The Stacks project

Lemma 20.27.5. Let $f : (X, \mathcal{O}_ X) \to (Y, \mathcal{O}_ Y)$ be a morphism of ringed spaces. Let $\mathcal{K}^\bullet $ and $\mathcal{M}^\bullet $ be complexes of $\mathcal{O}_ Y$-modules. The diagram

\[ \xymatrix{ Lf^*(\mathcal{K}^\bullet \otimes _{\mathcal{O}_ Y}^\mathbf {L} \mathcal{M}^\bullet ) \ar[r] \ar[d] & Lf^*\text{Tot}(\mathcal{K}^\bullet \otimes _{\mathcal{O}_ Y} \mathcal{M}^\bullet ) \ar[d] \\ Lf^*\mathcal{K}^\bullet \otimes _{\mathcal{O}_ X}^\mathbf {L} Lf^*\mathcal{M}^\bullet \ar[d] & f^*\text{Tot}(\mathcal{K}^\bullet \otimes _{\mathcal{O}_ Y} \mathcal{M}^\bullet ) \ar[d] \\ f^*\mathcal{K}^\bullet \otimes _{\mathcal{O}_ X}^\mathbf {L} f^*\mathcal{M}^\bullet \ar[r] & \text{Tot}(f^*\mathcal{K}^\bullet \otimes _{\mathcal{O}_ X} f^*\mathcal{M}^\bullet ) } \]

commutes.

Proof. We will use the existence of K-flat resolutions as in Lemma 20.26.7. If we choose such resolutions $\mathcal{P}^\bullet \to \mathcal{K}^\bullet $ and $\mathcal{Q}^\bullet \to \mathcal{M}^\bullet $, then we see that

\[ \xymatrix{ Lf^*\text{Tot}(\mathcal{P}^\bullet \otimes _{\mathcal{O}_ Y} \mathcal{Q}^\bullet ) \ar[r] \ar[d] & Lf^*\text{Tot}(\mathcal{K}^\bullet \otimes _{\mathcal{O}_ Y} \mathcal{M}^\bullet ) \ar[d] \\ f^*\text{Tot}(\mathcal{P}^\bullet \otimes _{\mathcal{O}_ Y} \mathcal{Q}^\bullet ) \ar[d] \ar[r] & f^*\text{Tot}(\mathcal{K}^\bullet \otimes _{\mathcal{O}_ Y} \mathcal{M}^\bullet ) \ar[d] \\ \text{Tot}(f^*\mathcal{P}^\bullet \otimes _{\mathcal{O}_ X} f^*\mathcal{Q}^\bullet ) \ar[r] & \text{Tot}(f^*\mathcal{K}^\bullet \otimes _{\mathcal{O}_ X} f^*\mathcal{M}^\bullet ) } \]

commutes. However, now the left hand side of the diagram is the left hand side of the diagram by our choice of $\mathcal{P}^\bullet $ and $\mathcal{Q}^\bullet $ and Lemma 20.26.5. $\square$


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