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FORMALLY REGULAR RINGS AND DESCENT OF REGULARITY

Part of: Local rings and semilocal rings Ring extensions and related topics Local theory Commutative algebra: Homological methods

Published online by Cambridge University Press:  02 January 2024

Javier Majadas Open the ORCID record for Javier Majadas [Opens in a new window] ,
Samuel Alvite Open the ORCID record for Samuel Alvite [Opens in a new window]  and
Nerea G. Barral
Javier Majadas*
Affiliation:
Departamento de Matemáticas, Facultad de Matemáticas, Universidad de Santiago de Compostela, E15782 Santiago de Compostela, Spain
Samuel Alvite
Affiliation:
Departamento de Matemáticas, Facultad de Matemáticas, Universidad de Santiago de Compostela, E15782 Santiago de Compostela, Spain (samuelalvite@gmail.com)
Nerea G. Barral
Affiliation:
Departamento de Matemáticas, Facultad de Matemáticas, Universidad de Santiago de Compostela, E15782 Santiago de Compostela, Spain (nereabarral@gmail.com)
*
j.majadas@usc.es
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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.

Keywords

regular ringperfectoid ringformal smoothness

MSC classification

Primary: 14B10: Infinitesimal methods 13H05: Regular local rings 13B40: Étale and flat extensions; Henselization; Artin approximation
Secondary: 13D03: (Co)homology of commutative rings and algebras (e.g., Hochschild, André-Quillen, cyclic, dihedral, etc.)

Information

Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 23 , Issue 5 , September 2024 , pp. 2279 - 2318
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Copyright
© The Author(s), 2024. Published by Cambridge University Press

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