Rocky Mountain Journal of Mathematics

On the rationality of Poincaré series of Gorenstein algebras via Macaulay's correspondence

Gianfranco Casnati, Joachim Jelisiejew, and Roberto Notari

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Abstract

Let $A$ be a local Artinian Gorenstein algebra with maximal ideal $\fM $, \[P_A(z) := \sum _{p=0}^{\infty } (\tor _p^A(k,k))z^p \] its Poicar\'{e} series. We prove that $P_A(z)$ is rational if either $\dim _k({\fM ^2/\fM ^3}) \leq 4 $ and $ \dim _k(A) \leq 16,$ or there exist $m\leq 4$ and $c$ such that the Hilbert function $H_A(n)$ of $A$ is equal to $ m$ for $n\in [2,c]$ and equal to $1$ for $n =c+1$. The results are obtained due to a decomposition of the apolar ideal $\Ann (F)$ when $F=G+H$ and $G$ and $H$ belong to polynomial rings in different variables.

Article information

Source
Rocky Mountain J. Math., Volume 46, Number 2 (2016), 413-433.

Dates
First available in Project Euclid: 26 July 2016

Permanent link to this document
https://projecteuclid.org/euclid.rmjm/1469537470

Digital Object Identifier
doi:10.1216/RMJ-2016-46-2-413

Mathematical Reviews number (MathSciNet)
MR3529076

Zentralblatt MATH identifier
06624467

Subjects
Primary: 13D40: Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series
Secondary: 13H10: Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.) [See also 14M05]

Keywords
Artinian Gorenstein local algebra rational Poincaré series

Citation

Casnati, Gianfranco; Jelisiejew, Joachim; Notari, Roberto. On the rationality of Poincaré series of Gorenstein algebras via Macaulay's correspondence. Rocky Mountain J. Math. 46 (2016), no. 2, 413--433. doi:10.1216/RMJ-2016-46-2-413. https://projecteuclid.org/euclid.rmjm/1469537470


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