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.


upd: 


1. The example of $x^2=y^2+y^3$ certainly counts, but can you suggest smth similar in the case of a locally irreducible analytic hypersurface?
2. Sorry I'm outsider in algebra. By surjectivity I meant smth like: every formal module over an analytic hypersurface arises from a locally analytic module.
(Or maybe weaker: if a formal module has a submodule of the same rank that arises from locally analytic category, then the initial formal module arises from locally analytic category.)


up2: Thanks to everybody, sorry for delay



Algebraic, analytic, formal modules
The most relevant result I am aware of is the following: suppose $R$ is a local Gorenstein ring, essentially of finite type over a field, with an isolated singularity (so certainly include your case) then the stable categories of maximal Cohen-Macaulay modules over $R$ and the completion $\hat R$ are equivalent up to direct summand via $\hat R \otimes\_R?$. See Proposition 1.6 of [this paper](https://arxiv.org/abs/0803.0720) by Keller-Murfet-Van den Bergh.


In everyday English, this means that every maximal Cohen-Macaulay (MCM) module over $\hat R$ (which are torsion-free, and include high enough syzygies of any given module) is a direct summand of a completion of some MCM module over $R$. So restricting to MCM modules, the functor: (MCM modules over) $\mathbb C[x\_1,\cdots,x\_n]\_{(x\_1,\cdots,x\_n)}/(f) \to \mathbb C[[x\_1,\cdots,x\_n]]/(f)$ is "surjective" up to direct summand.


In general, as BCnrd remarked, one can only hope to descend to the henselization of $R$. There is a big literature on this topic, see for example the references [here](http://arxiv.org/abs/math/0612311).


ADDED: The nodal curve example by Manish is discussed in example A.5 of the Keller-Murfet-Van den Bergh paper. The completion is isomorphic to $S=\mathbb C[[u,v]]/(uv)$ and each $S/(u), S/(v)$ is not extended, but their direct sum is.


For your updated question 1 and Manish's comment: I think one can build higher dimension examples from Manish's original example using a result known as Knörrer's periodicity theorem (see Invent. Math., 88 (1987), 153–-164), namely that there is an equivalence of the stable categories of MCM modules over $\mathbb C[[x\_1,\cdots,x\_n]]/(f)$ and $\mathbb C[[x\_1,\cdots,x\_n,u,v]]/(f+uv)$. In particular, there should be an example for $f=y^2-x^2-x^3+uv$, which will be analytically irreducible.


