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*}

Lemma 10.10.2. Let $R$ be a ring. Let $M$ be a finitely presented $R$-module. Let $N$ be an $R$-module.

  1. For $f \in R$ we have $\mathop{\mathrm{Hom}}\nolimits _ R(M, N)_ f = \mathop{\mathrm{Hom}}\nolimits _{R_ f}(M_ f, N_ f) = \mathop{\mathrm{Hom}}\nolimits _ R(M_ f, N_ f)$,

  2. for a multiplicative subset $S$ of $R$ we have

    \[ S^{-1}\mathop{\mathrm{Hom}}\nolimits _ R(M, N) = \mathop{\mathrm{Hom}}\nolimits _{S^{-1}R}(S^{-1}M, S^{-1}N) = \mathop{\mathrm{Hom}}\nolimits _ R(S^{-1}M, S^{-1}N). \]

Proof. Part (1) is a special case of part (2). The second equality in (2) follows from Lemma 10.9.7. Choose a presentation

\[ \bigoplus \nolimits _{j = 1, \ldots , m} R \longrightarrow \bigoplus \nolimits _{i = 1, \ldots , n} R \to M \to 0. \]

By Lemma 10.10.1 this gives an exact sequence

\[ 0 \to \mathop{\mathrm{Hom}}\nolimits _ R(M, N) \to \bigoplus \nolimits _{i = 1, \ldots , n} N \longrightarrow \bigoplus \nolimits _{j = 1, \ldots , m} N. \]

Inverting $S$ and using Proposition 10.9.12 we get an exact sequence

\[ 0 \to S^{-1}\mathop{\mathrm{Hom}}\nolimits _ R(M, N) \to \bigoplus \nolimits _{i = 1, \ldots , n} S^{-1}N \longrightarrow \bigoplus \nolimits _{j = 1, \ldots , m} S^{-1}N \]

and the result follows since $S^{-1}M$ sits in an exact sequence

\[ \bigoplus \nolimits _{j = 1, \ldots , m} S^{-1}R \longrightarrow \bigoplus \nolimits _{i = 1, \ldots , n} S^{-1}R \to S^{-1}M \to 0 \]

which induces (by Lemma 10.10.1) the exact sequence

\[ 0 \to \mathop{\mathrm{Hom}}\nolimits _{S^{-1}R}(S^{-1}M, S^{-1}N) \to \bigoplus \nolimits _{i = 1, \ldots , n} S^{-1}N \longrightarrow \bigoplus \nolimits _{j = 1, \ldots , m} S^{-1}N \]

which is the same as the one above. $\square$


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