Singularities in mixed characteristic via perfectoid big Cohen-Macaulay algebras

2018/06/25 by Ma, Linquan, Schwede, Karl · 5 citations
#11G99 #13A35 #13D22 #14B05 #14D10 #14F18 #Algebraic Geometry (math.AG) #Commutative Algebra (math.AC) #FOS: Mathematics

paper · doi:10.48550/arxiv.1806.09567

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.

Cited by

• Graded perfectoid rings
• Measuring birational derived splinters
• Quasi-F-splitting and smooth weak del Pezzo surfaces in mixed characteristic
• The Briançon-Skoda theorem for pseudo-rational and Du Bois singularities and uniformity in excellent rings
• Bounds on the plus-pure thresholds of some hypersurfaces in (ramified) regular rings

Related

• Singularities in mixed characteristic. The perfectoid approach
• Integral perfectoid big Cohen-Macaulay algebras via André's theorem
• Perfectoid signature, perfectoid Hilbert-Kunz multiplicity, and an application to local fundamental groups
• Homological Detection by Perfectoid Algebras
• A mixed characteristic analogue of the perfection of rings and its almost Cohen-Macaulay property