Calabi–Yau properties of nontrivial Noetherian DG down-up algebras

Mao, Xuefeng; He, Jiwen; Liu, M.; Xie, J.-F.

doi:10.1142/s0219498818500901

Cited by 12 publications

(11 citation statements)

References 29 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Recently, it is proved in [MHLX,Proposition 6.5] that a connected cochain DG algebra A is Calabi-Yau if H(A) = [⌈z 1 ⌉, ⌈z 2 ⌉] where z 1 ∈ ker(∂ 1 A ) and z 2 ∈ ker(∂ 2 A ). And a connected cochain DG algebra A is not Calabi-Yau if H(A) = [⌈z 1 ⌉, ⌈z 2 ⌉] where z 1 ∈ ker(∂ 1 A ) and z 2 ∈ ker(∂ 1 A ) (see [MH, Theorem B]).…”

Section: Introductionmentioning

confidence: 99%

DG polynomial algebras and their homological properties

et al. 2018

View full text Add to dashboard Cite

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.

show abstract

Section: Introductionmentioning

confidence: 99%

DG polynomial algebras and their homological properties

et al. 2018

View full text Add to dashboard Cite

show abstract

“…(3) A 3 with A # 3 = A(ξ − 1, ξ) and AS-Gorenstein by Proposition 5.7, and A is Calabi-Yau by [MHLX,Theorem 6.11]. Therefore, any DG down-up DG algebra A is homologically smooth and H(A) is AS-Gorenstein.…”

Section: Hence It Is Reasonable To Writementioning

confidence: 96%

“…A 3 ) = k[⌈(xy + yx) 3 ⌉]are all AS-Gorenstein. Furthermore, A 1 , A 2 and A 3 are Calabi-Yau DG algebras by[MHLX, Proposition 6.1, Corollary 6.2 and Proposition 6.4]. For Case 3, H(A) is…”

mentioning

confidence: 98%

Local cohomology for Gorenstein homologically smooth DG algebras

Mao¹,

Wang²

2021

Preprint

View full text Add to dashboard Cite

In this paper, we introduce the theory of local cohomology and local duality to Notherian connected cochain DG algebras. We show that the notion of local cohomology functor can be used to detect the Gorensteinness of a homologically smooth DG algebra. For any Gorenstein homologically smooth DG algebra A, we define a group homomorphism Hdet : Aut dg (A) → k × , called the homological determinant. As applications, we present a sufficient condition for the invariant DG subalgebra A G to be Gorensten, where A is a homologically smooth DG algebra such that H(A) is AS-Gorenstein and G is a finite subgroup of Aut dg (A). Especially, we can apply this result to nontrivial DG free algebras generated in two degree one elements, DG polynomial algebras and DG down-up algebras. introductionLocal cohomology is an indispensable tool in almost all branches of analytic and algebraic geometry as well as in commutative and combinatorial algebra. Since the flourishing of non-commutative algebraic geometry in 1990s, people have done a lot to develop a theory of local cohomology in the context of non-commutative cases. In [Yek], Yekutieli introduced balanced dualizing complexes for non-commutative graded algebras by using local cohomology functors. After his work, Jørgensen [Jor1] generalized the theory of local cohomology and local duality theorem to non-commutative positively graded Noetherian algebras, which are quotients of Artin-Schelter Gorenstein algebras. Later, Van den Bergh gave a more general local duality formula in [VDB]. In [JoZ], local cohomology functor is used to define homological determinants for AS-Gorenstein algebras. In the invariant theory of Hopf algebra action (and formerly group actions) on non-commutative graded algebras, the notion of homological determinant usually serves as an effective tool in precise analysis (cf. [CWZ, JoZ, JZ, KKZ1, LMZ1]).In this paper, we are concerned with connected cochain DG k-algebras, where k is an algebraically closed field with zero characteristic. Any connected graded algebra can be seen as a connected cochain DG algebra with zero differential. The motivation of this paper is to introduce an analogous local cohomology theory under the background of DG homological algebra. Let A be a connected cochain DG algebra with maximal DG ideal m = A ≥1 . For any DG A-module M , its subset Γ m (M ) = {m ∈ M | A ≥n m = 0, for some n ≥ 1} is a DG A-submodule of M . We actually define a left exact covariant A-linear functor Γ m (−) = lim − →

show abstract

“…x 2 y + (1 − ξ)xyx − ξyx 2 xy 2 + (1 − ξ)yxy − ξy 2 x , |x| = |y| = 1, ξ 3 = 1, ξ = 1 and the differential ∂ A is defined by ∂ A (x) = y 2 and ∂ A (y) = 0. We have H(A) = ⌈xy + yx⌉, ⌈y⌉ ξ⌈y⌉⌈xy + yx⌉ − ⌈xy + yx⌉⌈y⌉ ⌈y 2 ⌉ by [MHLX,Proposition 5.5]. One sees that H(A) is not Koszul and gl.dimH(A) = ∞.…”

Section: The Derived Picard Groups Of 4 Families Of Dg Algebrasmentioning

confidence: 99%

“…By [MHLX, Proposition 6.1], A is a Koszul Calabi-Yau DG algebra. From the proof of [MHLX,Proposition 6.1], one sees that A k admits a minimal semi-free resolution F with F # = A # ⊕ A # e y ⊕ A # e z and a differential ∂ F defined by ∂ F (Σe y ) = y, ∂ F (e t ) = x + yΣe y . By the minimality of F , we have H(Hom A (F, k)) = Hom A (F, k) = k · 1 * ⊕ k · (Σe y ) * ⊕ k · (Σe z ) * .…”

Section: The Derived Picard Groups Of 4 Families Of Dg Algebrasmentioning

confidence: 99%

Derived Picard groups of homologically smooth Koszul DG algebras

Mao

Yang

2019

Journal of Algebra

Self Cite

View full text Add to dashboard Cite

In this paper, we show that the derived Picard group of a homologically smooth Koszul connected cochain DG algebra is isomorphic to the opposite group of the derived Picard group of its finite dimensional local Ext-algebra. As applications, we compute the derived Picard groups of some important DG algebras such as trivial DG polynomial algebras, trivial DG free algebras and several non-trivial DG down-up algebras and DG free algebras.

show abstract

Calabi–Yau properties of nontrivial Noetherian DG down-up algebras

Cited by 12 publications

References 29 publications

DG polynomial algebras and their homological properties

DG polynomial algebras and their homological properties

Local cohomology for Gorenstein homologically smooth DG algebras

Derived Picard groups of homologically smooth Koszul DG algebras

Contact Info

Product

Resources

About