Injection into double dual is isomorphism away from codimension $\geq 2$ subscheme

Question

Let $X$ be a quasi-projective integral scheme and let $F$ be a torsion-free coherent sheaf over $X$. Then there is an injective map $$\phi\colon F\hookrightarrow F^{\vee\vee}.$$ Moreover, in my situation it is known that $F^{\vee\vee}$ is locally free. Is it true that $\phi$ is an isomorphism away from a closed subscheme of codimension $\geq 2$? Of course, this holds true if $X$ is normal, but is it also true for non-normal $X$?

Answer 1

I believe the following is true:

Proposition. Let $X$ be a locally noetherian scheme that is Gorenstein in codimension one and satisfies Serre's condition $S_1$. Then, for every torsion-free coherent sheaf $F$ on $X$, the natural morphism $$\phi\colon F \longrightarrow F^{\vee\vee}$$ is injective, and is an isomorphism away from a closed subscheme of codimension $\ge2$.

Proof. We first note that $\phi$ is injective since $X$ is Gorenstein in codimension zero and satisfies $S_1$, and hence "torsion-free" and "torsion-less" are equivalent [Vasconcelos 1968, Thm. A.1].

We now prove that $\phi$ is an isomorphism away from a closed subscheme of codimension $\ge 2$. Note that the non-isomorphism locus is closed since it is the support of $\operatorname{coker}(\phi)$, which is closed since $\operatorname{coker}(\phi)$ is coherent. It suffices to show that $\phi$ is an isomorphism in codimension one, and hence it suffices to show that if $R$ is a Gorenstein local ring, then all torsion-less $R$-modules are reflexive. But this follows from [Jans 1961, Cor. 1.3]. $\blacksquare$

Remark. The assumption that $X$ is Gorenstein in codimension one and satisfies $S_1$ is Marinari's condition $G_1$ [Marinari 1972, Def. 4.5]. I have not used this terminology in the statement because there is more than one definition known as $G_1$ in the literature (cf. [Ischebeck 1969, Def. 3.16] and [Reiten and Fossum 1970, p. 142]).