Singularities in mixed characteristic via perfectoid big Cohen–Macaulay algebras

By Linquan Ma, Karl Schwede. 2018

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We utilize recent results of Andre 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.

Subfields: Algebra and Number Theory, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/3419833401086ab0b65917545824918a791de793