Various local global principles for abelian groups

Publicacions Matemàtiques, Vol 38 (1994), 353-370 . VARIOUS LOCAL GLOBAL PRINCIPLE S FOR ABELIAN GROUP S GEORGE PESCHKE 1 AND PETER SYMONDS 2 A bstract  We discuss local global principies for abelian groups by examining the adjoint functor pair obtained by (left adjoint) sending a n abelian group A to the local diagram .C(A) _ { Z (p) ® A —> ~ ® A } and (right adjoint) applying the inverse limit functor to such diagrams ; p runs through all integer primes . We show that th e natural map A ---} lim ,C(A) is an isomorphism if A has torsion a t only finitely many primes . If A is fixed we answer the genus problem of identifying all those groups B for which the local diagram s .C(A) and £(B) are isomorphic . A similar analysis is carried ou t for the arithmetic systems S(A) -- {12 ® A —> ~ ® A A , A A } an d the local systems {Q 0 A --+ (I47L (p) ® A) 1--- Ii(7L (p ) ® A}} . The delicate relationship between the various adjoint functor pair s described aboye is explained . Introductxo n Given an abelian group A it is often easierto work with the localizatio n of A at a prime p than with A itself . Thus one would like to analyz e the localizations of A, one prime at a time, and then recover informatio n about A from its local data . For this reason we consider two methods o f deterrnining a group from local data . 1 . We associate to an abelian group Athe inverse local diagra m .C(A) :--- {r p : A ( p ) —> A é , where A (p) is the p -localization of A and r p rationalizes . Specializing a result of [7], we see that the natural ma p c : A ---} l l im ,C(A) is an isomorphism if and only if A has torsion at onl y finitely many primes . ~ S upported by NSERC of Canad a 2 Partially supported by NSERC of Canada 354  G . PESCHKE, P . SYMOND S II . We associate to an abelian group A the inverse local syste m £S(A) := {A 0 —> (HA () )~ 4--- (HA () )} . A result of [5] says that th e natural map c : A —> l { im ,CS(A) is always an isomorphism .  _ Using results likeI and II to obtain information about A has a lon g tradition ; see [4], [6], [10] . However, our foundation in I is much stronge r than its counterpart in earlier expositions where the abelian group A i s assumed to be finitely generated . As a consequence we obtain new insigh t into some old questions related to the notion of the "genus" of a group . We define the .C-genus of a group A to consist of all isomorphis m classes of groups B with £(A) ,C (B) . Thus the .C -genus measure s the extent to which the local data in.C(A) fail to determine A uniquely . In section 3, we show how the ,C-genus can be calculated in terms o f invariants familiar from homological algebra . If we compare the recovery results I and II we see that the extent t o which thelocal data determine a group depends crucially on the mean s used to splice the local data together . In section 6 we explain the relationship between the categories of local systems and local diagrams . I n particular we explain which additional structural ingredient makes loca l systems a more powerful recovery tool than local diagrams . Closely related to localization at a prime p is p -completion . In fact , both processes have the same kernel but completion has a more structur e enhancing quality . We carry out a similar development for completion s as well . Then we show that the categories of local systems and complet e systems are isomorphic . Many of our methods and resultscan be extended in severa" directions . Firstly, to nilpotent groups and, secondly to algebras over a Dedekin d domain . In section 7 we indicate, without proof, the nature of suc h extensions to Dedekind domains . O . Preliminarie s We begin by collecting the necessary definitions and properties of P - localization of groups. For details, see [3], [4], [fi] . Given a set of integra l primes P, let Zp denote the set of rational numbers whose denominato r is not divisible by any p E P . Thus 7L o denotes the rationals Q . If P consists of a single prime p we write 7L (p) for 7L p . A homomorphism of nilpotent groups e : G ~ H is said to P -localiz e if and only i f (i) H is a P-local group ; i .e . for every prime p P, the functio n H h ~--~ h P E H is a bijection ; (u) the kernel of e consists of all those torsion elements of G whos e arder has nodivisior in P ; LOCAL GLOBAL PRINCIPLES FOR ABELIAN GROUPS  35 5 (iii) for every h E H there exists an integer n with no divisor in P such that hn E im(e) . Given G, the P -localizing homomorphism e : G ~ H is completel y determined by P . So we write ep : G ~ G . The functor G G p is exact and idempotent . Moreover, ep has a nonincreasing effect o n the order of nilpotency of a given group . Every homomorphism fro m G to a P-local group factors uniquely through ep . This is called th e universal property of P -localization . In particular, if P C Q then e p factors uniquely through e Q . W e turn to formal completions as introduced by Sullivan [9] . Th e p -adic integers Z p A are given as the inverse limit of the system • • • ~ Z'p2 ~ Z/ p —> O . Formal p -completion of an abelian group A is give n by A p A := A o Z p . This is an exact functor because Z p A is flat . Th e p -adic rationals can be obtained as Qp = Zp . The completion o f Z is Z A = fl p Z p and the formal completion of an abelian group A i s AA = A 0 Z n From [S] we know that A p A is the inverse limit of th e system of quotients of A which are p -torsion and that A A is the invers e limit of the system of all torsion quotients of A . 1 . Properties of localizin g and completing homomorphism s We make frequent use of the fact that the natural homomorphism fro m an abelian group to the product of its p -localizations or its p -completion s is a monomorphism whose cokernel is a rational vector space . Results o f this nature are collected in this section . 1 .1 . Lemma . The homomorphism e : A—> li A (p) , whose p -th coordínate map is p -localizatíon, is a monomorphism whose cokernel is a rational vector space . Proof The monomorphism part follows from (u) in section o and i s well known ; see e .g . [6] . We investigate the cokernel of i using th e commutative diagram below . 356  G . PESCHKE, P . SYMOND S Tor(A, Z/p)  Tor(IIA (p) , 7L/p)  0 A  HA (p)  -» coker(i ) p J - I  P l  l p A   2  >  IIA (p)  » coker(i ) 1  1  1 A ® Z/p -"=--> (llA () ) ~ 7G/p ~ 0 As the rocas and the left two volums are exact, so is the right hand col - umn . Thus multiplication by any prime p is an isomorphism on coker(i) . So coker(i) is a rational vector space . ■ 1 .2 . Lemma . The natural maps Z (p)  and Z c —> Z A ^= J H p 7L p induce natural monomorphism s A(p)>--->Ap and A>—> A " whose cokernels are rational vector spaces . Proof : We show first that 7Lp / L (p) is a rational vector space . Conside r the commuting diagram below . o o ~ LOCAL GLOBAL PRINCIPLES FOR ABELIAN GROUPS  35 7 As both Z (p) and Z p A are 7L (p) - modules, so is the cokernel 7Lp /Z (p) . Th e diagram aboye shows that multiplication by p on this cokernel is a n isomorphism . Therefore we have the exact sequenc e A(p) —> A p —» Ap / A(p ) A 0 (7L'/Z () ) . The claim foliows because n / 7L (p ) is a Q - module . For A^ / A, the sam e method applied to Z ~--~ lI7Lp -» II7Lp / 7 L p Z >—> 117L p A  » Z / p o  Z/p ---- 4 shows that A A /A is a rational vector space . ■ 1 .3 . . Remark . The cokernel of the inclusion O p 7L p ^ , H p 7L p is a rational vector space . Proof The splitting n p 7L p " ^ J 7L p ^ ~ n q ~ p Zq shows that multiplicatio n by p on ~ 7L p ^ has cokernel Z/p . Now the proof can be completed as i n the argument of the previous result . ■ 2 . Facts about fiber square s Our main tool to identify pullback (fiber square) diagrams is the following lemma . 2 .1 . Lemma . Suppose f : A —> B is a homomorphism betwee n abelian groups whose kernel is a rational vector space and whose cokernel is torsion free . Then the diagram below is a pullback diagram i n the category of all groups . A  f } B rA l T D A 0 ---> B ~ fo 358  G . PESCHKE, P . SYMOND S Proof : The lemma will follow after we have shown that, for a E A o and b E B with f o(a) = r B (b), there exists a E A with f(a) = b and r A (a) = a . Let K = ker (f ) . This is a rational vector space . So we get a splittin g A = K e, A l , and the restriction of f to A 1 is an isomorphism to im (f ) . Now pick n ~ 1 and a ' E A such that na = r A (a ' ) . This yield s rBf(a ' ) = .ÍO r a ( a' ) = n f0 (a) = nr B (b) , and so nb — f (a ' } E ker(r B ) = T [Bl, the torsion subgroup of B . A s coker ( f) is torsion free, there exists a torsion element t E A l with f (t) = nb — f (a' ) . Thus we 'hav e r A (t + a ' ) = r A (a ' ) = n a f (t + a ' ) = f (t) + f (a ' ) = nb . Thus nb E im ( f) and, hence, b E im ( f) ; otherwise coker ( f) is not torsio n free . So b = f ( a l), far some a 1 E A l . On the other hand , h( a ) = r B( b ) = r s( .Í( a i)) = M r a( a i) ) and, hence, (a — r A (a l )) E ker( f o) = r A (K) . So let r A (k) = a — rA(a l ) , for some k E K and set a := k + a l . By design, f(a) = f (a l ) = b an d r A (a) = a . E 2 .2 . Corollary . For every abelian group A the diagram below is a ffber square . A  e  ' 11A (p ) eol e A o  >(HA (p) ) o a Mere e is the homomorphism whose p -th coordinate is the p -localization map of A . The rest of the diagram is obtained by rationalizing e . Proof : This follows from 1 .1 and 2 .1 . ■ 2 .3 . Remark . Corollary 2 .2 has been shown by Hilton and Misli n [5] by different arguments . In fact they also show that (D) is a push -ou t diagram in the category of abelian groups . (D) LOCAL GLOBAL PRINCIPLES FOR ABELIAN GROUPS  35 9 2 .4 . Proposition . Por every abelian group A, there arefiber square s A ( p )  } A A  P  A -> A A A0 -> (A) 0  A 0  (A A ) 0 Proof Use 1 .2 and 2 .1 . ■ 3 . The genus of an £-diagram of abelian group s Let ,£ denote the diagram of localizing functor s L 2  L 3  L 5 L 0 where L p denotes localization at the prime p and Lo , denotes rationalization . 3 .1 . Definition . A diagram A — {r p : A p ~ Ao } of abelian group s modeled on .0 is an £-diagram i f (i) A p is a p-local group ; (u) Ao is a rational vector space ; (iii) for each prime p, rp : Ap —> Ao rationalizes . The £-diagrams form a category ,CAZ3 . "Inverse limit" is a covarian t functor ,C,A B —> AB : it is right adjoint to the functor AB —> £AB whic h sends an abelian group A to the £-diagram £(A) := {LA ---} L 0 A} . The ,C-genus of an £-diagram A consists of all isomorphism classes o f abelian groups A with £(A) A, and two groups A and B belong t o the same £-genus if £(A) £(B) . In this section we relate the genus o f an abelian group to invariants familiar from homological algebra . 3 .2 . Theorem . Let A = {r p : A p —> Ao } be an ,C-diagram of abelia n groups . Put 1 := n Im(r p ) and suppose that A satis fies the conditio n (*)  1 c ---> Ao induces an isornorphism 10 —> Ao . 360  G . PESCHKE, P . SYMOND S Let T[A] be the direct sum of the torsion subgroups T[A p ] of A p an d denote the product of the T[A p ] by II[A] = l l im {T[A P ] - > Ao} . Set O[A] = l < im{A p /T[A P ] -> Ao} . Then the following hol d (i) The sequence II[A] ~ A --» O[A] is exact . (ii) An abelian group A belongs to the G-gens of A if and only i f there is a commutative diagram with exact rows . T [A]  A  H [A ] TI  I y II[AI  1 S imA — L - ~[A ] Q  > (limA)/A —> o t Proof : All claims pertaining to the top two rows of the diagram ar e special cases of 7 .2 in [7] . We sketch the argument for completeness ' sake . The sequence II [A] >--> lim A [A] is the inverse limit of the diagra m { of short exact sequences {(T[A] >-> Ap - - -» A p / T [A p ] ) -> Ad . . Thus th e inverse limit sequence is exact in the left hand and center positions, usin g general properties of inverse límits ; see [11 . Moreover, q is onto becaus e I = O [A] and lA -+ 1 is onto . Next show that, given an abelian group B, the image of the rationalizing map B -> Bo coincides with the intersection of the images B ( p) -> A ) . This implies the claim in the case where all the groups A p are torsio n free because c¢[A] = 1 . In general, if A satisifes £( A) A, we see that A / T [A] r =" ~[A] fro m the torsion free case . The maps from the top row to the middle ro w come from the universal property of inverse limits . Conversely, if A i s an extension as in the diagram we get commuting diagram s A  } A(p) .—> A o 4- - l lim A  Ao LOCAL GLOBAL PRINCIPLES FOR ABELIAN GROUPS  36 1 from the universal property of p-localization . Moreover, h, p restrict s to an isomorphism between torsion subgroups and induces an isomorphism between torsion free quotients . Thus h p is an isomorphism by th e 5 -lemma . In either case, the map t exists because Kerey} contains z (Ker (cx)} . The bottom row is seen to be exact by regarding the vertical arrows i n the diagram as an exact sequence of chain complexes . ■ W e use Theorem 3 .2 to relate the genus of an .C- diagram A to invariant s from homological algebra . The homomorphism •r : (T := T[A]) —> (II [A] _ : II} induces the ma p T * : Ext(0, T) —> Ext(0, II ) by taking the pushout as indicated . T  + K T  Y I I I ~ II,_-_— L Let a E Ext(0, II) denote the class corresponding to the extension II >-- > lim A --~ O . <- - 3 .3 . Theorem . If an .C-diagram of abelian groups satisfies condition (*) of 3 .2 then the elements of the ,C-genes of A are in bijective correspondence with (T * ) -1 ( A ) modulo the action of the subgrou p of Aut(0) x Aut(T) which stabilizes A . Proof : Every extension B of ~ by T with £(B) A represents an ele - ment in T(À) and every element in 7- . -1 (A) determines a group B wit h £(B) A ; by (3 .2) . Therefore, we determine which elements in T* 1 (a ) have isomorphic groups in the middle . Consider the exact sequenc e o -4 Hom (0, T) ~ Hom (0, II) —> Hom (0, Q) ~ --4 Ext (0, T) ~ Ext (q5, II) ----} Ext (c¢, Q) = o in which the last term is o because Q is rational vector space ( multiplica - tion by an integer n o in II is an isomorphismmodulo n-torsion) . Thu s we see that T~ is onto . Suppose , f : B 1 —> Bz is an isomorphism betwee n 0 -- » (Hom -Ext) 368  G . PESCHKE, P . SYMOND S 5 .5(CS) is satisfied because the map Bo --> 11(B P Q p ^) lifts to Bo — > (fl(B P ED 7G^ )) 0 according to the commuting diagram below .  + ( IIB p ) O —+ (llC) 0 Bo n(B P ~ Q) - - > (llCE~Q ) p ¿''' , Conversely, given a complete system  we construct a local syste m by setting Bo := Co and defining B p to be the pullback of Co —> (C) 0 < — B p ; compare 2 .4 . To see that condition 5 .1(LS) is satisfied, consider th e commuting diagram below . IIB p  ~--- =--- -~  HC p (nBp)o  ( nB p ) 0 l v Bo  } II(B p 11) r-> II ( C p e Q ) u The outer rectangle is a fiber square since it is a product of fiber squares . Its rationalization is the bottom rectangle,which is, therefore, also a fibe r square . Condition 5 .1(LS) now follows using the universal property o f pullbacks . ■ 7 . Further remark s 7 .1 . Remark . If we regard an abelian group as a 7L - module then ther e is an obvious generalization of this work to the case of modules over a Dedekind domain R, where localization, completion and torsion are al l with respect to the prime ideals of R . The appropriate generalization s of all our results and their proofs remain valid . Since our constructions are natural, most of them also apply to modules for an algebra A over R (localization etc. are still with respect t o primes of R) . Theorem 6 .1 remains true but (3 .4) may fail . The proble m is that T* 1 (a) in 3 .3 may be empty when there is torsion at infinitel y many primes . However, corollary 3 .4 remains valid if every A 0 - modul e is injective . This happens, for example, if A is the group algebra of a finite group . LOCAL GLOBAL PRINCIPLES FOR ABELIAN GROUPS  36 9 7 .2 . Example . Let C = (c) be an infinite cyclic group . Far eac h prime p, let A p = Z/p Z (p) be a Z (p) [ C]-- module with action c(x, y) - - (x + y, y), x E 7L / p, y E Z(p) and the image of y in 7L / p . Let A 0 =~ and let r p : A (p) ~ Q be the usual inclusion of Z (p) in Q . This loca l diagram has a unique realization as an abelian group, namely (e p Z/p ) Z . However, the action of C does not extend because c(9, 1) would hav e to have non -zero coordinate in every summand . ■ Theorem 6 .1 can be generalized to nilpotent groups . We state th e result . Its proof is completely parallel to our treatment of the abelia n case . An additional technical quibble arises, however, from the fact tha t nat every subgroup of a nilpotent group is normal . This is relevant, fo r example, in the nilpotent analogue of lemma 1 .1 . 7 .3 . Theorem . _U e is any of the categories aboye, let denote th e analogous category but with nilpotent objects . Then we have the followin g equivalences of categories . (i) 9't%' 91£0 ; (u) ~ ^ J the subcategory of 915 . .120 whose objects have a uniforrn boun d on their order of nilpotency ; (iii) fR 1 Tl f ^r YteC3 f 91 .Lt£C3 f . Reference s 1. BOURBAKI N ., "Theory of Sets, " Addison Wesley, Reading/Do n Mills, 1968 . 2. FUCHS L ., "Infinite Abelian Groups II," Academic Press, New York , London, 1973 . 3. HILTON P . J ., Localization and cohomology of nilpotent groups , Math . Z . 132 (1973), 265-286 . 4. HILTON P . J ., "Nilpotente Gruppen und nilpotente Rdume," Springer LNM 1053, Berlin/New York, 1984 . 5. HILTON P . J . AND MISLIN O ., Bicartesian Squares of Nilpoten t Groups, Comment . Math . H ely . 50 (1975), 477-491 . 6. HILTON P . J ., MISLIN G . AND ROITBERG J ., "Localization o f IVilpotent Groups and Spaces, " North - Holland, Amsterdam, 1975 . 7. PESCHKE G ., Localization and genus in group theory, Trans . AMS , to appear . 8. PESCHKE G . AND SYMONDS P ., Pro -torsion completions of abelia n groups, J . Pure and Applied Algebra 88 (1993), 187-197 . 9. SULLIVAN D ., Geometric Topology I and II, MIT Notes, 1970 . 370  G . PESCHKE, P . SYMOND S 10 . SULLIVAN D ., Genetics of homotopy theory and the .Adams conjecture, Ann . o f Math . 100 (1974), 1-79 . George Peschke :  Peter Symonds : Department of Mathematical Sciences  Department of Mathematic s University of Alberta  Northwestern Universit y Edmonton  Evanston, 111 . CANADA T6G 2G1  60208 U .S .A . Primera versió rebuda el 2 de Febrer de 1994 , darrera versis rebuda el 16 de Maig de 1994