Lemma 30.12.2. Let X be a Noetherian scheme. Let i : Z \to X be an integral closed subscheme. Let \xi \in Z be the generic point. Let \mathcal{F} be a coherent sheaf on X. Assume that \mathcal{F}_\xi is annihilated by \mathfrak m_\xi . Then there exist an integer r \geq 0 and a coherent sheaf of ideals \mathcal{I} \subset \mathcal{O}_ Z and an injective map of coherent sheaves
i_*\left(\mathcal{I}^{\oplus r}\right) \to \mathcal{F}
which is an isomorphism in a neighbourhood of \xi .
Proof.
Let \mathcal{J} \subset \mathcal{O}_ X be the ideal sheaf of Z. Let \mathcal{F}' \subset \mathcal{F} be the subsheaf of local sections of \mathcal{F} which are annihilated by \mathcal{J}. It is a quasi-coherent sheaf by Properties, Lemma 28.24.2. Moreover, \mathcal{F}'_\xi = \mathcal{F}_\xi because \mathcal{J}_\xi = \mathfrak m_\xi and part (3) of Properties, Lemma 28.24.2. By Lemma 30.9.5 we see that \mathcal{F}' \to \mathcal{F} induces an isomorphism in a neighbourhood of \xi . Hence we may replace \mathcal{F} by \mathcal{F}' and assume that \mathcal{F} is annihilated by \mathcal{J}.
Assume \mathcal{J}\mathcal{F} = 0. By Lemma 30.9.8 we can write \mathcal{F} = i_*\mathcal{G} for some coherent sheaf \mathcal{G} on Z. Suppose we can find a morphism \mathcal{I}^{\oplus r} \to \mathcal{G} which is an isomorphism in a neighbourhood of the generic point \xi of Z. Then applying i_* (which is left exact) we get the result of the lemma. Hence we have reduced to the case X = Z.
Suppose Z = X is an integral Noetherian scheme with generic point \xi . Note that \mathcal{O}_{X, \xi } = \kappa (\xi ) is the function field of X in this case. Since \mathcal{F}_\xi is a finite \mathcal{O}_\xi -module we see that r = \dim _{\kappa (\xi )} \mathcal{F}_\xi is finite. Hence the sheaves \mathcal{O}_ X^{\oplus r} and \mathcal{F} have isomorphic stalks at \xi . By Lemma 30.9.6 there exists a nonempty open U \subset X and a morphism \psi : \mathcal{O}_ X^{\oplus r}|_ U \to \mathcal{F}|_ U which is an isomorphism at \xi , and hence an isomorphism in a neighbourhood of \xi by Lemma 30.9.5. By Schemes, Lemma 26.12.4 there exists a quasi-coherent sheaf of ideals \mathcal{I} \subset \mathcal{O}_ X whose associated closed subscheme Y \subset X is the complement of U. By Lemma 30.10.5 there exists an n \geq 0 and a morphism \mathcal{I}^ n(\mathcal{O}_ X^{\oplus r}) \to \mathcal{F} which recovers our \psi over U. Since \mathcal{I}^ n(\mathcal{O}_ X^{\oplus r}) = (\mathcal{I}^ n)^{\oplus r} we get a map as in the lemma. It is injective because X is integral and it is injective at the generic point of X (easy proof omitted).
\square
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Comment #8385 by Ryo Suzuki on
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