Non-commutative generalisations of valuations and places
21 October 2016
UAntwerpen, Campus Drie Eiken, Promotiezaal Q0.02 - Universiteitsplein 1 - 2610 Antwerpen-Wilrijk (route: UAntwerpen, Campus Drie Eiken
Fred Van Oystaeyen
PhD defence Nikolaas Verhulst - Faculty of Science, Department of Mathematics and Computer Science
In this thesis we study several non-commutative generalisations of the concept of a valuation. We study the interplay between a.o. primes, partial valuation rings, total subrings, and Dubrovin valuation rings. In particular, we introduce arithmetical pseudo-valuations associated to Dubrovin valuation rings with non-idempotent Jacobson radicals.
Since localisations of bounded Krull orders at height-one prime ideals are such Dubrovin valuation rings, this allows us to develop a divisor theory in this context, including results which are analogous to the approximation theorems from commutative valuation theory. We also introduce groupoid valuations and groupoid valuation rings and show a correspondence between the two.
Finally, a connection between groupoid valuations on the one hand and Dubrovin valuation rings on the other is established.