Formally regular rings and descent of regularity

TL;DR

This paper extends the concept of regular local rings to non-noetherian rings like valuation and perfectoid rings, incorporating topology and proving descent theorems for regularity in this broader context.

Contribution

It introduces a new definition of regularity for non-noetherian rings that includes valuation and perfectoid rings, and proves descent results extending prior work.

Findings

Extended regularity to non-noetherian rings including valuation and perfectoid rings.

Proved descent of regularity along flat homomorphisms with finite flat dimension.

Generalized descent results from perfectoid rings to a broader class of rings.

Abstract

Valuation rings and perfectoid rings are examples of (usually non-noetherian) rings that behave in some sense like regular rings. We give and study an extension of the concept of regular local rings to non-noetherian rings so that it includes valuation and perfectoid rings and it is related to Grothendieck's definition of formal smoothness as in the noetherian case. For that, we have to take into account the topologies. We prove a descent theorem for regularity along flat homomorphisms (in fact for homomorphisms of finite flat dimension), extending some known results from the noetherian to the non-noetherian case, as well as generalizing some recent results in the non-noetherian case, such as the descent of regularity from perfectoid rings by B. Bhatt, S. Iyengar and L. Ma.