The Stacks project

Lemma 20.27.2. Let $f : X \to Y$ and $g : Y \to Z$ be morphisms of ringed spaces. Then $Lf^* \circ Lg^* = L(g \circ f)^*$ as functors $D(\mathcal{O}_ Z) \to D(\mathcal{O}_ X)$.

Proof. Let $E$ be an object of $D(\mathcal{O}_ Z)$. By construction $Lg^*E$ is computed by choosing a K-flat complex $\mathcal{K}^\bullet $ representing $E$ on $Z$ and setting $Lg^*E = g^*\mathcal{K}^\bullet $. By Lemma 20.26.8 we see that $g^*\mathcal{K}^\bullet $ is K-flat on $Y$. Then $Lf^*Lg^*E$ is given by $f^*g^*\mathcal{K}^\bullet = (g \circ f)^*\mathcal{K}^\bullet $ which also represents $L(g \circ f)^*E$. $\square$

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