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*}

20.38 Global derived hom

Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $K, L \in D(\mathcal{O}_ X)$. Using the construction of the internal hom in the derived category we obtain a well defined object

\[ R\mathop{\mathrm{Hom}}\nolimits _ X(K, L) = R\Gamma (X, R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (K, L)) \]

in $D(\Gamma (X, \mathcal{O}_ X))$. We will sometimes write $R\mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_ X}(K, L)$ for this object. By Lemma 20.36.1 we have

\[ H^0(R\mathop{\mathrm{Hom}}\nolimits _ X(K, L)) = \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}_ X)}(K, L), \quad H^ p(R\mathop{\mathrm{Hom}}\nolimits _ X(K, L)) = \mathop{\mathrm{Ext}}\nolimits _{D(\mathcal{O}_ X)}^ p(K, L) \]

If $f : Y \to X$ is a morphism of ringed spaces, then there is a canonical map

\[ R\mathop{\mathrm{Hom}}\nolimits _ X(K, L) \longrightarrow R\mathop{\mathrm{Hom}}\nolimits _ Y(Lf^*K, Lf^*L) \]

in $D(\Gamma (X, \mathcal{O}_ X))$ by taking global sections of the map defined in Remark 20.36.11.


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