Dubrovin Valuation Rings and Orders in Central Simple Algebras
by
Joachim Gräter
University of Potsdam, Germany

This talk presents a survey of a general valuation theory for finite-dimensional division algebras as well as their matrix rings, i.e., central simple algebras. It is addressed to non-specialists and only basic knowledge from commutative valuation theory is required.

A subring B of a division algebra D is called (total) valuation ring if x in B or x^-1 in B holds for all nonzero x in D. With this notation the invariant valuation rings are exactly those which arise with classical valuations in the sense of Schilling. Especially in the finite-dimensional case valuation rings are closely related to their centers and so the investigation of noncommutative valuation rings can be understood as the investigation of extensions of a given commutative valuation ring. Unfortunately, a general extension theorem does not exist, i.e., valuation rings of the center are not necessarily extendible to the entire division algebra. With a view to improving this situation Dubrovin introduced a new type of valuation ring for division algebras as well as their matrix rings. They have many properties which signify that Dubrovin valuation rings are the right object to study in central simple algebras. For example, if Q denotes a central simple algebra with center F then each valuation ring V of F can be extended to a Dubrovin valuation ring of Q where all extensions are conjugate. Furthermore, if V is discrete these extensions are precisely the maximal orders over V. Finally, if Q is in addition a division algebra then all extensions are total valuation rings if there is at least one total extension of V to Q, i.e., general Dubrovin valuation rings only appear if no total valuation rings are available.

If K is a commutative finite degree extension of F then it is well known that the integral closure T of V in K is a Bezout ring and that T is the intersection of all extensions of V to K. In the noncommutative case the integral closure of V in Q need not be a ring at all but it is proved that V can be extended to a Bezout order R in Q which is integral over V. This order R is unique up to inner automorphisms and the union of all its conjugates presents exactly the integral closure of V in Q. Furthermore, there exists a natural number n_V such that R is the intersection of n_V extensions of V where n_V also occurs in the so-called Defect Theorem as well as in connection with the Henselization of Q.

The talk ends with some general remarks concerning a global theory by means of Prüfer orders which are intersections of Dubrovin valuation rings and which can be comprehended as a natural generalization of Dedekind orders to the nonnoetherian case.

Date received: April 7, 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-21.