Vol. 7, No. 9, 2013

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Multiplicities associated to graded families of ideals

Steven Dale Cutkosky

Vol. 7 (2013), No. 9, 2059–2083

We prove that limits of multiplicities associated to graded families of ideals exist under very general conditions. Most of our results hold for analytically unramified equicharacteristic local rings with perfect residue fields. We give a number of applications, including a “ volume =  multiplicity” formula, generalizing the formula of Lazarsfeld and Mustaţă, and a proof that the epsilon multiplicity of Ulrich and Validashti exists as a limit for ideals in rather general rings, including analytic local domains. We prove a generalization of this to generalized symbolic powers of ideals proposed by Herzog, Puthenpurakal and Verma. We also prove an asymptotic “additivity formula” for limits of multiplicities and a formula on limiting growth of valuations, which answers a question posed by the author, Kia Dalili and Olga Kashcheyeva. Our proofs are inspired by a philosophy of Okounkov for computing limits of multiplicities as the volume of a slice of an appropriate cone generated by a semigroup determined by an appropriate filtration on a family of algebraic objects.

multiplicity, graded family of ideals, Okounkov body
Mathematical Subject Classification 2010
Primary: 13H15
Secondary: 14B05
Received: 20 July 2012
Revised: 11 October 2012
Accepted: 17 November 2012
Published: 18 December 2013
Steven Dale Cutkosky
Department of Mathematics
University of Missouri
Columbia, MO
United States