Abelian Groups - László Fuchs [Springer]

5 The Functor Pext 275
F Notes. Lemma 4.4 was published by Eklof–Fuchs [Annali Mat. Pura Appl. 90, 363–374
(1988)] for the special case (on valuation domain) needed there.
Let us call attention to the basic difference between Lemmas 4.3 and 4.4. In the former lemma,
the second argument of Ext was kept fixed, while in the latter lemma the second argument could
grow as large as necessary to match the size of the group in the first argument. This explains why
we needed the Diamond Principle in one case, but not in the other, this difference will be more
apparent in Sect. 7 in Chapter 13.
Exercises
(1) Suppose that A satisfies Ext.X; A/ D 0 for all rank 1 torsion-free groups X.
Then Ext.C; A/ D 0 for all torsion-free groups C.
(2) If Ext.Z.p/; A/ D 0 holds for A, then Ext.C; A/ D 0 for all p-groups C.
(3) If A satisfies Ext.Z.p/; A/ D 0 for all primes p,thenA is divisible.
(4) Derive Pontryagin’s Theorem 7.1 in Chapter 3 from Lemma 4.1.
(5) (Eklof–Huber) If C is a torsion-free group of countable rank such that
Ext.B; A/ D 0 holds for all finite rank subgroups B < C, then also
Ext.C; A/ D 0 for any A.
5 The Functor Pext
That the extensions of A by C in which A is a pure subgroup (we will call them pureextensions)
play a distinguished role does not seem to be obvious at the outset. The
truly significant thing here is that these extensions form a subgroup of Ext which
can be identified as the first Ulm subgroup of Ext.
Preliminary Lemma We recall that if ˛ W A ! A and W C ! C are
endomorphisms, then there are induced endomorphisms ˛ and of Ext.C; A/.
We begin with investigating the actions of ˛ and in Ext.C; A/. In the proof of
the next lemma, conveniently we may view A as a subgroup of B.
Lemma 5.1 (Baer [4]). Given the exact sequence
eW 0 ! A ! B ˇ
! C ! 0 (9.16)
and the endomorphisms ˛ W A ! A and W C ! C, we have:
(i) e 2 Im ˛ if and only if A=˛A is a summand of B=˛A;
(ii) e 2 Im if and only if A is a summand of ˇ 1 Ker ;
(iii) if Ker D 0,thene 2 Ker if and only if ˇ 1 Im Š A ˚ Im .