Regularity on varieties over non-noetherian valuation rings
by
Hagen Knaf
University of Heidelberg

Regularity of noetherian schemes plays an important role in Algebraic as well as in Diophantine Geometry. It is therefore natural to ask for a generalization of this notion to non-noetherian schemes. In 1971 J. Bertin gave such a generalization: The local ring O is called regular if the projective dimension of every finitely generated ideal of O is finite.

In the sequel coherent regular rings were investigated, showing the similiarity of this class of rings with the class of noetherian regular rings as well as significant differences. To give an example it is on the one hand possible to give a non- noetherian version of the Theorem of Auslander-Buchsbaum on the factoriality of noetherian regular local rings, on the other hand coherent regular local rings can in general neither be characterized by the finiteness of homological dimensions nor by the length of maximal regular sequences.

In the talk the local rings of points on an integral separated scheme of finite type over a Pruefer domain R - an R-variety for short - are considered: Let O be the local ring in a point x of such a scheme and let M be its maximal ideal. The following topics are discussed:

(1) Characterization of regularity by the finiteness of the weak homological dimension wdim(O).

In contrast to the general situation such a characterization is possible and a precise formula for wdim(O) involving the dimension of the local ring of x on the fibre passing through x is given. For Pruefer rings R satisfying certain cardinality conditions as a consequence of 1 one gets:

(2) Characterization of regularity by the finiteness of the global homological dimension gldim(O).

Both results are based on the theory of non-noetherian grade, which also allows the investigation of regular sequences in M and their connection to the weak dimension of O:

(3) Characterization of regularity in terms of maximal regular sequences in M and the embedding dimension of O.

Here the results strongly depend on weather M is finitely generated or not.

Throughout the course of the talk some finiteness properties of (normal) schemes of finite type over valuation rings will also be discussed.

Date received: April 30, 1999

Copyright © 1999 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cacv-36.