<i>F</i>
-signature of pairs: continuity, <i>p</i>
-fractals and minimal log discrepancies

Blickle, Manuel; Schwede, Karl; Tucker, Kevin

doi:10.1112/jlms/jds070

Cited by 9 publications

(16 citation statements)

References 23 publications

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“…Using 5) in Theorem 1.6, we know 1]. Moreover, if we know that ξ f (x) is continuous at 0 and 1 (see Theorem 1.6 1)), we obtain 3) in Theorem 1.6 immediately from Theorem 4.4 in [3].…”

Section: )mentioning

confidence: 85%

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A Function Determined by a Hypersurface of Positive Characteristic

Ohta¹

2017

Tokyo J. Math.

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Let R = k[[X 1 , . . . , X n+1 ]] be a formal power series ring over a perfect field k of prime characteristic p > 0, and let m = (X 1 , . . . , X n+1 ) be the maximal ideal of R. Suppose 0 = f ∈ m. In this paper, we introduce a function ξ f (x) associated with a hypersurface defined on the closed interval [0, 1] in R. The Hilbert-Kunz function and the F-signature of a hypersurface appear as the values of our function ξ f (x) on the interval's endpoints. The F-signature of the pair, denoted by s(R, f t ), was defined in [3]. Our function ξ f (x) is integrable, and the integral 1 t ξ f (x)dx is just s(R, f t ) for any t ∈ [0, 1]. 1991 Mathematics Subject Classification. 13F25.

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“…Using 5) in Theorem 1.6, we know 1]. Moreover, if we know that ξ f (x) is continuous at 0 and 1 (see Theorem 1.6 1)), we obtain 3) in Theorem 1.6 immediately from Theorem 4.4 in [3].…”

Section: )mentioning

confidence: 85%

“…Let (A, n) be an F-finite regular local ring, and let g ∈ A be a non-zero element. In [3], the F-signature of the pair (A, g t ) for any real number t ∈ [0, 1] is denoted as…”

Section: )mentioning

confidence: 99%

A Function Determined by a Hypersurface of Positive Characteristic

Ohta¹

2017

Tokyo J. Math.

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“…In this section, we consider the F -signature function 2 associated to an element f of an F -finite regular local ring (R, m). We begin by summarizing the needed theory, directing the interested reader to [BST12] and [BST13] for a complete development with historical Assuming that (R, f t ) is sharply F -pure, then we define the splitting prime P = P (R, f t ) to be the largest ideal such that…”

Section: The Left Derivative Of the F -Signature Function At The F -Pmentioning

confidence: 99%

“…which does not depend on the particular representation t = a/q [BST13, Proposition 4.1]. By [BST13], all one-sided derivatives of s(R, f t ) exist. In this section, we show that the left derivative at t = fpt(f ) equals zero whenever the F -pure threshold is "mild."…”

Section: The Left Derivative Of the F -Signature Function At The F -Pmentioning

confidence: 99%

On the behavior of singularities at the $F$-pure threshold

Canton¹,

Hernández²,

Schwede³

et al. 2016

Illinois J. Math.

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We provide a family of examples where the F -pure threshold and the log canonical threshold of a polynomial are different, but where p does not divide the denominator of the F -pure threshold (compare with an example of Mustaţȃ-Takagi-Watanabe). We then study the F -signature function in the case where either the F -pure threshold and log canonical threshold coincide or where p does not divide the denominator of the F -pure threshold. We show that the F -signature function behaves similarly in those two cases. Finally, we include an appendix which shows that the test ideal can still behave in surprising ways even when the F -pure threshold and log canonical threshold coincide.

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“…We borrow freely from these papers for some of the examples presented in this paper. We do not cover many new developments and calculations of the F-signature, for example see [BST1]- [BST2] and for toric rings see [S] and more recently [VK]. See [EY] for further extensions of Hilbert-Kunz multiplicity, and [Vr] for additional work.…”

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confidence: 99%

Hilbert–Kunz Multiplicity and the F-Signature

Huneke

2012

Commutative Algebra

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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 −...

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

Cited by 9 publications

References 23 publications

A Function Determined by a Hypersurface of Positive Characteristic

A Function Determined by a Hypersurface of Positive Characteristic

On the behavior of singularities at the $F$-pure threshold

Hilbert–Kunz Multiplicity and the F-Signature

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