Proceedings of the London Mathematical Society - recent issues

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

Connectivity of Knight's Graphs

Rhodes, F., Wilson, S. — 1993-09-01

Semiprimitivity of Group Algebras of Locally Finite Simple Groups

Passman, D. S., Zalesskii, A. E. — 1993-09-01

Let G be a locally finite simple group which is not a linear group. For any field K, we prove that the group algebra K[G] is semiprimitive. The argument is almost entirely group-theoretic in nature and ultimately rests upon the classification of the finite simple groups.

Cleft Extensions of Hopf Algebras, II

Byott, N. P. — 1993-09-01

Let K be a p-adic field, let G and U be groups of order p, and let H) (respectively A) be a Hopf order in the group algebra KG (respectively in the algebra of maps U->K). We use the algebraic machinery of [2] to determine the cleft Hopf algebra extensions of A by H, and investigate which of these cleft extensions are Hopf orders in a group algebra.

Minimal Index Surface Subgroups of Noneuclidean Crystallographic Groups

Izquierdo, M. — 1993-09-01

A Fixed Point Theorem for Groups Acting on Finite 2-Dimensional Acyclic Simplicial Complexes

Aschbacher, M., Segev, Y. — 1993-09-01

Random Times, Predictable Processes and Stochastic Integration in Finite Von Neumann Algebras

Barnett, C., Wilde, I. — 1993-09-01

Positive Braids are Visually Prime

Cromwell, P. R. — 1993-09-01

It is shown that positive braids represent split or non-prime links only in the obvious ways.

The Classification of Heegaard Splittings for (Compact Orient Able Surface) x S1

Schultens, J. — 1993-09-01

Affine Distance-Transitive Groups

Bon, J. V. — 1993-07-01

Dividing the Primes into Two Subsets with Nearly the Same Number of Products

Birch, B., Scourfield, E. — 1993-07-01

Geometric Quotients of Unipotent Group Actions

Greuel, G.-M., Pfister, G. — 1993-07-01

Presentations of Groups Involving More Generators than are Necessary

Evans, M. J. — 1993-07-01

Injective Modules, Induction Maps and Endomorphism Rings

Brookes, C. J. B., Brown, K. A. — 1993-07-01

Analogues of the vertex-source pairs of the modular representation theory of finite groups are introduced in order to study injective modules over infinite groups. A simplified analogue of the Green correspondence is established. The analogue of Clifford theory is developed by applying Moody's theorem on G₀ of crossed products to the endomorphism rings of induced modules.

Lieb-Thirring Inequalities on the N-Sphere and in the Plane, and Some Applications

Ilyin, A. A. — 1993-07-01

In this paper we prove the Lieb-Thirring inequalities for a family of scalar functions defined on a sphere Sⁿ, which are orthonormal in L₂(Sⁿ) and have zero mean value, for n ≥ 1. We give explicit values of all the constants involved. In the case of the two-dimensional sphere, we prove the Lieb-Thirring inequalities for an orthonormal family of non-divergent (or irrotational) vector fields with the explicit value of the constant as well. For non-divergent (or irrotational) vector fields defined on the plane R² we prove the Lieb-Thirring inequalities with the value of the constant less than was known before. Finally, the rate of growth of the constant is estimated, when a parameter p tends to its limit, and embeddings in the exponential Orlicz spaces are proved. Applications to the dimension of attractors are given.

Some Applications of Projective Resolutions of Identity

Deville, R., Godefroy, G. — 1993-07-01

We show that the unit ball K of the bidual of an Asplund space is a Corson compact or contains [0,₁], and that it has the Namioka property on separate-to-joint continuity. The same results are shown for K a Valdivia compact; a by-product is that all dyadic compacts have the Namioka property. Some connections with weakly compactly generated dual spaces and renormings are given.

A Graph-Theoretical Approach to Kleinian Groups

Reni, M. — 1993-07-01