On the categorical structure of H

The categorical structure of H² is shown to be a particular instance of the cokernel of a morphism between symmetric categorical groups.     

Abelian groups with a vanishing homology group

An abelian group A is called absolutely abelian, if in every central extension N ? G ? A the group G is also abelian. The abelian group A is absolutely abelian precisely when the Schur multiplicator H₂A vanished. These groups, and more generally groups with H_nA = 0 for some n, are characterized by elementary internal properties. (Here H₁A denotes the integral homology of A.) The cases of even n and odd n behave strikingly different. There are 2^?ο different isomorphism types of abelian groups A with reduced torsion subgroup satisfying H_2nA = 0. The major tools are direct limit arguments and the Lyndon-Hochschild-Serre (L-H-S) spectral sequence, but the treatment of absolutely abelian groups does not use spectral sequences. All differentials d^r for r ≥ 2 in the L-H-S spectral sequence of a pure abelian extension vanish. Included is a proof of the folklore theorem, that homology of groups commutes with direct limits also in the group variable, and a discussion of the L-H-S spectral sequence for direct limits.     

ON RELATIVE INJECTIVITY

Abstract

We investigate the σ-injectivity of O. Goldman and other related types of relative injectivity when σ is an arbitrary torsion radical, and when the associated filter satisfies some standard finiteness conditions.     

On F-inverse covers of inverse monoids

It is shown that each finite inverse monoid admits a finite F-inverse cover if and only if the same is true for each finite combinatorial strict inverse semigroup with an identity adjoined if and only if the same is true for the Margolis-Meakin expansion M(H) of each finite elementary abelian p-group H for some prime p. Additional equivalent conditions are given in terms of the existence of locally finite varieties of groups having certain properties. Ultimately, the problem of whether each finite inverse monoid admits a finite F-inverse cover, is reduced to a question concerning the Kostrikin-Zelmanov varieties K_n of all locally finite groups of exponent dividing n.     

Varieties of finite supersolvable groups with the M. Hall property

The varieties in the title are shown to be precisely the product varieties G_p*Ab(d) for some prime p and some positive integer d dividing p−1. Here G_p denotes the variety of all finite p-groups and Ab(d) the variety of all finite Abelian groups of exponent dividing d. It turns out that these are exactly those varieties H of supersolvable groups for which all finitely generated free pro-H groups are freely indexed in the sense of Lubotzky and van den Dries. Several alternative characterizations of these varieties are presented. Some applications to formal language theory and finite monoid theory are also given. Among these is the determination of all supersolvable solutions H to the equations PH = J*H and J*H = J H which is, to the present date, the most complete solution to a problem raised by Pin. Another consequence of our results is that for each such variety H the monoid variety PH = J*H = J H has decidable membership. The authors gratefully acknowledge the support of NSERC     

Higher cohomologies of commutative monoids

Extending the Eilenberg–Mac Lane approach, we introduce and explore higher-level cohomology theories for commutative monoids and compare them with pre-existing theories (Leech, Grillet, etc.). We offer a cohomological classification of symmetric monoidal groupoid structures and work out some explicit computations for cyclic monoids.     

When the heart of a faithful torsion pair is a module category

An abelian category with arbitrary coproducts and a small projective generator is equivalent to a module category (Mitchell (1964) [17]). A tilting object in an abelian category is a natural generalization of a small projective generator. Moreover, any abelian category with a tilting object admits arbitrary coproducts (Colpi et al. (2007) [8]). It naturally arises the question when an abelian category with a tilting object is equivalent to a module category. By Colpi et al. (2007) [8], the problem simplifies in understanding when, given an associative ring R and a faithful torsion pair (X,Y) in the category of right R-modules, the heartH(X,Y)of the t-structure associated with (X,Y) is equivalent to a category of modules. In this paper, we give a complete answer to this question, proving necessary and sufficient conditions on (X,Y) for H(X,Y) to be equivalent to a module category. We analyze in detail the case when R is right artinian.     

On varieties arising from the solution of the Restricted Burnside Problem

For a family of group-words w we prove that the class of all groups G satisfying the identity wⁿ≡1 and having the verbal subgroup w(G) locally nilpotent is a variety.     

Homotopical algebraic geometry I: topos theory

This is the first of a series of papers devoted to lay the foundations of Algebraic Geometry in homotopical and higher categorical contexts. In this first part we investigate a notion of higher topos.For this, we use S-categories (i.e. simplicially enriched categories) as models for certain kind of ∞-categories, and we develop the notions of S-topologies, S-sites and stacks over them. We prove in particular, that for an S-category T endowed with an S-topology, there exists a model category of stacks over T, generalizing the model category structure on simplicial presheaves over a Grothendieck site of Joyal and Jardine. We also prove some analogs of the relations between topologies and localizing subcategories of the categories of presheaves, by proving that there exists a one-to-one correspondence between S-topologies on an S-category T, and certain left exact Bousfield localizations of the model category of pre-stacks on T. Based on the above results, we study the notion of model topos introduced by Rezk, and we relate it to our model categories of stacks over S-sites.In the second part of the paper, we present a parallel theory where S-categories, S-topologies and S-sites are replaced by model categories, model topologies and model sites. We prove that a canonical way to pass from the theory of stacks over model sites to the theory of stacks over S-sites is provided by the simplicial localization construction of Dwyer and Kan. As an example of application, we propose a definition of étale K-theory of ring spectra, extending the étale K-theory of commutative rings.     

The Chase radical and reduced products

Let κ be a regular uncountable cardinal. We shall give a criterion for certain reduced products of torsion-free abelian groups to be ℵ₁-free. As an application we shall show that the norm of the Chase radical is ℵ₁ in ZFC, a result which was previously known only under the assumption of the continuum hypothesis 2^ℵ₀=ℵ₁.     

Some inequalities for the Baer-invariant of a pair of finite groups

In this paper we introduce the concept of Baer-invariant of a pair of groups with respect to a variety of groups v. Some inequalities for the Baer-invariant of a pair of finite groups are obtained, when v is considered to be the Schur-Baer variety. We also present a condition for which the order of the Baerinvariant of a pair of finite groups divides the order of the Baer-invariant of their factor groups. Finally, some inequalities for the Schur-multiplier of a pair of finite nilpotent groups and their factor groups are given.     

Twisted conjugacy classes in nilpotent groups

Let N be a finitely generated nilpotent group. We show that there is an algorithm that for any automorphism φ∈Aut(N) computes its Reidemeister number R(φ). It is proved that any free nilpotent group N_rc of rank r and class c belongs to class R_∞ if any of the following conditions holds: r=2 and c≥4; r=3 and c≥12; r≥4 and c≥2r.     

Pseudogroups and their étale groupoids

A pseudogroup is a complete infinitely distributive inverse monoid. Such inverse monoids bear the same relationship to classical pseudogroups of transformations as frames do to topological spaces. The goal of this paper is to develop the theory of pseudogroups motivated by applications to group theory, C^∗$C^{\ast }$-algebras and aperiodic tilings. Our starting point is an adjunction between a category of pseudogroups and a category of étale groupoids from which we are able to set up a duality between spatial pseudogroups and sober étale groupoids. As a corollary to this duality, we deduce a non-commutative version of Stone duality involving what we call boolean inverse semigroups and boolean étale groupoids, as well as a generalization of this duality to distributive inverse semigroups. Non-commutative Stone duality has important applications in the theory of C^∗$C^{\ast }$-algebras: it is the basis for the construction of Cuntz and Cuntz–Krieger algebras and in the case of the Cuntz algebras it can also be used to construct the Thompson groups. We then define coverages on inverse semigroups and the resulting presentations of pseudogroups. As applications, we show that Paterson’s universal groupoid is an example of a booleanization, and reconcile Exel’s recent work on the theory of tight maps with the work of the second author.     

Torsion theories and Galois coverings of topological groups

For any torsion theory in a homological category, one can define a categorical Galois structure and try to describe the corresponding Galois coverings. In this article we provide several characterizations of these coverings for a special class of torsion theories, which we call quasi-hereditary. We describe a new reflective factorization system that is induced by any quasi-hereditary torsion theory. These results are then applied to study various examples of torsion theories in the category of topological groups.     

Regular and normal closure operators and categorical compactness for groups

For a class of groupsF, closed under formation of subgroups and products, we call a subgroupA of a groupG F-regular provided there are two homomorphismsf, g: G » F, withF F, so thatA = {x G |f(x) =g(x)}.A is calledF-normal providedA is normal inG andG/A F. For an arbitrary subgroupA ofG, theF-regular (respectively,F-normal) closure ofA inG is the intersection of allF-regular (respectively,F-normal) subgroups ofG containingA. This process gives rise to two well behaved idempotent closure operators.A groupG is calledF-regular (respectively,F-normal) compact provided for every groupH, andF-regular (respectively,F-normal) subgroupA ofG × H, ₂(A) is anF-regular (respectively,F-normal) subgroup ofH. This generalizes the well known Kuratowski-Mrówka theorem for topological compactness.In this paper, theF-regular compact andF-normal compact groups are characterized for the classesF consisting of: all torsion-free groups, allR-groups, and all torsion-free abelian groups. In doing so, new classes of groups having nice properties are introduced about which little is known.