Direct-Sum Decomposition of Modules and Zero-Sum Theory

The study of Direct-Sum Decompositions of Modules is a long-established topic, and re- cent work by Facchini, Herbera, and Wiegand has paved the way to a new semigroup-theoretical approach to the subject in the case when the classical Krull-Remak-Schmidt-Azumaya Theorem fails to hold (to the effect that the decomposition is no longer essentially unique). The funda- mental insight of this approach is that, in some relevant cases (in fact, those on which we focus), problems on direct-sum decompositions can be shifted, by well-known transfer principles, to the monoid of zero-sum sequences of a suitable abelian group G with support in a certain set GP (in practice, G is the class group of an appropriate reduced Krull monoid and GP the set of classes of G containing prime divisors), which makes it possible to study direct-sum decompositions by methods of Additive Theory and, more specifically, Zero-Sum Theory. Zero-Sum Theory is a subfield of Additive (Group and Number) Theory, which has been receiving, as well as Additive Theory in general, a rapidly increasing attention for the last few decades. Sequences over abelian groups (a sequence here is a finite, unordered sequence allowing the repetition of elements), their sets of subsequence sums, and their structure under extremal conditions are some of the main objects of study in the area. In particular, problems dealing with sequences are often translated into problems with sets, and then studied via sumsets. Thus, addition theorems, along with polynomial methods and group rings, are key tools. Note that the set of zero-sum sequences over a set GP is a Krull monoid. This Project lies, in fact, in the overlap of Module Theory and Zero-Sum Theory, insofar as we aim to exploit methods from Additive Theory to derive explicit characterizations of various finiteness properties of direct-module decompositions in terms of G and GP . In this respect, it is worth remarking that the focus of Zero-Sum Theory has been so far on finite abelian groups, whereas the class groups stemming from module theory are mostly infinite (but finitely generated in many relevant cases). This suggests that entirely new methods will be necessary. Our goals include structure theorems for sets of lengths and their unions, as well as the study of the Davenport constant of GP and of further arithmetical invariants.

Factorization theory (FT) has been, for a long time, mainly devoted to the study of the nonuniqueness of factorizations of elements into irreducibles in commutative cancellative monoids. During the past decade and half, the field has seen two substantially new developments, pushed forward by the work of Facchini and Wiegand on direct-sum decompositions of modules, and of Baeth and Smertnig on (possibly non-commutative) domains. In the same spirit, one of the main contributions of the present project has been to further extend the boundaries of FT so as to cover new scenarios. A strong motivation for pursuing this line of research comes from Y. Fan and S. Tringali, Power monoids: A bridge between Factorization Theory and Arithmetic Combinatorics, Journal of Algebra (to appear), where the authors introduce a new class of non-cancellative monoids that provide an effective framework for arithmetic combinatorics and FT to benefit from the interaction with each other. Another contribution is related to the structure of sets of lengths and their unions. As an example, it was well known (before the start of the present project) that unions of sets of lengths are, in various cases, almost arithmetical progressions with global bounds on all parameters: This means that their beginning and end parts are merely subsets of an arithmetic progression, and there is some irregularity in their structure. But Theorem 1.2 in S. Tringali, Structural properties of subadditive families with applications to factorization theory, Israel Journal of Mathematics (to appear), showed that, for monoids with accepted elasticity, these irregularities repeat periodically. Further details can be found in the extended version of the final report. The results of this project are presented in seven papers (one of which is co-authored with the Austrian co-investigator) submitted for publication in international mathematical journals (five have already been accepted and two are currently under review). All publications can be found on the personal website of the grant holder (https://imsc.uni-graz.atringali/).