Algebraic, analytic, formal modules

Question

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

Answer 1

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 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.

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.

Answer 2

Dear Hailong (and all others reading his), I apologize for using `answer' for a comment, since my openid does not work (I am traveling) or I have forgotten it and MO does not give me a way to comment.

Yes, there are examples over any field. If my memory serves me right, the first example I saw was in Samuel's TIFR lecture notes on UFD's. There are also examples (as an aside) in my paper on rational double points which appeared in the 80s in the Inventiones.

For the question of Brian, in general class group is used for (over normal domains) rank one torsion free finitely generated modules upto isomorphism modulo free modules. If one wants to look at invertible modules, nowadays it seems more common (especially people with an alg. geom. bend) to use Picard group. The examples I mention above are `algebraic' and not Henselizations.

Answer 3

There are well studied hypersurfaces of dimension 2 which are UFDs whose completions are not. So, any ideal representing a non-trivial element in the class group of the completion will not come from the algebraic ring.