F-signature exists

Abstract

Suppose R is a Noetherian local ring with prime characteristic p>0. In this article, we show the existence of a local numerical invariant, called the F-signature, which roughly characterizes the asymptotic growth of the number of splittings of the iterates of the Frobenius endomorphism of R. This invariant was first formally defined by C. Huneke and G. Leuschke and has previously been shown to exist only in special cases. The proof of our main result is based on the development of certain uniform Hilbert-Kunz estimates of independent interest. Additionally, we analyze the behavior of the F-signature under finite ring extensions and recover explicit formulae for the F-signatures of finite quotient singularities.

19 pages. Substantially updated from a version which was circulated on a limited basis in the fall of 2010

References in corpus (3)

• Two theorems about maximal Cohen--Macaulay modules
• F-signature of pairs and the asymptotic behavior of Frobenius splittings
• When does the F-signature exist?

Cited by in corpus (44)

• A survey of test ideals
• Fundamental groups of $F$-regular singularities via $F$-signature
• F-signature and Hilbert-Kunz Multipicity: a combined approach and comparison
• Finite torsors over strongly $F$-regular singularities
• Valuations and Frobenius
• Upper semi-continuity of the Hilbert-Kunz multiplicity
• Conic divisorial ideals of Hibi rings and their applications to non-commutative crepant resolutions
• Frobenius Splitting in Commutative Algebra
• F-Signature of Affine Toric Varieties
• Equimultiplicity in Hilbert-Kunz theory
• F-signature of pairs and the asymptotic behavior of Frobenius splittings
• The number of torsion divisors in a strongly $F$-regular ring is bounded by the reciprocal of $F$-signature
• $F$-signature under birational morphisms
• Stability and deformation of F-singularities
• Seshadri Constants and Fujita's Conjecture via Positive Characteristic Methods
• Bertini Theorems for $F$-signature and Hilbert-Kunz multiplicity
• Finite F-type and F-abundant modules
• Uniform Symbolic Topologies in Normal Toric Rings
• On the behavior of $F$-signatures, splitting primes, and test modules under finite covers
• Hilbert-Kunz multiplicity of the powers of an ideal
• Relating F-Signature and F-Splitting Ratio of Pairs Using Left-Derivatives
• Local cohomology bounds and test ideals
• Hilbert-Kunz functions of surface rings of type ADE
• Global Frobenius Betti numbers and F-splitting ratio
• On the existence of $F$-thresholds and related limits
• $F$-singularities: applications of characteristic $p$ methods to singularity theory
• The Symmetric signature of cyclic quotient singularities
• Globalizing F-invariants
• Local Okounkov bodies and limits in prime characteristic
• F-singularities via alterations
• Decomposition Theory
• Inversion of adjunction for $F$-signature
• The F-signature Function on the Ample Cone
• Generalized F-signature of invariant subrings
• $F$-Invariants of Stanley-Reisner Rings
• Generalized F-signatures of Hibi rings
• $F$-Volumes
• Lech's conjecture in dimension three
• Varieties with ample Frobenius-trace kernel
• On the Finite F-representation type and F-signature of hypersurfaces
• Dual F-signature of special Cohen-Macaulay modules over cyclic quotient surface singularities
• Dual F-signature of Cohen-Macaulay modules over rational double points
• Limit $F$-signature functions of two-variable binomial hypersurfaces
• Lower bounds on Hilbert--Kunz multiplicities and maximal F-signatures