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An infinite series of Gorenstein local algebras failing the affine homogeneity property

Roman Avdeev, Yulia Zaitseva

Abstract

We provide an infinite series of commutative finite-dimensional Gorenstein local algebras $A_n$ for $n \ge 2$. We give an elementary proof that the maximal ideal of every algebra $A_n$ possesses a one-dimensional subspace that is different from the socle and invariant under the automorphism group of $A_n$. The latter implies that the algebras $A_n$ fail the affine homogeneity property. We also discuss some consequences concerning additive actions on projective hypersurfaces, related to the generalized Hassett-Tschinkel correspondence for these algebras.

An infinite series of Gorenstein local algebras failing the affine homogeneity property

Abstract

We provide an infinite series of commutative finite-dimensional Gorenstein local algebras for . We give an elementary proof that the maximal ideal of every algebra possesses a one-dimensional subspace that is different from the socle and invariant under the automorphism group of . The latter implies that the algebras fail the affine homogeneity property. We also discuss some consequences concerning additive actions on projective hypersurfaces, related to the generalized Hassett-Tschinkel correspondence for these algebras.

Paper Structure

This paper contains 2 sections, 8 theorems, 6 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The algebras A_n

Key Result

Theorem 1

Suppose that $n\geqslant 2$ and $\mathop{\mathrm{char}}\nolimits \mathbb{K}$ is either zero or coprime to both $n$ and $n-1$. Then the one-dimensional subspace $\langle y^{2n+1}\rangle \subseteq \mathfrak{m}_n$ is invariant with respect to the action of the automorphism group $\mathop{\mathrm{Aut}}\

Figures (1)

  • Figure 1: The algebra $A_n$

Theorems & Definitions (10)

  • Theorem 1
  • Corollary 1
  • Corollary 2
  • Corollary 3
  • Theorem 2
  • Proposition 1
  • proof
  • Corollary 4
  • proof
  • Proposition 2