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 1Suppose is a local domain which is a localization of a finitely generated -algebra for a field of characteristic zero, with residue field . Then is a regular local ring if and only if is a free -module of rank .

First, I claim 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 be the residue field of . I claim

Then, the theorem will follow from the next lemma:

Lemma 2Let be a finitely generated module over the local noetherian domain , with residue field and quotient field . Then is free iff (more…)