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
“”
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.
Keywords
multiplicity, graded family of ideals, Okounkov body
