Finally, we’re going to come to the Kahler criterion for regularity.  As far as algebraic geometry is concerned, it states that a variety over an algebraically closed field of characteristic zero is nonsingular precisely when the sheaf of differentials on it (to be defined shortly) is locally free of rank equal to the dimension.

Theorem 1 Suppose ${A}$ is a local domain which is a localization of a finitely generated ${k}$-algebra for ${k}$ a field of characteristic zero, with residue field ${k}$. Then ${A}$ is a regular local ring if and only if ${\Omega_{A/k}}$ is a free ${A}$-module of rank ${\dim A}$.

First, I claim ${\Omega_{A/k}}$ is finitely generated. This follows because the corresponding claim is true for a polynomial ring, we have a conormal sequence implying it for finitely generated algebras over a field, and taking differentials commutes with localization.

Let ${K}$ be the residue field of ${A}$. I claim

$\displaystyle \boxed{ \dim k \otimes \Omega_{A/k} = \dim( \mathfrak{m}/\mathfrak{m}^2 ) , \quad \dim (K \otimes \Omega_{A/k}) = \dim A.}$

Then, the theorem will follow from the next lemma:

Lemma 2

Let ${M}$ be a finitely generated module over the local noetherian domain ${A}$, with residue field ${k}$ and quotient field ${K}$. Then ${M}$ is free iff$\displaystyle \dim K \otimes M = \dim k \otimes M$ (more…)