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

By Linquan Ma, Karl Schwede. Inventiones Mathematicae, 2017

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Using perfectoid algebras we introduce a mixed characteristic analog of the multiplier ideal, respectively test ideal, from characteristic zero, respectively $$p > 0$$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.

Subfields: Algebra and Number Theory, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/d06e7b34f7cdcd27ae640dacd3e4953d3351b6b4