Homological properties of cochain Differential Graded algebras

Frankild, Anders; Jørgensen, Peter

doi:10.1016/j.jalgebra.2008.08.001

Cited by 11 publications

(5 citation statements)

References 13 publications

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“…We assume that the reader is familiar with basic definitions concerning DG homological algebra. If this is not the case, we refer to [AFH,FHT2,MW1,MW2,FJ2] for more details on them. We begin by fixing some notations and terminology.…”

Section: Notations and Conventionsmentioning

confidence: 99%

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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: Notations and Conventionsmentioning

confidence: 99%

“…In the literature, there are many papers on the research of homological properties of connected cochain DG algebras. For example, Gorenstein properties of DG algebras are studied in [FHT1,FIJ,FJ1,FJ2,FM,Gam,Jor,Mao,MW2,Sch]; He-Wu (cf. [HW]) introduce and study Koszul connected cochain DG algebras; and recently, the first author and J.-W.…”

Section: Introductionmentioning

confidence: 99%

DG polynomial algebras and their homological properties

et al. 2018

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“…Theorem 2.4 and a triangular inequality on the levels (Lemma 4.1) allow us to compare the (co)chain type levels of maps. This corollary is essentially a special case of [18,Proposition 2.3]; see also [2,Theorem 4.8] for other equivalence conditions for the level to be finite.…”

Section: Resultsmentioning

confidence: 96%

“…This corollary is essentially a special case of [18,Proposition 2.3]; see also [2,Theorem 4.8] for other equivalence conditions for the level to be finite.…”

Section: Resultsmentioning

confidence: 96%

The ghost length and duality on the chain and cochain type levels

Kuribayashi

2016

Homology, Homotopy and Applications

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Abstract. We establish equalities between cochain and chain type levels of maps by making use of exact functors which connect appropriate derived and coderived categories. Relevant conditions for levels of maps to be finite are extracted from the equalities which we call duality on the levels. Moreover, we give a lower bound of the cochain type level of the diagonal map on the classifying space of a Lie group by considering the ghostness of a shriek map which appears in derived string topology. A variant of Koszul duality for a differential graded algebra is also discussed.

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“…We assume that the reader is familiar with basics on DG homological algebra. If this is not the case, we refer to [AFH,FJ2,MW1,MW2] for more details on them.…”

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confidence: 99%

A note on homologically smooth connected cochain DG algebras

Mao,

Xie

2013

Preprint

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In this paper, we obtain two interesting results on homologically smooth connected cochain DG algebras. More precisely, we show that any Koszul DG module in D fg (A) is compact, when A is a homologically smooth connected cochain DG algebra with a Noetherian cohomology graded algebra H(A). And we prove that the homologically smoothness of A is equivalent toif A is a Koszul connected cochain DG algebra such that H(A) is a Noetherian graded algebra with a balanced dualizing complex.

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Homological properties of cochain Differential Graded algebras

Cited by 11 publications

References 13 publications

DG polynomial algebras and their homological properties

DG polynomial algebras and their homological properties

The ghost length and duality on the chain and cochain type levels

A note on homologically smooth connected cochain DG algebras

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