The Stacks project

Lemma 20.27.3. Let $f : (X, \mathcal{O}_ X) \to (Y, \mathcal{O}_ Y)$ be a morphism of ringed spaces. There is a canonical bifunctorial isomorphism

\[ Lf^*( \mathcal{F}^\bullet \otimes _{\mathcal{O}_ Y}^{\mathbf{L}} \mathcal{G}^\bullet ) = Lf^*\mathcal{F}^\bullet \otimes _{\mathcal{O}_ X}^{\mathbf{L}} Lf^*\mathcal{G}^\bullet \]

for $\mathcal{F}^\bullet , \mathcal{G}^\bullet \in \mathop{\mathrm{Ob}}\nolimits (D(X))$.

Proof. We may assume that $\mathcal{F}^\bullet $ and $\mathcal{G}^\bullet $ are K-flat complexes. In this case $\mathcal{F}^\bullet \otimes _{\mathcal{O}_ Y}^{\mathbf{L}} \mathcal{G}^\bullet $ is just the total complex associated to the double complex $\mathcal{F}^\bullet \otimes _{\mathcal{O}_ Y} \mathcal{G}^\bullet $. By Lemma 20.26.5 $\text{Tot}(\mathcal{F}^\bullet \otimes _{\mathcal{O}_ Y} \mathcal{G}^\bullet )$ is K-flat also. Hence the isomorphism of the lemma comes from the isomorphism

\[ \text{Tot}(f^*\mathcal{F}^\bullet \otimes _{\mathcal{O}_ X} f^*\mathcal{G}^\bullet ) \longrightarrow f^*\text{Tot}(\mathcal{F}^\bullet \otimes _{\mathcal{O}_ Y} \mathcal{G}^\bullet ) \]

whose constituents are the isomorphisms $f^*\mathcal{F}^ p \otimes _{\mathcal{O}_ X} f^*\mathcal{G}^ q \to f^*(\mathcal{F}^ p \otimes _{\mathcal{O}_ Y} \mathcal{G}^ q)$ of Modules, Lemma 17.15.4. $\square$


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