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.

A Gorenstein Ring with Larger Dilworth Number than Sperner Number

2000 ◽  
Vol 43 (1) ◽  
pp. 100-104 ◽  
Author(s):  
James S. Okon ◽  
J. Paul Vicknair
Keyword(s):  
Local Ring ◽  
Gorenstein Ring ◽  
Embedding Dimension ◽  
Gorenstein Rings

AbstractA counterexample is given to a conjecture of Ikeda by finding a class of Gorenstein rings of embedding dimension 3 with larger Dilworth number than Sperner number. The Dilworth number of is computed when A is an unramified principal Artin local ring.

scholarly journals Almost Gorenstein rings arising from fiber products

2020 ◽  
pp. 1-18
Author(s):  
Naoki Endo ◽  
Shiro Goto ◽  
Ryotaro Isobe
Keyword(s):  
Local Ring ◽  
The Other ◽  
Gorenstein Ring ◽  
Local Rings ◽  
Regular Local Ring ◽  
Gorenstein Rings ◽  
Fiber Product

Abstract The purpose of this paper is, as part of the stratification of Cohen–Macaulay rings, to investigate the question of when the fiber products are almost Gorenstein rings. We show that the fiber product $R \times _T S$ of Cohen–Macaulay local rings R, S of the same dimension $d>0$ over a regular local ring T with $\dim T=d-1$ is an almost Gorenstein ring if and only if so are R and S. In addition, the other generalizations of Gorenstein properties are also explored.

scholarly journals On the non-vanishing of Poincaré series

1980 ◽  
Vol 23 (2) ◽  
pp. 225-228 ◽  
Author(s):  
J. Lehner
Keyword(s):  
Complete System ◽  
Poincaré Series ◽  
Matrix Group ◽  
Poincare Series ◽  
Image Position

Let M = SL(2, Z) be the classical modular matrix group. One form of the Poincaré series on M ishere z ∈ H={z = x + iy: y >0}, q ≧ 2 and m ≧ 1 are integers, and the summation is over a complete system of matrices (ab: cd) in M with different lower row. The problem of the identical vanishing of the Poincaré series for different values of m and q goes back to Poincaré.

scholarly journals Motivic zeta functions for curve singularities

2010 ◽  
Vol 198 ◽  
pp. 47-75 ◽  
Author(s):  
J. J. Moyano-Fernández ◽  
W. A. Zúňiga-Galindo
Keyword(s):  
Functional Equation ◽  
Zeta Function ◽  
Local Ring ◽  
Algebraic Curve ◽  
Algebraic Curves ◽  
Zeta Functions ◽  
Poincaré Series ◽  
Poincare Series ◽  
Motivic Zeta Functions ◽  
Curve Singularities

AbstractLet X be a complete, geometrically irreducible, singular, algebraic curve defined over a field of characteristic p big enough. Given a local ring Op,x at a rational singular point P of X, we attached a universal zeta function which is a rational function and admits a functional equation if Op,x is Gorenstein. This universal zeta function specializes to other known zeta functions and Poincaré series attached to singular points of algebraic curves. In particular, for the local ring attached to a complex analytic function in two variables, our universal zeta function specializes to the generalized Poincaré series introduced by Campillo, Delgado, and Gusein-Zade.

scholarly journals The Poincaré series of a local ring IV

1971 ◽  
Vol 19 (1) ◽  
pp. 116-124 ◽  
Author(s):  
Jack Shamash
Keyword(s):  
Local Ring ◽  
Poincaré Series ◽  
Poincare 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 A note on the Poincaré series of a local ring

1969 ◽  
Vol 13 (2) ◽  
pp. 242-245 ◽  
Author(s):  
Tor H Gulliksen
Keyword(s):  
Local Ring ◽  
Poincaré Series ◽  
Poincare Series
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