Lemma 31.12.4 (0AY2)—The Stacks project

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Lemma 31.12.4. Let $X$ be an integral locally Noetherian scheme. Let $\mathcal{F}$ be a coherent $\mathcal{O}_ X$-module.

If $\mathcal{F}$ is reflexive, then $\mathcal{F}$ is torsion free.
The map $j : \mathcal{F} \longrightarrow \mathcal{F}^{**}$ is injective if and only if $\mathcal{F}$ is torsion free.

Proof. Omitted. See More on Algebra, Lemma 15.23.2. $\square$

Comments (2)

Comment #6273 by Matthieu Romagny on June 05, 2021 at 09:10

Add a point at the end of condition (2).

Comment #6395 by Johan on July 19, 2021 at 08:55

Thanks and fixed here.

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