$G$-prime and $G$-primary $G$-ideals on $G$-schemes

TL;DR

This paper introduces and analyzes $G$-prime and $G$-primary ideals on $G$-schemes, establishing their fundamental properties, decompositions, and a generalized Matijevic-Roberts theorem for graded rings with specific properties.

Contribution

It defines $G$-prime and $G$-primary ideals on $G$-schemes and proves key properties including minimal decompositions and associated primes, extending classical theorems.

Findings

Existence of minimal $G$-primary decomposition.

Well-definedness of $G$-associated $G$-primes.

Generalization of Matijevic-Roberts theorem for graded rings.

Abstract

Let $G$ be a flat finite-type group scheme over a scheme $S$, and $X$ a noetherian $S$-scheme on which $G$-acts. We define and study $G$-prime and $G$-primary $G$-ideals on $X$ and study their basic properties. In particular, we prove the existence of minimal $G$-primary decomposition and the well-definedness of $G$-associated $G$-primes. We also prove a generalization of Matijevic-Roberts type theorem. In particular, we prove Matijevic-Roberts type theorem on graded rings for $F$-regular and $F$-rational properties.