Observations on the F-signature of local rings of characteristic p
By Yongwei Yao. 2006
72 citations recorded by the source corpus.
Abstract Let ( R , m , k ) be a d-dimensional Noetherian reduced local ring of prime characteristic p such that R 1 / p e are finite over R for all e ∈ N (i.e. R is F-finite). Consider the sequence { a e q α ( R ) + d } e = 0 ∞ , in which α ( R ) = log p [ k : k p ] , q = p e , and a e is the maximal rank of free R-modules appearing as direct summands of R-module R 1 / q . Denote by s − ( R ) and s + ( R ) the liminf and limsup, respectively, of the above sequence as e → ∞ . If s − ( R ) = s + ( R ) , then the limit, denoted by s ( R ) , is called the F-signature of R. It turns out that the F-signature can be defined in a way that is independent of the module finite property of R 1 / q over R. We show that: ( 1 ) If s + ( R ) ⩾ 1 − 1 ( d ! p d ) , then R is regular; ( 2 ) If R is excellent such that R P is Gorenstein for every P ∈ Spec ( R ) ∖ { m } , then s ( R ) exists; (3) If ( R , m ) → ( S , n ) is a local flat ring homomorphism, then s ± ( R ) ⩾ s ± ( S ) and, if furthermore S / m S is Gorenstein, s ± ( S ) ⩾ s ± ( R ) s ( S / m S ) .
Subfields: Algebra and Number Theory, Geometry and Topology
View Paper: https://www.semanticscholar.org/paper/f36f1f893574971e80d78e29fa2b627d77198222
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Citations in the corpus
- F-signature and Hilbert–Kunz multiplicity : a combined approach and comparison (2016, unknown): “ Jwith finite colength [HL02, Theorem 15] (see Lemma 6.1). The infimum among these differences was studied independently by Watanabe and Yoshida [WY04], and the connection to F-signature was made by Yao [Yao06]. Moreover, Watanabe and Yoshida expected and formally questioned if the infimum of the relati”
- F-signature exists (2011, methodology): “In the notation of the previous result, Y. Yao has proposed lime→∞ be pe(d+α(R)) as a potential definition of F -signature for a finitely generated R-module M ; see [Yao06].”
- Quantifying singularities with differential operators (2018, background): “Remark 5.5 ([AE05, Yao06]).”