Groups with isomorphic Burnside rings

We prove that for some families of finite groups, the isomorphism class of the group is completely determined by its Burnside ring. Namely, we prove the following: if two finite simple groups have isomorphic Burnside rings, then the groups are isomorphic; if G is either Hamiltonian or abelian or a minimal simple group, and G is any finite group such that B(G) B(G), then G G.Received: 22 April 2004     

On the cohomology rings of small categories

Let C be a small category and R a commutative ring with identity. The cohomology ring of C with coefficients in R is defined as the cohomology ring of the topological realization of its nerve. First we give an example showing that this ring modulo nilpotents is not finitely generated in general, even when the category is finite EI. Then we study the relationship between the cohomology ring of a category and those of its subcategories and extensions. The main results generalize certain theorems in group cohomology theory.     

On cohomology rings of infinite groups

Let R be any ring (with 1), G a torsion free group and RG the corresponding group ring. Let be the cohomology ring associated with the RG-module M. Let H be a subgroup of finite index of G. The following is a special version of our main Theorem: Assume the profinite completion of G is torsion free. Then an element is nilpotent (under Yoneda’s product) if and only if its restriction to is nilpotent. In particular this holds for the Thompson group.There are torsion free groups for which the analogous statement is false.     

Stable cohomology over local rings

For a commutative noetherian ring R with residue field k stable cohomology modules have been defined for each n∈Z, but their meaning has remained elusive. It is proved that the k-rank of any characterizes important properties of R, such as being regular, complete intersection, or Gorenstein. These numerical characterizations are based on results concerning the structure of Z-graded k-algebra carried by stable cohomology. It is shown that in many cases it is determined by absolute cohomology through a canonical homomorphism of algebras . Some techniques developed in the paper are applicable to the study of stable cohomology functors over general associative rings.     

Relative cohomology theory for profinite groups

In this paper we define and develop the theory of the cohomology of a profinite group relative to a collection of closed subgroups. Having made the relevant definitions we establish a robust theory of cup products and use this theory to define profinite Poincaré duality pairs. We use the theory of groups acting on profinite trees to give Mayer–Vietoris sequences, and apply this to give results concerning decompositions of 3-manifold groups. Finally we discuss the relationship between discrete duality pairs and profinite duality pairs, culminating in the result that profinite completion of the fundamental group of a compact aspherical 3-manifold is a profinite Poincaré duality group relative to the profinite completions of the fundamental groups of its boundary components.     

Coclass and cohomology

Recently, it was proved by Leedham-Green and others that with a finite number of exceptions, every p-group of coclass r is a quotient of one of only a finite number of p-adic uniserial space groups. In this paper we use that structure to demonstrate that there are only finitely many isomorphism classes of cohomology rings of 2-groups of coclass r with coefficients in any fixed field k of characteristic 2. In addition, there is experimental evidence indicating that in many cases successive quotients of the uniserial space groups have isomorphic cohomology rings.     

Extensions for finite Chevalley groups III: Rational and generic cohomology

Let G be a connected reductive algebraic group and B   be a Borel subgroup defined over an algebraically closed field of characteristic p>0$p>0$. In this paper, the authors study the existence of generic G-cohomology and its stability with rational G-cohomology groups via the use of methods from the authors' earlier work. New results on the vanishing of G and B  -cohomology groups are presented. Furthermore, vanishing ranges for the associated finite group cohomology of G(_Fq)$G({\mathbb{F}}_q)$ are established which generalize earlier work of Hiller, in addition to stability ranges for generic cohomology which improve on seminal work of Cline, Parshall, Scott and van der Kallen.     

Generalized Burnside rings and group cohomology

In this paper we give by a unified formula the classification of exceptional compact simple Kantor triple systems defined on tensor products of composition algebras corresponding to realifications of exceptional simple Lie algebras.     

Constructing Witt-Burnside rings

The Witt-Burnside ring of a profinite group G over a commutative ring A generalizes both the Burnside ring of virtual G-sets and the rings of universal and p-typical Witt vectors over A. The Witt-Burnside ring of G over the monoid ring Z[M], where M is a commutative monoid, is proved isomorphic to the Grothendieck ring of a category whose objects are almost finite G-sets equipped with a map to M that is constant on G-orbits. In particular, if A is a commutative ring and A_× denotes the set A as a monoid under multiplication, then the Witt-Burnside ring of G over Z[A_×] is isomorphic to Graham's ring of “virtual G-strings with coefficients in A.” This result forms the basis for a new construction of Witt-Burnside rings and provides an important missing link between the constructions of Dress and Siebeneicher [Adv. in Math. 70 (1988) 87-132] and Graham [Adv. in Math. 99 (1993) 248-263]. With this approach the usual truncation, Frobenius, Verschiebung, and Teichmüller maps readily generalize to maps between Witt-Burnside rings.     

Lazy cohomology: An analogue of the Schur multiplier for arbitrary Hopf algebras

We propose a detailed systematic study of a group associated, by elementary means of lazy 2-cocycles, to any Hopf algebra A. This group was introduced by Schauenburg in order to generalize Kac's exact sequence. We study the various interplays of lazy cohomology in Hopf algebra theory: Galois and biGalois objects, Brauer groups and projective representations. We obtain a Kac-Schauenburg-type sequence for double crossed products of possibly infinite-dimensional Hopf algebras. Finally, the explicit computation of for monomial Hopf algebras and for a class of cotriangular Hopf algebras is performed.     

Tornehave morphisms, II: The lifted Tornehave morphism and the dual of the Burnside functor

We introduce the lifted Tornehave morphism , an inflation Mackey morphism for finite groups, π being a set of primes, K the kernel of linearization, and B^∗ the dual of the Burnside functor. For p-groups, is unique up to scalar multiples. It induces two morphisms of biset functors, one with a codomain associated with a subgroup of the Dade group, the other with a codomain associated with a quotient of the Burnside unit group.     

Ideals associated with realcompactness in pointfree function rings

Abstract

Let RL denote the ring of continuous real-valued functions on a com- pletely regular frame L. The support of an α ∈ RL is the closed quotient ↑(coz α)?. We show that if supports are coz-quotients in L, then the set of functions with realcompact support is an ideal. If L satisfies the stronger condition that supports are C-quotients, then this ideal is the intersection of pure parts of the free maximal ideals of RL. The set of functions whose cozeroes are realcompact is always an ideal, which is free if and only if L is locally realcompact if and only if L is (isomorphic to) an open quotient of υL. Further, this ideal is prime if and only if it is a free real maximal ideal if and only if υL → L is a one-point extension of L.     

Jumps in cohomology and free group actions

A discrete group G has periodic cohomology over R if there is an element in a cohomology group cup product with which it induces an isomorphism in cohomology after a certain dimension. Adem and Smith showed that if , then this condition is equivalent to the existence of a finite dimensional free-G-CW-complex homotopy equivalent to a sphere. It has been conjectured by Olympia Talelli, that if G is also torsion-free then it must have finite cohomological dimension. In this paper we use the implied condition of jump cohomology over R to prove the conjecture for -groups and solvable groups. We also find necessary conditions for free and proper group actions on finite dimensional complexes homotopy equivalent to closed, orientable manifolds.     

Comments on monoreflections in unital archimedean lattice-ordered groups and rings

In quite general concrete categories, the monoreflections which are H-closed (closed under homomorphic image) are the “naturally occuring” ones, since each is comprised of objects closed under some family of functorial implicit operations. In the specific categories under consideration here, we have earlier shown the class of H-closed monoreflections to be a set, indeed, relatively small. In spite, or because of that very few non-H-closed monoreflections have been exhibited, in fact, none in the categories considered here. We shall exhibit many and assuming no measurable cardinal, a proper class.     

Topological methods for complex-analytic Brauer groups

Using methods from algebraic topology and group cohomology, I pursue Grothendieck's question on equality of geometric and cohomological Brauer groups in the context of complex-analytic spaces. The main result is that equality holds under suitable assumptions on the fundamental group and the Pontrjagin dual of the second homotopy group. I apply this to Lie groups, Hopf manifolds, and complex-analytic surfaces.     

The Green rings of pointed tensor categories of finite type

In this paper, we compute the Clebsch–Gordan formulae and the Green rings of connected pointed tensor categories of finite type.