Compact DG modules and Gorenstein DG algebras

Mao, Xuefeng; Wu, Q.-S.

doi:10.1007/s11425-008-0175-z

Cited by 29 publications

(30 citation statements)

References 22 publications

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“…By the assumption H(A) = [⌈y 1 ⌉, · · · , ⌈y m ⌉] is a Koszul, Gorenstein and Noetherian graded algebra with gl.dimH(A) = m < ∞. Hence A is Koszul, Gorenstein and homologically smooth by [HW,Proposition2.3], [Gam,Proposition 1] and [MW2,Corollary 3.7], respectively. Now, lets consider the Calabi-Yau properties of A.…”

Section: Calabi-yau Properties Of Polynomial Dg Algebrasmentioning

confidence: 99%

DG polynomial algebras and their homological properties

et al. 2018

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In this paper, we introduce and study differential graded (DG for short) polynomial algebras. In brief, a DG polynomial algebra A is a connected cochain DG algebra such that its underlying graded algebra A # is a polynomial algebra [x 1 , x 2 , · · · , xn] with |x i | = 1, for any i ∈ {1, 2, · · · , n}.We describe all possible differential structures on DG polynomial algebras; compute their DG automorphism groups; study their isomorphism problems; and show that they are all homologically smooth and Gorestein DG algebras. Furthermore, it is proved that the DG polynomial algebra A is a Calabi-Yau DG algebra when its differential ∂ A = 0 and the trivial DG polynomial algebra (A, 0) is Calabi-Yau if and only if n is an odd integer.2010 Mathematics Subject Classification. Primary 16E45, 16E65, 16W20,16W50.

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Section: Calabi-yau Properties Of Polynomial Dg Algebrasmentioning

confidence: 99%

DG polynomial algebras and their homological properties

et al. 2018

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“…However, we have the following proposition which is proved in [15], as a corollary of some homological identities over DG algebras. For completeness we give a direct proof here.…”

Section: Proposition 21 Letmentioning

confidence: 99%

“…However, if the Ext-algebra is infinitedimensional, little is known about A. As shown in [15] (see also Proposition 2.2), Ak is not compact if H(A) is finite-dimensional. In this paper, it is proved that the Koszul duality theorem also holds when H(A) is finite-dimensional by using Foxby duality.…”

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confidence: 99%

Koszul differential graded algebras and BGG correspondence II

2009

Chin. Ann. Math. Ser. B

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The concept of Koszul differential graded (DG for short) algebra is introduced in [8]. Let A be a Koszul DG algebra. If the Ext-algebra of A is finite-dimensional, i.e., the trivial module Ak is a compact object in the derived category of DG A-modules, then it is shown in [8] that A has many nice properties. However, if the Ext-algebra is infinitedimensional, little is known about A. As shown in [15] (see also Proposition 2.2), Ak is not compact if H(A) is finite-dimensional. In this paper, it is proved that the Koszul duality theorem also holds when H(A) is finite-dimensional by using Foxby duality. A DG version of the BGG correspondence is deduced from the Koszul duality theorem.

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“…If this is not the case, we refer to [3,5,11,14,15] for more details. We begin by fixing some notation and terminology.…”

Section: Preliminariesmentioning

confidence: 99%

DG algebra structures on AS-regular algebras of dimension 2

Mao

2011

Sci. China Math.

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Let A be a connected cochain DG algebra, whose underlying graded algebra is an Artin-Schelter regular algebra of global dimension 2 generated in degree 1. We give a description of all possible differential of A and compute H(A). Such kind of DG algebras are proved to be strongly Gorenstein. Some of them serve as examples to indicate that a connected DG algebra with Koszul underlying graded algebra may not be a Koszul DG algebra defined in He and We (J Algebra, 2008(J Algebra, , 320: 2934(J Algebra, -2962. Unlike positively graded chain DG algebras, we give counterexamples to show that a bounded below DG A-module with a free underlying graded A # -module may not be semi-projective.

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Compact DG modules and Gorenstein DG algebras

Cited by 29 publications

References 22 publications

DG polynomial algebras and their homological properties

DG polynomial algebras and their homological properties

Koszul differential graded algebras and BGG correspondence II

DG algebra structures on AS-regular algebras of dimension 2

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