Perfectoid multiplier/test ideals in regular rings and bounds on symbolic powers

Abstract

Using perfectoid algebras, we introduce a mixed characteristic analog of the multiplier ideal, respectively test ideal, from characteristic zero, respectively $p>0$, in the case of a regular ambient ring. We prove several properties about this ideal such as subadditivity. We then use these techniques to derive a uniform bound on the growth of symbolic powers of radical ideals in all excellent regular rings. The analogous result was shown in equal characteristic by Ein-Lazarsfeld-Smith and Hochster-Huneke.

35 pages, numerous clarified proofs and expositional improvements throughout. To appear in Inventiones Mathematicae

References in corpus (1)

• Desingularization of quasi-excellent schemes in characteristic zero

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