Compatible ideals in Gorenstein rings

TL;DR

This paper investigates compatible ideals in certain Gorenstein rings and demonstrates that such ideals can be expressed as sums of images of maps from finite extensions, linking ideal theory with module extensions.

Contribution

It establishes a new characterization of compatible ideals in Gorenstein rings via module finite extensions and images of linear maps.

Findings

Compatible ideals are sums of images of maps from finite extensions.

The result applies to $Q$-Gorenstein, $F$-finite, and $F$-pure rings in prime characteristic.

Provides a new perspective on the structure of compatible ideals in Gorenstein rings.

Abstract

Suppose $R$ is a $\mathbb{Q}$-Gorenstein $F$-finite and $F$-pure ring of prime characteristic $p>0$. We show that if $I\subseteq R$ is a compatible ideal (with all $p^{-e}$-linear maps) then there exists a module finite extension $R\to S$ such that the ideal $I$ is the sum of images of all $R$-linear maps $S\to R$.