F-signature and Hilbert–Kunz multiplicity : a combined approach and comparison

Abstract

We present a unified approach to the study of F -signature, Hilbert-Kunz multiplicity, and related limits governed by Frobenius and Cartier linear actions in positive characteristic commutative algebra. We introduce general techniques that give vastly simplified proofs of existence, semicontinuity, and positivity. Furthermore, we give an affirmative answer to a question of Watanabe and Yoshida allowing the F -signature to be viewed as the infimum of relative differences in the Hilbert-Kunz multiplicites of the cofinite ideals in a local ring.

F-signature function of quotient singularities

“…For the F-signature function, it is known that F S(e) = s(R)p de + O(p (d−1)e ) (see [Tuc12], [PT16]). In their recent work, Polstra and Tucker ask whether a second coefficient for the F-signature function exists as well [PT16,Question 7.4].…”

“…For the F-signature function, it is known that F S(e) = s(R)p de + O(p (d−1)e ) (see [Tuc12], [PT16]). In their recent work, Polstra and Tucker ask whether a second coefficient for the F-signature function exists as well [PT16,Question 7.4]. We thank Polstra for pointing out to us that this is known to be true for some classes of rings, including rings that are Q-Gorenstein on the punctured spectrum and affine semigroup rings, as a consequence of the existence of a second coefficient for Hilbert-Kunz functions with respect to m-primary ideals [HMM04].…”

-Signature Under Birational Morphisms

“…This limit exists by [Tuc12] and [DPY16], also see [PT18]. Furthermore, by [ It is clear that 0 s(R) 1 and it is a fact that s(R) = 1 if and only if R is regular by [HL02] and [DPY16].…”

“…where λ R (•) denotes the length of the module •. We refer the reader to [HL02,Pol18,PT18,Tuc12] for additional properties of F -signature.…”

Local Okounkov bodies and limits in prime characteristic

“…is studied in [29], and is defined in [13] as the F -signature of R, provided the limit exists. Subsequent works showed that this number captures sensitive information about the ring, e.g., s(R) (as a limit superior) is equal to 1 if and only if R is regular [13], is greater than zero if and only if R is strongly F -regular [1], and is equal to the infimum of differences of Hilbert-Kunz multiplicities of a pair of nested m-primary ideals [37,38,25].…”