TITLE: Is a torsion free sheave of rank one on a reducible curve the pushforward of a line bundle on a normalization?
QUESTION [7 upvotes]: Let $X$ be a nodal curve, possibly reducible. Then can any torsion free sheaf of rank one on $X$ be expressed as $\pi_*(L)$, where $L$ is a line bundle on a partial normalization of $X$? This looks standard, but I just can not find a reference that precisely state this conclusion.

REPLY [3 votes]: Short proof: A given rank $1$, torsion-free sheaf $F$ is the direct image of a sheaf $L$ on the partial normalization $Y := \operatorname{Spec} \operatorname{Hom}(F,F)$.  When $X$ is nodal, $Y$ is Gorenstein, and a theorem of Vasconcelos implies that $L$ is a line bundle.
In more detail, I first claim the sheaf $\operatorname{Hom}(F,F)$ of endomorphisms is a sheaf of finite commutative $\mathcal{O}_{X}$-algebras.  Since $F$ is torsion-free, $\operatorname{Hom}(F,F)$ injects into its generic fiber which is just the ring of rational functions since $F$ is generically trivial of rank $1$.  We conclude that $\operatorname{Hom}(F, F)$ is commutative.  The $\mathcal{O}_{X}$-algebra is also finitely generated (fix generators and track where they go) and integral (use Hamilton-Cayley to produce a polynomial that kills a given local section), so we conclude that it is finite.
Essentially by construction, $Y := \operatorname{Spec} \operatorname{Hom}(F,F)$ is a partial normalization of $X$ that carries a sheaf $L$ whose direct image is $F$. Furthermore, $L$ satisfies $\operatorname{Hom}(L,L)=\mathcal{O}_{y}$.  Since $X$ was nodal, $Y$ is nodal as well, and we conclude by Theorem (3.1) of Vasconcelos's paper "Reflexive modules over Gorenstein rings".
Note: This argument applies to a curve with arbitrary singularities, although we can only conclude that $L$ is a line bundle when $Y$ is Gorenstein (otherwise we can only assert that $\operatorname{Hom}(L,L) = \mathcal{O}_{Y}$).  The partial normalization $Y$ is always Gorenstein when $X$ has at worst $A_{n}$ singularities, but otherwise $X$ admits sheaves that are not direct images of line bundles.  For example, if $X$ is the Zariski closure of the curve $\{x^3-y^4=0\}$, then the maximal ideal $(x,y)$ is not the direct image of a line bundle.