Lemma 21.18.3 (0D6D)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 21: Cohomology on Sites
Section 21.18: Derived pullback
Lemma 21.18.3 (cite)

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Lemma 21.18.3. Consider morphisms of ringed topoi $f : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}_\mathcal {C}) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}_\mathcal {D})$ and $g : (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}_\mathcal {D}) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{E}), \mathcal{O}_\mathcal {E})$ . Then $Lf^* \circ Lg^* = L(g \circ f)^*$ as functors $D(\mathcal{O}_\mathcal {E}) \to D(\mathcal{O}_\mathcal {C})$ .

Proof. Let $E$ be an object of $D(\mathcal{O}_\mathcal {E})$ . We may represent $E$ by a K-flat complex $\mathcal{K}^\bullet$ with flat terms, see Lemma 21.17.11. By construction $Lg^*E$ is computed by $g^*\mathcal{K}^\bullet$ , see Lemma 21.18.2. By Lemma 21.18.1 the complex $g^*\mathcal{K}^\bullet$ is K-flat with flat terms. Hence $Lf^*Lg^*E$ is represented by $f^*g^*\mathcal{K}^\bullet$ . Since also $L(g \circ f)^*E$ is represented by $(g \circ f)^*\mathcal{K}^\bullet = f^*g^*\mathcal{K}^\bullet$ we conclude. $\square$

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