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15 September 2021 Singularities in mixed characteristic via perfectoid big Cohen–Macaulay algebras
Linquan Ma, Karl Schwede
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Duke Math. J. 170(13): 2815-2890 (15 September 2021). DOI: 10.1215/00127094-2020-0082

Abstract

We utilize recent results of André and Gabber on the existence of weakly functorial, integral perfectoid big Cohen–Macaulay (BCM) algebras to study singularities of local rings in mixed characteristic. In particular, we introduce a mixed characteristic BCM-variant of rational/F-rational singularities, of log terminal/F-regular singularities, and of multiplier/test ideals of divisor pairs. We prove a number of results about these objects including a restriction theorem for perfectoid BCM multiplier/test ideals and deformation statements for perfectoid BCM-regular and BCM-rational singularities. As an application, we obtain results on the behavior of F-regular and F-rational singularities in arithmetic families.

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Linquan Ma. Karl Schwede. "Singularities in mixed characteristic via perfectoid big Cohen–Macaulay algebras." Duke Math. J. 170 (13) 2815 - 2890, 15 September 2021. https://doi.org/10.1215/00127094-2020-0082

Information

Received: 5 July 2018; Revised: 1 July 2020; Published: 15 September 2021
First available in Project Euclid: 27 August 2021

Digital Object Identifier: 10.1215/00127094-2020-0082

Subjects:
Primary: 13D22 , 14B05 , 14D10
Secondary: 11G99 , 13A35 , 14F18

Keywords: big Cohen–Macaulay , F-rational , F-regular , log terminal , multiplier ideal , perfectoid , rational , singularities , test ideal

Rights: Copyright © 2021 Duke University Press

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Vol.170 • No. 13 • 15 September 2021
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