Abelian Groups - László Fuchs [Springer]

4 Pure-Injective Hulls 201
subgroup, then the identity map 1 G can be extended to a monomorphism W
A ! A 0 .IfA 0 happens to be also minimal, then necessarily A D A 0 . Thus
is then an isomorphism.
ut
We next prove a useful criterion for the pure-injective hull.
Lemma 4.3. A pure-injective group A containing G as a pure subgroup is the pureinjective
hull of G exactly if
(a) the maximal divisible subgroup of A is the injective hull of G 1 ; and
(b) the factor group A=G is divisible.
Proof. First, assume A is a pure-injective hull of G. Write A D D ˚ C with D
divisible and C complete. In view of the purity of G in A,
G 1 D\ n nG D\ n .G \ nA/ D G \ A 1 D G \ D:
Since every non-zero summand of D must intersect G, D hastobeaninjective
hull of G 1 .DefineE A such that E=G is the first Ulm subgroup of A=G. Then
E D D ˚ .C \ E/, andC=.C \ E/ Š .C C E/=E D A=E Š .A=G/=.E=G/ has
trivial Ulm subgroup. Thus C \ E is closed in C, so it is complete. Consequently, E
is pure-injective containing G, soE D A by minimality. Hence A=G coincides with
its own Ulm subgroup, which means that it is divisible.
To prove sufficiency, assume (a) and (b) hold for the pure-injective A containing
G as a pure subgroup. There is a pure-injective hull E of G contained in A.Fromthe
first part of the proof it follows that E=G is divisible, so E is pure in A, and hence it
is a summand. It cannot be a proper summand, so A D E.
ut
It is now easy to describe how to obtain the pure-injective hull of a group.
Theorem 4.4. The pure-injective hull of a group G is isomorphic to the direct sum
of the injective hull of G 1 and the completion QGofG.
Proof. Let D denote the injective hull of G 1 ,andlet W G ! D be an extension
of the inclusion map G 1 ! D. If W G ! QG stands for the canonical map, then
consider the map W G ! D ˚ QG which is the composite of the diagonal map
G ! G ˚ G followed by ˚ . The purity of G in QG guarantees that Im is pure
in D ˚ QG. From Lemma 4.3 the assertion follows at once.
ut
The last theorem makes it possible to determine the complete system of invariants
for the pure-injective hull of a group G in terms of certain invariants of G;
cf. Theorem 3.1 in Chapter 4 and Theorem 3.2.
Pure-Essential is Not Transitive The analogy of ‘essential’ with ‘pure-essential’
subgroups breaks down when transitivity is considered.
Lemma 4.5 (Fuchs–Salce–Zanardo [1]).
(i) The property of being a “pure-essential extension” is not transitive.
(ii) However, it is transitive for p-local groups.