Annihilators of Artinian modules compatible with a Frobenius map

TL;DR

This paper develops a method to identify prime annihilators of submodules in Artinian modules over power series rings with Frobenius maps, extending previous algorithms to a broader class of modules.

Contribution

It introduces a new approach for finding prime annihilators in Artinian modules with Frobenius actions, generalizing earlier algorithms for specific submodules.

Findings

Provides a method to find all prime annihilators preserved by Frobenius

Extends existing algorithms to a wider class of modules

Solves dual problems related to radical annihilators in the F-finite case

Abstract

In this paper we consider Artinian modules over power series rings endowed with a Frobenius map. We describe a method for finding the set of all prime annihilators of submodules which are preserved by the given Frobenius map and on which the Frobenius map is not nilpotent. This extends the algorithm by Karl Schwede and the first author, which solved this problem for submodules of the injective hull of the residue field. The Matlis dual of this problem asks for the radical annihilators of quotients of free modules by submodules preserved by a given Frobenius near-splitting, and the same method solves this dual problem in the $F$-finite case.