# An algebraic proof of Mumford's smoothness criterion for surfaces?

(Disclaimer: I'm a beginner in this area, so welcome corrections.)

Let $$(X,x)$$ be a germ of a complex surface (i.e. locally the zero set of some holomorphic functions) and assume that $$x$$ an isolated singular point. Mumford proved that if the local fundamental group of $$X$$ at $$x$$ is trivial, then in fact $$x$$ is smooth.

All the critters in the above paragraph have algebraic analogues, and the conversion was carried out (I believe) by Flenner: Let $$A$$ be a two-dimensional complete local normal domain containing an algebraically closed field of characteristic zero; if the 'etale fundamental group of [EDIT: the punctured spectrum of] $$A$$ is trivial, then $$A$$ is regular.

However, Flenner's proof is essentially by reduction to Mumford's theorem [as far as I, a non-German-speaker, can tell], rather than a new algebraic (or algebro-geometric) proof. So:

Does there exist a purely algebraic or algebro-geometric proof of Mumford's theorem?

Motivations include: (1) Mumford's proof is completely opaque to me; (2) No, I mean really really opaque; (3) I'm curious about extensions of the theorem to non-isolated singularities [which should probably be another question].

I found what I think is the answer, in a paper by Cutkosky and Srinivasan called "Local fundamental groups of surface singularities in characteristic $$p$$". They prove, as Corollary 5: Suppose that $$(A, m)$$ is a complete normal local domain of dimension two, with algebraically closed residue field $$k$$ of characteristic zero. (Slightly surprising, given the title of the paper.) Then $$\pi_1(\operatorname{Spec} A -m)=0$$ if and only if $$A$$ is smooth over $$k$$. They say that this gives "an arithmetic proof of the theorem of Mumford and Flenner."
The proof apparently uses Flenner's paper, but I don't think it uses Mumford's result. They get an expression for the local fundamental group in terms of a tree, and appeal to Flenner's Theorem 2.7 to know that the group is trivial iff $$A$$ is smooth. I haven't tried to read that section of Flenner's paper yet, but it seems to be independent of Mumford.
Helene Esnault (and Eckart Viehweg) have a recent preprint on the arxiv: https://arxiv.org/abs/1002.0024 for a characteristic $$p$$ version of Mumford's theorem. Perhaps your question is answered in the Flenner reference quoted there.