The Stacks project

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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