Abstract
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 the local cohomology functor can be used to detect the Gorensteinness of a homologically smooth DG algebra. For any Gorenstein homologically smooth locally finite DG algebra \({\cal A}\), we define a group homomorphism \({\rm{Hdet}}:{\rm{Au}}{{\rm{t}}_{dg}}\left( {\cal A} \right) \to {k^ \times }\), called the homological determinant. As applications, we present a sufficient condition for the invariant DG subalgebra \({{\cal A}^G}\) to be Gorenstein, where \({\cal A}\) is a homologically smooth DG algebra such that \(H\left( {\cal A} \right)\) is a Noetherian AS-Gorenstein graded algebra and G is a finite subgroup of \({\rm{Au}}{{\rm{t}}_{dg}}\left( {\cal A} \right)\). Especially, we can apply this result to DG down-up algebras and non-trivial DG free algebras generated in two degree-one elements.
Similar content being viewed by others
References
Avramov L L, Foxby H-B. Locally Gorenstein homomorphisms. Amer J Math, 1992, 114: 1007–1047
Benkart G, Roby T. Down-up algebras. J Algebra, 1998, 209: 305–344
Benkart G, Roby T. Addendum: “Down-up algebras”. J Algebra, 1999, 213: 378–378
Chan K, Walton C, Zhang J J. Hopf actions and Nakayama automorphisms. J Algebra, 2014, 409: 26–53
Dwyer W G, Greenlees J P C, Iyengar S. Duality in algebra and topology. Adv Math, 2006, 200: 357–402
Félix Y, Halperin S, Thomas J-C. Gorenstein spaces. Adv Math, 1988, 71: 92–112
Frankild A, Iyengar S, Jørgensen P. Dualizing differential graded modules and Gorenstein differential graded algebras. J Lond Math Soc (2), 2003, 68: 288–306
Frankild A, Jørgensen P. Gorenstein differential graded algebras. Israel J Math, 2003, 135: 327–353
Gammelin H. Gorenstein space with nonzero evaluation map. Trans Amer Math Soc, 1999, 351: 3433–3440
He J-W, Wu Q-S. Koszul differential graded algebras and BGG correspondence. J Algebra, 2008, 320: 2934–2962
Hinič V A. On the Gorenstein property of the ring of invariants of a Gorenstein ring. Izv Akad Nauk SSSR Ser Mat, 1976, 10: 47–53
Jing N, Zhang J J. Gorensteinness of invariant subrings of quantum algebras. J Algebra, 1999, 221: 669–691
Jørgensen P. Local cohomology for non-commutative graded algebras. Comm Algebra, 1997, 25: 575–591
Jørgensen P. Duality for cochain DG algebras. Sci China Math, 2013, 56: 79–89
Jørgensen P, Zhang J J. Gourmet’s guide to Gorensteinness. Adv Math, 2000, 151: 313–345
Kirkman E, Kuzmanovich J, Zhang J J. Gorenstein subrings of invariants under Hopf algebra actions. J Algebra, 2009, 322: 3640–3669
Kriz I, May J P. Operads, algebras, modules and motives. Astérisque, 1995, 1995: 1–109
Lv J-F, Mao X-F, Zhang J J. Nakayama automorphism and applications. Trans Amer Math Soc, 2017, 369: 2425–2460
Mao X-F, He J-W, Liu M, et al. Calabi-Yau properties of nontrivial Noetherian DG down-up algebras. J Algebra Appl, 2018, 17: 1850090
Mao X-F, Wang X-T, Zhang M-Y. DG Algebra structures on the quantum affine n-space \({{\cal O}_{ - 1}}\left( {{k^n}} \right)\). J Algebra, 2022, 594: 389–482
Mao X-F, Wu Q-S. Homological invariants for connected DG algebras. Comm Algebra, 2008, 36: 3050–3072
Mao X-F, Wu Q-S. Compact DG modules and Gorenstein DG algebras. Sci China Ser A, 2009, 52: 648–676
Mao X-F, Wu Q-S. Cone length for DG modules and global dimension of DG algebras. Comm Algebra, 2011, 39: 1536–1562
Mao X-F, Xie J-F, Yang Y-N, et al. Isomorphism problem and homological properties of DG free algebras. Comm Algebra, 2019, 47: 4031–4060
van den Bergh M. Existence Theorems for dualizing complexes over Non-commutative graded and filtered rings. J Algebra, 1997, 195: 662–679
Watanabe K. Certain invariants subrings are Gorenstein. Osaka J Math, 1974, 11: 1–8
Watanabe K. Certain invariants subrings are Gorenstein II. Osaka J Math, 1974, 11: 379–388
Yekutieli A. Dualizing complexes over noncommutative graded algebras. J Algebra, 1992, 153: 41–84
Acknowledgements
This work was supported by National Natural Science Foundation of China (Grant No. 11871326).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Mao, X., Wang, H. Local cohomology for Gorenstein homologically smooth DG algebras. Sci. China Math. 66, 1161–1176 (2023). https://doi.org/10.1007/s11425-021-2003-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11425-021-2003-2