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The Stacks project

Lemma 18.27.6. Let \mathcal{C} be a category. Let \mathcal{O} be a presheaf of rings.

  1. Let \mathcal{F}, \mathcal{G}, \mathcal{H} be presheaves of \mathcal{O}-modules. There is a canonical isomorphism

    \mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O} (\mathcal{F} \otimes _{p, \mathcal{O}} \mathcal{G}, \mathcal{H}) \longrightarrow \mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O} (\mathcal{F}, \mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(\mathcal{G}, \mathcal{H}))

    which is functorial in all three entries (sheaf Hom in all three spots). In particular,

    \mathop{\mathrm{Mor}}\nolimits _{\textit{PMod}(\mathcal{O})}( \mathcal{F} \otimes _{p, \mathcal{O}} \mathcal{G}, \mathcal{H}) = \mathop{\mathrm{Mor}}\nolimits _{\textit{PMod}(\mathcal{O})}( \mathcal{F}, \mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(\mathcal{G}, \mathcal{H}))
  2. Suppose that \mathcal{C} is a site, \mathcal{O} is a sheaf of rings, and \mathcal{F}, \mathcal{G}, \mathcal{H} are sheaves of \mathcal{O}-modules. There is a canonical isomorphism

    \mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O} (\mathcal{F} \otimes _\mathcal {O} \mathcal{G}, \mathcal{H}) \longrightarrow \mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O} (\mathcal{F}, \mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(\mathcal{G}, \mathcal{H}))

    which is functorial in all three entries (sheaf Hom in all three spots). In particular,

    \mathop{\mathrm{Mor}}\nolimits _{\textit{Mod}(\mathcal{O})}( \mathcal{F} \otimes _\mathcal {O} \mathcal{G}, \mathcal{H}) = \mathop{\mathrm{Mor}}\nolimits _{\textit{Mod}(\mathcal{O})}( \mathcal{F}, \mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(\mathcal{G}, \mathcal{H}))

Proof. This is the analogue of Algebra, Lemma 10.12.8. The proof is the same, and is omitted. \square


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