The Stacks project

\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

18.27 Internal Hom

Let $\mathcal{C}$ be a category and let $\mathcal{O}$ be a presheaf of rings. Let $\mathcal{F}$, $\mathcal{G}$ be presheaves of $\mathcal{O}$-modules. Consider the rule

\[ U \longmapsto \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_ U}(\mathcal{F}|_ U, \mathcal{G}|_ U). \]

For $\varphi : V \to U$ in $\mathcal{C}$ we define a restriction mapping

\[ \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_ U}(\mathcal{F}|_ U, \mathcal{G}|_ U) \longrightarrow \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_ V}(\mathcal{F}|_ V, \mathcal{G}|_ V) \]

by restricting via the relocalization morphism $j : \mathcal{C}/V \to \mathcal{C}/U$, see Sites, Lemma 7.25.8. Hence this defines a presheaf $\mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(\mathcal{F}, \mathcal{G})$. In addition, given an element $\varphi \in \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}|_ U}(\mathcal{F}|_ U, \mathcal{G}|_ U)$ and a section $f \in \mathcal{O}(U)$ then we can define $f\varphi \in \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}|_ U}(\mathcal{F}|_ U, \mathcal{G}|_ U)$ by either precomposing with multiplication by $f$ on $\mathcal{F}|_ U$ or postcomposing with multiplication by $f$ on $\mathcal{G}|_ U$ (it gives the same result). Hence we in fact get a presheaf of $\mathcal{O}$-modules. There is a canonical “evaluation” morphism

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

Lemma 18.27.1. If $\mathcal{C}$ is a site, $\mathcal{O}$ is a sheaf of rings, $\mathcal{F}$ is a presheaf of $\mathcal{O}$-modules, and $\mathcal{G}$ is a sheaf of $\mathcal{O}$-modules, then $\mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(\mathcal{F}, \mathcal{G})$ is a sheaf of $\mathcal{O}$-modules.

Proof. Omitted. Hints: Note first that $\mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(\mathcal{F}, \mathcal{G}) = \mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(\mathcal{F}^\# , \mathcal{G})$, which reduces the question to the case where both $\mathcal{F}$ and $\mathcal{G}$ are sheaves. The result for sheaves of sets is Sites, Lemma 7.26.1. $\square$

Lemma 18.27.2. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{F}, \mathcal{G}$ be sheaves of $\mathcal{O}$-modules. Then formation of $\mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(\mathcal{F}, \mathcal{G})$ commutes with restriction to $U$ for $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$.

Proof. Immediate from the definition. $\square$

In the situation of Lemma 18.27.1 the “evaluation” morphism factors through the tensor product of sheaves of modules

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

Lemma 18.27.3. Internal hom and (co)limits. Let $\mathcal{C}$ be a category and let $\mathcal{O}$ be a presheaf of rings.

  1. For any presheaf of $\mathcal{O}$-modules $\mathcal{F}$ the functor

    \[ \textit{PMod}(\mathcal{O}) \longrightarrow \textit{PMod}(\mathcal{O}) , \quad \mathcal{G} \longmapsto \mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(\mathcal{F}, \mathcal{G}) \]

    commutes with arbitrary limits.

  2. For any presheaf of $\mathcal{O}$-modules $\mathcal{G}$ the functor

    \[ \textit{PMod}(\mathcal{O}) \longrightarrow \textit{PMod}(\mathcal{O})^{opp} , \quad \mathcal{F} \longmapsto \mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(\mathcal{F}, \mathcal{G}) \]

    commutes with arbitrary colimits, in a formula

    \[ \mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(\mathop{\mathrm{colim}}\nolimits _ i \mathcal{F}_ i, \mathcal{G}) = \mathop{\mathrm{lim}}\nolimits _ i \mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(\mathcal{F}_ i, \mathcal{G}). \]

Suppose that $\mathcal{C}$ is a site, and $\mathcal{O}$ is a sheaf of rings.

  1. For any sheaf of $\mathcal{O}$-modules $\mathcal{F}$ the functor

    \[ \textit{Mod}(\mathcal{O}) \longrightarrow \textit{Mod}(\mathcal{O}) , \quad \mathcal{G} \longmapsto \mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(\mathcal{F}, \mathcal{G}) \]

    commutes with arbitrary limits.

  2. For any sheaf of $\mathcal{O}$-modules $\mathcal{G}$ the functor

    \[ \textit{Mod}(\mathcal{O}) \longrightarrow \textit{Mod}(\mathcal{O})^{opp} , \quad \mathcal{F} \longmapsto \mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(\mathcal{F}, \mathcal{G}) \]

    commutes with arbitrary colimits, in a formula

    \[ \mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(\mathop{\mathrm{colim}}\nolimits _ i \mathcal{F}_ i, \mathcal{G}) = \mathop{\mathrm{lim}}\nolimits _ i \mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(\mathcal{F}_ i, \mathcal{G}). \]

Proof. Let $\mathcal{I} \to \textit{PMod}(\mathcal{O})$, $i \mapsto \mathcal{G}_ i$ be a diagram. Let $U$ be an object of the category $\mathcal{C}$. As $j_ U^*$ is both a left and a right adjoint we see that $\mathop{\mathrm{lim}}\nolimits _ i j_ U^*\mathcal{G}_ i = j_ U^* \mathop{\mathrm{lim}}\nolimits _ i \mathcal{G}_ i$. Hence we have

\begin{align*} \mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(\mathcal{F}, \mathop{\mathrm{lim}}\nolimits _ i \mathcal{G}_ i)(U) & = \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_ U}(\mathcal{F}|_ U, \mathop{\mathrm{lim}}\nolimits _ i \mathcal{G}_ i|_ U) \\ & = \mathop{\mathrm{lim}}\nolimits _ i \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_ U}(\mathcal{F}|_ U, \mathcal{G}_ i|_ U) \\ & = \mathop{\mathrm{lim}}\nolimits _ i \mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(\mathcal{F}, \mathcal{G}_ i)(U) \end{align*}

by definition of a limit. This proves (1). Part (2) is proved in exactly the same way. Part (3) follows from (1) because the limit of a diagram of sheaves is the same as the limit in the category of presheaves. Finally, (4) follow because, in the formula we have

\[ \mathop{Mor}\nolimits _{\textit{Mod}(\mathcal{O})}( \mathop{\mathrm{colim}}\nolimits _ i \mathcal{F}_ i, \mathcal{G}) = \mathop{Mor}\nolimits _{\textit{PMod}(\mathcal{O})}( \mathop{\mathrm{colim}}\nolimits ^{PSh}_ i \mathcal{F}_ i, \mathcal{G}) \]

as the colimit $\mathop{\mathrm{colim}}\nolimits _ i \mathcal{F}_ i$ is the sheafification of the colimit $\mathop{\mathrm{colim}}\nolimits ^{PSh}_ i \mathcal{F}_ i$ in $\textit{PMod}(\mathcal{O})$. Hence (4) follows from (2) (by the remark on limits above again). $\square$

Lemma 18.27.4. 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,

    \[ \mathop{Mor}\nolimits _{\textit{PMod}(\mathcal{O})}( \mathcal{F} \otimes _{p, \mathcal{O}} \mathcal{G}, \mathcal{H}) = \mathop{Mor}\nolimits _{\textit{PMod}(\mathcal{O})}( \mathcal{F}, \mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(\mathcal{G}, \mathcal{H})) \]
  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,

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

Proof. This is the analogue of Algebra, Lemma 10.11.8. The proof is the same, and is omitted. $\square$

Lemma 18.27.5. Tensor product and colimits. Let $\mathcal{C}$ be a category and let $\mathcal{O}$ be a presheaf of rings.

  1. For any presheaf of $\mathcal{O}$-modules $\mathcal{F}$ the functor

    \[ \textit{PMod}(\mathcal{O}) \longrightarrow \textit{PMod}(\mathcal{O}) , \quad \mathcal{G} \longmapsto \mathcal{F} \otimes _{p, \mathcal{O}} \mathcal{G} \]

    commutes with arbitrary colimits.

  2. Suppose that $\mathcal{C}$ is a site, and $\mathcal{O}$ is a sheaf of rings. For any sheaf of $\mathcal{O}$-modules $\mathcal{F}$ the functor

    \[ \textit{Mod}(\mathcal{O}) \longrightarrow \textit{Mod}(\mathcal{O}) , \quad \mathcal{G} \longmapsto \mathcal{F} \otimes _\mathcal {O} \mathcal{G} \]

    commutes with arbitrary colimits.

Proof. This is because tensor product is adjoint to internal hom according to Lemma 18.27.4. See Categories, Lemma 4.24.5. $\square$

Lemma 18.27.6. Let $\mathcal{C}$ be a category, resp. a site Let $\mathcal{O} \to \mathcal{O}'$ be a map of presheaves, resp. sheaves of rings. Then

\[ \mathop{\mathrm{Hom}}\nolimits _\mathcal {O}(\mathcal{G}, \mathcal{F}) = \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}'}(\mathcal{G}, \mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(\mathcal{O}', \mathcal{F})) \]

for any $\mathcal{O}'$-module $\mathcal{G}$ and $\mathcal{O}$-module $\mathcal{F}$.

Proof. This is the analogue of Algebra, Lemma 10.13.4. The proof is the same, and is omitted. $\square$

Lemma 18.27.7. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$. For $\mathcal{G}$ in $\textit{Mod}(\mathcal{O}_ U)$ and $\mathcal{F}$ in $\textit{Mod}(\mathcal{O})$ we have $j_{U!}\mathcal{G} \otimes _\mathcal {O} \mathcal{F} = j_{U!}(\mathcal{G} \otimes _{\mathcal{O}_ U} \mathcal{F}|_ U)$.

Proof. Let $\mathcal{H}$ be an object of $\textit{Mod}(\mathcal{O})$. Then

\begin{align*} \mathop{\mathrm{Hom}}\nolimits _\mathcal {O}(j_{U!}(\mathcal{G} \otimes _{\mathcal{O}_ U} \mathcal{F}|_ U), \mathcal{H}) & = \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_ U}(\mathcal{G} \otimes _{\mathcal{O}_ U} \mathcal{F}|_ U, \mathcal{H}|_ U) \\ & = \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_ U}(\mathcal{G}, \mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ U}(\mathcal{F}|_ U, \mathcal{H}|_ U)) \\ & = \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_ U}(\mathcal{G}, \mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(\mathcal{F}, \mathcal{H})|_ U) \\ & = \mathop{\mathrm{Hom}}\nolimits _\mathcal {O}(j_{U!}\mathcal{G}, \mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(\mathcal{F}, \mathcal{H})) \\ & = \mathop{\mathrm{Hom}}\nolimits _\mathcal {O}(j_{U!}\mathcal{G} \otimes _\mathcal {O} \mathcal{F}, \mathcal{H}) \end{align*}

The first equality because $j_{U!}$ is a left adjoint to restriction of modules. The second by Lemma 18.27.4. The third by Lemma 18.27.2. The fourth because $j_{U!}$ is a left adjoint to restriction of modules. The fifth by Lemma 18.27.4. The lemma follows from this and the Yoneda lemma. $\square$

Remark 18.27.8. Let $\mathcal{C}$ be a site. Let $\mathcal{F}$ be a sheaf of sets on $\mathcal{C}$ and consider the localization morphism $j : \mathop{\mathit{Sh}}\nolimits (\mathcal{C})/\mathcal{F} \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C})$. See Sites, Definition 7.30.4. We claim that (a) $j_!\mathbf{Z} = \mathbf{Z}_\mathcal {F}^\# $ and (b) $j_!(j^{-1}\mathcal{H}) = j_!\mathbf{Z} \otimes _\mathbf {Z} \mathcal{H}$ for any abelian sheaf $\mathcal{H}$ on $\mathcal{C}$. Let $\mathcal{G}$ be an abelian on $\mathcal{C}$. Part (a) follows from the Yoneda lemma because

\[ \mathop{\mathrm{Hom}}\nolimits (j_!\mathbf{Z}, \mathcal{G}) = \mathop{\mathrm{Hom}}\nolimits (\mathbf{Z}, j^{-1}\mathcal{G}) = \mathop{\mathrm{Hom}}\nolimits (\mathbf{Z}_\mathcal {F}^\# , \mathcal{G}) \]

where the second equality holds because both sides of the equality evaluate to the set of maps from $\mathcal{F} \to \mathcal{G}$ viewed as an abelian group. For (b) we use the Yoneda lemma and

\begin{align*} \mathop{\mathrm{Hom}}\nolimits (j_!(j^{-1}\mathcal{H}), \mathcal{G}) & = \mathop{\mathrm{Hom}}\nolimits (j^{-1}\mathcal{H}, j^{-1}\mathcal{G}) \\ & = \mathop{\mathrm{Hom}}\nolimits (\mathbf{Z}, \mathop{\mathcal{H}\! \mathit{om}}\nolimits (j^{-1}\mathcal{H}, j^{-1}\mathcal{G})) \\ & = \mathop{\mathrm{Hom}}\nolimits (\mathbf{Z}, j^{-1}\mathop{\mathcal{H}\! \mathit{om}}\nolimits (\mathcal{H}, \mathcal{G})) \\ & = \mathop{\mathrm{Hom}}\nolimits (j_!\mathbf{Z}, \mathop{\mathcal{H}\! \mathit{om}}\nolimits (\mathcal{H}, \mathcal{G})) \\ & = \mathop{\mathrm{Hom}}\nolimits (j_!\mathbf{Z} \otimes _\mathbf {Z} \mathcal{H}, \mathcal{G}) \end{align*}

Here we use adjunction, the fact that taking $\mathop{\mathcal{H}\! \mathit{om}}\nolimits $ commutes with localization, and Lemma 18.27.4.


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