onp-local abelian groups with decomposition bases

Math. Z. 202, 129-141 (1989) Mathematische Zeitschrift

�9 Springer-Verlag 1989

On p-Local Abelian Groups with Decomposition Bases

Mark Lane Massachusetts Institute of Technology, Lincoln Laboratory, Lexington, MA 02173-0073, USA

I. Introduction

P. Hill and C. Megibben have posed the following problem: determine the structure of a reduced p-local abelian group G which possesses a nice decomposition basis X such that G/(X) is an A-group. Can we at least determine that G can be embedded as an isotype submodule of a Warfield group? Of course, the difficult aspect of this problem is the complex structure of the cokernel, even though there is a good deal of information about the A-groups [5]. Indeed, it appears that the usual techniques involved when G/(X) is totally projective are not applicable in the more general situation seeing that, in the latter case, G/(X) has a nice composition series and one can employ the infinite combina- torics associated with Hill's third axiom of countability to lift simple extensions of height preserving isomorphisms [10, 7]. For a complete analysis of these groups (called Warfield groups), the reader is encouraged to consult [10, 11, 9], and [-7]. In the more complicated situation at hand, there is not known to be an abundance of nice subgroups in G/(X), nor is there known to be a third axiom of countability characterization for the A-groups of Hill [5] or the S-groups of Warfield [18].

In this note, we will consider the problem where G/(X) is an S-group, and we will develop techniques that will determine that this problem does not add any increased difficulty. That is, we will show that in this case G must decompose into the direct sum of a Warfield group and an S-group, and we will thereby be able to discern that a summand of a group which is a direct sum of a Warfield group and an S-group is also a direct sum of a Warfield group and an S-group. The latter result may be considered by some as the dominant motivation behind the original problem.

All groups in this note are p-local and abelian, and this implies that they

Z { m . are modules over the ring of integers localized at a fixed prime p, v = n "

neZ, (n, p) = 1~. An element x of a group G is said to have height ~ provided m,

xsp~G\p~+lG and height oo if x~p~G for every ordinal e. We will use the symbol Ix[ to denote the height of x in G. A direct sum @A~ in a group

130 M. Lane

G is called a valuated coproduct provided [~ ail = min lai[ for aieAi (with finitely many of the ai non-zero). If X is a collection {xi} of elements in G such that each xi has infinite order and (X> = (~ (x i> is a valuated coproduct, then we will call X a valuated basis in G, and if, in addition, G/(X> is torsion, then X is called a decomposition basis for G [10].

Suppose X is a valuated basis with each x e X having the property that [pkx[ = I Xl + k for every k < co. This property is also referred to by saying that the height sequence of each x e X is free of gaps, and X is called a free-valuated basis in G. A free-valuated decomposition basis is also known as a K-basis [19], and the p-local balanced projectives are precisely those p-local groups G which possess a K-basis X with G/<X> totally projective. These groups were studied extensively by Warfield in [18] and [19], and then later by the author in [14]. A useful property of balanced projectives is that they always decompose into a direct sum of submodules of torsion-free rank at most one. Moreover, as the name indicates, balanced projectives enjoy the projective property with respect to all balanced exact sequences; that is, those short exact sequences 0 ~ A ~ B ~ C ~ 0 such that 0 ~ p~ A --+ p~ B ~ p~ C ~ 0 is also exact for each ordinal ~. The kernel A of such a sequence is nice in the sense that p~(B/A)=(p~B +A)/A for every ordinal 0r and isotype in the sense that p~A=Ac~p'B for every ordinal cr An important characterization of nice submodules is that each coset b + A in B/A contains an element which has maximal height [3].

If X is a decomposition basis for a group G such that (X> is a nice subgroup of G, then we will say that X is a nice decomposition basis for G, and G is a Warfield group provided it possesses a nice decomposition basis X with G/<X> totally projective. The Warfield groups are characterized as summands of simply presented groups [10], and their torsion parts are known separately as the S-groups. Warfield [18] introduced the S-groups as those groups which could be represented as the torsion part of some balanced projective group, and it was later shown that the torsion part of a Warfield group is also an S-group [12]. The balanced projective groups, Warfield groups, and S-groups all have a structure theory developed in terms of a complete set of numerical invariants, which become the foundation for both uniqueness and existence theorems [18, 10, 11]. If we allow the term SW-module to denote a group which is a direct sum of a Warfield group and an S-group and the term SKT-module to denote a group which is a direct sum of a balanced projective group and an S-group, then it will be appropriate to briefly mention previous work related to our problem at hand. Stanton did a considerable amount of work on the summand theorem for SKT-modules in [16] and for SW-modules in [17], and Wick proved a special case of the summand theorem for SKT-modules in [20]. The author proved the general summand theorem for SKT-modules in [15], and it is the techniques introduced in that paper which will be expanded upon here. It is important to note that these techniques were in large part inspired by the insight- ful paper of Hunter and Walker [12].

This work was sponsored with the support of the Depar tment of the Air Force under contract F 19628-85-C-0002. The views expressed are those of the author and do not reflect the official policy or position of the US Government.

On p-Local Abelian Groups with Decomposition Bases 131

2. Cotorsion Envelopes and Constructions

As usual in the study of S-groups, the cotorsion envelope will play a particularly important role. Following the definition given in Sect. 54 of [2], a group G is said to be cotorsion provided Ext(J, G)=O for any torsion-free group J ; that is, any extension of G by a torsion-free group splits. A few useful observations from this definition, which are listed in [2], are summarized below:

1. an epimorphie image of a cotorsion group is cotorsion, 2. a submodule H of a reduced cotorsion group G is itself cotorsion if and

only if G/H is reduced, and 3. if H is a subgroup of G such that both H and G/H are cotorsion, then

G is also cotorsion. If G is any reduced group, then G can be embedded as an isotype submodule

of a reduced cotorsion group c(G) such that c(G)/G is torsion-free and divisible. It is customary to refer to c(G) as the cotorsion envelope (or cotorsion hull) of G, and there is a natural isomorphism from c(G) onto Ext(Z(p~), G). The interested reader is encouraged to consult Fuchs' thorough treatment of these points [2]. For computational reasons, it is desirable (and the main purpose of this section) to make Lemmas 2.1 and 2.2 in [18] more precise for our application.

Consider G a reduced group with c(G) a fixed reduced cotorsion group containing G such that c(G)/G is torsion-free and divisible. If N is any submodule of G with G/N reduced, then the quotient group c(G)/N splits into its reduced and divisible parts

c(G)/N=Cr/N@C/N,

where C/N is the maximal divisible subgroup of c(G)/N. Since G/N is reduced and c(G)/G is torsion-free and divisible, it follows that C/N is torsion-free and divisible with G/N c_ G/N and C,/N/G/N torsion-free and divisible. Moreover, since c (G)/C ~ C,/N is reduced and c (G) is cotorsion, it follows that C is cotorsion and GIN is cotorsion (being an epimorphic image of c(G)). We will define c(N), the cotorsion envelope of N, to be identical to C, and we will define G/N-c (G/N) to be the cotorsion envelope of G/N, noting the natural inclusion G/N ~ GIN. Furthermore, the natural map ~ + N ~ , + c ( N ) from GIN onto c(G)/c(N) is an isomorphism. When reference to the submodule N is clear, we will use the notation G to denote this submodule of c(G) throughout the paper.

One can now recognize that if C' is any other reduced and cotorsion submodule of c(G) which contains N such that C'/N is torsion-free and divisible, then C'= C. Indeed, it is clear that C'___ C since C/N is the maximal divisible submodule of c(G)/N, and if C/C' is not zero, then at least C/C' is torsion-free. But C' is cotorsion, and so the extension C of C' splits. Therefore we can write C=C'OC with C reduced and cotorsion and ClaN=O, and this contradicts C/N torsion-free and divisible.

Now if ~ is any ordinal, then G/p ~ G, G/(p ~ G + N), and G/(p ~ G n N) are also reduced, and so the reduced cotorsion groups c (p~ G), c (p~ G + N), and c (p~ G f~ N) can (and will) all be thought of in the same manner as c(N) (as submodules of c(G)). Moreover, since (p~G + N)/N is reduced, it follows that

c (p~ G + N ) / N : G./N 4G c (N)/N,

132 M. Lane

where Go]N is necessarily the cotorsion envelope of (p~G + N)/N. It is useful to note that G~/N = GIN n c ( f G + N)/N since c (p" G + N)/N and c (G)/N both have the same maximal divisible submodule with c(p'G + N)/N c_ c(G)/N.

Proposition 1. I f G is a reduced group which contains a submodule N with GIN reduced, then c (p" G + N) = c (p" G) + c (N) and c ( f G n N) = c ( f G) n c ( N) for each ordinal ~.

Proof We first must observe that c(p~G)+ e(N) and c ( f G ) n c(N) are cotorsion. Since c (N) is cotorsion and {c(p~ G) + c (N)}/c (N) ~ e (p" G)/{c (p" G) n c (N)} are both cotorsion, it immediately follows that c ( f G)+ c(N) is cotorsion, and since { c ( f G) + c(N)}/c(N) ~p~(c(G)/c(N)) is reduced, it is also clear that c(p ~ G) n c(N) is cotorsion. F rom the introductory remarks of this section, it remains to prove that (c (p~ G) + c (N))/(p ~ G + N) and (c (p~ G) n c (N))/(p ~ G n N) are both torsion-free and divisible.

Since both c(p~G)/p~G and c(N)/N are divisible, it easily follows that (c (p~ G) + c (N))/(p ~ G + N) is divisible, so suppose p x ep~ G + N, where x e c (p" G) +c(N). Since c(G)/G is torsion-free, we at least know that x e G. Now construct a sequence of elements gke c (p~ G) + c (N) with gk + P~ G + N = p [gk + 1 "~ P~ G + N] and x + p ' G + N = p [ g x +p~G+N-] using the fact that ( c ( f G ) + c ( N ) ) / ( f G + N ) is divisible. But xeG, and so each gk~G since c(G)/G is torsion-free. Hence the element x + p~ G + N is a divisible element of G/(p ~ G + N), and since G/(p ~ G + N ) is reduced, it follows that x s p ' G + N .

It is clear that (c(p~G)nc(N))/(p'GnN) is torsion-free because both e(p~G)/p~G and c(N)/N are torsion-free, and so we need only show that

(c (p~ G) n c (N))/(p ~ G n N) = p [(c (p~ G) n c ( N))/(p ~ G n N)].

Towards this goal, let yec(p~G)nc(N) and write y = g , + p x ~ = n + p z , where g, ep~ G, x~E c (p" G), n e N, and z ~ c (N). Thus x , - z + p~ G + N is a torsion element of (c (p~ G)+ c (N))/(p~G + N), and so by the argument in the preceding paragraph, x~- zep~G+N. If we write x~-z=g ' ,+n ' for g'~ep~G and n'eN, then n - p n ' = g , + p ( x , - z - n ' ) e p ' G n N , and so

y = n + p z = ( n - p n ' ) + p(z +n')

= ( n - p n ' ) + p ( x ~ - g ' ) ~ p ~ G n N +p[c(p~G)nc(N)]. []

In order to see that the hypothesis requiring G/N reduced is essential, consider a 2-elementary balanced projective group G with N the torsion part of G. Then GIN ~ Q and c (pZ G n N) = 0 =~ c (px G) = c (p~ G) n c (N). Moreover, G/(p ~ GG N) ~ Z (p oo), and so c (p~ G) + c (N) = c (G) :~ c (pZ G �9 N).

Proposition 2. I f G is a reduced group containing a submodule N with G/N reduced, then the extension p" c ( G) n c(N) of c ( f G n N) splits for each ordinal ~.

Proof Since c (p~GnN)=c(p 'G)nc(N) is cotorsion, it suffices to prove that ( f c(G) c~ c(N))/(c(fG) n c(N)) is torsion-free. By Lemma 1 in [12], we can write p~ c (G) -- c (p" G)(~X~, where X, ~ p" c (G/p ~ G) is reduced, torsion-free, and cotorsion. Hence if y ~p~ c(G) n c (N) with p y ~ c (p~ G) n c(N), then for y written y = c~


+ x~ with c~ ~ c (p'G) and x~ e X, , it is clear that p x, = 0. Since X~ is torsion-flee, we conclude that x , = 0 , and so y=c~ec(p~G)nc(N) . []

The X~ in the above proof was the key ingrediant in a fruitful characterization of S-groups in [12], and it is necessary at this time to introduce some of the terminology from that paper. First recall that an ordinal e is said to have cofinality co provided there exists a countable collection of ordinals/~, < e with sup ft, = e. We will use the standard notation cof(e) = co to denote that an ordinal has cofinality co and cof(e) > co to denote that an ordinal does not have cofinality co. If X = @X~, where the sum is over all those ~ with cof(~)>co, then X is called a Incwo subgroup of c(G). Evidently the lncwo stands for "limits not cofinal with omega". Since each X, is reduced, torsion-flee, and cotorsion, it follows that p~X~=O [2]. Now it is not difficult to see that for any ordinal ~, p c (p" G) is reduced and cotorsion with p c (p~ G)/p ~ + 1G torsion-free and divisible since c(p ~ G)/p~G is torsion-free and divisible and G is isotype in c(G). Hence, by the introductory remarks to this section, we deduce that p c(p~G)= c(p ~ + 1 G) and we write p~ + 1 c (G) = p c (p~ G) | X~. Furthermore, if we select a decomposition p ' e ( G ) = c ( p ' G ) | for each limit ordinal ~ as prescribed by Lemma 1 in [12], then if/~ = c~ + n for any n < co, then pa c (G) = p" c (p" G)@p" X~ is a decomposition of p~c(G) prescribed by Lemma 1 in [12]. F rom this simple analysis, it is not difficult to see that X,___ n c(pPG) for each limit ordinal ~. Indeed,

(#<~) suppose x e X , , and for any limit ordinal /~<~, write x = c a + x a for caec(paG) and xpeXp. But then xaep"Xp for every n<co, and so xaEp~Xa=O, and we conclude that x = cp~c(p ~ G) for every fl < ~. This observation will actually turn out to be very important in the next section, but first we would like to narrow further our selection of a particular decomposition of p~ c (G).

Lemma 1. Suppose G is a reduced group which contains a submodule N with GIN reduced, and suppose that, for ~ a limit ordinal, a f ixed decomposition p ~ e ( G ) n c ( N ) = c ( p ' G n N ) G D ~ is chosen as prescribed by Proposition2. I f (p~ c (G) + c (N))/c (p" G + N) is torsion-free, then the extension p~ c (G) of c (p" G) OD, splits.

Proof Since p~c (G) is cotorsion and p~ c (G)/(p ~ c (G) n c (N)) ~ (p~ c (G) + c (N))/c (N) is reduced, it follows that p" c (G)n c(N) is cotorsion, and hence, so is D, cotorsion. Moreover, c(p" G) n D~ = 0 since 0 = c(p" G n N) ~ D~ = c(p ~ G) n D~ by Prop- osition L Now, by the modular law, we can represent

c(p~ G)O D~= c(p" G) + [p" c(G) n c(N)] = p~ c( G) c~ [c(N) + c(p~ G)] ,

and so

p~c(G)/[c(p ~ G)O D j = p~ c(Gff[p ~ c(G) n (c(N) + c(p ~ G))]

[p~ c (G) + c (N)]/c (p~ G + N)

by the Noether isomorphism and Proposition 1. Since the latter quotient is torsion-flee by hypothesis and c(p~G)@D~ is cotorsion, the extension splits. [ ]

134 M. Lane

We will use the following notation to refer to this lemma (keeping in mind that all of this analysis is centered around a specific submodule N of G). For

a limit ordinal, we will write p~c(G)=c(p~G)GD~GM~, where X~=D,OM~ is reduced, torsion-free, and cotorsion. For fl = e + n with c~ a limit ordinal and n < o, we will recognize pP c(G)=p" c(p ~ G)Gp"D~Op"M~. A particularly useful property of the M~ is that if M = @ M ~ is summed over any combination of limit ordinals e, then MOc(N) is a valuated coproduct, and this will be important in the sequel where N = ( X ) is generated by a nice decomposition basis. Indeed, the hypothesis requiring (p~ c(G) + c(N))/c(p ~ G + N) torsion-free is satisfied for each ordinal e when N is nice in G since p~ c(G/N)/e(p'(G/N)) is torsion- free by Lemma 1 in [12] and p~(G/N)=(p~G + N)/N.

Two representations for the cotorsion hull of a reduced quotient group GIN have been introduced, depending upon whether we wish to speak of elements in G/N as g + N or as g+c (N) for g~G. These two were GIN and c(G)/c(N), respectively. It is the purpose of our next lemma to show that, if N is nice in G, then a Incwo subgroup for c(G/N) can be chosen irrespective of the representation.

Lemma 2. I f N is a nice submodule of the reduced group G, then for each limit ordinal ~, there exists a submodule N~ of G with N~ ~ p~ c (G/(p~G + N)) such that

and p~ [c (G)/c (N)] = (c (p~ G) + c (N))/c (N) G (N~(~ c (N))/c (N)

p" [ G/N] = GJN G (N~ G N)/N.

Proof Since G/N= c(G/N), by Lemma 1 in [12] we can decompose

p'(G/N) = GJNO(Y~ + N)/N,

where G~/N -= c (p~ (G/N)) = c ((p~ G + N)/N) and

(Y~ + N)/N ,~ p~ c(G/N/p~(G/N)) ,,~ p~ c(G/(p ~ G + N)).

Noting the isomorphism GIN ,,~ c(G)/c(N) and Proposition 1, it follows that

p~ [c (G)/c (N)] = (c (p" G) + c (N))/c (N) @ ( Y~ + c (N))/c (N).

Now express p~ [c (G)/c (p~ G + N)] = ZJc (p~ G + N) with Z~_ c (G). Since this epi- morphism of Z , is torsion-free and c(p~G+N) is cotorsion, the extension Z~ of c (p" G + N) splits, and we can write Z~ = c (p~ G + N)@ N,, where N~ _ G. From the observations following Proposition 2, it is evident that if y~E Y,, then for each fl<c~ there exists some zt~ec(N ) with y,+zpec(pr and so y~+c(p ~ + N)eZJc (p~G + N). From this we conclude that

(Y~ + c (N))/c (N) ~_ (c (p~ G) 4- c (N))/c (N) �9 (N~, �9 c (N))/c (N),

and so the first equality in the statement of the lemma holds. Moreover, since both Y~ and N~ are in G and GdN=c(p~G+N)/Nc~G/N, we also have Y~ ~ G~GN~, and so the second equality in the statement of the lemma holds. []


We make the observation that if GIN is reduced, then c(p'G +N)/e(N)c_(p~c(G)+c(N))/c(N) and G~JN~_(p'G+N)/N, and so if N is a nice submodule of a reduced group G, then it is the structure of N~ which determines whether c(N) is nice in c(G) or N is nice in G. Indeed, the direct sum N~@c(N) is not necessarily a valuated coproduct, and we will give an example shortly confirming the intuition that it is not enough for N to be nice in G to insure that N is nice in G.

Proposition 3. I f N is a nice submodule of the reduced group G, then N is nice in G if and only if c(N) is nice in c(G).

Proof We will actually prove that p" [c(G)/c(N)] = (p~ c(G) + c(S))/c(N) if and only if p'[G/N]=(p'G+N)/N for each limit ordinal ~. Towards this end, let

be a fixed limit ordinal and construct N, as in Lemma 2. Define two submodules of N~ as follows:

and N~ = {weN,: w + zep~ c(G) for some zec(N)}

N~= {w~N~: w+n~p~Gfor some n~N}.

Clearly c(N) is nice in c(G) at the ordinal c~ if and only if N~=N~ and N is nice in G at the ordinal e if and only if N~ = N,, and so it remains to prove that N~=b~,. Since one direction is trivial, suppose that w~N~ such that there exists z e c (N) with w + z ep~ c (G). Using Lemma 1 as in the after-discussion, we can write w+z=c~+d~+m~ for c~ec(p~G), d~D, , and m~eM~. Since m~ and w are both in G, it follows that

w-m~ + N =c, +d~--z + N~G/N nc(p~G + N)/N =G~/N,

and so w--m~+Ne(p~G+N)/N. Since m~ep~c(G)nG=p'G, it follows that w+N~(p~G+N)/N. []

We will now give an example of a nice subgroup N of a reduced group G such that N is not nice in G. This example was first given by Wick in [21], and because of Corollary 2.3 in that paper, one can pinpoint that N fails to be nice at limit ordinals which are not co final with on.

Example Wick [21]. There exists a nice submodule N of a reduced group G such that N is not nice in G.

Proof We can use our constructions in this section to prove the following observation: if N is a balanced (i.e. nice and isotype) submodule of a reduced group G, then c(N) is isotype in c(G). The proof of this claim is by induction on the ordinals. Suppose that p y =z, where it is assumed that y ep ~ c(G), z~c(N), and p~c(G)nc(N)=p'c(N). It must be demonstrated that py=pz ' for some z'~p~c(N). By Lemma 1, we can write y=c~+d~+m~, where c~ec(p~G), d~eD~, and m~eM~. Since pyec(N) and M, is torsion-free, it can first be seen that m~ = 0. Moreover,

p c~ec(p ~+1 G) n c(N)=c(p ~+1 G n N)=c(p~+ l N)= p c(p~N)

136 M. Lane

since N is isotype in G. Hence p y = p(c'~ + d~), where c'~ + d~ep ~ c(N) by induction, and so c(N) is isotype in c(G) as claimed.

Proceeding with the construction, let S be a reduced S-group which is not totally projective and construct for S a balanced projective resolution

E: O ~ N ~ G ~ S ~ O .

Now N is balanced in G, but if c(N) was balanced in c(G), then Wick has shown in 1-21] that S would have to project onto E, and so S would be isomorphic to a summand of the totally projective group G. This is impossible since S is not totally projective, and we conclude that e(N) cannot be balanced in c(G). From our observation in the preceding paragraph, we see that c(N) must not be nice in e(G) since it is necessarily isotype, and so N cannot be nice in G by Proposition 3. []

One final note must be made concerning the subgroup N~ constructed in Lemma 2. If e is any limit ordinal and N, = N~ (in the language of Proposition 3), then we can replace N~ by Ms so that advantage can be made of the valuated coproduct M~@c(N). This substitution will always be assumed whenever possi- ble. If N~ = N" for every limit ordinal c~ with cof(c~) > co, then the above observation implies that

@ [(M~e~/N]

(where the sum is over all limit ordinals c~ with cof(e)> co) is a Incwo subgroup of G/N.

3. Applications to Groups with Nice Decomposition Bases

In this section, we will apply the constructions of the previous section to a submodule N which is generated by a nice decomposition basis. Following Hill in [4], we will call a submodule H of a group G an No-separable submodule provided for each g~G, the supremum sup{]g+hl: h~H}=~ can be realized by an (at most) countable subset in H, ~ = sup {]g + h,,[: h,,EH, m < co}. Although Sect. 11 in [-10] goes into great detail on when a decomposition basis is nice and examples are given which show that a decomposition basis need not be nice, a valuated basis has an important structural property in the valuated coproduet. The main argument in the results of the referenced work centers around supremums of the type above with cofinality co, and of course, if the supremum is approached but not attained, then the submodule cannot be nice; however, this is as bad as the situation can get since our next result shows that any valuated basis is (at least) N0-separable. This becomes an especially important observation since cosets of infinite order are allowed.

Theorem 1. I f X is a valuated basis of a reduced group G, then ( X ) is No-separable in G.


Proof Suppose that a is a limit ordinal and g~G satisfies a = s u p { l g + x l : x ~ ( X ) } , For every x e ( X ) , we can write x in a unique way as x=x l +x2 + ... + x , , where n depends on x and for l<j<n, we can write xj=tuxij-[-tzjXzj + . . . + t,,j x,,j such that tueZ p, xueX, and ]qj x~jl = ]t2j x2j] . . . . . ]t~j x~j]. More- over, we may order the decomposition so that ]x~[<lx2]<. . . <]x,]. When x is written in this way, we will say that x is expressed in its standard representation. Now suppose that x and x' are two elements of ( X ) such that, in their standard representations, x = xl and x' = x~, and suppose that [g[ < [g + xl and ]g] < ]g + x'l. Since ( X ) is a valuated coproduct over the xffs, it is evident that if we express x and x' further in their standard representations as

and X = t l l X~I q- t 2 1 X21 -~- . . . 71- Zml Xml

Xt / t t ! ! t = t l i x H +t21Xza + . . . +tklXkl,

then k=m and, for each 1 <_i<m, X i l = X l i l and p divides (tia-t'il) (because Ix - x ' [ > Ix[ = [x'D. We conclude from this observation that there are at most coun- tabty many distinct x e ( X ) with x = x l in its standard representation and Igl < ] g + x l .

Now if there exists an x e ( X ) such that ] g +x ]=~ , then there is nothing to prove, so we may suppose that there exists a collection of elements {x~} in ( X ) with ]g + xp[ = fl < ~ with sup {fl} = ~. Considering each x~ in its standard representation, it is clear that we may restrict our attention to those xp which can be expressed as

X f l = X 1 -J- X 2 "Jv . . . -~- Xn(fl ) and

Igl < Ig + xll < Ig+x~ +x21 < ... < Ig+xl +x2 + ... + x,(B)l �9

Since the argument in the preceding paragraph applies to any element in G, there are at most a countable number of distinct xl 's sueh that ]g l<]g+xl l , and for each x l , there are at most countably many x2's such that ]g+xl] <]g + x~ + x2]. Extending this argument to any positive integer n, it is now apparent that the set {xa} must be at most countable, and so ( X ) is N0-separable in G. []

Suppose that N = ( X ) is generated by a nice decomposition basis for a reduced group G, and suppose further that for each limit ordinal 0~, a decomposition of p'c(G) has been chosen as prescribed by L e m m a l , p'c(G) --c(p'G)@D,@M~. If for each ordinal ~ with cof(~)>co, we select a p-basic submodule of M~ and denote i t /~ r , then the submodule

is generated by a valuated basis in G, and the submodule M = @ / ~ r is generated by a free-valuated basis in G. Also, if we follow the convention of [12] and write [Y:G] to denote the submodule of G consisting of all gEG with pkgeY

138 M. Lane

for some k< co, then Y is generated by a decomposition basis for [Y: G]. Since G/[Y:G] is torsion-free, we note that [Y:G] is isotype in G and [Y:G]/N is isotype in GIN.

Lemma 3. I f N = ( X ) is generated by a nice decomposition basis for a reduced group G, then the submodule Y'= @]fI,~) N is nice in [ Y: G].

Proof Since Y/N is generated by a K-basis for [Y:G]/N, it suffices to prove that N is nice in [Y:G] by Theorem 31 in [10] and Proposition 3.2 in [3]. To this end, it suffices to show that if ~ + N is an element of p~([Y':G]/N), where ~ is a limit ordinal, then f ,+nep~G for some n~N. First suppose that cof(~)=c0. If ~eG, then simply pick an n~N with ~+n of maximal height. If ~r then there must be some non-negative integer k such that p k ~ , , and this implies that pk~+N~p'+~'(G/N) since cof(a)=co. Thus we can write pk~, + N = p k c , + N with c~p 'G, and since c(G)/G is torsion-free, it follows that ~,-c ,+Nep'(G/N)=(p~G+N)/N. We can then use the niceness of N in G to construct an n e N such that ~ + n~p ~ G. Now suppose that cof(c 0 > co. By Lem- ma 2, we see that either ~,+N~(p~G+N)/N or else sup{l~+nl: neN}=a. But since N is No-separable in G by Theorem 1, it follows that this supremum must actually be attained by some n~N by Proposition 1 in [4] (because c~ is not cofinal with co). []

Theorem 2. A reduced group G is a direct sum of a Warfield group and an S-group if and only if G contains a nice decomposition basis X such that G / (X) is an S-group.

Proof The stated conditions are clearly necessary since any Warfield group possesses a nice decomposition basis with totally projective cokernel, so suppose the hypotheses are satisfied and we will proceed to prove that G has the desired decomposition. Since N = (X) is No-separable in G by Theorem 1, the observation at the end of the previous section is applicable to insure that

Y / N - @ [ M ~ O N ] / N

(where the sum is over all those limit ordinals �9 with cof(c0>co ) is a Incwo subgroup of GIN. By Theorem 5 in [12 l, the quotient [Y/N: G/N1/Y/N is totally projective, and hence, so is [Y: G]/Y totally projective. Therefore, if M~ denotes a p-basic submodule of M~ and Y= @M~GN, then Y is generated by a nice decomposition basis for [Y:G] by Lemma 3. Moreover, by Sect. 2(B) in [12], the quotient [Y:G]/Y~[Y:G]/Y, and so [Y:G] is a p-local Warfield group. Now, by Theorem 45 in [10], we can write [Y-:G]--B| where B is balanced projective and the torsion part of C is totally projective. But /~---@/fir is free valuated in [Y: G] with each element in M generated by an element with height a limit ordinal which is not cofinal with co. It follows that C is a Warfield group with a decomposition basis comprised of elements entirely from X. More- over, since B is balanced projective, it splits over its K-basis [19], and so we may decompose [Y: G] further as

[Y:G] = B ~ G B 2 0 T t ~ C ,


where T is totally projective, B1 is balanced projective with a K-basis comprised entirely of elements in M, and B2 is balanced projective with a K-basis comprised entirely from elements in X. Finally, if t B1 represents the torsion part of B1, we see that G=tBI(~Bz@TOC, and so G is a direct sum of a Warfield group and an S-group. []

It is curious to note that, historically, summand theorems for various classes of Abelian groups have not usually fallen from techniques which were developed to originally study the class itself. Several examples which support this statement are: the class of direct sums of countable groups [13], the class of Warfield groups [-1], and the class of S-groups [12]. For each of the classes mentioned above, new machinery was developed to prove that a summand of a group in a particular class also belonged to that class. The summand theorem of SW-modules is an immediate consequence of Theorem 2, but the author knows of no direct method of proving this result using other techniques associated with the study of Warfield groups and S-groups.

Corollary 1. A summand of an SW-module is itself an SW-module.

Proof Suppose G= WGS=AOB, where W is a Warfield group and S is an S-group. We must show that A and B are SW-modules. By Theorem 6.6 in [1], we may assume there exist decomposition bases XA for A and XB for B, which are contained in W with XAUXB a decomposition basis for W. By Corollary 9.4 in 1-9], we may assume (Xa u Xn) is nice in W with simply presented cokernel, and this implies that (XA) is nice in A and (Xn) is nice in B with A/(XA) and B/(XB) summands of an S-group. By Theorem 7 in [12], A/(XA) and B/(XB) are S-groups, and so A and B are SW-modules by Theorem 2. []

Since any Warfield module which has a K-basis is balanced projective, Theo- rem 3.1 and Corollary 3.2 in [15] for SKT-modules are readily seen to be special cases of Theorem 2 and Corollary 1, respectively. As a final application to our main result, we will provide yet another characterization of SW-modules; this time in the spirit of Theorem 5.1 in 1-6] and Theorem 3.4 in [15]. Recall that if H is a submodule of a group G, then the coset valuation on G/H is defined as

[g+Hl-=sup{lg+h[+l : h~H}

for each g~G. The reader can see from this formulation, that the power of this concept is in its ability to distinguish limit ordinals which are approached by the supremum Ig+h[ for h~H from those which are actually attained by some heH. In this regard, the coset valuation has been a major tool in the study of isotype subgroups of totally projective groups [6] and in the study of isotype submodules of p-local balanced projective groups [15]. However, it has recently been pointed out by Hill and Megibben in 1-8] that the coset valuation is not enough to study isotype submodules of p-local Warfield groups in the same capacity, but for torsion cokernels, this difficulty is not encountered. In fact, the proof of our characterization relies only on Theorem 2 and an appli-

140 M. Lane

cat ion of the equivalence theorem for isotype subgroups of totally projective p-groups [6], and all of the difficulty with the elements of infinite order is taken care of by the decomposi t ion basis.

Theorem 3. A p-local group H is an SW-module if and only if H appears as an isotype submodule of a p-local Warfield group G so that G/H is a coproduct of countable torsion groups when G/H is endowed with the coset valuation.

Proof. The p roof of necessity does not present a p rob lem since, if H has the desired decompos i t ion as H = W O S , then we can select G=WGT, where T is a totally projective g roup containing the S-group S as an isotype subgroup with T/S a coproduc t of countable groups when it is endowed with the coset valuation. Suppose now that H can be embedded in a Warfield g roup G with G/H a coproduc t of countable groups when it is endowed with the coset valuation. Recall that a subordinate decomposition basis is a subset of a decomposi t ion basis with every element a multiple of some element in that basis. I t is noted in [-10] that if X is a nice decomposi t ion basis for the Warfield g roup G with G/(X) total ly projective, then any subordinate X ' of X is also a nice decomposi- t ion basis of G with G/(X') totally projective. Since G/H is tors ion and G is a Warfield group, we m a y assume there exists a nice decompos i t ion basis X for G which is conta ined in H with G/(X) totally projective. It is easy to verify that since H is isotype in G, X is also a nice decompos i t ion basis for H and H / ( X ) is isotype in G/(X). (Note that it is crucial here that ( X ) is nice in G; for a more general s tatement concerning this claim, see L e m m a 5.2 in [73.) Moreover , since G/H is a coproduc t of countable groups when G/H is endowed with the coset valuat ion and ( X ) is nice in G, it follows that G / ( X ) / H / ( X ) is also a coproduc t of countable groups under its coset valuation. Finally, since G/(X) is totally projective, we can use the applicat ion of the equivalence theorem for isotype subgroups of totally projectives embodied in Theorem 5.1 in [-6] to conclude that H I ( X ) is an S-group, and so H is a direct sum of a Warfield g roup and an S-group by Theorem 2. [ ]

References

1. Arnold, D., Hunter, R., Richman, F.: Global Azumaya theorems in additive categories. J. Pure Appl. Algebra 16, 223-242 (1980)

2. Fucbs, L.: Infinite abelian groups, vol. I. New York: Academic Press 1970 3. Hill, P.: On the classification of abelian groups (photocopied manuscript). 1967 4. Hill, P.: Isotype subgroups of totally projective groups. In: G6bel, R., Walker, E. (eds.) Abelian

group theory. Proceedings, Oberwolfach 1981. (Lect. Notes Math., vol. 874, pp.305-321) Berlin Heidelberg New York: Springer 1981

5. Hill, P.: On the structure of abelian p-groups. Trans. Am. Math. Soc. 288, 505-525 (1985) 6. Hill, P., Megibben, C.: On the theory and classification of abelian p-groups. Math. Z. 190, 17-38

(1985) 7. Hill, P., Megibben, C.: Axiom 3 modules. Trans. Am. Math. Soc. 295, 715-734 (1987) 8. Hill, P., Megibben, C.: The local equivalence theorem. Proceedings, Perth 1987 (to appear in

Am. Math. Soc. 1989) 9. Hunter, R., Richman, F.: Global Warfield groups. Trans. Am. Math. Soc. 266, 555 572 (1981)

10. Hunter, R., Richman, F., Walker, E.: Warfield modules. In: Arnold, D., Hunter, R., Walker, E. (eds.) Abelian group theory. Proceedings, New Mexico 1976. (Lect. Notes Math., vol. 616, pp. 87-123) Berlin Heidelberg New York: Springer 1976


11. Hunter, R., Richman, F., Walker, E.: Existence theorems for Warfield groups. Trans. Am. Math. Soc. 235, 345-362 (1978)

12. Hunter, R., Walker, E.: S-groups revisited. Proc. Am. Math. Soc. 82, 13-18 (1981) 13. Kaplansky, I.: Projective modules. Ann. Math. 68, 372-377 (1958) i4. Lane, M.: A new characterization for p-local balanced projective groups. Proc. Am. Math. Soc.

96, 379-386 (1986) 15. Lane, M.: Isotype submodules of p-local balanced projective groups. Trans. Am. Math. Soc.

301, 313-325 (1987) 16. Stanton, R.: S-groups. (preprint) 17. Stanton, R.: Warfield groups and S-groups. (preprint) 18. Warfield, R.: A classification theorem for abelian p-groups. Trans. Am. Math. Soc. 210, 149-168

(1975) 19. Warfield, R.: Classification theory of abelian groups. I. Balanced projectives. Trans. Am. Math.

Soc. 222, 33-63 (1976) 20. Wick, B.: Classification theorems for infinite abelian groups. PhD thesis, University of Washington,

1972 21. Wick, B.: A projective characterization for SKT-modules. Proc. Am. Math. Soc. 80, 3943 (1980)

Received March 7, 1988

onp-local abelian groups with decomposition bases

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