Cotorsion group

An Abelian group $ C $ is a cotorsion group if $ { \mathop{\rm Ext} } ( G,C ) = 0 $ for all torsion-free Abelian groups $ G $, i.e. every extension of $ C $ by a torsion-free group $ G $ splits (cf. also Extension of a group). For $ C $ to be a cotorsion group it suffices to assume that $ { \mathop{\rm Ext} } ( \mathbf Q,C ) = 0 $. The importance of cotorsion groups lies in the facts that $ { \mathop{\rm Ext} } ( B,A ) $ is a cotorsion group for all Abelian groups $ A $ and $ B $, and that they have several nice features.

Cotorsion groups can also be characterized by their injective property with respect to those exact sequences $ 0 \rightarrow G \rightarrow H \rightarrow K \rightarrow 0 $ of Abelian groups which split when $ K $ is restricted to its torsion part $ t ( K ) $( cf. also Exact sequence).

Epimorphic images of cotorsion groups are cotorsion, and so are the extensions of cotorsion groups by cotorsion groups. A direct product of groups is cotorsion if and only if each summand is cotorsion.

Examples of cotorsion groups are: 1) divisible (i.e., injective) Abelian groups, like $ \mathbf Q $, $ \mathbf Z ( p ^ \infty ) $( cf. also Divisible group); and 2) algebraically compact groups, like finite groups and the additive group of the $ p $- adic integers (for any prime $ p $); cf. also Compact group. A torsion Abelian group is cotorsion if and only if it is a direct sum of a divisible group and a bounded group (the Baer–Fomin theorem), and a torsion-free Abelian group is cotorsion exactly if it is algebraically compact. Ulm subgroups of cotorsion groups are cotorsion, and the Ulm factors of cotorsion groups are algebraically compact.

For a reduced cotorsion group $ C $, there is a natural isomorphism $ { \mathop{\rm Ext} } ( \mathbf Q/ \mathbf Z,C ) \cong C $. This fact is relevant in showing that every Abelian group $ A $ can be imbedded as a subgroup in a cotorsion group $ C $ such that the cokernel $ C/A $ is torsion-free and divisible. If $ A $ is reduced, then $ C $ can be chosen as $ { \mathop{\rm Ext} } ( \mathbf Q/ \mathbf Z,A ) $; this is the smallest cotorsion group in which $ A $ can be imbedded in this manner. It is called the cotorsion hull of $ A $, and is unique up to isomorphism over $ A $.

A cotorsion group is said to be adjusted if it is reduced and contains no non-trivial torsion-free summand. The cotorsion hull of a reduced torsion group is adjusted, and the correspondence $ T \mapsto { \mathop{\rm Ext} } ( \mathbf Q/ \mathbf Z,T ) $ between the class of reduced torsion groups $ T $ and the class of adjusted cotorsion groups is a bijection; its inverse is the formation of the torsion part. As a consequence, the classification of reduced torsion groups and that of adjusted cotorsion groups are equivalent problems. The Harrison structure theorem [a2] states that every cotorsion group $ C $ is a direct sum of three groups: $ C = D \oplus G \oplus A $, where $ D $ is a divisible group, $ G $ is a reduced torsion-free algebraically compact group and $ A $ is an adjusted cotorsion group. Such a decomposition of $ C $ is unique up to isomorphism.

Some authors use "cotorsion" as "cotorsion in the above sense + reduced" .

A general reference is [a1]. See [a3] for a generalization to cotorsion modules over commutative domains.

See also Cotorsion-free group.

References

[a1]	L. Fuchs, "Infinite abelian groups" , 1 , Acad. Press (1970)
[a2]	D.K. Harrison, "Infinite abelian groups and homological methods" Ann. of Math. , 69 (1959) pp. 366–391
[a3]	E. Matlis, "Cotorsion modules" , Memoirs , 49 , Amer. Math. Soc. (1964)