On $G$-invariant Gorenstein ideals

TL;DR

This paper proves that under certain conditions, the Gorenstein property of a $G$-invariant quotient ring is preserved when passing to the ring of invariants, extending understanding of invariant theory in algebraic geometry.

Contribution

It establishes that if a $G$-invariant ideal yields a Gorenstein quotient, then the invariant quotient also remains Gorenstein, assuming no non-trivial one-dimensional representations of $G$ over $k$.

Findings

Gorenstein property is preserved under invariants for certain group actions.

No non-trivial one-dimensional representations condition is crucial.

Results apply to finite groups acting linearly on polynomial rings.

Abstract

Let $k$ be a field and $G \subseteq Gl_n(k)$ be a finite group with $|G|^{-1} \in k$. Let $G$ act linearly on $A = k[X_1, \ldots, X_n]$ and let $A^G$ be the ring of invariant's. Suppose there does not exist any non-trivial one-dimensional representation of $G$ over $k$. Then we show that if $Q$ is a $G$-invariant homogeneous ideal of $A$ such that $A/Q$ is a Gorenstein ring then $A^G/Q^G$ is also a Gorenstein ring.