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 27.22.1. Let $j : U \to X$ be a quasi-compact open immersion of schemes.

  1. Any quasi-coherent sheaf on $U$ extends to a quasi-coherent sheaf on $X$.

  2. Let $\mathcal{F}$ be a quasi-coherent sheaf on $X$. Let $\mathcal{G} \subset \mathcal{F}|_ U$ be a quasi-coherent subsheaf. There exists a quasi-coherent subsheaf $\mathcal{H}$ of $\mathcal{F}$ such that $\mathcal{H}|_ U = \mathcal{G}$ as subsheaves of $\mathcal{F}|_ U$.

  3. Let $\mathcal{F}$ be a quasi-coherent sheaf on $X$. Let $\mathcal{G}$ be a quasi-coherent sheaf on $U$. Let $\varphi : \mathcal{G} \to \mathcal{F}|_ U$ be a morphism of $\mathcal{O}_ U$-modules. There exists a quasi-coherent sheaf $\mathcal{H}$ of $\mathcal{O}_ X$-modules and a map $\psi : \mathcal{H} \to \mathcal{F}$ such that $\mathcal{H}|_ U = \mathcal{G}$ and that $\psi |_ U = \varphi $.

Proof. An immersion is separated (see Schemes, Lemma 25.23.8) and $j$ is quasi-compact by assumption. Hence for any quasi-coherent sheaf $\mathcal{G}$ on $U$ the sheaf $j_*\mathcal{G}$ is an extension to $X$. See Schemes, Lemma 25.24.1 and Sheaves, Section 6.31.

Assume $\mathcal{F}$, $\mathcal{G}$ are as in (2). Then $j_*\mathcal{G}$ is a quasi-coherent sheaf on $X$ (see above). It is a subsheaf of $j_*j^*\mathcal{F}$. Hence the kernel

\[ \mathcal{H} = \mathop{\mathrm{Ker}}(\mathcal{F} \oplus j_* \mathcal{G} \longrightarrow j_*j^*\mathcal{F}) \]

is quasi-coherent as well, see Schemes, Section 25.24. It is formal to check that $\mathcal{H} \subset \mathcal{F}$ and that $\mathcal{H}|_ U = \mathcal{G}$ (using the material in Sheaves, Section 6.31 again).

The same proof as above works. Just take $\mathcal{H} = \mathop{\mathrm{Ker}}(\mathcal{F} \oplus j_* \mathcal{G} \to j_*j^*\mathcal{F})$ with its obvious map to $\mathcal{F}$ and its obvious identification with $\mathcal{G}$ over $U$. $\square$


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