Frobenius-Perron Theory of Endofunctors

Chen, Jianmin; Gao, Zhibin; Wicks, Elizabeth; Zhang, James Jian; Zhang, Xiaohong; Zhu, Hong

doi:10.48550/arxiv.1711.06670

Cited by 1 publication

(9 citation statements)

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“…Note that Theorem 0.1 is a "weighted" version of [CG,Proposition 6.5(1,2)]. By Lemma 2.1(2), we obtain the Frobenius-Perron dimension of the bounded derived category of finite dimensional representations of acyclic quivers of A D E type.…”

Section: Introductionmentioning

confidence: 90%

“…Note that (E2.8.1)-(E2.8.2) are in fact the transpose of the usual Hom and Ext-matrices. By [CG,Lemma 3.7] (by considering the opposite category), Lemma 2.10 holds for the transpose matrices of (E2.8.1)-(E2.8.2) too.…”

Section: Frobenius-perron Theory Of Weighted Projective Linesmentioning

confidence: 99%

“…Definition 1.5. [CG,Definition 2.7] (1) Let A be an abelian category. The Frobenius-Perron dimension of A is defined to be fpd…”

Section: Preliminaries and Definitionsmentioning

confidence: 99%

“…Lemma 1.6. [CG,Lemma 6.1] Let n be a positive integer. Suppose that B has a σ-decomposition {B λ } λ∈Λ .…”

Section: Preliminaries and Definitionsmentioning

confidence: 99%

“…Question 2.2. [CG,Question 7.11] Let X be a weighted projective line. What is the exact value of fpd n D b (coh(X)), for n ≥ 1, in terms of other invariants of X?…”

Section: Frobenius-perron Theory Of Weighted Projective Linesmentioning

confidence: 99%

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Frobenius-Perron theory for projective schemes

Chen¹,

Gao²,

Wicks³

et al. 2019

Preprint

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The Frobenius-Perron theory of an endofunctor of a k-linear category (recently introduced in [CG]) provides new invariants for abelian and triangulated categories. Here we study Frobenius-Perron type invariants for derived categories of commutative and noncommutative projective schemes.In particular, we calculate the Frobenius-Perron dimension for domestic and tubular weighted projective lines, define Frobenius-Perron generalizations of Calabi-Yau and Kodaira dimensions, and provide examples. We apply this theory to the derived categories associated to certain Artin-Schelter regular and finite-dimensional algebras.

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Section: Introductionmentioning

confidence: 90%

Section: Frobenius-perron Theory Of Weighted Projective Linesmentioning

confidence: 99%

“…Definition 1.5. [CG,Definition 2.7] (1) Let A be an abelian category. The Frobenius-Perron dimension of A is defined to be fpd…”

Section: Preliminaries and Definitionsmentioning

confidence: 99%

“…Lemma 1.6. [CG,Lemma 6.1] Let n be a positive integer. Suppose that B has a σ-decomposition {B λ } λ∈Λ .…”

Section: Preliminaries and Definitionsmentioning

confidence: 99%

“…Question 2.2. [CG,Question 7.11] Let X be a weighted projective line. What is the exact value of fpd n D b (coh(X)), for n ≥ 1, in terms of other invariants of X?…”

Section: Frobenius-perron Theory Of Weighted Projective Linesmentioning

confidence: 99%

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