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I was reading Nakajima's "Lectures on Hilbert schemes of points on surfaces". In Chapter 2 he says that if $E$ is a torsion-free sheaf then $E^{**}$ is locally free. I guess that by `torsion-free' sheaf he meant on every open sets $U \subset X = \mathbb{P}^2$, $E(U)$ is a torsion-free $\mathscr{O}_X(U)$-module. I also think that duality is defined by $E^*= \text{Hom}_{\mathscr{O}_X}(E,\mathscr{O}_X)$. But then how do I show that $E^{**}$ is locally free?

Additional confusion on the matter: I'm not sure if I'm allowed to assume that $E$ is quasi-coherent or coherent here. He mentioned at the start of the chapter that $\text{rank} E = r$, which suggest that $E$ has to be coherent, but by talking about the rank of $E$ don't we already assume $E$ is locally free? How can we define a rank on a module that is not free?

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    $\begingroup$ Usually the rank of a not necessarily free module is defined as the rank of $M \otimes K(R)$, where $K(R)$ is the fraction field of the ring. $\endgroup$
    – Fredrik Meyer
    Commented Aug 3, 2016 at 9:10
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    $\begingroup$ Rank $1$ reflexive sheaves on a regular scheme are invertible sheaves. This should be in Hartshorne's article Stable reflexive sheaves. I don't think this has any hope of being true for higher ranks. $\endgroup$
    – Hoot
    Commented Aug 3, 2016 at 9:24
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    $\begingroup$ In that book $X$ is a surface, where it is true that $E$ torsionfree $=>$ $E^{**}$ is locally free. This follows from that fact that $E^{**}$ is reflexive, hence not locally free on a closed subset of codimension $>=$ 3 on $X$, hence nowhere. $\endgroup$
    – Bernie
    Commented Aug 3, 2016 at 13:11

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Regarding the question about what type of sheaves are under consideration, the answer is coherent sheaves. The corresponding algebra concepts are from the theory of finitely generated modules over Noetherian rings (and especially, from the theory of depth and related concepts).

As noted in comments, the property of reflexivity implying local freeness is special to surfaces. On a variety, a reflexive sheaf will be $S_2$ (where $S_n$ are the depth conditions that appear in the definition of Cohen--Macaulay: on a $n$-dimensional variety, being Cohen--Macaulay is the same as being $S_n$).

A theorem of Auslander--Buchsbaum shows that on a smooth variety, a coherent sheaf that is Cohen--Macaulay is locally free. So for smooth surfaces, reflexive equals $S_2$ equals Cohen--Macaulay equals locally free.

If $X$ is a smooth three-fold, if $\mathcal I$ is the ideal sheaf of a point, and if $\mathcal K$ denotes the kernel of a surjection $\mathcal F \to \mathcal I$, with $\mathcal F$ locally free, then $\mathcal K$ is reflexive, i.e. $S_2$, but not locally free.

Incidentally, torsion-free is the same as $S_1$, and can be thought of geometrically as saying that the sheaf admits no non-zero sections whose support is a proper closed subvariety. (More generally, if you are on a variety or scheme that is allowed to be reducible, then the condition is that the support of a non-zero section should be a union of irreducible components.)

Regarding the use of rank in the non-locally free context: the generic fibre of a torsion-free coherent sheaf is a finite rank vector space over the function field of $X$, and this is how its rank is defined.

Finally: I've found these notes of Karl Schwede helpful in the past when trying to think about this material.

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