The Stacks project

Lemma 52.6.1. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $f$ be a global section of $\mathcal{O}$.

  1. For $L, N \in D(\mathcal{O}_ f)$ we have $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(L, N) = R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ f}(L, N)$. In particular the two $\mathcal{O}_ f$-structures on $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(L, N)$ agree.

  2. For $K \in D(\mathcal{O})$ and $L \in D(\mathcal{O}_ f)$ we have

    \[ R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(L, K) = R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ f}(L, R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(\mathcal{O}_ f, K)) \]

    In particular $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(\mathcal{O}_ f, R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(\mathcal{O}_ f, K)) = R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(\mathcal{O}_ f, K)$.

  3. If $g$ is a second global section of $\mathcal{O}$, then

    \[ R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(\mathcal{O}_ f, R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(\mathcal{O}_ g, K)) = R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(\mathcal{O}_{gf}, K). \]

Proof. Proof of (1). Let $\mathcal{J}^\bullet $ be a K-injective complex of $\mathcal{O}_ f$-modules representing $N$. By Cohomology on Sites, Lemma 21.20.10 it follows that $\mathcal{J}^\bullet $ is a K-injective complex of $\mathcal{O}$-modules as well. Let $\mathcal{F}^\bullet $ be a complex of $\mathcal{O}_ f$-modules representing $L$. Then

\[ R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(L, N) = R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(\mathcal{F}^\bullet , \mathcal{J}^\bullet ) = R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ f}(\mathcal{F}^\bullet , \mathcal{J}^\bullet ) \]

by Modules on Sites, Lemma 18.11.4 because $\mathcal{J}^\bullet $ is a K-injective complex of $\mathcal{O}$ and of $\mathcal{O}_ f$-modules.

Proof of (2). Let $\mathcal{I}^\bullet $ be a K-injective complex of $\mathcal{O}$-modules representing $K$. Then $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(\mathcal{O}_ f, K)$ is represented by $\mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(\mathcal{O}_ f, \mathcal{I}^\bullet )$ which is a K-injective complex of $\mathcal{O}_ f$-modules and of $\mathcal{O}$-modules by Cohomology on Sites, Lemmas 21.20.11 and 21.20.10. Let $\mathcal{F}^\bullet $ be a complex of $\mathcal{O}_ f$-modules representing $L$. Then

\[ R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(L, K) = R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(\mathcal{F}^\bullet , \mathcal{I}^\bullet ) = R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ f}(\mathcal{F}^\bullet , \mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(\mathcal{O}_ f, \mathcal{I}^\bullet )) \]

by Modules on Sites, Lemma 18.27.6 and because $\mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(\mathcal{O}_ f, \mathcal{I}^\bullet )$ is a K-injective complex of $\mathcal{O}_ f$-modules.

Proof of (3). This follows from the fact that $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(\mathcal{O}_ g, \mathcal{I}^\bullet )$ is K-injective as a complex of $\mathcal{O}$-modules and the fact that $\mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(\mathcal{O}_ f, \mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(\mathcal{O}_ g, \mathcal{H})) = \mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(\mathcal{O}_{gf}, \mathcal{H})$ for all sheaves of $\mathcal{O}$-modules $\mathcal{H}$. $\square$


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