Consider torsion free modules over the germ of a fixed isolated algebraic hypersurface singularity {$f=0$}$\subset\mathbb{C}^n$. There are natural functors (using categories of finitely generated modules):

modules over $\mathbb{C}[x_1,..,x_n]_{(x_1,\dots,x_n)}/(f)$--> modules over $\mathbb{C}${$x_1,..,x_n$}$/(f)$--> modules over $\mathbb{C}[[x_1,..,x_n]]/(f)$.

Are they faithful, surjective? I know they are not surjective for an arbitrary local ring, but isolated hypersurface singularity is quite special.