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 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 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 the union of all these sheaves also reflexive?



Criterions for Reflexiveness of sheaves and a special case
I'll add another example to the mix even in the ring setting (as opposed to the sheaf setting).
Fix a DVR $(R, \langle x \rangle)$. Then the fraction field $K(R) = \bigcup\_n (x^{-n}R)$, is an ascending union of free (reflexive) modules. But clearly $K(R)$ is not itself reflexive.


On the other hand, under mild conditions on the scheme $X$, for example if it is normal, then for a *coherent* module being reflexive is equivalent to being S2 (Serre's second condition). The S2 condition behaves fairly well for quasi-coherent modules and might be useful for you. For example, see the paper(s) of Hartshorne: *"Generalized divisors ..."*


