A local-global correspondence for perfectoid purity

TL;DR

The paper introduces a global variant of perfectoid purity called lim-perfectoid splitting, establishing a correspondence with lim-perfectoid purity of Gorenstein rings and providing new examples of lim-perfectoid pure rings.

Contribution

It develops the concept of lim-perfectoid splitting and demonstrates its connection to lim-perfectoid purity, expanding the class of known lim-perfectoid pure rings.

Findings

Established a correspondence between lim-perfectoid splitting and lim-perfectoid purity.

Constructed new examples of lim-perfectoid pure rings beyond classical cases.

Abstract

We introduce (lim-)perfectoid splitting, which is a global variant of (lim-)perfectoid purity. Our main result establishes a correspondence between the lim-perfectoid splitting of projective schemes and the lim-perfectoid purity of their Gorenstein section rings. As an application, we construct a new supply of examples of lim-perfectoid pure rings that go beyond the previously known complete intersection or splinter-type cases.