Let \mathcal{C} be a site. Let f : X \to Y be a morphism of \mathcal{C}. Then we obtain a morphism of topoi
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
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
See Modules on Sites, Lemma 18.19.5.
Lemma 21.21.2. Let (\mathcal{C}, \mathcal{O}) be a ringed site. Let
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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