QUOTIENT DIVISIBLE ABELIAN GROUPS All Groups Will Be Additive and Abelian. the Torsion-Free Rank of a Group G Is the Rank Of

PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 126, Number 1, January 1998, Pages 45–52 S 0002-9939(98)04230-0

QUOTIENT DIVISIBLE ABELIAN GROUPS

A. FOMIN AND W. WICKLESS

(Communicated by Ronald M. Solomon)

Dedicated to the memory of Ross A. Beaumont

Abstract. An abelian group G is called quotient divisible if G is of finite torsion-free rank and there exists a free subgroup F G such that G/F is divisible. The class of quotient divisible groups contains⊂ the torsion-free finite rank quotient divisible groups introduced by Beaumont and Pierce and essen- tially contains the class of self-small mixed groups which has recently been investigated by several authors.G We construct a duality from the category of quotient divisible groups and quasi-homomorphisms to the category of torsion- free finite rank groups and quasi-homomorphisms. Our duality when restricted to torsion-free quotient divisible groups coincides with the duality of Arnold and when restricted to coincides with the duality previously constructed by the authors. G

All groups will be additive and abelian. The torsion-free rank of a group G is the rank of G/T (G), where T (G) is the torsion subgroup of G. The p-rank of G is

the rank of G/pG. For a group G and prime p we let Gp be the p-adic completion of G/pωG and regard G/pωG G . A marks the end of a proof. ⊂ p Definition 1. An abelian group G is quotient divisibleb (qd) if G is of finite torsion-free rank and there existsb a free subgroup F G with G/F a divisible group. ⊂ We remark that the torsion-free qd groups are precisely the classical quotient divisible groups introduced by Beaumont and Pierce in 1961 [BP]. The following proposition is elementary. Proposition 2. For an abelian group G of finite torsion-free rank the following are equivalent: (1) G is qd. (2) There exists a free subgroup F G such that, for each positive integer m, G = F + mG. ⊂ (3) There exists a free subgroup F G such that, for each prime p, G = F +pG. ⊂ Definition 3. For each prime p let M(p) be a module over the ring of p-adic

integers Zp. A subgroup A p M(p) satisfies the projection condition if there is a free subgroup of finite⊂ rank F A such that, for every p, the natural Q ⊂ projectionb of F into M(p) generates M(p)asaZp- module. We call a free subgroup F of a group G full if G/F is torsion. b Received by the editors June 14, 1996. 1991 Mathematics Subject Classification. Primary 20K21, 20K40.

c 1998 American Mathematical Society

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Theorem 4. Let A be reduced of torsion-free rank n. Then A is qd if and only if A is (can be embedded as) a pure subgroup of a direct product of p-adic modules A M(p) such that A satisfies the projection condition. ⊂ p Proof.QSuppose A is reduced qd of torsion-free rank n. Let F A be free of rank ⊂ n with A/F divisible. By Proposition 2, if Tp is the p-torsion part of A, then Tp cannot have n + 1 nonzero cyclic summands. Hence each Tp is finite. For each p 1 ω write A = T A0 where A0 has no p-torsion. The first Ulm subgroup A = p A p ⊕ p p p is contained in A since an element of infinite p-height cannot have a nonzero T p p0 T p component in any of the previous decompositions. We claim that for each a A1 T 1 ∈ and prime p, there exists a b A such that a = pb. Plainly, we can choose b Ap0 with a = pb. If q = p is prime∈ then b will have infinite q-height since a does.∈ For 6 k+1 k each positive integer k there exists bk Ap0 with p bk = a. Hence p(b p bk)=0 k k ∈ − and, since (b p b ) A0 , (b p b )=0.Thus b has infinite p-height so that − k ∈ p − k b A1. We have shown that A1 is divisible. Since A is reduced, A1 =0. ∈It follows that A is embedded as a pure subgroup of its Z-adic completion A = ω p M(p)whereM(p)=Ap,the p-adic completion of A/p A. (See [Fu], Sections 39 and 40.) It remains to show that A satisfies the projection condition. In ourb Q ω context each projection πpbis the factor map A A/p A followed by inclusion A/pωA M(p). Thus each M(p)/π (A) is divisible.→ But A/F is divisible so that → p πp(A)/πp(F ) is also divisible. Hence M(p)/πp(F ) is divisible. Since Zp[πp(F )] is a finitely generated submodule, hence a summand, of the p-adic module M(p), and M(p) is reduced, it follows that M(p)=Zp[πp(F)] for each prime p. b Conversely, suppose that A is a pure subgroup of p M(p) such that A satisfies the projection condition. Let F = Zx A be a full free subgroup such that jb⊂ Q M(p)=Z[πp(F)] for each prime p. Let a A and q be a fixed prime. Then p L ∈ πq(a)= αjπqxj for q-adic integers αj. For each j let aj Z be congruent to αj mod q. Then a a x is divisible by q in M(p), hence∈ divisible by q in A. b − j j p We haveP shown that A/F is q-divisible for each q and the proof is complete. P Q Definition 5. Let be the category with objects qd groups with reduced torsion part and maps quasi-homomorphisms;QD that is,

Hom (A, A0)=Q Z HomZ(A, A0). QD ⊗ Let 0 be the full subcategory of with objects torsion-free quotient divisible groups.QD Let be the categoryQD with objects torsion-free ﬁnite rank groups and maps quasi-homomorphisms.QT F Let be the category with objects self-small mixed abelian groups G such that G/TQG(G) is ﬁnite rank divisible and maps quasi- homomorphisms. The category has been investigated by several authors ([AGW], [FiW], [FoW], [GW], [VW],QG [Wi]). The relation between and is described in the following proposition. QG QD

Proposition 6. The category can be identiﬁed with the full subcategory t of whose objects H satisfy theQG additional restriction that the p-adic modulesQDM(p) ofQD Theorem 4, deﬁned with respect to A = H/V, V the maximal divisible subgroup of H, are torsion for each p. Proof. It follows from Corollary 2.4 of [AGW] that every reduced group A in QG is a pure subgroup of Tp(A), where Tp(A)isthep-torsion subgroup of A, with Q

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the projection condition holding for some full free subgroup F A for almost all primes p. Let S be the finite subset of primes such that Z [π (⊂F )] = T (A). For p p 6 p p/S,Tp(A) will be finite of rank less than or equal to rank F = torsion-free rank A.∈In order for A to be self-small the groups T (A),p S, also must necessarily be p ∈ b finite [AGW]. Thus any object G of is a direct sum G = V A = V (B A0) QG ⊕ ⊕ ⊕ where V is finite rank torsion-free divisible, B = p S Tp(A) is a finite group ∈ and A0 = A p/S Tp(A) is a pure subgroup of Tp(A0) with the projection ∩ ∈ L condition. By Theorem 4, A0 is in . Hence each object G of is a direct sum Q QD Q QG G = B (V A0) of a finite group and an object H = V A0 of .Sincewe are working⊕ in⊕ quasi-homomorphism categories the correspondence⊕ QDG H is an → embedding of as a full subcategory of t. To show thatQG the embedded copy of QDin is exactly ,letAbe a pure QG QD QDt subgroup of a product of torsion p-adic modules p M(p) with the projection condition. We show that, for each fixed prime q, M(q)=Tq(A) and, hence, by Corollary 2.4 of [AGW], that A is a group in .LetqQbe fixed and m M(q). There is QG k∈ an f =(fp) F, a full free subgroup of A, with fq = m. If q M(q)=0(M(q) is finitely generated)∈ then qkf M(p) A. Since elements of M(p) ∈ p=q ∩ p=q have infinite q-height and A is pure6 in M(p) it follows that qkf = 6q2ka for Q p Q some a A. Thus qk(f m)=q2ka=qk(qka).Since division by q in M(p) ∈ − Q p=q is unique we have f m = qka A. Thus m A and M(q) coincides with6 the Q q-torsion subgroup of−A. The proof∈ is complete. ∈ For the reader’s convenience, we summarize constructions that have been useful in the study of torsion-free finite rank groups. For additional details see [Fo1]. Let n G be torsion-free of rank n with free subgroup F = i=1 Zxi. For each p let rp be the p-rank of G. Then the torsion group G/F has the form: L rp n k (1) G/F = [G/F ] = [ Z(p ip ) Z(p∞)], 0 k < . p ⊕ ≤ ip ∞ p p i=1 i=r +1 M M M Mp Fix a prime p. For each of the first rp cyclic summands in (1) there is a collection p p kip n p p of integers aij , 1 j n, such that for yi = p− j=1 aij xj , yi +F is a generator ≤ ≤ p p of that summand. If α = a + pkip Z then, for 1 i r , we obtain a relation ij ij P≤ ≤ p n p kip kip kip j=1 αij xj =0inG/p G. Here we regard each G/p G as a Z/p Z module kip in the natural way and abuse notation by writing xj for xj + p G.(Ifkip =0we P n choose the zero relation j=1 0xj =0.) For each of the n rp quasi-cyclic summands in (1) we choose a set of generators p − P p yi (k)+F :1 k< of that summand with yi (1)+ F of order p and, for k 2, { p ≤ p ∞} ≥p p[yi (k)+F]=yi(k 1) + F. Exactly as in the previous paragraph, each yi (k) − n p k determines a relation j=1 αij (k)xj =0inG/p G. For each fixed j the sequence p p αij (k) k determines a p-adic integer αij . We obtain, for each i with rp

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matrix with the i-th row consisting of elements of the ring Z/pkip Z for i r and ≤ p elements of the ring Zp for rp

rp n k [ Z/p jp Z] [ (Z(p∞)) ] ⊕ j j=1 j=r +1 M Mp as n 1 column matrices and thereby represent ϕ(a) as a matrix product. × k n n Lemma 9. With notation as above, an element (α1, ..., αn) of (Z/p Z) or (Zp) k p is a p -relation or a p∞-relation if and only if (ϕ1, ..., ϕn)MG =(α1, ..., αn) for p some (ϕ1, ..., ϕn). Furthermore (ϕ1, ..., ϕn)MG =(0, ..., 0) only if (ϕ1, ..., ϕn)=b (0, ..., 0). p Proof. These results follow from the fact that the rows of each MG are the relations obtained from generators of G/F which are linearly independent. For details see [Fo1], Lemma 2. Theorem 10. The categories and are dual. QD QT F Proof. We construct a contravariant equivalence d from to . Let G be an object of .ThenGis a qd group such that its maximalQD divisibleQT F subgroup D is torsion-free.QD Hence A = G/D is a reduced qd group which, by Theorem 4, can be

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regarded as a pure subgroup A A = A with the projection condition. Let ⊂ p p F = n Zx be a full free subgroup such that G/ n Zx is divisible. For each i=1 i Q i=1 i prime p, let πp: G Ap be the naturalb mapbG A followed by projection of A L → →L into Ap. Because of the projectionb condition each Ap is generated over Zp by the set sp kjp n πp(x1b), ..., πp(xn) . It follows that Ap = [ j=1 Z/p Z] [ (Zp)j]. Here, { } ∼ ⊕ j=sp+1 for each p, the sp are integers with 0 sp bn and the kjp,1 jb n, are a ≤ L ≤ L ≤ ≤ non-decreasing sequence of nonnegativeb integers. For each fixed i we havebπp(xi)= (αi1, ..., αin) where the first sp entries are in the appropriate residue class rings Z/pkjp Z and the remaining entries are p-adic integers We obtain, for each p,an G n nmatrix Mp = M(p, G, x1, ..., xn )whosei-th row is πp(xi). For j sp × G { } kjp ≤ the j-th column of Mp will contain elements of the ring Z/p Z. The remaining G columns of Mp will contain p-adic integers. Of course, just as in the construction p of MG, we allow the possibilities sp =0(Ap has no finite part) and sp = n (Ap has no torsion-free part) as well as the possibility that some or all kjp =0. We first construct the object dG.LetQG be the divisible hull of G/T (G). We n b b identify QG with Qx . Let x∗, ..., x∗ be the basis of QG∗ = Hom(QG, Q) i=1 i { 1 n} dual to x1, ..., xn . Then dG is the subgroup of Hom(QG, Q) that is generated by n { }L i=1 Zxi∗ and the following set. For each prime p and 1 j sp we include the kjp ≤ ≤ generator (a1jx1∗ + ... + anjxn∗ )/p . Here the aij are integers such that αij = aij + Lkjp G p Z, 1 i n,isthej-th column of Mp . For each p-adic column (α1j, ..., αnj ) G ≤ ≤ of Mp (j>sp) and for each positive integer k, we include a generator (a1j(k)x1∗ + k k ... + anj (k)xn∗ )/p where aij (k) is an integer congruent to αij modulo p . Next, let G, H and f : G H be a quasi-homomorphism. In particular f can be regarded∈QD as a map from QG→ to QH. The standard dual transformation f ∗ = Hom(f,Q)mapsQH∗ to QG∗. Put df = f ∗ . | dH We prove that df is a quasi-homomorphism from dH to dG. Let y1, ..., ym be the maximal independent subset of H used in the construction of {dH and let} L f(x1) y1 be the n m rational matrix defined by the equation =L . ×  ····   ··  f(xn) ym If g G has coordinate row (q1, ..., qn) with respect to x1, ..., xn thenf(g)has ∈ { } coordinate row (q1, ..., qn)L with respect to y1, ..., ym . By replacing f with a suitable integral multiple{ of}f we can assume that f is a homomorphism and that L has integral entries. We show that for this new f, df will be a homomorphism from dH to dG. Let B be H factored by its maximal divisible subgroup. Then f induces a homomorphism from A to B and hence a Z -homomorphism f : A B for each p. Write p p p → p

sp n b Ab =[b (bZ/pkjp Z)e ] [ (Z )e ], p j ⊕ p j j=1 j=s +1 M Mp b sp0 m b k0 B =[ (Z/p jp Z)e0 ] [ (Z )e0 ]. p j ⊕ p j j=1 M j=Msp0 +1 b b k k0 Here, for 1 j s (resp.1 j s0 ), order e = p jp (resp. order e0 = p jp ), and ≤ ≤ p ≤ ≤ p j j the ej (resp. ej0 ) are of inﬁnite order for j>sp (resp. j>sp0).Note that some or

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kjp kjp0 all of the ej or ej0 and hence the accompanying coefficient ring Z/p Z or Z/p Z can be zero. fp(e1) e10 Let ∆ be the matrix defined by the equation =∆ .If p  ···  p ···  fpb(en) em0 a A has coordinate row (α1, ..., αn) with respect to the decomposition ofA ∈ p p then fp(a) has coordinate row (α1, ..., αn)∆p with respectb to the decomposition of Bp. b b H Let πbp0 : H Bp be the map employed in the definition of the matrix Mp , exactly → G asbπp was used to define Mp . Then πp0 f = fpπp for each p. It is a simple exercise H G in linear algebra tob verify the matrix equalities LMp = Mp ∆p. By construction of G H dG, (x1∗, ..., xn∗ )Mp =(0, ..., 0) for each primebp. It follows that (x1∗, ..., xn∗ )LMp = f∗(y1∗) x1∗ (0, ..., 0) for each p. Since f is the dual of f, =Lt where Lt ∗  ···   ···  f∗(ym∗ ) xn∗ is the transpose of L. Transposing this matrix equation gives (f∗(y1∗),...,f ∗(ym∗ )) = H (x1∗, ..., xn∗ )L. Thus (f ∗(y1∗),...,f ∗(ym∗ ))Mp =(0, ..., 0) for each p. By construction H p of dH the matrix Mp is the transpose of the matrix MdH. Thus, we can appply Lemma7toconcludethatf∗restricted to dH is a homomorphism into dG. We have shown that df is a homomorphism from dH to dG, as desired. Let G be a torsion-free finite rank group with maximal independent set x1, ..., xn . To construct the inverse functor d0 for d we start with the associ- { } p ated set of matrices MG . Then we construct a quotient divisible group d0G such { } d0G that the divisible hull of d0G/T (d0G)isequaltoHom(QG, Q)andsuchthatMp p is equal to the transpose of the matrix MG for each p. More precisely, given G and x1, ..., xn consider the columns of the matrix p { } MG . Each of these columns can be regarded as an element of the Zp-module r n M(p)=[ p (Z/pkjp Z)] [ (Z ) ]. If M(p) is not generated over Z j=1 j=rp +1 p j p p ⊕ by the columns of MG, we can obtain a nonzero Zp-module homomorphismb ϕ = L L p (ϕ1, ..., ϕn): M(p) Z/pZ mapping all theb columns to zero. But (ϕ1, ..., ϕn)M b → G =(0, ..., 0), contradicting Lemma 9. For 1 i n,letb vi =(vip) be the element of ≤ ≤p p M(p) such that vip is the i-th column of MG.LetAbe the pure subgroup of p M(p) generated by the torsion subgroup of p M(p)andtheset v1, ..., vn . If Q p { } MG is the zero matrix for all p (G = Zxi)thenA=0. Q n Q Let F ∗ = Zx∗ G∗ = Hom(QG, Q). The assignment x∗ vi, 1 i=1 i ⊂ L i → ≤ i n, determines a map g : Hom(QG, Q) Q A. Let d0G = A ker(g). ≤ L → ⊗ ⊕ Since v :1 i n generates M(p)overZ for each p, then d0G satisfies the { ip ≤ ≤ } p projection condition with respect to F = Zxi0, where x10 , ..., xn0 are chosen such that xi0 = vi + di,di ker(g). Thus d0G is an objectb of . Let f : G H be∈ a homomorphism ofL torsion-free finiteQD rank groups. We need → to construct a quasi-homomorphism d0f : d0H d0G. Let f ∗ : H∗ G∗ be the → → standard dual map. Since, for any torsion-free finite rank G, we can identify G∗ with Q d0G via x∗ 1 x0 , we regard f ∗ as a map from Q d0H to Q d0G. ⊗ i → ⊗ i ⊗ ⊗ We put d0f = f ∗ and claim that

d0f = f ∗ Q Hom(d0H, d0G). ∈ ⊗ As before, we can assume the matrix L of f with respect to the distinguished bases x , ..., x for G and y , ..., y for H has integral entries. By Lemma 7 { 1 n} { 1 m }

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f(x1) 0 y1 0 M p = for all p. It follows that M p L = for G  ····   ··  G  ··   ··  f(xn) 0 ym 0 p all p.By Lemma 9, applied to each row of MGL,foreachpthere exists  a matrix of p p homomorphisms ∆p with MGL =∆pMH.It follows that the matrix ∆p determines a Zp-homomorphism δp : Bp Ap.Hered0H=B ker(g0) has been constructed → ⊕ p p inthesamemannerasd0G=A ker(g). The matrix equality M L =∆ M can ⊕ G p H be rewrittenb (v1p,...,vnp)L =(b δp(wb1p), ..., δp(wmp)) where vip and wjp are the i-th p p column of MG and the j-th column of MH , respectively. Let δ = δ : B A . Then (v ,...,v )L =(δ(w), ..., δ(w )). Since p p p p → p p 1 n 1 m v ,...,v Aand L has integral entries, δ(w ) A for 1 i m. By deﬁnition { 1 n}⊂Q Q Q i ∈ ≤ ≤ of B, it follows that theb restrictionb of δ to B is a homomorphism from B to A. Let i : Q B Q d0H be the natural embedding and π : Q d0G Q A ⊗ → ⊗ ⊗ → ⊗ be projection. Then, under our identiﬁcations, πf∗i =1 δsince (x∗,...,x∗ )L = ⊗ 1 n (f ∗(y1∗), ..., f ∗(ym∗ )). Thus πf∗i is a quasi-homomorphism. But then, since A, B are reduced and ker(g), ker(g0) are divisible,

f ∗ :(Q B) ker(g0) (Q A) ker(g) ⊗ ⊕ → ⊗ ⊕ is a quasi-homomorphism as well. It is easy to check that d and d0 satisfy d0d 1 and dd0 1 . This completes the proof of the theorem. ∼ QD ∼ QT F

One can check directly that d restricted to the full subcategory 0 of torsion- free finite rank qd groups coincides with Arnold duality [A]. ItQD is also not too difficult to check that d restricted to the embedded copy t of in (Propo- sition 6) coincides with the duality between the categoryQG andQG theQD category of locally free torsion-free finite rank groups and quasi-homomorphisms,QG introduced by the authors in [FoW]. It follows from the construction that d, d0 preserve torsion- free rank. Since, under our identifications, both df and d0f are defined to be the classical vector space dual map f ∗ it follows that the functors d, d0 are additive as well. As an alternate method of proof of our main theorem, we could have constructed a category equivalence from to the category of linear mappings to finitely presented modules over the ringQD of universal numbers.L Then we could have applied the duality of [Fo2] from to . The composite of this equivalence and duality would be the same as ourL duality.QT F To make our paper reasonably self-contained, we chose to construct d directly.

Acknowledgment We thank the referee for his or her careful reading of the paper.

References

[A] D. Arnold, A duality for quotient divisible abelian groups of ﬁnite rank, Pac. J. Math. 42 (1972), 11-15. MR 47:361 [AGW] U. Albrecht, H.P. Goeters and W. Wickless, The ﬂat dimension of mixed abelian groups as E-modules, Rocky Mt. J. Math. 25(2) (1995), 569-90. MR 96f:20086 [BP] R. Beaumont and R. Pierce, Torsion-free rings, Ill. J. Math. 5 (1961), 61-98. MR 26:6212 [FiW] S. Files and W. Wickless, The Baer-Kaplansky Theorem for a class of global mixed abelian groups, Rocky Mt. J. Math. 26(2) (1996), 593-613. CMP 96:17

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[Fo1] A. Fomin, The category of quasi-homomorphisms of abelian torsion-free groups of finite rank, Cont. Math. 131 (1992), 91-111. MR 93j:20108 [Fo2] , Finitely presented modules over the ring of universal numbers, Cont. Math. 171 (1994), 109-120. MR 95i:20074 [FoW] A. Fomin and W. Wickless, Categories of mixed and torsion-free abelian groups, Abelian Groups and Modules, Kluwer, Boston, 1995, 185-92. MR 97c:20083 [Fu] L. Fuchs, Infinite Abelian Groups I, II, Academic Press, New York, 1970, 1973. MR 41:333; MR 50:2362 [GW] S. Glaz and W. Wickless, Regular and principal projective endomorphism rings of mixed abelian groups, Comm. in Algebra 22(4) (1994), 1161-76. MR 95a:20060 [VW] C. Vinsonhaler and W. Wickless, Realizations of finite dimensional algebras over the rationals, Rocky Mt. J. Math.24(4) (1994), 1553-65. MR 96e:20089 [Wi] W. Wickless, A functor from mixed groups to torsion-free groups, Cont. Math. 171 (1995), 407-19. MR 95k:20090

Algebra Department, Moscow State Pedagogical University, Moscow, Russia E-mail address: [email protected] Department of Mathematics, University of Connecticut, Storrs, Connecticut 06269 E-mail address: [email protected]

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