Remark 20.42.12 (0G7A)—The Stacks project

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

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

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 $\mathcal{O}_ Y$).

Comments (2)

Comment #10608 by Dinglong Wang on July 24, 2025 at 19:33

Minor typos: every instance of $M$ in $\cdots \otimes^{\mathbf{L}} Rf_* M$ and $\cdots \otimes^{\mathbf{L}} M$ should be replaced by $K$ .

$L$ is not used in the remark.

If we want to stay consistent with the previous notations, we should change all instances like $\mathrm{Hom}(K, M) \otimes M$ to $\mathrm{Hom}(L, K) \otimes L$ , update the bottom right corner to $Rf_* K$ and remove $M$ in "let $K$ , $L$ , $M$ be objects of ...$.

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2 comment(s) on Section 20.42: Internal hom in the derived category

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