Proceedings of the London Mathematical Society - current issue

A Construction of Many Uncountable Rings Using SFP Domains and Aronszajn Trees

Hodkinson, I., Shelah, S. — 1993-11-01

The paper is in two parts. In Part I we describe a construction of a certain kind of subdirect product of a family of rings. We endow the index set of the family with the partial order structure of an SFP domain, as introduced by Plotkin, and provide a commuting system of homomorphisms between those rings whose indices are related in the ordering. We then take the subdirect product consisting of those elements of the direct product having finite support in the sense of this domain structure. In the special case where the homomorphisms are isomorphisms of a fixed ring S, our construction reduces to taking the Boolean power of S by a Boolean algebra canonically associated with the SFP domain.

We examine the ideals of a ring obtainable in this way, showing for instance that each ideal is determined by its projections onto the factor rings. We give conditions on the underlying SFP domain that ensure that the ring is atomless. We examine the relationship between the L-theory of the ring and that of the SFP domain.

In Part II we prove a ‘non-structure theorem’ by exhibiting 2^N₁ pairwise non-embeddable L-equivalent rings of cardinality N₁ with various higher-order properties. The construction needs only ZFC, and uses Aronszajn trees to build many different SFP domains with bases of cardinality N₁.

Deformation des Syzygies et Theorie de Brill-Noether

Voisin, C. — 1993-11-01

Grothendieck Groups of Invariant Rings: Filtrations

Brown, K. A., Lorenz, M. — 1993-11-01

We investigate the Grothendieck group G₀(R) of finitely generated modules over the ring of invariants R = S^G of the action of a finite group G on an FBN ring S under the assumption that the trace map from S to R is surjective. Using a certain filtration of G₀(R) that is defined in terms of (Gabriel-Rentschler) Krull dimension, properties of G₀(R) are derived to a large extent from the connections between the sets of prime ideals of S and R. A crucial ingredient is an equivalence relation ~ on Spec R that was introduced by Montgomery [25]. For example, we show that

rank G₀(R) ≤ rank G₀(S)_G + $$\sum _{\Omega }\left(\#\Omega -1\right)$$

where runs over the ~-equivalence classes in Spec R and (·)_G denotes G-coinvariants. The torsion subgroup of G₀(R) is also considered. We apply our results to group actions on the Weyl algebra in positive characteristics, the quantum plane, and the localized quantum plane for roots of unity.

Primitive Permutation Groups with a Regular Non-Abelian Normal Subgroup

Baddeley, R. W. — 1993-11-01

On the Geometry of Normal Forms in Discrete Groups

Bridson, M. R. — 1993-11-01

We study normal forms in finitely generated groups from the geometric viewpoint of combings. We introduce notions of combability considerably weaker than those commonly in use. We prove that groups which satisfy these conditions are finitely presented and satisfy isoperimetric and isodiametric inequalities of a controlled nature.

Projective and Inductive Limits of Hypergroups

Voit, M. — 1993-11-01

The purpose of this paper is to introduce projective and inductive limits of hypergroups and to discuss their basic properties in a systematic way. In particular, we study the behaviour of these limits with a view to forming coset and double coset hypergroups, and determine the dual spaces of these limits. As a consequence, some duality results for certain commutative hypergroups can be extended to wider classes. Our results will be illustrated by hypergroups associated with Gelfand pairs for which some of the results are well known.

Inequalities for Non-Moderate Functions of a Pair of Stochastic Processes

Jacka, S. D., Yor, M. — 1993-11-01