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A new proof of non-Cohen-Macaulayness of Bertin's example

Takuma Seno

TL;DR

The paper addresses Bertin's classic example of a Noetherian UFD that is not Cohen-Macaulay and provides a new ring-theoretic proof based on Hilbert-series techniques and Gorenstein criteria. It leverages Stanley's palindromic condition for Hilbert series and Braun's invariant-theory results to derive a contradiction under the assumption of Cohen-Macaulayness in characteristic two. The main contribution is a structurally different proof of non-Cohen-Macaulayness for Bertin's invariant ring and a generalization to a broader class of permutation-group actions with no pseudoreflections. This work clarifies the role of Gorensteinness and modular phenomena in invariant theory and offers a general framework for detecting CM failures via Hilbert-series palindromicity.

Abstract

Bertin's example is famous as the first known Noetherian UFD that is not Cohen-Macaulay. In the example, she employed a ring of invariants and proved that the ring is not Cohen-Macaulay by calculating a homogeneous system of parameter and generators of it. In this paper, we give a new proof by arguments on ring theoretic properties.

A new proof of non-Cohen-Macaulayness of Bertin's example

TL;DR

The paper addresses Bertin's classic example of a Noetherian UFD that is not Cohen-Macaulay and provides a new ring-theoretic proof based on Hilbert-series techniques and Gorenstein criteria. It leverages Stanley's palindromic condition for Hilbert series and Braun's invariant-theory results to derive a contradiction under the assumption of Cohen-Macaulayness in characteristic two. The main contribution is a structurally different proof of non-Cohen-Macaulayness for Bertin's invariant ring and a generalization to a broader class of permutation-group actions with no pseudoreflections. This work clarifies the role of Gorensteinness and modular phenomena in invariant theory and offers a general framework for detecting CM failures via Hilbert-series palindromicity.

Abstract

Bertin's example is famous as the first known Noetherian UFD that is not Cohen-Macaulay. In the example, she employed a ring of invariants and proved that the ring is not Cohen-Macaulay by calculating a homogeneous system of parameter and generators of it. In this paper, we give a new proof by arguments on ring theoretic properties.

Paper Structure

This paper contains 4 sections, 15 theorems, 29 equations.

Table of Contents

  1. Introduction
  2. From commutative algebra
  3. From invariant theory
  4. Bertin's example

Key Result

Proposition 2.2

Let $R$ be an $\mathbb N$-graded Cohen-Macaulay ring with $R_0 = K$, and $x_1 , \dots x_d$ be an h.s.o.p. of $R$. Then, for $i=1 , \dots d$, where $a_i = \deg x_i$.

Theorems & Definitions (32)

  • Definition 2.1
  • Proposition 2.2
  • Definition 2.3
  • Theorem 2.4
  • Definition 2.5
  • Proposition 2.6
  • proof
  • Remark 2.7
  • Theorem 3.1
  • Corollary 3.2
  • ...and 22 more