Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2024-12-24T13:55:54.000Z Has data issue: false hasContentIssue false
  • English
  • Français

NAKAYAMA AUTOMORPHISMS OF ORE EXTENSIONS OVER POLYNOMIAL ALGEBRAS

Part of: Homological methods Rings and algebras arising under various constructions Rings and algebras with additional structure

Published online by Cambridge University Press:  17 June 2019

LIYU LIU Open the ORCID record for LIYU LIU [Opens in a new window]  and
WEN MA
LIYU LIU
Affiliation:
School of Mathematical Sciences, Yangzhou University, No. 180 Siwangting Road, 225002 Yangzhou, Jiangsu, China e-mail: [email protected]; [email protected]
WEN MA
Affiliation:
School of Mathematical Sciences, Yangzhou University, No. 180 Siwangting Road, 225002 Yangzhou, Jiangsu, China e-mail: [email protected]; [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Nakayama automorphisms play an important role in the fields of noncommutative algebraic geometry and noncommutative invariant theory. However, their computations are not easy in general. We compute the Nakayama automorphism ν of an Ore extension R[x; σ, δ] over a polynomial algebra R in n variables for an arbitrary n. The formula of ν is obtained explicitly. When σ is not the identity map, the invariant EG is also investigated in terms of Zhang’s twist, where G is a cyclic group sharing the same order with σ.

MSC classification

Primary: 16E40: (Co)homology of rings and algebras (e.g. Hochschild, cyclic, dihedral, etc.)
Secondary: 16S36: Ordinary and skew polynomial rings and semigroup rings 16W22: Actions of groups and semigroups; invariant theory
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 62 , Issue 3 , September 2020 , pp. 518 - 530
Copyright
© Glasgow Mathematical Journal Trust 2019

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Bell, J. and Zhang, J. J., Zariski cancellation problem for noncommutative algebras, Selecta Math. (N.S.) 23 (2017), 1709–1737.CrossRefGoogle Scholar
Brown, K. and Gilmartin, P., Quantum homogeneous spaces of connected Hopf algebras, J. Algebra 454 (2015), 400–432.CrossRefGoogle Scholar
Brown, K. A. and Zhang, J. J., Dualising complexes and twisted Hochschild (co)homology for Noetherian Hopf algebras, J. Algebra 320 (2008), 1814–1850.CrossRefGoogle Scholar
Ginzburg, V., Calabi–Yau Algebras, preprint, arXiv:math/0612139, 2006.Google Scholar
Goodman, J. and Kr¨ahmer, U., Untwisting a twisted Calabi-Yau algebra, J. Algebra 406 (2014), 272–289.10.1016/j.jalgebra.2014.02.018CrossRefGoogle Scholar
He, J.-W., Van Oystaeyen, F. and Zhang, Y., Skew polynomial algebras with coefficients in Koszul Artin–Schelter Gorenstein algebras, J. Algebra 390 (2013), 231–249.10.1016/j.jalgebra.2013.05.023CrossRefGoogle Scholar
Jørgensen, P. and Zhang, J. J., Gourmet’s guide to Gorensteinness, Adv. Math. 151 (2000), 313–345.CrossRefGoogle Scholar
Kr¨ahmer, U., On the Hochschild (co)homology of quantum homogeneous spaces, Israel J. Math. 189 (2012), 237–266.CrossRefGoogle Scholar
Liu, L., Wang, S. and Wu, Q., Twisted Calabi–Yau property of Ore extensions, J. Noncommut. Geom. 8 (2014), 587–609.10.4171/JNCG/165CrossRefGoogle Scholar
Lü, J.-F., Mao, X.-F. and Zhang, J. J., Nakayama automorphism and applications, Trans. Am. Math. Soc. 369 (2017), 2425–2460.CrossRefGoogle Scholar
Lü, J.-F., Mao, X.-F. and Zhang, J. J., Nakayama automorphisms of a class of graded algebras, Israel J. Math. 219 (2017), 707–725.CrossRefGoogle Scholar
McConnell, J. C. and Robson, J. C., Noncommutative noetherian rings (John Wiley & Sons Ltd., Chichester, 1987).Google Scholar
Reyes, M., Rogalski, D. and Zhang, J. J., Skew Calabi–Yau algebras and homological identities, Adv. Math. 264 (2014), 308–354.CrossRefGoogle Scholar
Shen, Y. and Lu, D., Nakayama automorphisms of PBW deformations and Hopf actions, Sci. China Math. 59 (2016), 661–672.CrossRefGoogle Scholar
Yekutieli, A., Dualizing complexes over noncommutative graded algebras, J. Algebra 153 (1992), 41–84.CrossRefGoogle Scholar
Yekutieli, A., The rigid dualizing complex of a universal enveloping algebra, J. Pure Appl. Algebra 150 (2000), 85–93.10.1016/S0022-4049(99)00032-8CrossRefGoogle Scholar
Zhang, J. J., Twisted graded algebras and equivalences of graded categories, Proc. London Math. Soc. 72(3) (1996), 281–311.10.1112/plms/s3-72.2.281CrossRefGoogle Scholar
Zhu, C., Van Oystaeyen, F. and Zhang, Y., Nakayama automorphisms of double Ore extensions of Koszul regular algebras, Manuscripta Math. 152 (2017), 555–584.CrossRefGoogle Scholar