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.Comment: 19 pages. Substantially updated from a version which was circulated on a limited basis in the fall of 201

F -signature of pairs: continuity, p -fractals and minimal log discrepancies

“…Recently, the third author has shown that this limit exists in general [24]. The F -signature of the pair (R, f t ) is defined similarly after restricting which direct summands are taken into account; setting a f t e to be the maximal rank of an R-free direct summand of F e * R R ⊕a f t e ⊕ M f t e where the associated projections F e * R − → R factor through multiplication by F e * f t(p e −1) on F e * R, the F -signature of the pair (R, f t ) is the limit…”

“…It is a real number between 0 and 1, s(R) = 1 characterizing regular rings [HL02,Yao06], and s(R) > 0 characterizing F -regular rings [AL03]. More generally, this rather subtle invariant is thought to encode various arithmetic and geometric properties of R; for example, the F -signature recovers the group order of tame finite quotient singularities ([HL02, Example 18], [Yao06,Remark 4.7], [Tuc,Corollary 4.13]), and is closely related to the theory of Hilbert-Kunz multiplicity.…”

Dual F-Signature of Cohen-Macaulay Modules Over Rational Double Points

“…The existence of the F -signature of R was shown by K. Tucker [21]. By Kunz's theorem, R is regular if and only if e R is a free R-module of rank p ed [11].…”

-Signature Under Birational Morphisms

“…The F -signature is a measure of singularities that simply states the percentage of F e * R that is free (measured in terms of a rank of a maximal free summand). F -signature was implicitly introduced by K. Smith and M. Van Den Bergh [SVdB97] and formally defined by C. Huneke and G. Leuschke in [HL02], although it wasn't shown to exist until [Tuc12].…”