Category Of Abelian Groups Research Articles

Operator K-Theory and Functor N0. Some Applications

In this paper, we present a general introduction to the K-theory of C *-algebras and survey of our previous papers, where the functor N 0 from the category of von Neumann algebras to the category of Abelian groups was defined. We investigate the properties of this functor (in particular, its interrelation with the functor K 0) and point out some applications of the functor N 0 in noncommutative geometry. In addition, we recall facts of theory of C *-algebras, von Neumann algebras, and Hilbert C *-modules.

Characterization of the inessential endomorphisms in the category of abelian groups

An endomorphism f of an Abelian group A is said to be inessential (in the category of Abelian groups) if it can be extended to an endomorphism of any Abelian group which contains A as a subgroup. In this paper we show that f is as above if and only if (f − v idA)(A) is contained in the maximal divisible subgroup of A for some v ∈ Z.

Characterization of the automorphisms having the lifting property in the category of abelian p‐groups

Let p be a prime. It is shown that an automorphism α of an abelian p‐group A lifts to any abelian p‐group of which A is a homomorphic image if and only if α = π idA, with π an invertible p‐adic integer. It is also shown that if A is torsion group or torsion‐free p‐divisible group, then idA and −idA are the only automorphisms of A which possess the lifting property in the category of abelian groups.

Quadratic categories and square rings

We consider quadratic categories which generalize the classical additive categories. An additive category A is a category for which morphism sets are abelian groups and the composition fg is bilinear, and for which sums exist in A . A quadratic category Q is slightly more general in the sense that morphism sets are groups and the composition fg is linear in g and quadratic in f. This implies that morphism sets are groups of nilpotency degree 2. We describe below many examples of quadratic categories in algebra and topology which motivate the systematic study of quadratic categories started here; it may be considered as an extension of the investigation of quadratic functors in [4]. The properties of a quadratic category and its subcategories lead to the new notion of a “square ring” which is exactly the quadratic analogue of the classical notion of a “ring”. Indeed each object X in an additive category A yields an endomorphism ring given by all morphisms X → X in A; similarly each object in a quadratic category yields the endomorphism square ring End( X) of X. Hence the connection between square rings and quadratic categories is similar to the relation of rings and additive categories studied by Mitchell [18]. The initial object in the category of rings is the ring Z of integers for which the category of modules is the category of abelian groups. We here determine the initial object Z nil in the category of square rings for which the category of modules is the category of groups of nilpotency degree 2. We compute various square rings explicitly, for example, the endomorphism square rings of the suspended projective planes ΣR P 2 and ΣC P 2. This yields as an application an algebraic description of the homotopy category of all Moore spaces M( V,2) where V is a Z/2-vector space; in fact this category is equivalent to the full category of free objects in the category of 2-restricted nil(2)-groups. There has been recently a lot of interest in operads [9]. In fact, operads O = {O n } with O n = 0 for n ≥ 3 are the same as special square rings. Therefore the theory of square rings shows naturally how the theory of operads has to be modified in order to deal with nilpotent groups.

Quasigroups, right quasigroups and category coverings

The category of modules over a fixed quasigroup in the category of all quasigroups is equivalent to the category of representations of the fundamental groupoid of the Cayley diagram of the quasigroup in the category of abelian groups. The corresponding equivalent category of coverings, and the generalization to the right quasigroup case, are also described.

Modularity and descent

We show that a left exact category E with finite coproducts is affine, i.e. it is a slice of an additive category with kernels, if and only if the forgetful functor from the category of abelian groups in E is monadic and the corresponding monad T is nullary, i.e. the canonical maps T(0) + X → T( X) are isomorphisms, and the adjoint is comonadic (and then also the corresponding comonad is conullary). This result should be compared with the well-known characterization of additive categories as those for which the same forgetful functor is an equivalence.

On rational equivariantH-spaces

Let O G be the category of canonical orbits of G. The objects of O G are the orbit spaces G/H, H a subgroup of G, and the morphisms are the equivariant maps between them, where G acts on G/H by left multiplication. By a coefficient system for G we mean a contravariant functor from O a into the category of abelian groups. A contravariant functor from O a into the category of q-vec tor spaces will be called a rational coefficient system. The coefficient systems of either sort and the natural transformations between them form abelian categories with sufficiently many projectives and injectives [1, 1.10]. For a G-space X, the coefficient systems n n (X) and -//n (X) can be defined by _~, (X) (G/H) = _n, (Xn), _/J, (X) (G/H) = H, (Xn), n > 1, where H, ( ) denotes the singular homology with Z-coefficients. By naturality of the Hurewicz homomorphism, we get a morphism of coefficient systems h,: _n, (X) -~ Un (X), which we shall also call the Hurewicz homomorphism.

Z/kZ-finiteness for certain S¹-spaces

Let G-ZFIDCW denote the category of G-spaces having the G-homotopy type of a finitely dominated G-CW complex for a compact Lie group G. Luck [8] has introduced a functor Wa from G-EF^C^ into the category of abelian groups and has realized the equivariant finiteness obstriction as the element w(X) in Wa(X). That is, a finitely dominated G-CW complex X is G-homotopy equivalent to a finite G-CW complex if and only if w(X)=Q. When G is the trivial group, there is an isomorphism from Wa(X) to the reduced projective group KQ(Z[πι(X)]) which sends the element w (X) to the WalΓs finiteness obstruction ([14]). Anderson [1] and Ehrlich [4] have studied a sufficient condition for w(E)= 0 for some fibration E-^B with fiber S. Munkholm, Pedresen [11], Luck [6, 7, 9] and others have studied the transfer map KQ(Z[^(B)])-^K:Q(Z[^(E)]). The purpose of this paper is to get a sufficient condition for w(X)=0 for a S^space X and a finite cyclic group L. We call G-maps /0: YQ-^X and /0: Y4-*X equivalent if there exists a commutative diagram

Finlteness Obstructions of Equivariant Fibrations

Let G be a compact Lie group and E->B a G-fibration. We define a homomorphism Wafi(B)®U^JB) into Wa<*(E)@U®(E) sending the pair of the finiteness obstruction of B and the equivariant Euler characteristic of B to that of E. Here Wa® is the functor from the Ghomotopy category of finitely dominated G-CW complexes into the category of abelian groups given by W. Luck. By making use of this, we show that if H and K are closed subgroups with HOT K normal such that W(HK) is not finite, GxBX is X-homotopy equivalent to a finite K-CW complex.

Injectivity and modelling in the blass topos

A very obvious general question concerning the modelling of any familiar mathematical notion cfJ, with associated relevant maps, in some topos E, is how properties of the 'usual' category determined by e:p relate to the properties of the corresponding category of cfJ-objects in E. This paper is an illustration of what can happen in this context with regard to injectivity in the categories concerned. Blass [4,6] showed, for a certain topos B, that the category of abelian groups in B has no non-zero injectives although the 'usual' category of abelian groups enjoys the familiar property of having enough injectives, meaning that any of its objects can be embedded into an injective one. We refer to this occurrence as a breakdown of injectivity (in B). In the following, we exhibit further such breakdowns, using the same topos as Blass [4] and appropriate variants of his method of proof. The notions we deal with are: modules over any ring, Boolean algebras, and distributive lattices. Regarding the latter two, it might be recalled that, in the usual setting, the injective Boolean algebras are exactly the complete ones [12, p. 112] and the injective distributive lattices the complete Boolean algebras [3]. The existence of enough injectives then follows fairly easily in either case. By way of contrast, we also discuss three notions for which the existence of enough injectives is retained when they are modelled in the topos considered here, thus showing that the breakdown of injectivity in the earlier situations is by no means a general feature of that topos. The three concepts of this second type are partially ordered sets, semilattices, and normed linear spaces, the maps being, respectively, the order preserving maps, the homomorphisms, and the linear contractions; here, injectivity is understood with respect to embeddings (rather than mere monomorphisms) in the first and third case. We note that, in the usual situation, the injectives are, respectively, the complete partially ordered sets [2], the locales, i.e. the complete partially ordered sets in which the distribution law X/ V xa '= V (xvxJ holds for binary meets (1\) and arbitrary joins (V) [7], and the

The Geometric Finiteness Obstruction

The purpose of this paper is to develop a geometric approach to Wall's finiteness obstruction. We will do this for equivariant CW-complexes. The main advantage will be that we can derive all the formal properties of the equivariant finiteness obstruction easily from this geometric description. Namely, the obstruction property, homotopy invariance, the sum and product formulas, and the restriction formula can be stated and proved in a simple manner. Also a characterization of the finiteness obstruction by a universal property is quickly available. This geometric approach is similar to the treatment of Whitehead torsion by Cohen in [3]. In the first section we define a functor Wa from the category of G-spaces to the category of abelian groups. We assign to a finitely dominated G-CW-complex X an element w(X) e Wa(X) called its finiteness obstruction. The finiteness obstruction vanishes if and only if X is G-homotopic to a finite G-CW-complex and satisfies a sum formula and is homotopy invariant. The notion of a universal functorial additive invariant is introduced in the second section where its existence and uniqueness are proved. Product and restriction formulas for the universal additive invariant are obtained by abstract nonsense. We define equivariant Euler characteristics in the third section generalizing the notion of the Euler characteristic of a finite CW-complex. The goal of the fourth section is to prove that the equivariant Euler characteristic and finiteness obstruction determine the universal functorial additive invariant for finite, respectively finitely dominated, G-CW-complexes. The fifth section contains some algebraic computations of Wa in terms of reduced projective class groups of certain integral group rings. In the nonequivariant case Wall's algebraic approach and our geometric one agree. Finally, in the sixth section, the results of the second and fourth sections are used to state an abstract product formula, a restriction formula, and a diagonal product formula. We make some remarks about the simple-homotopy approach to the finiteness obstruction due to Ferry. The treatment by Ferry in [7] is extended by Kwasik in [14] to the equivariant case. In § 6 we construct geometrically an injection I(Y): W(Y)^>Wh(YxS) into the equivariant Whitehead group of Y x S sending our geometric finiteness obstruction to that of Kwasik. A compact Lie group is denoted by G.

Characteristic classes (purities) of short exact sequences in the category of Abelian groups

Characteristic classes (purities) of short exact sequences in the category of Abelian groups

BIVARIANT LONG EXACT SEQUENCES I

Abstract Given a pair of short exact sequences in an abelian category A, extending results due to Pressman [7]. We construct nine bivariant long exact sequences that involve the groups extn(δ,β), extn(δ,α) and extn(γ,α) that interlock with the univariant hom-ext sequences determined by (i). Two applications of these sequences in the category of abelian groups are given. Firstly we describe an isomorphism that yields a “constructive” way to compute the adjusted cotorsion part of a reduced cotorsion group and secondly, given a torsion-free, reduced group G with no free summands, we indicate how to determine the p-ranks of the divisible group Ext(G,Z).

On the cotorsion hull in torsion theory

The concept of cotorsion was first introduced in the category of Abelian groups (Fuchs [l] ). Matlis [5], studied the cotorsion modules over integral domains. Henderson and Orzech [4], Fuchs [2], and Mines [6], replaced the classical notion of torsion by a torsion theory (T,F) on R-mod, where R is not necessarily commutative ring. In this paper we find conditions on the torsion theory in order to get a T-cotorsion hull for every module. This generalizes the result of Fuchs [2].

A characterization of free abelian groups

In the category of abelian groups, being free is equivalent to having a discrete norm.

On Evaluation Subgroups of Generalized Homotopy Groups

AbstractG(A, X) consists of all homotopy classes of cyclic maps from a space A to another space X. If A is an H-cogroup, then G(A, X) is a group. G(A, X) preserves products in the second variable and is a contravariant functor of A from the full subcategory of H-cogroups and maps into the category of abelian groups and homomorphisms. If X is an H-cogroup, then G(X, X) is a ring.

Functorial radicals and non-abelian torsion, II

The object of this paper is to complete and continue some matters in [1].In [1], Section 2, the torsion and torsion-free functors, whose operation on the category of abelian groups are well known, were extended to the category of all groups as follows. For a group A, put t0(A)= 0 and t1(A) = the subgroup of A generated by the torsion elements of A. Inductively define tn+1(A)/tn(A)=t1(A)/tn(A)), for every positive integer n. Then T(A)=∪ntn(A) is the smallest subgroup H of A such that A/H is torsion-free, [1], Th. 2.2. A group A satisfying T(A) = A was called a pre-torsion group. In [1], 2.12 an example was constructed of a group A satisfying t1(A)≠t2(A)=A. The question was posed whether for every positive integer n there exist groups A, satisfying tn–1(A)≠tn(A)=A. Here we give an affirmative answer. In fact, such groups will be constructed, as well as pre-torsion groups A with tk(A)≠A for every positive integer k, see Section 1.

Twisted Homological Stability for General Linear Groups

Suppose that R is a principal ideal domain and that, for each n, p is a module over the general linear group GL, (R). The major purpose of this paper is to show that for very general choices of {p,} the homology groups Hk(GL, (R), pa) stabilize with respect to n, i.e., for each fixed k assume a constant value as n becomes large. A secondary purpose is to show that this stability phenomenon has interesting consequences for Waldhausen's algebraic K-theory of topological spaces and ultimately for geometric topology. The main stability theorem (Theorem 2.2) is explained in detail at the beginning of Section 2; it has an inductive statement too elaborate to summarize here. A sample application of 2.2 is the following. Let Ab be the category of abelian groups, and let Ad, (R) be the abelian group of n x n matrices over R, considered as a GL,, (R)-module by conjugation. Note that if T: Ab -> Ab is a functor, there is a natural action of GLUM (R) on T(AdJ(R)). 1.1. PROPOSITION. If T: Ab -* Ab is a functor of finite degree (see ? 3 and [31) then the homology groups Hk(GL, (R), T(Adn(R))) stabilize with respect to n.

Indecomposable injective functors from an abelian category to the category of abelian groups

Indecomposable injective functors from an abelian category to the category of abelian groups

Thick Groups and Essentially Finitely Indecomposable Groups

In this article, we examine the relationship between the notions of thick groups and that of essentially finitely indecomposable (e.f.i.) groups. These groups are indecomposable objects in the quotient category of the category of abelian groups induced by the Serre-class of bounded groups.