Dualizing complexes over noncommutative graded algebras

Yekutieli, Amnon

doi:10.1016/0021-8693(92)90148-f

Cited by 170 publications

(141 citation statements)

References 3 publications

Supporting

Mentioning

139

Contrasting

Order By: Relevance

“…Here I simply note that one key idea is to show ([SVdB08, Proposition 2.6]) that if Λ is homologically homogeneous of dimension d then ω Λ = Hom R (Λ, R) is an invertible Λ-module, and furthermore the shift ω Λ [d] is a dualizing complex for Λ in the sense of Yekutieli [Yek92]. This result has been extended [Mac10, Theorem 5.1.12] to remove the hypothesis of finite global dimension (so Λ is assumed to be "injectively homogeneous") and the hypotheses on the field k.…”

Section: Lemma O2 Let R Be a CM Normal Affine K-algebra Where K Ismentioning

confidence: 99%

Non-commutative Crepant Resolutions: Scenes From Categorical Geometry

Leuschke

2012

Progress in Commutative Algebra 1

View full text Add to dashboard Cite

ABSTRACT. Non-commutative crepant resolutions are algebraic objects defined by Van den Bergh to realize an equivalence of derived categories in birational geometry. They are motivated by tilting theory, the McKay correspondence, and the minimal model program, and have applications to string theory and representation theory. In this expository article I situate Van den Bergh's definition within these contexts and describe some of the current research in the area.

show abstract

Section: Lemma O2 Let R Be a CM Normal Affine K-algebra Where K Ismentioning

confidence: 99%

Non-commutative Crepant Resolutions: Scenes From Categorical Geometry

Leuschke

2012

Progress in Commutative Algebra 1

View full text Add to dashboard Cite

show abstract

“…The existence of dualizing complexes over various graded rings was studied in [10] and some of which are listed in Corollary 5 below. The proof of Theorem 1 is given at the end.…”

Section: Theorem 1 Let a Be A Locally Finite ‫-ގ‬Graded Noetherianmentioning

confidence: 99%

“…First we recall some definitions and results due to Yekutieli [10] and Van den Bergh [6]. Other definitions, results and notations can be found in [6], [7] and [10].…”

Section: Theorem 1 Let a Be A Locally Finite ‫-ގ‬Graded Noetherianmentioning

confidence: 99%

See 1 more Smart Citation

Gorenstein Property of Hopf Graded Algebras

Wu¹,

Zhang²

2003

Glasgow Math. J.

View full text Add to dashboard Cite

Abstract. Let A be a locally finite, ‫-ގ‬graded, noetherian algebra with a balanced dualizing complex. If A is a Hopf algebra, then A has finite injective dimension.2000 Mathematics Subject Classification. 16E10, 16W30, 16W50.In [2] and [3] Brown and Goodearl showed that a noetherian affine polynomial identity (PI) Hopf algebra with finite injective dimension has various good homological properties. They verified that many examples of noetherian affine PI Hopf algebras, some of which are quantum groups at roots of unity, have finite injective dimension. In [2, Question A] Brown asks if every noetherian affine PI Hopf algebra has finite left and right injective dimension. Recently we gave an affirmative answer to Brown's question in [9, 0.1].In this short note we are trying to provide some evidence that a noetherian affine Hopf algebra, not necessarily PI, has finite injective dimension. We prove the following statement. THEOREM 1. Let A be a locally finite, ‫-ގ‬graded, noetherian algebra with a balanced dualizing complex. If A is a Hopf algebra, then A has finite left and right injective dimension and satisfies the AS-Gorenstein condition.

show abstract

“…A balanced Cohen-Macaulay algebra is a connected algebra A having a balanced dualizing complex ω A [d] in the sense of Yekutieli (1992) for some integer d and some graded A-A bimodule ω A . We study some homological properties of a balanced Cohen-Macaulay algebra.…”

mentioning

confidence: 99%

Homological properties of balanced Cohen-Macaulay algebras

Mori

2002

Trans. Amer. Math. Soc.

View full text Add to dashboard Cite

Abstract. A balanced Cohen-Macaulay algebra is a connected algebra A having a balanced dualizing complex ω A [d] in the sense of Yekutieli (1992) for some integer d and some graded A-A bimodule ω A . We study some homological properties of a balanced Cohen-Macaulay algebra. In particular, we will prove the following theorem:Theorem 0.1. Let A be a Noetherian balanced Cohen-Macaulay algebra, and M a nonzero finitely generated graded left A-module. Then:1. M has a finite resolution of the formwhere H is a finitely generated maximal Cohen-Macaulay graded left A-module. M has finite injective dimension if and only if M has a finite resolution of the formAs a corollary, we will have the following characterizations of AS Gorenstein algebras and AS regular algebras: Corollary 0.2. Let A be a Noetherian balanced Cohen-Macaulay algebra. A is AS Gorenstein if and only if ω A has finite projective dimension as a graded left A-module. 2. A is AS regular if and only if every finitely generated maximal Cohen-Macaulay graded left A-module is free. Hyperhomological algebrasThroughout this paper, we fix a field k. A connected algebra is a graded algebra of the formIn this first section, we will fix terminology and notation, and collect some elementary results on hyperhomological algebras over connected algebras.Let A, B, C be connected algebras. The category of graded left A-modules and graded left A-module homomorphisms of degree 0 is denoted by GrMod A. For

show abstract

Dualizing complexes over noncommutative graded algebras

Cited by 170 publications

References 3 publications

Non-commutative Crepant Resolutions: Scenes From Categorical Geometry

Non-commutative Crepant Resolutions: Scenes From Categorical Geometry

Gorenstein Property of Hopf Graded Algebras

Homological properties of balanced Cohen-Macaulay algebras

Contact Info

Product

Resources

About