian diagram of $\mathcal{C}$. Then we have $j_{Y'/Y}^* \circ Rj_{X/Y, *} = Rj_{X'/Y', *} \circ j_{X'/X}^*$ as functors $D(\mathcal{O}_ X) \to D(\mathcal{O}_{Y'})$.

Proof. Since $j_{Y'/Y}^{-1}\mathcal{O}_ Y = \mathcal{O}_{Y'}$ we have $j_{Y'/Y}^* = Lj_{Y'/Y}^* = j_{Y'/Y}^{-1}$. Similarly we have $j_{X'/X}^* = Lj_{X'/X}^* = j_{X'/X}^{-1}$. Thus by Lemma 21.20.7 it suffices to prove the result on derived categories of abelian sheaves which we did in Lemma 21.21.1. $\square$

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