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The Poincaré Series Of Stretched Cohen-Macaulay Rings

Published online by Cambridge University Press:  20 November 2018

Judith D. Sally
Judith D. Sally*
Affiliation:
Northwestern University, Evanston, Illinois
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There are relatively few classes of local rings (R, m) for which the question of the rationality of the Poincaré series

where k = R/m, has been settled. (For an example of a local ring with non-rational Poincaré series see the recent paper by D. Anick, “Construction of loop spaces and local rings whose Poincaré—Betti series are nonrational”, C. R. Acad. Sc. Paris 290 (1980), 729-732.) In this note, we compute the Poincaré series of a certain family of local Cohen-Macaulay rings and obtain, as a corollary, the rationality of the Poincaré series of d-dimensional local Gorenstein rings (R, m) of embedding dimension at least e + d – 3, where e is the multiplicity of R. It follows that local Gorenstein rings of multiplicity at most five have rational Poincaré series.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 32 , Issue 5 , 01 October 1980 , pp. 1261 - 1265
Copyright
Copyright © Canadian Mathematical Society 1980

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