Hilbert–Kunz Multiplicity and the F-Signature

Abstract

This paper is a much expanded version of two talks given in Ann Arbor in May of 2012 during the computational workshop on F-singularities. We survey some of the theory and results concerning the Hilbert-Kunz multiplicity and F-signature of positive characteristic local rings.Dedicated to David Eisenbud, on the occasion of his 65th birthday. IntroductionThroughout this paper (R, m, k) will denote a Noetherian local ring of prime characteristic p with maximal ideal m and residue field k. We let e be a varying non-negative integer, and let q = p e . By I [q] we denote the ideal generated by x q , x ∈ I. If M is a finite R-module, M/I [q] M has finite length. We will use λ(−) to denote the length of an R-module. We assume knowledge of basic ideas in commutative algebra, including the usual Hilbert-Samuel multiplicity, Cohen-Macaulay, regular, and Gorenstein rings. The basic question this paper studies is how λ(M/I [q] M) behaves as a function on q, and how understanding this behavior leads to better understanding of the singularities of the ring R. In a seminal paper which appeared in 1969, [Ku1], Ernst Kunz introduced the study of this function as a way to measure how close the ring R is to being regular.The Frobenius homomorphism is the map F : R −→ R given by F (r) = r p . We say that R is F-finite if R is a finitely generated module over itself via the Frobenius homomorphism. It Date: September 2, 2014. 2010 Mathematics Subject Classification. 13-02, 13A35, 13C99, 13H15. 1 2 CRAIG HUNEKE is not difficult to prove that if (R, m, k) is a complete local Noetherian ring of characteristic p, or an affine ring over a field k of characteristic p, then R is F-finite if and only if [k 1/p : k]is finite. When R is reduced we can identify the Frobenius map with the inclusion of R into R 1/p , the ring of pth roots of elements of R. If M is an R-module, we will usually write M 1/q to denote what is more commonly denoted F e * (M), where q = p e , the module which is the same as M as abelian groups, but whose R-module structure is coming from restriction of scalars via e-iterates of the Frobenius map. This is an exact functor on the category of R-modules. Notice that F e * (R) can be naturally identified with R 1/q . If the residue field k of R is perfect then the lengths of the R-modules R 1/q /IR 1/q and R/I [q] are the same. If k is not perfect, but R is F-finite, then we can adjust by [k 1/q : k]. We define α(R) := log p ([k 1/p : k]), so that we can write [k 1/q : k] = q α(R) . With this notation,More broadly, the two numbers we will study, namely the Hilbert-Kunz multiplicity and the F-signature, are characteristic p invariants which give information about the singularities of R, and lead to many interesting issues concerning how to use characteristic p methods to study singularities. There are four basic facts about characteristic p which make things work.Those facts are first that (r + s) p = r p + s p for elements in a ring of characteristic p (i.e., the Frobenius is an endomorphism); second, that the map from R −...

Frobenius-Poincaré function and Hilbert-Kunz multiplicity

“…Let S be a ring containing a field of characteristic p > 0; recall that x ∈ S is said to be in the tight closure of an ideal J, if there is a c ∈ S, not in any minimal prime of S such that, c.x p n ∈ J [p n ] for all large n (see Definition 3.1, [HH90]). The theory of Hilbert-Kunz multiplicity is related to the theory of tight closure-see Proposition 5.4, Theorem 5.5 of [Hun13] and Theorem 8.17, [HH90]. A similar relation between tight closure of an ideal and the corresponding Frobenius-Poincaré function is the content of the next result.…”

“…Hilbert-Kunz multiplicity is a multiplicity theory in positive characteristic. We refer readers to [Hun13] for a survey of this theory. In this subsection, k is a field of characteristic p > 0.…”

“…The assertion on the growth of λ S (N/J [p n ] N ) is standard, for example see Lemma 3.5 of [Hun13] for a proof. For the other assertion, we present a simplified version of the argument in Lemma 7.2 of [Hun13]. Let K • be the Koszul complex of f 1 , f 2 , .…”

Uniform bounds in F-finite rings and lower semi-continuity of the F-signature

“…In particular, the length of the prime filtration of R q γ(R) ⊆ R 1/q has length no more than C|S(R)|q γ(R) . This result, for local domains whose residue field is perfect, is exercise 10.4 in [9], whose proof is given in the second appendix by Karen Smith, and this result is explicitly stated and proved in [8] as Lemma 4.…”

Generalized Hilbert–kunz Function in Graded Dimension 2

“…If is an -primary ideal, then and coincide with the classical Hilbert–Kunz function and multiplicity. For a survey on the classical Hilbert–Kunz function and multiplicity, see [10].…”