The Hilbert series of some graded algebras and the Poincaré series of some local rings.

In this paper, we investigate the Koszul behavior of finitely semi-graded algebras by the distributivity of some associated lattice of ideals. The Hilbert series, the Poincaré series, and the Yoneda algebra are defined for this class of algebras. Moreover, the point modules and the point functor are introduced for finitely semi-graded rings. Finitely semi-graded algebras and rings include many important examples of non- N -graded algebras coming from mathematical physics that play a very important role in mirror symmetry problems, and for these concrete examples, the Koszulity will be established, as well as the explicit computation of its Hilbert and Poincaré series.

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The Poincaré Series Of Stretched Cohen-Macaulay Rings

Canadian Journal of Mathematics ◽

10.4153/cjm-1980-094-0 ◽

1980 ◽

Vol 32 (5) ◽

pp. 1261-1265 ◽

Cited By ~ 6

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.

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A criterion for modules over Gorenstein local rings to have rational Poincaré series

Pacific Journal of Mathematics ◽

10.2140/pjm.2020.305.165 ◽

2020 ◽

Vol 305 (1) ◽

pp. 165-187

Author(s):

Anjan Gupta

Keyword(s):

Poincaré Series ◽

Local Rings ◽

Poincare Series

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Massey operations and the Poincaré series of certain local rings

Journal of Algebra ◽

10.1016/0021-8693(72)90143-3 ◽

1972 ◽

Vol 22 (2) ◽

pp. 223-232 ◽

Cited By ~ 26

Author(s):

Tor H Gulliksen

Keyword(s):

Poincaré Series ◽

Local Rings ◽

Poincare Series

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

Nagoya Mathematical Journal ◽

10.1017/nmj.2018.20 ◽

2018 ◽

Vol 238 ◽

pp. 47-85 ◽

Cited By ~ 1

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.

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