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An [[Abelian group|Abelian group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110460/c1104601.png" /> is a cotorsion group if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110460/c1104602.png" /> for all torsion-free Abelian groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110460/c1104603.png" />, i.e. every extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110460/c1104604.png" /> by a torsion-free group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110460/c1104605.png" /> splits (cf. also [[Extension of a group|Extension of a group]]). For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110460/c1104606.png" /> to be a cotorsion group it suffices to assume that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110460/c1104607.png" />. The importance of cotorsion groups lies in the facts that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110460/c1104608.png" /> is a cotorsion group for all Abelian groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110460/c1104609.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110460/c11046010.png" />, and that they have several nice features.
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Cotorsion groups can also be characterized by their injective property with respect to those exact sequences <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110460/c11046011.png" /> of Abelian groups which split when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110460/c11046012.png" /> is restricted to its torsion part <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110460/c11046013.png" /> (cf. also [[Exact sequence|Exact sequence]]).
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An [[Abelian group|Abelian group]]  $  C $
 +
is a cotorsion group if  $  { \mathop{\rm Ext} } ( G,C ) = 0 $
 +
for all [[Torsion-free group|torsion-free]] Abelian groups  $  G $,
 +
i.e. every extension of  $  C $
 +
by a torsion-free group  $  G $
 +
splits (cf. also [[Extension of a group|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|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.
 
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110460/c11046014.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110460/c11046015.png" /> (cf. also [[Divisible group|Divisible group]]); and 2) algebraically compact groups, like finite groups and the additive group of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110460/c11046016.png" />-adic integers (for any prime <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110460/c11046017.png" />); cf. also [[Compact group|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.
+
Examples of cotorsion groups are: 1) divisible (i.e., injective) Abelian groups, like $  \mathbf Q $,  
 +
$  \mathbf Z ( p  ^  \infty  ) $(
 +
cf. also [[Divisible group|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|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110460/c11046018.png" />, there is a natural isomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110460/c11046019.png" />. This fact is relevant in showing that every Abelian group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110460/c11046020.png" /> can be imbedded as a subgroup in a cotorsion group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110460/c11046021.png" /> such that the cokernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110460/c11046022.png" /> is torsion-free and divisible. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110460/c11046023.png" /> is reduced, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110460/c11046024.png" /> can be chosen as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110460/c11046025.png" />; this is the smallest cotorsion group in which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110460/c11046026.png" /> can be imbedded in this manner. It is called the cotorsion hull of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110460/c11046027.png" />, and is unique up to isomorphism over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110460/c11046028.png" />.
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110460/c11046029.png" /> between the class of reduced torsion groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110460/c11046030.png" /> 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 [[#References|[a2]]] states that every cotorsion group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110460/c11046031.png" /> is a direct sum of three groups: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110460/c11046032.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110460/c11046033.png" /> is a divisible group, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110460/c11046034.png" /> is a reduced torsion-free algebraically compact group and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110460/c11046035.png" /> is an adjusted cotorsion group. Such a decomposition of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110460/c11046036.png" /> is unique up to isomorphism.
+
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 [[#References|[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" .
 
Some authors use  "cotorsion"  as  "cotorsion in the above sense + reduced" .

Latest revision as of 17:31, 5 June 2020


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)
How to Cite This Entry:
Cotorsion group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cotorsion_group&oldid=18282
This article was adapted from an original article by L. Fuchs (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article