Hochster's small MCM conjecture for three-dimensional weakly F-split rings

TL;DR

This paper proves Hochster's small MCM conjecture for three-dimensional complete F-pure rings by extending the notion of F-purity to weakly F-split rings, and conjectures this property for all complete rings.

Contribution

It establishes the conjecture for three-dimensional complete F-pure rings using a new criterion based on weakly F-split rings.

Findings

Proves Hochster's small MCM conjecture for 3D complete F-pure rings.

Introduces weakly F-split rings as a key concept.

Conjectures all complete rings are weakly F-split.

Abstract

We prove Hochster's small MCM conjecture for three-dimensional complete F-pure rings. We deduce this from a more general criterion, and show that only a weakening of the notion of F-purity is needed, to wit, being weakly F-split. We conjecture that any complete ring is weakly F-split.