Some properties and applications of $F$-finite $F$-modules

TL;DR

This paper explores the properties of $F$-finite $F$-modules, demonstrating their applications in understanding Frobenius maps, near-splittings, and morphisms, with new results on their structure and compatibility.

Contribution

It introduces new insights into the structure of morphisms and Frobenius near-splittings of $F$-finite $F$-modules, extending previous results to broader contexts.

Findings

Morphisms of $F$-finite $F$-modules have a simple form.

Certain Frobenius near-splittings have finitely many compatible submodules.

Generalization of a result to the non-$F$-finite case.

Abstract

The purpose of this paper is to describe several applications of finiteness properties of $F$-finite $F$-modules recently discovered by M. Hochster to the study of Frobenius maps on injective hulls, Frobenius near-splittings and to the nature of morphisms of $F$-finite $F$-modules. Among the results in the paper we show that morphisms of $F$-finite $F$-modules have a particularly simple form, and we show that certain Frobenius near-splittings have finitely many compatible submodules, thus generalizing a result of M. Blickle and G. B\"ockle to the non-$F$-finite case.