Globalizing F-invariants

Stefani, Alessandro De; Polstra, Thomas; Yao, Yongwei

doi:10.48550/arxiv.1608.08580

Cited by 6 publications

(16 citation statements)

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“…We also prove that the strong F-regularity of a pair (R, D), where D is a Cartier algebra, is equivalent to the positivity of the global F-signature s(R, D) of the pair. This extends a result proved in [DSPY16], by removing an extra assumption on the Cartier algebra. is part of a free resolution of F e * (M s ) over the ring R s .…”

supporting

confidence: 84%

“…In the local case, it was proved in [BST12] that the F-signature of a Cartier algebra D on R is positive if and only if the pair (R, D) is strongly F-regular. These authors were able to recover the same result in the global setting, provided the Cartier algebra D satisfies certain additional assumptions [DSPY16,Theorem 2.24]. We are able to remove these extra conditions:…”

Section: Introductionsupporting

confidence: 52%

“…A key ingredient that is used in [DSPY16] for developing a global theory of Hilbert-Kunz multiplicity and F-signature are certain semicontinuity results. In particular, to relate the global Hilbert-Kunz multiplicity to the invariants in the localization, the upper semicontinuity of the functions λ e : Spec(R)…”

Section: Uniform Convergence and Upper Semi-continuity Resultsmentioning

confidence: 99%

“…(1) there is a containment of R-modules R ⊕p eγ(R) ⊆ F e * R, (2) which has a prime filtration with prime factors isomorphic to R/Q, where Q ∈ S(R), (3) and for each Q ∈ S(R), the prime factor R/Q appears no more than Cp eγ(R) times in the chosen prime filtration of R ⊕p eγ(R) ⊆ F e * R. Lemma 3.2 is used by the second author in [Pol18] to establish the presence of strong uniform bounds for all F-finite rings, and in [DSPY16] to show the existence of global Hilbert-Kunz multiplicity and F-signature. Lemma 3.3.…”

Section: Uniform Convergence and Upper Semi-continuity Resultsmentioning

confidence: 99%

“…The goal of this article is to extend to the global setting numerical invariants related to Frobenius, that are typically only defined for local rings. This follows the path started in [DSPY16], where these authors extended the definitions of Hilbert-Kunz multiplicity and F-signature to rings that are not necessarily local.…”

Section: Introductionmentioning

confidence: 97%

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Global Frobenius Betti numbers and F-splitting ratio

De Stefani,

Polstra,

Yao

2018

Preprint

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We extend the notion of Frobenius Betti numbers and F-splitting ratio to large classes of finitely generated modules over rings of prime characteristic, which are not assumed to be local. We also prove that the strong F-regularity of a pair (R, D), where D is a Cartier algebra, is equivalent to the positivity of the global F-signature s(R, D) of the pair. This extends a result proved in [DSPY16], by removing an extra assumption on the Cartier algebra. is part of a free resolution of F e * (M s ) over the ring R s . In particular, by localizing at any prime Q ∈ Spec(R) not containing s, the complex is still exact, and it becomes a free resolution of F e * (M Q ) over R Q . However, it may not be minimal. That is,∈ Q}, we therefore have that β i (e, Q, M) β i (e, P, M) for all Q ∈ D(s). This shows dense upper semi-continuity of the function P → β i (e, P, M). We now focus on the function P → χ i (e, P, M). Let P, ϕ j , ψ j and s ∈ R P be as above. Let Q ∈ D(s), and denote Ω j = ker((ψ j−1 ) Q ), for all j = 0, . . . , i.

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supporting

confidence: 84%

Section: Introductionsupporting

confidence: 52%

Section: Uniform Convergence and Upper Semi-continuity Resultsmentioning

confidence: 99%

Section: Uniform Convergence and Upper Semi-continuity Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 97%

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Global Frobenius Betti numbers and F-splitting ratio

De Stefani,

Polstra,

Yao

2018

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F-signature function of quotient singularities

Caminata

Stefani

2019

Journal of Algebra

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We study the shape of the F-signature function of a d-dimensional quotient singularity k x 1 , . . . , x d G , and we show that it is a quasi-polynomial. We prove that the second coefficient is always zero and we describe the other coefficients in terms of invariants of the finite acting group G ⊆ Gl(d, k). When G is cyclic, we obtain more specific formulas for the coefficients of the quasi-polynomial, which allow us to compute the general form of the function in several examples.

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The uniform symbolic topology property for diagonally F-regular algebras

Carvajal-Rojas

Smolkin

2020

Journal of Algebra

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Let k be a field of positive characteristic. Building on the work of [Smo18], we define a new class of k-algebras, called diagonally F -regular algebras, for which the so-called Uniform Symbolic Topology Property (USTP) holds effectively. We show that this class contains all essentially smooth k-algebras. We also show that this class contains certain singular algebras, such as the affine cone over P r k × P s k , when k is perfect. By reduction to positive characteristic, it follows that USTP holds effectively for the affine cone over P r C × P s C and more generally for complex varieties of diagonal F -regular type.holds for our choice of t. Following [Smo18], we will construct t using the so-called diagonal Cartier algebras. Namely, we setwhere D (n) is the n-th diagonal Cartier algebra; see Definition 3.1. Then Proposition 3.4(c) demonstrates that containment (2 ′ ) holds for any reduced k-algebra essentially of finite type, while (1 ′ ) holds whenever D (n) is F -regular. When this is the case for all n, we say our ring is diagonally F -regular as a k-algebra. This sketches the proof of our main theorem:1 If k is perfect, then essentially smooth k-algebras are the same as regular k-algebras essentially of finite type. 2 Also known as "Cartesian products," as in [Har77, Ch. II, Exc. 5.11]. 3 Ein-Lazarsfeld-Smith's original argument uses multiplier ideals, which are only known to exist in characteristic 0. Their argument was adapted to positive characteristic rings by N. Hara [Har05] and to mixedcharacteristic rings by L. Ma and K. Schwede [MS17]. Hara and Ma-Schwede achieved this by using positive characteristic and mixed characteristic analogs of multiplier ideals, respectively. * R, R). More generally, given a finite R-module M we may define a Cartier algebra C M over R as R in degree zero and as C e,M := Hom R (F e * M, M) in higher degrees. The ring multiplication of C M is defined by the rule ϕ e · ϕ d := ϕ e • F e * ϕ d for all ϕ e ∈ C e,M , ϕ d ∈ C d,M . Furthermore, the left R-module structure of C M is the usual one given by post-multiplication,whereas the right R-module structure is given by pre-multiplication by elements of F e * R. More precisely, if ϕ ∈ C e,M and r ∈ R, then (ϕ · r)(−) = ϕ(F e * r · −) It is worth mentioning we are primarily concerned with Cartier subalgebras of C R in this work.Definition 2.2 (Cartier Modules). Given a ring R and a Cartier R-algebra C, we define a Cartier C-module to be a finite R-module M equipped with a homomorphism C − → C M of Cartier R-algebras. This is the same as saying M is a left C-module with coherent underlying R-module structure [BS16, Lemma 5.2]. A morphism of Cartier C-modules is defined to be a morphism of left C-modules.Let R be a ring and C a Cartier R-algebra. Under the assumption R is essentially of finite type over k, Blickle and Stäbler constructed a covariant functor τ = τ(−, C) : left-C-mod − → R-mod, from the category of Cartier C-modules to the category of R-modules. This functor is an additive subfunctor of the forgetful functor between ...

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Globalizing F-invariants

Cited by 6 publications

References 24 publications

Global Frobenius Betti numbers and F-splitting ratio

Global Frobenius Betti numbers and F-splitting ratio

F-signature function of quotient singularities

The uniform symbolic topology property for diagonally F-regular algebras

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