The behavior of Nil-groups under localization and the relative assembly map

In this article, we introduce the notion of a functor on coarse spaces being\ncoarsely excisive- a coarse analogue of the notion of a functor on topological\nspaces being excisive. Further, taking cones, a coarsely excisive functor\nyields a topologically excisive functor, and for coarse topological spaces\nthere is an associated coarse assembly map from the topologically exicisive\nfunctor to the coarsely excisive functor.\n We conjecture that this coarse assembly map is an isomorphism for uniformly\ncontractible spaces with bounded geometry, and show that the coarse isomorphism\nconjecture, along with some mild technical conditions, implies that a\ncorrespoding equivariant assembly map is injective. Particular instances of\nthis equivariant assembly map are the maps in the Farrell-Jones conjecture, and\nin the Baum-Connes conjecture.\n

In this chapter, we discuss several special types of modules that will make frequent appearances later. These include semisimple modules, artinian modules, and torsionfree modules, which have prototypes in vector spaces, finite abelian groups, and torsionfree abelian groups, respectively. That special types of modules have useful roles in the study of arbitrary modules may be seen already in the case of abelian groups (i.e., ℤ-modules). In studying an arbitrary abelian group, an almost reflexive first step is to look at its torsion part (i.e., the torsion subgroup) and its torsionfree part (i.e., the factor group modulo the torsion subgroup), since entirely different techniques are available (and needed) for dealing with torsion groups and torsionfree groups. On the torsionfree side, vector spaces make an appearance due to the facts that the torsionfree divisible abelian groups are exactly the vector spaces over ℚ and that every torsionfree abelian group can be embedded in a divisible one. On the torsion side, many questions can be reduced to the case of finite abelian groups, since every torsion abelian group is a directed union of finite subgroups. In studying torsion abelian groups, one also reduces to the case of p -groups for various primes p . Vector spaces make another appearance here, since in an abelian p -group the set of elements of order p (together with 0) forms a vector space over ℤ/ p ℤ.

In this chapter we shall investigate several types of abelian extension fields. First, we shall consider cyclotomic fields over the field of rational numbers and we shall determine their dimensionalities and Galois groups. Next we shall consider Kummer extensions, which are obtained by adjoining the roots of a finite number of pure equations x m = α to a field containing m distinct m-th roots of 1. Finally, we shall study the so-called abelian p-extensions, which are defined to be abelian extensions of p f dimensions of a field of characteristic p ≠ 0. The theory of characters of finite commutative groups is a basic tool for the investigation of Kummer extensions and abelian p-extensions. Besides this, our study of abelian p-extensions will be based on a certain type of ring, a ring of Witt vectors which can be constructed from any commutative algebra U over a field of characteristic p ≠ 0. For any such U and integer m = 1, 2, ···, we have a ring of Witt vectors M m(U) of characteristic p m . In the theory of valuations it is useful to pass to the limit as m → ∞ and to consider also rings M(U) of infinite Witt vectors. This will be considered in Chapter V.

The study of Hopficity in Abelian groups has been largely motivated by the fundamental results of Baumslag, who proved that torsion groups are always Hopfian regardless of their cardinality, but left several questions open concerning torsion-free groups. Later, Corner addressed some of these questions by providing counterexamples showing that a direct sum of two Hopfian groups can be non-Hopfian, and that a group with an automorphism group of order two does not guarantee Hopficity. These results highlighted the need for new constructions to explore Hopficity in torsion-free Abelian groups. Our work introduces a new approach based on divisibility techniques, as it contributes to the understanding of free-torsion groups with respect to the Hopficity property, providing new insights into their structural properties and implications within group theory. Our analysis also demonstrates how divisibility properties, as well as the introduction of totally invariant subgroups and homomorphisms, can be used to establish Hopficity in specific Abelian groups, particularly those that are free-torsion. In order to reach all of this, we start by taking a group defined as an infinite direct sum of cyclic groups; then we construct a specific subgroup generated by two particular families of elements; and finally we show that this group is Hopfian through results from the theory of divisible subgroups.

Let R be an associative ring with a unit and N be a left R-module. The set MR(N) = {f: N → N | f(rx) = rf(x), r ∈ R, x ∈ N} is a near-ring with respect to the operations of addition and composition and contains the ring ER(N) of all endomorphisms of the R-module N. The R-module N is endomorphic if MR(N) = ER(N). We call an Abelian group endomorphic if it is an endomorphic module over its endomorphism ring. In this paper, we find endomorphic Abelian groups in the classes of all separable torsion-free groups, torsion groups, almost completely decomposable torsion-free groups, and indecomposable torsion-free groups of rank 2.

In this paper, we prove that the hyperfocal subalgebra of a block with an abelian defect group and a cyclic hyperfocal subgroup is Rickard equivalent to the group algebra of the semidirect of the hyperfocal subgroup by the inertial quotient of the block. In particular, the hyperfocal subalgebra is a Brauer tree algebra, which is analogous to the structure of blocks with cyclic defect groups. As a consequence, we show that Broué's abelian defect group conjecture holds for blocks with cyclic hyperfocal subgroups.

Two operators or two subgroups of a group will be called I-conjugate whenever they are conjugate under the group of isomorphisms of this group. It is well known that all the cyclic subgroups of highest order contained in any abelian group are I-conjugate. A necessary and sufficient condition that all the cyclic subgroups of any other order contained in an abelian group G of order pmm, p being a prime number, are I-conjugate is that all the invariants of G are equal to each other. When this condition is satisfied we shall prove, in particular, that the number of the subgroups of order pa, a c m , contained in G is equal to the number of its subgroups of index pa. It is known that the ?-subgroup of G is composed of the pth power of every operator of G. The concept of +-subgroup in connection with abelian groups can readily be extended by considering the characteristic subgroup of G composed of the path power of every operator of G. This subgroup may be called the Oa-subgroup of G. It is composed of all the operators of G which are found in each one of its subgroups of index pa, and all the operators of highest order in a 4.-subgroup of G are I-conjugate. In particular, the 0&-subgroup generated by operators of order p is the fundamental characteristic subgroup of G. t In fact, when p > 2 the 0a-subgroup which is generated by operators of order pr is contained in every characteristic subgroup of G which involves operators of order pr. The quotient group of G corresponding to a 4a-subgroup may be called a 0,fi-quotient group. This quotient group is simply isomorphic with the subgroup of G generated bv all its operators whose orders divide p> Hence the following: THEOREM. The number of the subgroups of index pa contained in any abelian

A problem of Călugăreanu, Chekhlov and Krylov asks which abelian groups have the property that all of their pure subgroups that are endomorphic images are necessarily summands. Complete answers are given for the groups in certain classes (for example, the torsion groups). On the other hand, examples are constructed that show a complete solution to the general problem is likely to be quite difficult.

Introduction. A nil group is an (additively written) abelian group G such that every associative ring R with G as its additive group has trivial multiplication, i.e., xy = 0 for every x, y ER. Szele [2] has shown that every nil group is either a torsion group (every element of finite order) or a torsion-free group (every element except the identity of infinite order), and he has completely characterized the torsion nil groups. In [3 ], Szele discusses some questions related to nil groups, but is unable to find conditions for the existence of torsionfree nil groups. An additive group G is said to be strongly nil if there is no nontrivial ring, associative or nonassociative, which has G as its additive group. It is the purpose of this note to characterize those nil and strongly nil groups which are (isomorphic to) a weak direct sum of subgroups of the additive group of rationals, and to show that every such nil group is also strongly nil. The authors would like to express thanks to the referee for helpful suggestions.

We prove that the Waldhausen nilpotent class group of an injective index 2\namalgamated free product is isomorphic to the Farrell-Bass nilpotent class\ngroup of a twisted polynomial extension. As an application, we show that the\nFarrell-Jones Conjecture in algebraic K-theory can be sharpened from the family\nof virtually cyclic subgroups to the family of finite-by-cyclic subgroups.\n

On groups with an interval isomorphic to a projective geometry

Introduction. In the investigation of functors on the category of preschemes, one is led, by Grothendieck [3], to consider the following situation.Let A be a complete noetherian local ring, u, its maximal ideal, and k = A/u.the residue field.(In most applications A is k itself, or a ring of Witt vectors.)Let C be the category of Artin local A-algebras with residue field k.A covariant functor F from C to Sets is called pro-representable if it has the formwhere R is a complete local A-algebra such that R/mn is in C, all n.(m is the maximal ideal in R.)In many cases of interest, F is not pro-representable, but at least one may find an R and a morphism Hom(7?, )-> F of functors such that Hom(.R, A) -> F(A) is surjective for all A in C. If R is chosen suitably "minimal" then R is called a "hull" of F; R is then unique up to noncanonical isomorphism.Theorem 2.11, 2, gives a criterion for F to have a hull, and also a simple criterion for pro-representability which avoids the use of Grothendieck's techniques of nonflat descent [3], in some cases.Grothendieck's program is carried out by Levelt in [4].3 contains a few geometric applications of these results.To avoid awkward terminology, I have used the word " pro-representable " in a more restrictive sense than Grothendieck [3] has.He considers the category of A-algebras of finite length and allows R to be a projective limit of such rings.The methods of this paper are a simple extension of those used by David Mumford in a proof (unpublished) of the existence of formal moduli for polarized Abelian varieties.I am indebted to Mumford and to John Tte for many valuable suggestions.1.The category CA.Let A be a complete noetherian local ring, with maximal ideal u. and residue field k = A/u.. We define C= CA to be the category of Artinian local A-algebras having residue field k. (That is, the "structure morphism" A -> ,4 of such a ring A induces a trivial extension of residue fields.)Morphisms in C are local homomorphisms of A-algebras.

Abstract.A tuple (or subgroup) in a group is said to degenerate to another if the latter is an endomorphic image of the former. In a countable reduced abelian group, it is shown that if tuples (or finite subgroups) degenerate to each other, then they lie in the same automorphism orbit. The proof is based on techniques that were developed by Kaplansky and Mackey in order to give an elegant proof of Ulm's theorem. Similar results hold for reduced countably-generated torsion modules over principal ideal domains. It is shown that the depth and the description of atoms of the resulting poset of orbits of tuples depend only on the Ulm invariants of the module in question (and not on the underlying ring). A complete description of the poset of orbits of elements in terms of the Ulm invariants of the module is given. The relationship between this description of orbits and a very different-looking one obtained by Dutta and Prasad for torsion modules of bounded order is explained.

In this work, the assembly map in L-theory for the family of finite subgroups is proven to be a split injection for a class of groups. Groups in this class, including virtually polycyclic groups, have universal spaces that satisfy certain geometric conditions. The proof follows the method developed by Carlsson-Pedersen to split the assembly map in the case of torsion free groups. Here, the continuously controlled techniques and results are extended to handle groups with torsion.

§1. Introduction.The motivation of this work comes from two different directions: infinite abelian groups, and ordered algebraic structures. A challenging problem in both cases is that of classification. In the first case, it is known for example (cf. [KA]) that the classification of abelian torsion groups amounts to that of reducedp-groups by numerical invariants called theUlm invariants(given by Ulm in [U]). Ulm's theorem was later generalized by P. Hill to the class of totally projective groups. As to the second case, let us consider for instance the class of divisible ordered abelian groups. These may be viewed as ordered ℚ-vector spaces. Their theory being unstable, we cannot hope to classify them by numerical invariants. On the other hand, being o-minimal, the theory enjoys several good model theoretic properties (cf. [P-S]), so the search for some reasonable invariants is well motivated. The common denominator of the two cases, as well as of many others, is valuation theory. Indeed given an ordered vector space, one can consider it as a valued vector space, endowed with the natural valuation. Also, the socleG[p] of a reduced abelianp-groupG, endowed with the height functionhG, is a valued vector space over(the prime field of characteristicp) with values in the ordinals (cf. [F]).