Global generation of test ideals in mixed characteristic and applications

2021/06/27 by Hacon, Christopher, Lamarche, Alicia, Schwede, Karl · 3 citations
#13A35 #14B05 #14C20 #14F17 #14F18 #Algebraic Geometry (math.AG) #Commutative Algebra (math.AC) #FOS: Mathematics

paper · doi:10.48550/arxiv.2106.14329

Abstract

Suppose that X is an integral scheme (quasi-)projective over a complete local ring of mixed characteristic. Using ideas of Takamatsu-Yoshikawa and Bhatt-Ma-et. al, we define a notion of a +-test ideal on X, including for divisors and linear series. We obtain global generation results in this setting that generalize the well known global generation results obtained via multiplier ideal sheaf techniques in characteristic 0 and via test ideals in characteristic p>0. We also obtain applications to the order of vanishing of linear series and to the diminished base locus in mixed characteristic similar to results of Ein-Lazarsfeld-Mustata-Nakamaye-Popa, Nakayama, and Mustata in the equal characteristic case.

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• A Criterion for Perfectoid Purity and the Rationality of Thresholds
• Quasi-F-splitting and smooth weak del Pezzo surfaces in mixed characteristic
• Bounds on the plus-pure thresholds of some hypersurfaces in (ramified) regular rings

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