Algebra & Number Theory

Multiplicities associated to graded families of ideals

Steven Cutkosky

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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.

Article information

Algebra Number Theory, Volume 7, Number 9 (2013), 2059-2083.

Received: 20 July 2012
Revised: 11 October 2012
Accepted: 17 November 2012
First available in Project Euclid: 20 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 13H15: Multiplicity theory and related topics [See also 14C17]
Secondary: 14B05: Singularities [See also 14E15, 14H20, 14J17, 32Sxx, 58Kxx]

multiplicity graded family of ideals Okounkov body


Cutkosky, Steven. Multiplicities associated to graded families of ideals. Algebra Number Theory 7 (2013), no. 9, 2059--2083. doi:10.2140/ant.2013.7.2059.

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