# Reflexive sheaves on stable curves

Let $C$ be a stable curve over an algebraically closed field of positive characteristic and $\mathcal{F}$ be a reflexive sheaf on $C$. Is $\mathcal{F}$ locally free?

EDIT Is the projective dimension of $\mathcal{F}$ less than or equal to $1$?

Equivalently: is every f.g. reflexive module $M$ over $R=k[[x,y]]/(xy)$ projective? The answer is no: let $M = R/xR$, then ${\rm Hom}_R(M, R) = \{f\in R\,:xf=0\} = yR \cong M$. Thus $M$ is reflexive, but is not projective.
• The $R$-module $M$ in Piotr's answer has infinite projective dimension, with a resolution $$\ldots R \xrightarrow{\times\, x}R\xrightarrow{\times\, y}R \xrightarrow{\times\, x}R\rightarrow M\rightarrow 0\ .$$ – abx Apr 23 '15 at 9:09