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A Remark on Colimits

Published online by Cambridge University Press:  20 November 2018

Barbara L. Osofsky
Barbara L. Osofsky*
Affiliation:
Rutgers University, New Brunswick, New Jersey
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Let MR be a right module over the associative ring R (with 1). Assume one has an expression for M as a colimit (direct limit) of a system

over the (directed) poset D. A natural way to get M as a colimit of the family ﹛F∞ —” Fβ|∞, fβ ∈ E﹜ for some subset £ of D is to take E cofinal in D. However, if one is concerned about the cardinality of the set E, cofinal subsets may be too large. Let us look at a specific example. Lazard [3] has shown that any flat MR is a direct limit of finitely generated free R-modules. The cardinality of his indexing set depends on the cardinality of M.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 27 , Issue 3 , 01 June 1975 , pp. 496 - 499
Copyright
Copyright © Canadian Mathematical Society 1975

References

1. Kaplansky, I., Projective modules, Ann. of Math. 68 (1958), 372–377.Google Scholar
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3. Lazard, D., Sur les modules plats, C. R. Acad Sci. Paris 258 (1964), 6313–6316.Google Scholar
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6. Eilenberg, S. and Steenrod, N., Foundations of algebraic topology (Princeton University Press, Princeton, 1952).Google Scholar