I am reading "Ample divisors on fine moduli spaces on the Projective plane" by Stromme. In the proof of Proposition 2.4, he seems to claim that if $E$ is a torsion free sheaf of projective dimension at most 1 on a smooth variety and the singular locus of $E$ has codimension at least $3$, then $E$ is reflexive. Of course the converse is true without the projective dimension assumption. 


Does anyone know an explanation or reference for this fact?



Criterion for a sheaf to be reflexive
In general the converse is not true --- take for example the ideal of a point on 3-fold (its reflexive hull is the structure sheaf). On the other hand, if you know that $pd(F) \le 1$ then there is an exact sequence
$$
0 \to E\_1 \stackrel{f}\to E\_2 \to F \to 0
$$
with $E\_1$ and $E\_2$ locally free. Taking the dual one gets
$$
0 \to F^\* \to E\_2^\* \stackrel{f^\*}\to E\_1^\* \to F\_1 \to 0,
$$
where the last sheaf $F\_1 := \mathcal{Ext}^1(F,O)$ is supported in codimension 3. In particular, $\mathcal{Ext}^i(F\_1,O) = 0$ for $i \le 2$. It follows that $\mathcal{Ext}^i(F\_1,O)$ do not contribute into $F^{\*\*}$ and so there is an exact sequence
$$
0 \to E\_1^{\*\*} \stackrel{f^{\*\*}}\to E\_2^{\*\*} \to F^{\*\*} \to 0,
$$
and so since $E\_i$ are locally free, it follows that $F^{\*\*} = F$.


