Cotorsion group

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In abelian group theory, an abelian group is said to be cotorsion if any extension of it by a torsion-free abelian group splits. If the group is M {\displaystyle M} {\displaystyle M}, this says that the Ext-group Ext Z ⁡ ( F , M ) {\displaystyle \operatorname {Ext} _{\mathbb {Z} }(F,M)} {\displaystyle \operatorname {Ext} _{\mathbb {Z} }(F,M)} is zero for all torsion-free groups F {\displaystyle F} {\displaystyle F}. Since any such F {\displaystyle F} {\displaystyle F} embeds into a direct sum of copies of Q {\displaystyle \mathbb {Q} } {\displaystyle \mathbb {Q} }, it suffices to check the condition on the group of rational numbers.[1]

Every divisible group or injective group (in particular the group of rational numbers Q {\displaystyle \mathbb {Q} } {\displaystyle \mathbb {Q} }) is cotorsion. For any two abelian groups A {\displaystyle A} {\displaystyle A} and B {\displaystyle B} {\displaystyle B}, the group of extensions Ext Z ⁡ ( A , B ) {\displaystyle \operatorname {Ext} _{\mathbb {Z} }(A,B)} {\displaystyle \operatorname {Ext} _{\mathbb {Z} }(A,B)} is cotorsion.[2]

Properties of cotorsion groups

The class of cotorsion groups is closed under extensions, direct products, and quotients.[1]

A countable cotorsion group is a direct sum of a divisible group and a bounded group, that is a group of bounded exponent.[1]

The Baer--Fomin Theorem states that a torsion group is cotorsion if and only if it is a direct sum of a divisible group and a bounded group.[3][4][5]

A torsion-free abelian group is cotorsion if and only if it is algebraically compact.[6][7] Such groups are precisely the direct summands of direct products of p {\displaystyle p} {\displaystyle p}-adic integers.[6]

Ulm subgroups of cotorsion groups are cotorsion and Ulm factors of cotorsion groups are algebraically compact.[8]

In ring theory

A right module M over a ring R is said to be a cotorsion module if Ext R 1 ⁡ ( F , M ) = 0 {\displaystyle \operatorname {Ext} _{R}^{1}(F,M)=0} {\displaystyle \operatorname {Ext} _{R}^{1}(F,M)=0} for all flat (right) modules F.[9] When R {\displaystyle R} {\displaystyle R} is the ring of integers Z {\displaystyle \mathbb {Z} } {\displaystyle \mathbb {Z} }, this reduces to the previous definition of cotorsion abelian groups.

The ring R {\displaystyle R} {\displaystyle R} is said to be (right) cotorsion if the regular module R R {\displaystyle R_{R}} {\displaystyle R_{R}} is cotorsion. [10]

References

  1. Fuchs, László (2015). "Chapter 9.6". Abelian Groups. Springer Monographs in Mathematics. Cham: Springer International Publishing. doi:10.1007/978-3-319-19422-6. ISBN 978-3-319-19421-9.
  2. Fuchs, László (2015). "Theorem 6.5". Abelian Groups. Springer Monographs in Mathematics. Cham: Springer International Publishing. p. 284. doi:10.1007/978-3-319-19422-6. ISBN 978-3-319-19421-9.
  3. Baer, Reinhold (1936). "The Subgroup of the Elements of Finite Order of an Abelian Group". Annals of Mathematics. 37 (4): 766–781. doi:10.2307/1968617. ISSN 0003-486X. JSTOR 1968617.
  4. Fomin, Sergei (1937). "Über periodische Untergruppen der unendlichen Abelschen Gruppen". Matematicheskii Sbornik. 2 (44): 1007–1009.
  5. Griffith, Phillip (2003-03-01). "The Baer splitting problem in the twentyfirst century". Illinois Journal of Mathematics. 47 (1–2): 1. doi:10.1215/ijm/1258488150. ISSN 0019-2082.
  6. Van Leeuwen, L. C. A (1969-01-01). "On torsion-free cotorsion groups". Indagationes Mathematicae (Proceedings). 72 (4): 388–393. doi:10.1016/1385-7258(69)90041-9. ISSN 1385-7258.
  7. Fuchs, L. (1963-03-01). "Notes on abelian groups. II". Acta Mathematica Academiae Scientiarum Hungarica. 11 (1): 117–125. doi:10.1007/BF02020629. ISSN 1588-2632.
  8. Fuchs, László (1959). "Notes on abelian groups. I.". Annales Universitatis Scientiarum Budapestinensis. 2: 5–23.
  9. Mao, Lixin; Ding, Nanqing (October 2006). "Cotorsion modules and relative pure-injectivity". Journal of the Australian Mathematical Society. 81 (2): 225–244. doi:10.1017/S1446788700015858. ISSN 1446-7887.
  10. Asensio, Pedro A. Guil; Herzog, Ivo (May 2004). "Left Cotorsion Rings". Bulletin of the London Mathematical Society. 36 (3): 303–309. doi:10.1112/S0024609303002844. ISSN 0024-6093.