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 20.8.1. Let $X$ be a ringed space. Let $U \subset X$ be an open subspace.

  1. If $\mathcal{I}$ is an injective $\mathcal{O}_ X$-module then $\mathcal{I}|_ U$ is an injective $\mathcal{O}_ U$-module.

  2. For any sheaf of $\mathcal{O}_ X$-modules $\mathcal{F}$ we have $H^ p(U, \mathcal{F}) = H^ p(U, \mathcal{F}|_ U)$.

Proof. Denote $j : U \to X$ the open immersion. Recall that the functor $j^{-1}$ of restriction to $U$ is a right adjoint to the functor $j_!$ of extension by $0$, see Sheaves, Lemma 6.31.8. Moreover, $j_!$ is exact. Hence (1) follows from Homology, Lemma 12.26.1.

By definition $H^ p(U, \mathcal{F}) = H^ p(\Gamma (U, \mathcal{I}^\bullet ))$ where $\mathcal{F} \to \mathcal{I}^\bullet $ is an injective resolution in $\textit{Mod}(\mathcal{O}_ X)$. By the above we see that $\mathcal{F}|_ U \to \mathcal{I}^\bullet |_ U$ is an injective resolution in $\textit{Mod}(\mathcal{O}_ U)$. Hence $H^ p(U, \mathcal{F}|_ U)$ is equal to $H^ p(\Gamma (U, \mathcal{I}^\bullet |_ U))$. Of course $\Gamma (U, \mathcal{F}) = \Gamma (U, \mathcal{F}|_ U)$ for any sheaf $\mathcal{F}$ on $X$. Hence the equality in (2). $\square$


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