Arithmetic of one-dimensional domains

Research project P 14440 Arithmetic of one-dimensional integral domains Franz HALTER-KOCH 26.6.2000 In a noetherian integral domain R, every non-zero non-unit has a representation as a product of irreducible elements. If R is factorial, this representation is unique up to the order of the factors and up to associates. If R is not factorial, then the problem of a description and classification of the phenomena of non-uniqueness arises. The set of possible lengths of factorizations of an element from R is called its set of lengths. The hitherto best investigated invariant of non-unique factorizations is the system of sets of lengths. The domain R is called half-factorial if all sets of lengths consist of single element. If R is the ring of integers of an algebraic number field or more generally a Krull domain, then factorization problems in R may be translated into combinatorial problems for the divisor class group. If, as in the case of number fields, every class contains a prime divisor, then the structure of the sets of lengths is well known. If R is not integrally closed, good results are up to now only available under the additional hypothesis that R is weakly Krull and its integral closure is a finitely generated R-module. In this research project, we shall continue the investigation of the arithmetic of one-dimensional noetherian domains. In particular, the following questions will be addressed: 1) Investigation of the algebraic and arithmetical structure of analytically ramified one-dimensional local noetherian domains; 2) Investigation of the algebraic structure of finitely ) primary monoids (v-ideal theory, realization theorems, analoga to the Theorem of Eakin- Nagata) and of suitable generalizations of arithmetical interest; 3) Algebraic and arithmetical structure of congruence monoids defined from Dedekind domains; 4) Calculation of the elasticity and formulation of efficient criteria for an order in a global field to be half-factorial; 5) Determination of the microstructure of the sets of lengths for one-dimensional integral domains (in particular for orders in algebraic number fields).

Our research provided us with a better and broader understanding of phenomena of non-unique fac-torizations than we had before. In particular, we succeeded in proving finiteness theorems for factorizations in analytically ramified one-dimensional noetherian domains and in congurence monoids defined in Dedekind do-mains. To arrive at these results, it was necessary to investigate the structure and ideal theory of commutative monoids in a general frame. In a noetherian integral domain, every non-zero non-unit has a factorization into a product of irreducible elements. Unless the domain is factorial, there are many different such factoriza-tions, and the problem of describing and understanding the various phenomena of non-unicity arises. The theory of non-unique factorizations was developed during the last few decades as a discipline using methodes from commuatitve algebra, monoid theory and combinatorics. When we started our investigations, there were already good results for Krull monoids with a finite divisor class group, and also for one-dimensional noetherian domains, provided that the integral closure is a finite module. Among the most important finiteness theorems in the theory of non-unique factorizations are local tameness, finiteness of the catenary degree and the structure theorem for sets of lenths. We succeeded in proving them for one-dimensional ananlytically ramified domains and for conguence monoids defined in Dedekind domains. To arrive at these results, it was necessary to develop an algebraic theory of finitary monoids and of their tame and complete ideals. Another important monoid-theoretical tool for the investigation of the arithmetical properties of one-dimensional integral domains is the the theory of finitely primary monoids. We were able to construc exampes of finitely primary monoids which are not v-noetherian and even to realize some or them in integral domains. In the context of abstract congruence monoids, we also proved criteral for finitely primary monoids to be v-noetherian. It was for the first time that we could definitely establish irregularities for factorizations in Krull domains with infinite class group unless every class contains prime divisors. We also proved criteria for the finiteness of the elasticity for a new class of rings. For orders in alge-braic number fields we found several new arithmetical properties which are equivalent to the finiteness of the elasticity. In any case, either the elasticity is finite, or there is a universal bound for the length of the shortest factorization. Although we know this phenomenon, we do not yet fully understand it.