A category of matrices representing two categories of Abelian groups

Journal of Mathematical Sciences, Vol. 154, No. 3, 2008

A CATEGORY OF MATRICES REPRESENTING TWO CATEGORIES OF ABELIAN GROUPS A. A. Fomin

UDC 512.541

Abstract. Every τ -adic matrix represents both a quotient divisible group and a torsion-free, finite-rank group. These representations are an equivalence and a duality of categories, respectively.

1. Introduction Great interest in the class of torsion-free, finite-rank Abelian groups first appeared when L. S. Pontryagin discovered its duality [20] in 1934. Then A. G. Kurosh [18], A. I. Malcev [19], and D. Derry [4] obtained a matrix description of these groups in 1937–38. The concepts of quasi-isomorphism and quasi-homomorphism were introduced by B. Jonsson [17] in 1957. The category QT F of torsion-free, finite-rank Abelian groups with quasi-homomorphisms as morphisms is a classical subject of investigations. The class G of mixed self-small groups G such that the quotient G/T (G) is divisible of finite rank was actively investigated by many authors [1,2,5,9,12,14,16,21] in the 1990s. The notion of quotient divisible group was introduced in [13] in 1998. The class of quotient divisible groups contains the class G as well as the classical quotient divisible torsion-free groups introduced by R. Beaumont and R. Pierce [3] in 1961. The category QD of quotient divisible groups with quasi-homomorphisms as morphisms was considered in [13]. Two mutually inverse functors of duality of d

d


RM



-

categories QD −→ QT F and QT F −→ QD were introduced there. In the present paper, a new category RM of special matrices with τ -adic entries is introduced and it is proved that this category is equivalent to the category QD and is dual to the category QT F. Thus, we obtain the following commutative diagram of dualities and equivalences of categories:

QD 

b
c

d d


b

-



c


-

QT F.

The dualities b and b
can be considered as a new version of the Kurosh–Malcev–Derry description. Herewith note that our approach is closer to the approach of Malcev [19], which is different from the more well known approach of Kurosh and Derry. In the last section, a new interpretation of the duality functors d and d
is given without using matrices (Theorem 6). The connection between this duality and the Pontryagin duality is found as well. Note that all basic constructions of this paper can be generalized to modules over Dedekind rings in the manner of [11]. The author expresses his acknowledgment to Prof. Otto Mutzbauer for fruitful discussions and for his essential contribution to this paper. ˆ p denote the ring of integers, the field of ˆ p , and Q All groups will be additive Abelian groups. Z, Q, Z ˆ p is ˆ =
Z rationals, the ring of p-adic integers, and the field of p-adic numbers, respectively. The ring Z p ˆ is called the ring of universal numbers. the Z-adic completion of Z, and K = Q ⊗ Z Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 13, No. 3, pp. 223–244, 2007.

430

c 2008 Springer Science+Business Media, Inc. 1072–3374/08/1543–0430 

We use characteristics and types in the same manner as in [15], denoting the zero characteristic and the zero type by 0. Let τ and σ be types containing the characteristics (mp ) and (kp ), respectively. We define the operations τ ∧ σ = [(mp ) ∧ (kp )] = [(min{mp , kp })], τ ∨ σ = [(mp ) ∨ (kp )] = [(max{mp , kp })], τ + σ = [(mp ) + (kp )] = [(mp + kp )]. If (mp ) ≥ (kp ), then τ − σ = [(mp ) − (kp )] = [(mp − kp )], assuming herewith ∞ − ∞ = 0. We assume that the operations + and − are weaker than the operations ∧ and ∨, whence the expression τ − σ ∧ δ denotes τ − (σ ∧ δ). T (A) and Tp (A) denote the torsion part and the p-primary component of the torsion part of the group A, respectively. If {x1 , . . . , xn } ⊂ A, then x1 , . . . , xn is the subgroup generated by these elements and x1 , . . . , xn ∗ is the pure hull of these elements, i.e., the subgroup consisting of all elements that are in the subgroup x1 , . . . , xn after multiplication by some nonzero integers. In particular, T (A) ⊆ x1 , . . . , xn ∗ . If the group A is also a module over a ring R, then x1 , . . . , xn R denotes the submodule generated by these elements. A set of elements x1 , . . . , xn of a group or of a module is called linearly independent over Z if the equality a1 x1 + · · · + an xn = 0 with integer coefficients implies a1 = · · · = an = 0. All other notations are standard and are adopted from [15]. 2. τ -Adic Numbers ˆ p if mp = ∞. We define For a characteristic χ = (mp ), let Kp = Z/pmp Z if mp < ∞ and Kp = Z
h p the ring Zχ = Kp . For an element (α) ∈ Zχ , we denote by p the greatest power of p such that php p

divides αp in the ring Kp and pgp is the least power of p such that pgp αp = 0. If mp = ∞ and αp = 0, then 0 ≤ hp < ∞ and gp = ∞. In any case, hp + gp = mp for all prime numbers p. Since Q ⊗ Zχ = Q ⊗ Zµ for equivalent characteristics χ and µ, the following definition makes sense. Definition 1. Let τ be a type. The Q-algebra Q(τ ) = Q ⊗ Zχ for a characteristic χ ∈ τ is called ˆ the ring of τ -adic numbers. The ring or Z-module Zχ is said to be associated with Q(τ ). An element (αp ) ∈ Zχ is called a representative of the τ -adic number α = 1 ⊗ (αp ). The types type(α) = [(hp )] and cotype(α) = [(gp )] are called the type of α and the cotype or dual type of α, respectively. The ring Q(τ ) can be considered as a generalization of the ring Zpm . The type and cotype of α are well defined; they are analogues of the height and the order of α, respectively. In particular, Q(τ ) = K if τ = type(Q). Every τ -adic number α can be represented in the form α = rs ⊗ (αp ) with 0 = rs ∈ Q relative to an associated ring Zχ . If β = sr11 ⊗ (βp ) is the representation of the τ -adic number β relative to another associated ring, then α = β if and only if rs1 αp = r1 sβp for all prime numbers p with mp = ∞ and for almost all, i.e., all but finitely many, primes with mp < ∞. In particular, the rings Kp also coincide for this set of primes. Hence Q(τ ) depends on the type and not on the characteristic. Let χ = (mp ) ∈ τ be a characteristic with corresponding type τ . Then      ˆp ⊂ L : = ˆ p. Z Q Kp = Kp ⊕ Kp ⊕ Zχ = p

mp