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The Stacks project

Lemma 28.22.1. 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 26.23.8) 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 26.24.1 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 26.24. 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).

Part (3) is proved in the same manner as (2). 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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