On a positive equicharacteristic variant of the $p$-curvature conjecture

TL;DR

This paper formulates and proves a weak version of the $p$-curvature conjecture in positive characteristic and demonstrates a counterexample to its strong form, advancing understanding in algebraic geometry.

Contribution

It introduces a weak form of the $p$-curvature conjecture in equal characteristic $p>0$ and provides a counterexample to the strong form, clarifying the conjecture's limitations.

Findings

Proved a weak form of the $p$-curvature conjecture in characteristic p

Constructed a counterexample to the strong form of the conjecture

Enhanced understanding of the conjecture's scope in algebraic geometry

Abstract

Our aim is to formulate and prove a weak form in equal characteristic $p>0$ of the $p$-curvature conjecture. We also show the existence of a counterexample to a strong form of it.