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A Remark on Colimits

  • Barbara L. Osofsky (a1)

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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 ﹛FFβ|∞, fβE﹜ 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.

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References

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1. Kaplansky, I., Projective modules, Ann. of Math. 68 (1958), 372377.
2. Kaplansky, I., Fields and rings (University of Chicago Press, Chicago, 1969).
3. Lazard, D., Sur les modules plats, C. R. Acad Sci. Paris 258 (1964), 63136316.
4. Osofsky, B., Upper bounds on homologuai dimensions, Nagoya Math. J. 32 (1968), 315322.
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).
6. Eilenberg, S. and Steenrod, N., Foundations of algebraic topology (Princeton University Press, Princeton, 1952).
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A Remark on Colimits

  • Barbara L. Osofsky (a1)

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