Reflexive sheaves on a regular quasi-projective variety can be characterized by the following property that they are the kernel of a surjection from a vector bundle to a torsion-free sheaf. I wonder what the class of reflexive sheaves that are kernel of a surjection from a vector bundle to a reflexive sheaf consists of? They are clearly reflexive but can any reflexive sheaf written in that form? Or under what conditions every reflexive sheaf is a kernel of a surjection from a vector bundle to another reflexive sheaf? Is there any other way to characterize these specific reflexive sheaves?
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If the ambient variety is smooth, the locus of points where a reflexive sheaf is not locally free has codimension at least 3. And for a sheaf which is a kernel of a surjection from locally free to reflexive, this sheaf has codimension at least 4. So, the classes are different.
In general, one can consider so-called locally $m$-syzygy sheaves --- these are sheaves $F$ for which there is a resolution $$ 0 \to F \to E_1 \to E_2 \to \dots \to E_m $$ (not necessarily exact in the rightmost term) with $E_i$ locally free. For $m = 1$ these are torsion-free sheaves, for $m = 2$ reflexive, and for $m = 3$ your class.
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$\begingroup$ Is there a reference that proves this? Is this still true of the base field is not algebraically closed? $\endgroup$ Commented Dec 23, 2020 at 4:42
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2$\begingroup$ I learnt this stuff from the book of Okonek-Schneider-Spindler. $\endgroup$– SashaCommented Dec 23, 2020 at 5:16
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1$\begingroup$ As an answer to the question I asked above. The converse is not true i.e. if a reflexive sheaf is a vector bundle on the complement of a codimension 4 subvariety, it doesn't need to be locally 3-syzygy sheaf. I figured this out through an unexpected contradiction, there might be an easier way of showing it. $\endgroup$ Commented Dec 23, 2020 at 23:50