Abelian Groups - László Fuchs [Springer]

8 More on Ext 293
Ext is exactly the p-socle of Ext.A; Z/, thus the number of summands Z.p 1 / in
Ext.A; Z/ is given by the dimension of the elementary p-group Ext.A=pA; Z/ as a
Z=pZ-vector space. This is equal to the dimension of A=pA if this is finite, otherwise
it is of the power of the continuum if A=pA is countably infinite.
If A ¤ 0 and Hom.A; Z/ D 0, thenA contains a finite rank pure subgroup B
which is not free (Theorem 7.1 in Chapter 3), i.e. B contains a finitely generated
free subgroup F such that B=F is infinite. From the exact sequence 0 ! F !
B ! B=F ! 0 we derive the exact sequence Hom.F; Z/ ! Ext.B=F; Z/ !
Ext.B; Z/ ! 0: Here Hom is finitely generated free and the torsion-free rank of the
first Ext is of the power of the continuum (see, e.g., Sect. 7(e) in Chapter 13), so
the last Ext must have the same torsion-free rank. As Ext.B; Z/ is an epic image of
Ext.A; Z/, the proof is complete.
ut
F Notes. For more detailed information about the Ulm subgroups and Ulm factors of the
groups Ext.Q=Z; T/, we refer to Harrison [3].
The groups Ext.G; Z/ have been studied by several authors. The study splits into the cases
according as G is torsion or torsion-free. Ext.G; Z/ is isomorphic to the character group of G if
G is torsion (so it is a compact group). For countable torsion-free G, see Theorem 8.6. Forlarger
torsion-free G, see the Notes to Sect. 3. Additional information: Hiller–Huber–Shelah [1] show in
L that Hom.G; Z/ D 0 implies that Ext.G; Z/ admits a compact topology.
Schultz [5] calls G a splitter if Ext.G; G/ D 0 (e.g., torsion-free algebraically compact groups).
His study includes groups whose infinite direct sums or products are also splitters. See Göbel–
Shelah [3] for more on splitters, using new ideas and methods.
Exercises
(1) For a reduced p-group T, these conditions are equivalent:
(a) Ext.Z.p 1 /; T/ is algebraically compact;
(b) Ext.Z.p 1 /; T/ Š QT;
(c) T is torsion-complete.
(2) Find non-isomorphic reduced cotorsion groups with 2 Ulm factors such that the
corresponding Ulm factors are isomorphic. [Hint: B the standard basic, T pure
in torsion-completion B such that jB W Tj DjT W Bj D2 @ 0
; compare the groups
Ext.Z.p 1 /; B/ and Ext.Z.p 1 /; T/.]
(3) (a) Let G denote a reduced cotorsion group. There is a cotorsion group A such
that A 1 Š G.
(b) (Kulikov) Every cotorsion group G can be realized as G Š Pext.C; A/ for
suitable A; C.
(4) Let A be a reduced torsion-free algebraically compact group, and C an
algebraically compact subgroup. Show that there is a transfinite well-ordered
descending chain of algebraically compact subgroups A from A down to B
such that all the factors A =A C1 are algebraically compact. (At limit ordinals,
intersections are taken.) [Hint: A=C is cotorsion.]
(5) (Hiller–Huber–Shelah) Let A be torsion-free such that Hom.A; Z/ D 0. Then
the rank of the p-component of Ext.A; Z/ is either finite or of the form 2 for an
infinite cardinal .