The Stacks project

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. $\square$

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