On the localization theorem for F-pure rings

TL;DR

This paper addresses Grothendieck's localization problem for specific rings in tight closure theory, utilizing the relative Frobenius map to establish new localization results.

Contribution

It provides a solution to the localization problem for F-pure rings by leveraging the relative Frobenius map, advancing the understanding of tight closure theory.

Findings

Solved Grothendieck's localization problem for certain F-pure rings

Developed techniques involving the relative Frobenius map

Enhanced the theoretical framework of tight closure theory

Abstract

We solve Grothendieck's localization problem for certain class of rings arising from the tight closure theory. The idea of the proof depends heavily on the study of the relative version of the Frobenius map.