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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.)

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

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.

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.)

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

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

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.