Homomorphisms on infinite direct products of groups, rings and monoids

We study properties of a group, abelian group, ring, or monoid $B$ which (a) guarantee that every homomorphism from an infinite direct product $\prod_I A_i$ of objects of the same sort onto $B$ factors through the direct product of finitely many ultraproducts of the $A_i$ (possibly after composition with the natural map $B\to B/Z(B)$ or some variant), and/or (b) guarantee that when a map does so factor (and the index set has reasonable cardinality), the ultrafilters involved must be principal. A number of open questions, and topics for further investigation, are noted.Comment: 26 pages. Copy at http://math.berkeley.edu/~gbergman/papers may be updated more frequently than arXiv copy. Version 2 has minor revisions in wording etc. from version

Factorizations of Elements in Noncommutative Rings: A Survey

We survey results on factorizations of non zero-divisors into atoms (irreducible elements) in noncommutative rings. The point of view in this survey is motivated by the commutative theory of non-unique factorizations. Topics covered include unique factorization up to order and similarity, 2-firs, and modular LCM domains, as well as UFRs and UFDs in the sense of Chatters and Jordan and generalizations thereof. We recall arithmetical invariants for the study of non-unique factorizations, and give transfer results for arithmetical invariants in matrix rings, rings of triangular matrices, and classical maximal orders as well as classical hereditary orders in central simple algebras over global fields.Comment: 50 pages, comments welcom

G-Global Homotopy Theory and Algebraic K-Theory

We develop the foundations of $G$-global homotopy theory as a synthesis of classical equivariant homotopy theory on the one hand and global homotopy theory in the sense of Schwede on the other hand. Using this framework, we then introduce the $G$-global algebraic $K$-theory of small symmetric monoidal categories with $G$-action, unifying $G$-equivariant algebraic $K$-theory, as considered for example by Shimakawa, and Schwede's global algebraic $K$-theory. As an application of the theory, we prove that the $G$-global algebraic $K$-theory functor exhibits the category of small symmetric monoidal categories with $G$-action as a model of connective $G$-global stable homotopy theory, generalizing and strengthening a classical non-equivariant result due to Thomason. This in particular allows us to deduce the corresponding statements for global and equivariant algebraic $K$-theory.Comment: Several minor corrections and changes to exposition, some additional material; v + 236 page

An inner automorphism is only an inner automorphism, but an inner endomorphism can be something strange

The inner automorphisms of a group G can be characterized within the category of groups without reference to group elements: they are precisely those automorphisms of G that can be extended, in a functorial manner, to all groups H given with homomorphisms G --> H. Unlike the group of inner automorphisms of G itself, the group of such extended systems of automorphisms is always isomorphic to G. A similar characterization holds for inner automorphisms of an associative algebra R over a field K; here the group of functorial systems of automorphisms is isomorphic to the group of units of R modulo units of K. If one substitutes "endomorphism" for "automorphism" in these considerations, then in the group case, the only additional example is the trivial endomorphism; but in the K-algebra case, a construction unfamiliar to ring theorists, but known to functional analysts, also arises. Systems of endomorphisms with the same functoriality property are examined in some other categories; other uses of the phrase "inner endomorphism" in the literature, some overlapping the one introduced here, are noted; the concept of an inner {\em derivation} of an associative or Lie algebra is looked at from the same point of view, and the dual concept of a "co-inner" endomorphism is briefly examined. Several questions are posed.Comment: 20 pages. To appear, Publicacions Mathem\`{a}tiques. The 1-1-ness result in the appendix has been greatly strengthened, an "Overview" has been added at the beginning, and numerous small rewordings have been made throughou

On noncommutative bounded factorization domains and prime rings

A ring has bounded factorizations if every cancellative nonunit a ∈ R can be written as a product of atoms and there is a bound λ ( a ) on the lengths of such factorizations. The bounded factorization property is one of the most basic finiteness properties in the study of non-unique factorizations. Every commutative noetherian domain has bounded factorizations, but it is open whether such a result holds in the noncommutative setting. We provide sufficient conditions for a noncommutative noetherian prime ring to have bounded factorizations. Moreover, we construct a (noncommutative) finitely presented semigroup algebra that is an atomic domain but does not satisfy the ascending chain condition on principal right or left ideals (ACCP), whence it does not have bounded factorizations

The total coordinate ring of a wonderful variety

We study the cone of effective divisors and the total coordinate ring of wonderful varieties, with applications to their automorphism group. We show that the total coordinate ring of any spherical variety is obtained from that of the associated wonderful variety by a base change of invariant subrings.Comment: Final version, to appear in Journal of Algebr

About GKM- and non-abelian Hamiltonian actions

This thesis revolves around two different, but not entirely unrelated topics. The first is the realization problem in GKM theory, the second is the topic of multiplicity free manifolds. Regarding the realization problem, we first show that a large class of GKM graphs is in fact a restriction of a torus graph. This involves realizable GKM_4-graphs, so in particular realizable graphs in general position with valence at least 5. The corresponding GKM manifolds were studied first by Ayzenberg and later also Masuda in [A18] and [AM19]. Then, we give a sufficient criterion for when a T^2-manifold of dimension 6 is equivariantly formal, and use this, building on [GKZ22], to show that every orientable, 3-valent GKM graph is realizable as an equivariantly formal T^2-manifold. After that, we switch to the realization of certain GKM fiber bundles, as first studied in [GKZ20]. More precisely, we characterize which GKM fiber bundles Γ -] Γ' -] B are realizable. Here B is the GKM graph of a quasitoric manifold of dimension 4, and Γ is the GKM graph of a generalized flag manifold of the form G/T, where T ⊂ G is a maximal torus. At the end, we also construct many non-trivial examples of such GKM fiber bundles. The last chapter is essentially the article [GSW22], where we study multiplicity free U(2)-manifolds. Multiplicity free manifolds naturally generalize the class of toric manifolds as studied in [Del88] to non-abelian Lie groups. Friedrich Knop [Kno11] was able to classify those in terms of their principal isotropy type and their invariant momentum polytope, building directly on work of Losev [Los09]. We restrict ourselves to the group U(2) and explicitly give the equivariant diffeomorphism types as well as the symplectic form of certain multiplicity free U(2)-manifolds, including those whose momentum image is a triangle. We also give an easy-to-check characterization of when a multiplicity free U(2)-manifold admits a compatible U(2)-invariant Kähler structure. This turns out to be the case if and only if the corrsponding action of T^2 ⊂ U(2) admits an invariant Kähler structure