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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

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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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5. Osofsky, B., Homological dimensions of modules, Conference Board of the Mathematical Sciences Regional Conference Series in Mathematics, Number 12 (American Math. Soc, Providence, 1973). Google Scholar
6. Eilenberg, S. and Steenrod, N., Foundations of algebraic topology (Princeton University Press, Princeton, 1952).Google Scholar