Ordered Grothendieck groups for valuation domains
by
Peter Vámos
University of Exeter

Additive (and non-negative) invariants are fundamental in all areas of mathematics. The favourite trick of an algebraist is to study just one such invariant from which all others can be derived: this generic additive (and non-negative) invariant takes its values in the Grothendieck group (resp.ordered Grothendieck group).

Let M(R) be the Serre category generated by finitely generated R-modules for a ring R. The modules in M(R) for which the generic invariant is zero form a Serre subcategory of M(R) and the process of taking the generic invariant can be repeated for this category. This results in a descending sequence of Serre categories and its attendant sequence of ordered invariants; this sequence encodes and reflects important properties of the ring.

The most intensely studied commutative local rings are those which are either Noetherian or valuation rings. For a (general) commutative Noetherian ring this sequence of invariants is known. In this talk we'll consider these invariants for a valuation domain R. We will show how the valuation of R can be used to determine these invariants.

Date received: July 15, 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-82.