Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Anjan Gupta

Abstract

We prove that modules over an Artinian Gorenstein local ring $R$ have rational Poincaré series sharing a common denominator if $R/\soc(R)$ is a Golod ring. If $R$ is a Gorenstein local ring with square of the maximal ideal being generated by at most two elements, we show that modules over $R$ have rational Poincaré series sharing a common denominator. By a result of \c Sega, it follows that $R$ satisfies the Auslander-Reiten conjecture. We provide a different proof of a result of Rossi and \c Sega concerning rationality of Poincaré series of modules over compressed Gorenstein local rings. We also give a new proof of the fact that modules over Gorenstein local rings of codepth at most three have rational Poincaré series sharing a common denominator, which is originally due to Avramov, Kustin and Miller.

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

Abstract

We prove that modules over an Artinian Gorenstein local ring have rational Poincaré series sharing a common denominator if is a Golod ring. If is a Gorenstein local ring with square of the maximal ideal being generated by at most two elements, we show that modules over have rational Poincaré series sharing a common denominator. By a result of \c Sega, it follows that satisfies the Auslander-Reiten conjecture. We provide a different proof of a result of Rossi and \c Sega concerning rationality of Poincaré series of modules over compressed Gorenstein local rings. We also give a new proof of the fact that modules over Gorenstein local rings of codepth at most three have rational Poincaré series sharing a common denominator, which is originally due to Avramov, Kustin and Miller.
Paper Structure (8 sections, 20 theorems, 22 equations)

This paper contains 8 sections, 20 theorems, 22 equations.

Table of Contents

  1. introduction
  2. The main result
  3. Tate resolutions
  4. Golod algebras
  5. Proof of Theorem \ref{['intro2']}
  6. Stretched and almost stretched rings
  7. Proof of Theorem \ref{['intro3.5']}
  8. Revisiting known results

Key Result

Theorem I

Let $R$ be an Artinian Gorenstein local ring of embedding dimension $n \geq 2$ such that $R/ \mathop{\mathrm{socle}}\limits(R)$ is a Golod ring. Let $\eta : Q \rightarrow R$ be a minimal Cohen presentation, $\mathfrak{n}$ denote the maximal ideal of $Q$ and $I = \ker(\eta) \subset \mathfrak{n}^2$. T

Theorems & Definitions (36)

  • Theorem I
  • Theorem II
  • Theorem 2.1
  • Theorem 2.2
  • proof
  • Lemma 3.1
  • proof
  • Definition 3.2
  • Theorem 3.3
  • Theorem 3.4
  • ...and 26 more