Artin-Schelter regularity of twisted tensor products

Abstract

We prove an Artin-Schelter regularity result for the method of twisted tensor products under a certain form. Such twisted tensor products, whose twisting maps are determined by the action on the generators, include Ore extensions and double Ore extensions. It is helpful to construct highdimensional Artin-Schelter regular algebras.

Keywords

Artin-Schelter regular (AS-regular algebra) algebra / twisted tensor product / pure resolution with respect to length

References

[1] Artin M, Schelter W. Graded algebras of global dimension 3: Adv Math, 1987, 66(2): 171–216

[2] Artin M, Tate J, van den Bergh M. Some algebras associated to automorphisms of elliptic curves. In: Cartier P, Illusie L, Katz N M, Laumon G, Manin Yu I, Ribet K A, eds. The Grothendieck Festschrift, Vol I. Progr Math, Vol 86. Basel: Birkhauser, 1990, 33–85

[3] Artin M, Tate J, van den Bergh M. Modules over regular algebras of dimension 3: Invent Math, 1991, 106(2): 335–388

[4] Caenepeel S, Ion B, Militaru G, Zhu S. The factorization problem and the smash biproduct of algebras and coalgebras. Algebr Represent Theory, 2000, 3(1): 19–42

[5] Čap A, Schichl H, VanČura J. On twisted tensor products of algebras. Comm Algebra, 1995, 23(12): 4701–4735

[6] Kirkman E, Musson I M, Passman D S. Noetherian down-up algebras. Proc Amer Math Soc, 1999, 127(11): 3161–3167

[7] Li H S. Gröbner Bases in Ring Theory. Singapore: World Scientific Publishing Co Pte Ltd, 2012

[8] Shen Y, Zhou G S, Lu D M. Homogeneous PBW-deformation for Artin-Schelter regular algebras. Bull Aust Math Soc, 2015, 91(1): 53–68

[9] Van Daele A, Van Keer S. The Yang-Baxter and pentagon equation. Compos Math, 1994, 91(2): 201–221

[10] Zhang J J,Zhang J. Double Ore extension. J Pure Appl Algebra, 2008, 212(12): 2668{2690

[11] Zhang J J, Zhang J. Double extension regular algebras of type (14641): J Algebra, 2009, 322(2): 373–409

RIGHTS & PERMISSIONS

2018 Higher Education Press and Springer-Verlag GmbH Germany, part of Springer Nature