An algorithm for computing compatibly Frobenius split subvarieties

TL;DR

This paper presents an algorithm to compute all ideals compatible with a given Frobenius-related map in prime characteristic rings, with implementations in Macaulay2, extending to non-surjective cases.

Contribution

It introduces a novel algorithm for finding all $\, ext{ extphi}$-compatible ideals in rings of prime characteristic, including non-surjective maps, with practical implementation.

Findings

Algorithm successfully computes compatible ideals.

Implementation available in Macaulay2.

Extends to non-surjective maps.

Abstract

Let $R$ be a ring of prime characteristic $p$, and let $F^e_* R$ denote $R$ viewed as an $R$-module via the $e$th iterated Frobenius map. Given a surjective map $\phi : F^e_* R \to R$ (for example a Frobenius splitting), we exhibit an algorithm which produces all the $\phi$-compatible ideals. We also explore a variant of this algorithm under the hypothesis that $\phi$ is not necessarily a Frobenius splitting (or even surjective). This algorithm, and the original, have been implemented in Macaulay2.