Additive Combinatorics and Arithmetic of Krull monoids

Factorization Theory with a Focus on Krull Monoids. Let H be an atomic monoid. Then every nonunit can be written as a finite product of atoms (irreducible elements). In general, there are many (essentially) distinct factorizations, and the main objective is to describe and to classify the various phenomena of non-uniqueness. If a = u1 . . . uk is such a factorization of an element into atoms, then k is called the length of the factorization, and the set L(a) of all possible factorization lengths is the set of lengths of a. If H is a Krull monoid, then sets of lengths in H are finite and nonempty and they depend only on the subset GP of classes in the class group, which contain prime divisors. In particular, sets of lengths of H can be studied in the monoid of zero-sum sequences over GP . Additive Combinatorics with a Focus on Zero-Sum Theory. Zero-Sum Theory is a subfield of Additive Combinatorics, or say of Additive and Combinatorial Number Theory. In particular, during the last decade, this field, as well as Additive Combinatorics more generally, has been in rapid development. A main object of study are subsequence sums and zero-sums of sequences over abelian groups, where a sequence here is a finite, unordered sequence allowing the repetition of elements. Problems dealing with sequences are often translated into problems with sets, and then they are studied via sumsets. Thus addition theorems are of central importance, but also polynomial methods and group rings are key tools. Note that the set of zero-sum sequences over a set G P (with concatenation as operation) is a Krull monoid. This Project is in the overlap of the above areas, and it is inspired by recent developments in them. We study Krull monoids stemming from Number Theory which have a finite class group G and prime divisors in all classes (such as holomorphy rings in global fields), and Krull monoids stemming from Module Theory. Indeed, if C is a class of modules (closed under finite direct sums, direct summands, and isomorphisms) such that all endomorphism rings EndR(M ) are semilocal, then the set of isomorphism classes of modules is a Krull monoid. In many relevant cases the class group G is infinite, and the set GP of classes containing prime divisors is a proper subset. We focus on zero-sum problems over such subsets GP G . The goal is to establish abstract finiteness results for arithmetical invariants (such as sets of lengths) as well as to derive precise values in case where GP = G is finite.

Let H be an atomic monoid. Then every nonunit can be written as a finite product of atoms (irreducible elements). In general, there are many (essentially) distinct factorizations, and the main objective of factorization theory is to describe and to classify the various phenomena of non-uniqueness. If a = u1 . . .uk is such a factorization of an element into atoms, then k is called the length of the factorization, and the set L(a) of all possible factorization lengths is the set of lengths of a. Suppose that H is a Krull monoid with finite class group G and suppose that every class contains a prime divisor (such as holomorphy rings in global fields). Then sets of lengths depend only on the class group and can be studied with methods from Additive Combinatorics. It was well-known (before the start of the present project) that sets of lengths are AAMPs (almost arithmetical multiprogression) with global bounds on all parameters, and that the set * (G) of all differences occurring in long AAMPs is a finite set. In a joint paper (entitled The set of minimal distances in Krull monoids, published in Acta Arithmetica 173 (2016), 97 120,DOI: 10.4064/aa7906-1-2016) the grantholder and the Austrian co-researcher could give a precise formula for its maximum. Indeed, we have max * (G) = max{exp(G) - 2, r(G) - 1} , where exp(G) denotes the exponent of the group and r(G) its rank. This result was most crucial for all project goals. Among others it made possible substantial progress in the Characterization Problem. More information can be found in the long version of the final report. The results of this project are published in seven papers (four of them are co-authored with the Austrian co-researcher) which appeared in international mathematical journals. All publications can be found on the personal website of the grantholder: http://qinghai-zhong.weebly.com/.