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A coherent sheaf $V$ on a say noetherian scheme is called reflexive if the canonical map $V \rightarrow \mathcal Hom_{\mathcal O_X}(\mathcal Hom_{\mathcal O_X}(V,\mathcal O_X),\mathcal O_X)$ is an isomorphism of sheaves.

In principle, one can define this notion also for quasicoherent sheaves, and this is what my question is about: does one have criterions criteria about when a quasicoherent module is reflexive? Or is the question of reflexiveness in general very hard to answer?

What I am particularly interested in as a special case:

if

If one has an infinite chain of inclusions $V_1 \subset V_2 \subset ...$ reflexive sheaves, then one knows of course that each $V_i$ is reflexive. But is also the union of all these sheaves also reflexive?
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A coherent sheaf $V$ on a say noetherian scheme is called reflexive if the canonical map $V \rightarrow \mathcal Hom_{\mathcal O_X}(\mathcal Hom_{\mathcal O_X}(V,\mathcal O_X),\mathcal O_X)$ is an isomorphism of sheaves.

In principle, one can define this notion also for quasicoherent sheaves, and this is what my question is about: does one have criterions about when a quasicoherent module is reflexive? Or is the question of reflexiveness in general very hard to answer?

What I am particularly interested in as a special case:

if one has an infinite chain of inclusions $V_1 \subset V_2 \subset ...$ of vector bundles (each of finite rank)reflexive sheaves, then one knows of course that each $V_i$ is reflexive. But is also the union of all these sheaves (direct limit) reflexive?
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Criterions for Reflexiveness of sheaves and a special case

A coherent sheaf $V$ on a say noetherian scheme is called reflexive if the canonical map $V \rightarrow \mathcal Hom_{\mathcal O_X}(\mathcal Hom_{\mathcal O_X}(V,\mathcal O_X),\mathcal O_X)$ is an isomorphism of sheaves.

In principle, one can define this notion also for quasicoherent sheaves, and this is what my question is about: does one have criterions about when a quasicoherent module is reflexive? Or is the question of reflexiveness in general very hard to answer?

What I am particularly interested in as a special case:

if one has an infinite chain of inclusions $V_1 \subset V_2 \subset ...$ of vector bundles (each of finite rank), then one knows of course that each $V_i$ is reflexive. But is also the union of all these sheaves (direct limit) reflexive?