The Stacks project

Remark 20.42.12. Let $h : X \to Y$ be a morphism of ringed spaces. Let $K, L, M$ be objects of $D(\mathcal{O}_ Y)$. The diagram

\[ \xymatrix{ Rf_*R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(K, M) \otimes _{\mathcal{O}_ Y}^\mathbf {L} Rf_*M \ar[r] \ar[d] & Rf_*\left(R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(K, M) \otimes _{\mathcal{O}_ X}^\mathbf {L} M\right) \ar[d] \\ R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ Y}(Rf_*K, Rf_*M) \otimes _{\mathcal{O}_ Y}^\mathbf {L} Rf_*M \ar[r] & Rf_*M } \]

is commutative. Here the left vertical arrow comes from Remark 20.42.11. The top horizontal arrow is Remark 20.28.7. The other two arrows are instances of the map in Lemma 20.42.5 (with one of the entries replaced with $\mathcal{O}_ X$ or $\mathcal{O}_ Y$).

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