VARIOUS LOCAL GLOBAL PRINCIPLE S FOR ABELIAN GROUP

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 the natural map A ---} lim ,C(A) is an isomorphism if A has torsion at 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 —> ~ ® AA , AA} and 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 .


Introductxon
Given an abelian group A it is often easier to 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 A the inverse local diagram .C(A) :---{rp : A (p ) -> A é , where A (p) is the p-localization of A an d rp rationalizes .Specializing a result of [7], we see that the natural ma p c : A ---} l lim ,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 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 the natural map c : A -> l { im ,CS(A) is always an isomorphism ._ Using results like I 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 the local 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 .In particular we explain which additional structural ingredient makes local 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 structure 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 results can be extended in severa" directions .
Firstly, to nilpotent groups and, secondly to algebras over a Dedekind domain .In section 7 we indicate, without proof, the nature of suc h extensions to Dedekind domains .

O . Preliminaries
We begin by collecting the necessary definitions and properties of Plocalization of groups .For details, see [3], [4] H h ~--~ h P E H is a bijection ; (u) the kernel of e consists of all those torsion elements of G whose arder has no divisior in P ; (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 Gp is exact and idempotent .Moreover, ep has a nonincreasing effect on 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 the universal property of P-localization .In particular, if P C Q then e p factors uniquely through e Q .
We turn to formal completions as introduced by Sullivan [9] .The p-adic integers Zp A are given as the inverse limit of the system we know that Ap A is the inverse limit of th e system of quotients of A which are p -torsion and that AA is the invers e limit of the system of all torsion quotients of A .

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-completions is a monomorphism whose cokernel is a rational vector space .Results of this nature are collected in this section .
Tor(A, Z/p) Tor(IIA (p) , 7L/p) 0 A HA (p) -» coker(i ) p J- As the rocas and the left two volums are exact, so is the right hand column .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 7Lp induce natural monomorphism s
Proof: We show first that 7Lp / L (p) is a rational vector space .Conside r the commuting diagram below .o o ~ As both Z (p) and Zp A are 7L (p) -modules, so is the cokernel 7Lp /Z (p) .The diagram aboye shows that multiplication by p on this cokernel is an isomorphism .Therefore we have the exact sequence The claim foliows because n / 7L (p ) is a Q-module .For A^/A, the same shows that AA /A is a rational vector space .■ 1.3. .Remark .The cokernel of the inclusion Op 7Lp^, Hp 7L p is a rational vector space .
Proof The splitting n p 7Lp " Ĵ 7Lp ^~n q p Zq shows that multiplication by p on ~ 7Lp^has cokernel Z/p .Now the proof can be completed as in the argument of the previous result .■

. Facts about fiber square s
Our main tool to identify pullback (fiber square) diagrams is the following lemma .Let K = ker (f ) .This is a rational vector space .So we get a splitting 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) = nrB (b) ,

A f } B
Thus nb E im ( f) and, hence, b E im ( f) ; otherwise coker ( f) is not torsion free .So b = f ( a l), far some a 1 E A l .On the other hand , and, hence, (a -rA (a l )) E ker( fo) = r A (K) .So let r A (k) = a -rA(a l ) , for some k E K and set a := k + a l .Proof: This follows from 1 .1 and 2 .1 .■ (u) Ao is a rational vector space ; (iii) for each prime p, rp : Ap -> Ao rationalizes .

.3 . Remark . Corollary .has been shown by Hilton and Mislin [5] by different arguments . In fact they also show that (D) is a push-out diagram in the category of abelian groups .
The £-diagrams form a category ,CAZ3 ."Inverse limit" is a covariant functor ,C,AB -> AB : it is right adjoint to the functor AB -> £AB which sends an abelian group A to the £-diagram £(A) := {LA ---} L0 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.

Let T[A] be the direct sum of the torsion subgroups T[A p ] of Ap an d denote the product of the T[Ap ] by II[A] = l l im {T[AP ] -> Ao} . Set O[A] = l < im{Ap /T[AP ] -> Ao} . Then the following hold (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 if
there is a commutative diagram with exact rows .

T [A] A
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 diagram { of short exact sequences {(T[A] >-> Ap ---» Ap / T [Ap ] ) -> Ad.. Thus the 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 because I = O [A] and l A -+ 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 torsion free because c¢[A] = 1 .We call an ,C-diagram A realizable if it is of the form .C(A) for som e abelian group A .From the proof of 3 .3we see

.4 . Corollary. An £-diagram A is realizable if and only if A satisfies condition 3.2(*) . ■
The information pertaining to the .C-genus, which is encoded in the Hom -Ext sequence aboye, can be reinterpreted "geometrically" using an analogy with principie bundle classification .Let us consider all abelian groups whose torsion subgroup is a fixed group T .Associated with T is the universal T-extension oT~II-> o where II is the product of the p-prirnary subgroups of T .

.6 . Corollary . Suppose the element A E Ext(0, II) in the previou s proposition is 0 . Then the genus of A is in bijective correspondence wit h the cokernel of Hom(, II) -> Hom(, Q) modulo the action of Aut(0) x
Aut(T) .■ 3 .7 .Remark .It is well known that Ext (0, T) = o for all torsio n free ~ if and only if T is the direct sum of a divisible group D and a group T' of finite exponent ( i .e .kT ' = 0, for a suitable k E N) ; se e [2J .Consequently, a group A whose torsion group T is the direct su m of a divisible group and a group of finite exponent is itself isomorphic t o T ~ A/T .In particular, A has trivial £-genus .Q} is in bijective correspondence with the power set of the set of al l primes a sub j ect to the equivalence relation that X (-Y if and only if the symmetric difference of X and Y is finite . Proof: Let T be the sum of the groups Z/p and let II be the produc t of the groups 7L /p .Then the Hom-Ext sequence in the proof of 3 .3reads o= Hom(Q, II) ---} Hom(Q, Q) --> Ext(Q, T) ~ Ext(Q, II) .
One group in the genus of A is Te Q .So a = O .Moreover, Hom(Q, II) = o since no nonzero element of II is divisible .Consequently, the elements in the genus of A correspond bijectively with the orbits of Hom(Q, Q ) Q under the action of Aut(Q) x Aut(T), where Aut(Q) acts as scalar multiplication on Q and Aut(T) = ~ Aut(Z/p) acts on T and on II , hence on their quotient Q .We calculate these orbits : It can be see n that each Aut(Q)-orbit is contained in an Aut(T)-orbit .Thus it suffices to consider the action of Aut(T) on Q .Represent w E Q by x E H .
Using the action of Aut(T), we can alter x so that all its non-zero coordinates are 1 .Let X denote the set of primes corresponding to non-zero coordinates of x .Another element y E II represents an element in th e Aut(T)-orbit of w if and only if the symmetric difference of X and Y is finite .This implies the claim .
The argument just given is constructive in that it allows us to describe the group Ax in the £-genus of A corresponding to a given set X of primes ; compare 3 .5: Let x E II be the element with 1's as th e coordinates in X and O's everywhere else .Then x represents an element in Q and, hence, a homomorphism cx : Q -> Q .So Ax is obtained via the pullback of the sequence T >--3 II ~Q along cs .Consequently, Ax is isomorphic to the subgroup cx -1 (im (cs ) } of II .■ 4 .2 .Example .Let Q be a set of primes and pu t T(Q) := 7L /p and II(Q has th e same ,C-genus as 11(Q) .To see this, note that E(Q) is a rational vector space .Further, if p E Q, we get an isomorphism II(Q) (p ) r -" 7L /p E(Q ) from the splitting II(Q) ^--' 7L / p e IIgE(Q_{p } )7L/q• ■ 4 .3 .Remark.Two groups A and B are customarily defined to be in the same genus if A (p ) ^-' B (p) for all primes p .This notion corresponds to the 1C-genus associated with the diagram of localizing functor L 2 L 3 L 5 . . .(no arrows) .The 1C-genus differs from the r-genus in that no coherence conditions are imposed amongst the localizing functors involved .
As a consequence, it is rarely possible to recover an infinite group from its 1C-diagram .This is illustrated in the followin g 4.4 .Example .The number of isomorphism classes of abelian group s in the same 1C-genus as Z is uncountable .To see this, choose integer s rp > o for each prime p, and let A,. be the subgroup of ~ generated by {p -T P } .Then ( A,-) (p ) ^--' Z (p) for all p, but A T r-" A5 if and only if r and s differ at only finitely many primes ; see [2] .The groups Ar represent the entire incoherent 1C-genus of Z .

. r-diagrams, .C-systems
In this section we explain the relationship between various local global principies involving completion or localization .5 .1 .Definition .A local (arithmetic) system ,CS consists of a Z (p)module Bp , for each prime p, and a Q-module Bo together with rational isomorphisms kp : Bp -> Bo such that For the proof we need the followin g 5 .7 .Lemma .For any nmodule Cp the natural homomorphis m m : (C) p A b~r~--}rbECp has as its kernel a vector space over Q p " .
Proof: The kernel is a ZP-module .To see that it is also a Q-modul e notice that m is split by the inclusion Cp ~(C) p of 1 .2,whose cokerne l is a rational vector space .■ Proof of 5 .6 : There is a commutative diagram AA (HC) p - The three squares are fiber squares : the first because it is the completion of a fiber square and the other two by 2 .1 .Therefore the outer rectangle is a fiber square .Its bottom row is an isomorphism .So its top row is an isomorphism as well .We see that Ao Co by rationalizing the pullback diagram defining A .Together these isomorphism give an isomorphis m of diagrams .■ With the same methods we prove 5 .8 .Lemma.lf A is defined as the pullback oía-: Bo ~(B) 0 <-Bp , a part of a complete system, then Ao (A) 0 #-AP is isomorphic to cr .■

. Summary of equivalences of categorie s
In this section we provide a list of categories which are isomorphi c with respect to functors used in a local global principie .
The remainder of this section will be devoted to proving 6 .1 .All th e functors giving these equivalences preserve the f-subcategories .So 6 .1 .iii will follow from the fact that ,AB f £ f , which is a corollary of 3 .2 .
Identify Ao with its image in H(A) 0 under the diagonal map .To see that realizability implies 5 .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 remarks 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 generalizations 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 problem is that T* 1 (a) in 3 .3may be empty when there is torsion at infinitel y many primes .However, corollary 3 .4remains valid if every A 0 -modul e is injective .This happens, for example, if A is the group algebra of a finite group .Z .However, the action of C does not extend because c(9, 1) would have to have non -zero coordinate in every summand .■ Theorem 6 .1 can be generalized to nilpotent groups .We state the result .Its proof is completely parallel to our treatment of the abelian case .An additional technical quibble arises, however, from the fact that nat every subgroup of a nilpotent group is normal .This is relevant, fo r example, in the nilpotent analogue of lemma 1 .1 .

.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.
rA l T D A0 ---> Bf o Proof: The lemma will follow after we have shown that, for a E A o and b E B with fo(a) = r B (b), there exists a E A with f(a) = b an d r A (a) = a .
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
Mere e is the homomorphism whose p -th coordinate is the p -localizatio n map of A .The rest of the diagram is obtained by rationalizing e .

A 3 .
Proposition .Por every abelian group A, there are fiber square s The genus of an £-diagram of abelian groups Let ,£ denote the diagram of localizing functor s L2 L p denotes localization at the prime p and Lo , denotes rationalization .3 .1 .Definition .A diagram A -{rp : Ap ~ Ao } of abelian group s modeled on .0 is an £-diagram if (i) Ap is a p-local group ;

L
[A]  from the torsion free case.The maps from the top row to the middle row come from the universal property of inverse limits .Conversely, if A is an extension as in the diagram we get commuting diagram sA } A(p) .->A o 4-l lim AAo 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 hp 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 in the diagram as an exact sequence of chain complexes .■We use Theorem 3 .2to relate the genus of an .Cdiagram A to invariant s from homological algebra .The homomorphism •r : (T := T[A]) -> (II [A] _ : II} induces the map T* : Ext(0, T) -> Ext(0, II ) by taking the pushout as indicated .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 subgroup of Aut(0) x Aut(T) which stabilizes A .Proof: Every extension B of ~by T with £(B) Arepresents an element 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 ( multiplication by an integer n o in II is an isomorphism modulo n-torsion) .Thus we see that T~is onto .Suppose ,f : B 1 -> Bz is an isomorphism between 0 --» (Hom-Ext) two extensions of ~by T .Then f induces automorphisms T[f] : T -> T and O[f] : -} 0 which make the diagram below commute Hence B 1 is isomorphic to B2 if and only if ~7 = ( 0[f]_ l )* o T[f]E .This implies the claim .■ . Remark .The genus of the £-diagram A = {Z/p ~Pr°je °t }

7 . 2 .
Example .Let C = (c) be an infinite cyclic group .Far each prime p, let Ap = 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 A0 = ãnd let rp : A (p) ~Q be the usual inclusion of Z (p) in Q .This local diagram has a unique realization as an abelian group, namely (epZ/p) ,[fi].Given a set of integral 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 if (i) H is a P-local group ; i .e .for every prime p P, the functio n ~Z/ p -> O .Formal p-completion of an abelian group A is give n by A p A := A o Zp .This is an exact functor because Zp A is flat .Th e p-adic rationals can be obtained as Qp =