# 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.