Abelian Groups - László Fuchs [Springer]

296 9 Groups of Extensions and Cotorsion Groups
Proof. Applying the Hom-Ext exact sequence with Q=Z to e, the exactness of e is
immediate. Its splitting will follow once we can establish that it is torsion-splitting.
Since tC D tC, from the commutative diagram
:0 −−−−→ A −−−−→ B −−−−→ C −−−−→ 0
⏐ ⏐ ⏐
↓ ↓ ↓
• :0−−−−→ A • −−−−→ B • −−−−→ C • −−−−→ 0
it is routine to derive the projective property of torsion groups relative to the exact
sequence e .
ut
The cotorsion hull of a mixed group can be computed easily from those of
the torsion and torsion-free parts. In fact, from Lemma 7.2 it follows that for any
reduced group A,wehave
A Š .tA/ ˚ .A=tA/ :
Example 9.5. Let B D ˚nB n with B n D ˚Z.p n / for each n 2 N. ThenB is equal to the
subgroup G Q n B n such that G=B is the largest torsion-free divisible subgroup in . Q n B n /=B.
Example 9.6. For a reduced torsion-free group A, A Š QA.
Torsion-Free Cover The class of cotorsion groups (and especially their generalizations)
has attracted much attention in recent years, because they form a so-called
cotorsion pair along with the class of torsion-free groups. This means that G is
cotorsion if and only if Ext.A; G/ D 0 for every torsion-free A, andvice versa, A
is torsion-free if and only if Ext.A; G/ D 0 holds for every cotorsion G. Thereis
a very rich, fast growing literature available on this subject for modules. One of
the objects of research is the existence of covers and envelopes (or hulls) in the
respective classes; in our case: torsion-free covers and cotorsion hulls. The latter
has been settled above, so let us move to the question of torsion-free covers.
Needless to say, every group is an epic image of a torsion-free group, for instance,
of a free group. We wonder if there is a minimal one among such torsion-free groups.
Better yet, if there is a unique minimal one. Enochs [2] succeeded in showing that
there is always a minimal one that is even unique up to isomorphism. To make this
fact precise, we have to clarify what “minimality” should mean.
Let W F ! A be a homomorphism of the torsion-free group F into A. is called
a torsion-free cover of A if
(i) for every torsion-free group G and homomorphism W G ! A there is a map
W G ! F such that D , and
(ii) Ker contains no non-zero pure subgroup of F.
Condition (i) says, in more prosaic terms, that maps from torsion-free groups into
A must factor through . Hence it follows that the map has to be surjective (since