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?



Kernels of surjections from a vector bundle to a reflexive sheaf
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.


