On cosmall Abelian groups

B. Goldsmith; O. Kolman

doi:10.1016/j.jalgebra.2007.05.009

On cosmall Abelian groups

B. Goldsmith, O. Kolman

Research output: Contribution to journal › Article › peer-review

Abstract

It is a well-known homological fact that every Abelian group G has the property that Hom (G, -) commutes with direct products. Here we investigate the 'dual' property: an Abelian group G is said to be cosmall if Hom (-, G) commutes with direct products. We show that cosmall groups are cotorsion-free and that no group of cardinality less than a strongly compact cardinal can be cosmall. In particular, if there is a proper class of strongly compact cardinals, then there are no cosmall groups.

Original language	English
Pages (from-to)	510-518
Number of pages	9
Journal	Journal of Algebra
Volume	317
Issue number	2
DOIs	https://doi.org/10.1016/j.jalgebra.2007.05.009
Publication status	Published - 15 Nov 2007
Externally published	Yes

Keywords

Abelian groups
Homological algebra
Set theory

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10.1016/j.jalgebra.2007.05.009

https://arrow.tudublin.ie/scschmatart/40Licence: CC BY-NC-SA

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AB - It is a well-known homological fact that every Abelian group G has the property that Hom (G, -) commutes with direct products. Here we investigate the 'dual' property: an Abelian group G is said to be cosmall if Hom (-, G) commutes with direct products. We show that cosmall groups are cotorsion-free and that no group of cardinality less than a strongly compact cardinal can be cosmall. In particular, if there is a proper class of strongly compact cardinals, then there are no cosmall groups.

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KW - Set theory

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On cosmall Abelian groups

Abstract

Keywords

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