F-signature exists

By Kevin Tucker. 2011

109 citations recorded by the source corpus.

Suppose R is a d-dimensional reduced F-finite Noetherian local ring with prime characteristic p>0 and perfect residue field. Let $R^{1/p^{e}}$ be the ring of pe-th roots of elements of R for e∈ℕ, and let ae denote the maximal rank of a free R-module appearing in a direct sum decomposition of $R^{1/p^{e}}$. We show the existence of the limit $s(R) := \lim_{e \to\infty} \frac{a_{e}}{p^{ed}}$, called the F-signature of R. This invariant—which can be extended to all local rings in prime characteristic—was first formally defined by C. Huneke and G. Leuschke (in Math. Ann. 324(2), 391–404, 2002) 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 arbitrary finite quotient singularities.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/9b3d043801c3181745b85fef3f51b856ec9d9f3a

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Citations in the corpus

F-signature and Hilbert–Kunz multiplicity : a combined approach and comparison (2016, unknown): “Building upon the philosophy introduced in [Tucker 2012], we further attempt to carefully track the uniform constants controlling convergence in the proof and throughout this article.”
F-signature of pairs and the asymptotic behavior of Frobenius splittings (2011, methodology): “(Theorem 3.11) and is called the F-signature of (R,D). Furthermore, s(R,D) >0 is positive if and only if (R,D) is F-regular (Theorem 3.18). The existence statement in the theorem above generalizes [Tuc11] using related methods; however, it is quite distinct in that we make no appeal to Hilbert-Ku”
Quantifying singularities with differential operators (2018, background): “tors by Smith and Van den Bergh [SVdB97] (see also [Sei97]) and formally deﬁned by Huneke and Leuschke [HL02], provided that the limit exists. After partial results [HL02, Yao06, AE05, Sin05], Tucker [Tuc12] showed that the F-signature exists as a limit, rather than just a limsup, in general. Yao [Y”
A survey of test ideals (2011, background): “swer to a conjecture of Monsky implies the existence of local rings with irrational F-signature, see [Mon08]. The heart of the proof of Theorem 8.20 lies in the following technical lemma. Lemma 8.21. [Tuc11] Let (R,m,k) be a local ring of dimension d. If {Je}e∈N is any sequence of m-primary ideals s”