Return to Answer

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.

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.

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 $\mathbb C[x_1,\cdots,x_n]_{(x_1,\cdots,x_n)}/(f) \to \mathbb C[[x_1,\cdots,x_n]]/(f)$ 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' comment: I think one can build higher dimension examples from Manish 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 by analytically irreducible.

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.

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 $\mathbb C[x_1,\cdots,x_n]_{(x_1,\cdots,x_n)}/(f) \to \mathbb C[[x_1,\cdots,x_n]]/(f)$ 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.

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 $\mathbb C[x_1,\cdots,x_n]_{(x_1,\cdots,x_n)}/(f) \to \mathbb C[[x_1,\cdots,x_n]]/(f)$ 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' comment: I think one can build higher dimension examples from Manish 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 by analytically irreducible.