Poincare Series of Modules Over Local Rings

1978 ◽  
Vol 72 (1) ◽  
pp. 6 ◽  
Author(s):  
Gerson Levin
Keyword(s):  
Poincaré Series ◽  
Local Rings ◽  
Poincare Series

The Poincaré Series Of Stretched Cohen-Macaulay Rings

1980 ◽  
Vol 32 (5) ◽  
pp. 1261-1265 ◽  
Author(s):  
Judith D. Sally
Keyword(s):  
Local Ring ◽  
Poincaré Series ◽  
Local Rings ◽  
Embedding Dimension ◽  
Loop Spaces ◽  
Gorenstein Rings ◽  
Poincare Series ◽  
Image Position

There are relatively few classes of local rings (R, m) for which the question of the rationality of the Poincaré serieswhere 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.

scholarly journals DETECTING KOSZULNESS AND RELATED HOMOLOGICAL PROPERTIES FROM THE ALGEBRA STRUCTURE OF KOSZUL HOMOLOGY

2018 ◽  
Vol 238 ◽  
pp. 47-85 ◽  
Author(s):  
AMANDA CROLL ◽  
ROGER DELLACA ◽  
ANJAN GUPTA ◽  
JUSTIN HOFFMEIER ◽  
VIVEK MUKUNDAN ◽  
...  
Keyword(s):  
Poincaré Series ◽  
Koszul Complex ◽  
Local Rings ◽  
Common Denominator ◽  
Algebra Structure ◽  
Koszul Algebra ◽  
Multiplicative Structure ◽  
Poincare Series ◽  
Koszul Homology ◽  
Finitely Generated Modules

Let $k$ be a field and $R$ a standard graded $k$-algebra. We denote by $\operatorname{H}^{R}$ the homology algebra of the Koszul complex on a minimal set of generators of the irrelevant ideal of $R$. We discuss the relationship between the multiplicative structure of $\operatorname{H}^{R}$ and the property that $R$ is a Koszul algebra. More generally, we work in the setting of local rings and we show that certain conditions on the multiplicative structure of Koszul homology imply strong homological properties, such as existence of certain Golod homomorphisms, leading to explicit computations of Poincaré series. As an application, we show that the Poincaré series of all finitely generated modules over a stretched Cohen–Macaulay local ring are rational, sharing a common denominator.

scholarly journals Poincaré series of modules over local rings of small embedding codepth or small linking number

1988 ◽  
Vol 118 (1) ◽  
pp. 162-204 ◽  
Author(s):  
Luchezar L Avramov ◽  
Andrew R Kustin ◽  
Matthew Miller
Keyword(s):  
Poincaré Series ◽  
Local Rings ◽  
Poincare Series ◽  
Linking Number

scholarly journals Poincaré series of modules over compressed Gorenstein local rings

2014 ◽  
Vol 259 ◽  
pp. 421-447 ◽  
Author(s):  
Maria Evelina Rossi ◽  
Liana M. Şega
Keyword(s):  
Poincaré Series ◽  
Local Rings ◽  
Poincare Series
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