Dualizing Complexes, Morita Equivalence and the Derived Picard Group of a Ring

Yekutieli, Amnon

doi:10.1112/s0024610799008108

Cited by 113 publications

(106 citation statements)

References 13 publications

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“…The above Lemma is easily proved by the following Lemma taken from [Y,Lemma 2.1]. (See also [ML,Theorem XII.…”

Section: Theorem 22 Let a Be A Right Noetherian K-algebra Of Finitementioning

confidence: 99%

Ampleness of two-sided tilting complexes

Minamoto

2011

International Mathematics Research Notices

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In this paper we define the notion of ampleness for two-sided tilting complexes over finite dimensional algebras and prove its basic properties.We call a finite dimensional k-algebra A of finite global dimension Fano if (A * [−d]) −1 is ample for some d ≥ 0. For example geometric algebras in the sense of Bondal-Polishchuk are Fano. We give a characterization of representation type of a quiver from a noncommutative algebro-geometric view point, that is, a finite acyclic quiver has finite representation type if and only if its path algebra is fractional Calabi-Yau, and a finite acyclic quiver has infinite representation type if and only if its path algebra is Fano. IntroductionLet X be a nonsingular projective variety over a field k and let ω X be its canonical bundle. Then the functor. By this fact, from a noncommutative (or categorical) algebro-geometric view point, one thinks of a triangulated category T as the derived category of coherent sheaves of some "space" X and of the Serre functor S T of T (if exists) as the derived tensor product of " dim X"-shifted "canonical bundle" ω X . From this view point, the notion of Calabi-Yau algebra ( and Calabi-Yau category ) is defined and studied extensively by many researchers.In this paper we introduce the notion of ampleness for two-sided tilting complexes over finite dimensional k-algebras. Let A be a finite dimensional k-algebra of finite global dimension. Definition 0.1 (Definition 2.6). A two-sided tilting complex σ over A is called very ample if Hi (σ) = 0 for i ≥ 1 and σ n is pure for n 0. σ is called ample if σ n is pure for n 0.In Section 2, we justify this definition by using the theory of noncommutative projective schemes due to Artin-Zhang [AZ] and Polishchuk [Po]. In the theory of noncommutative projective schemes , for a graded coherent ring R over k we attach an imaginary geometric object proj R = ( cohproj R, R, (1) ) . An abelian category cohproj R is considered as the category of coherent sheaves on proj R. (See Section 1.) In Section 2 we show that the following facts hold. If σ is a very ample tilting complex over A, then the tensor algebra T := T A (H 0 (σ)) of H 0 (σ) over A is a graded connected coherent ring over A and there is a t-structure D σ defined by σ in Perf A and its heart H σ is equivalent to cohproj T . Moreover the following Theorem holds. where T := T A (H 0 (σ)) is the tensor algebra of H 0 (σ) over A.

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“…The above Lemma is easily proved by the following Lemma taken from [Y,Lemma 2.1]. (See also [ML,Theorem XII.…”

Section: Theorem 22 Let a Be A Right Noetherian K-algebra Of Finitementioning

confidence: 99%

Ampleness of two-sided tilting complexes

Minamoto

2011

International Mathematics Research Notices

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“…So far as we are aware, it remains possible that every affine noetherian k-algebra of finite GKdimension has a rigid dualizing complex. When a noetherian ring A does possess a rigid dualizing complex R, then R is unique up to a unique isomorphism, [37, Proposition 8.2(1)], [42,Theorem 5.2]. Here is the first of two "generalised Cohen-Macaulay conditions" which will feature in this discussion.…”

Section: Corollary 54 Let R Be a Ring Which Is A Finite Module Over mentioning

confidence: 99%

The Cohen Macaulay Property for Noncommutative Rings

Brown

Macleod

2017

Algebr Represent Theor

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Let R be a noetherian ring which is a finite module over its centre Z(R). This paper studies the consequences for R of the hypothesis that it is a maximal Cohen Macaulay Z(R)-module. A number of new results are proved, for example projectivity over regular commutative subrings and the direct sum decomposition into equicodimensional rings in the affine case, and old results are corrected or improved. The additional hypothesis of homological grade symmetry is proposed as the appropriate extra lever needed to extend the classical commutative homological hierarchy to this setting, and results are proved in support of this proposal. Some speculations are made in the final section about how to extend the definition of the Cohen-Macaulay property beyond those rings which are finite over their centres.

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“…Note that this type of isomorphism is known for the classical Picard group of the ring. (Compare with [2, Proposition 5.4, Chapter 2] and its non-commutative analogue in [11].) However, an original point of this paper is that we can define the Picard group for any additive full subcategory of module category over A whenever it contains A as an object.…”

Section: Aut K (Cm(a)) ∼ = Aut K-alg (A) (D = 2) Aut K-alg (A) C (A) mentioning

confidence: 99%

Automorphism groups and Picard groups of additive full subcategories

Hiramatsu

Yoshino²

2010

MATH. SCAND.

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We study category equivalences between additive full subcategories of module categories over commutative rings. And we are able to define the Picard group of additive full subcategories. The aim of this paper is to study the properties of the Picard groups and show that the automorphism group of an additive full subcategory is a semi-direct product of the Picard group with the group of algebra automorphisms of the ring.

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Dualizing Complexes, Morita Equivalence and the Derived Picard Group of a Ring

Cited by 113 publications

References 13 publications

Ampleness of two-sided tilting complexes

Ampleness of two-sided tilting complexes

The Cohen Macaulay Property for Noncommutative Rings

Automorphism groups and Picard groups of additive full subcategories

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