Kyoto Journal of Mathematics

Graded Morita equivalences for generic Artin-Schelter regular algebras

Kenta Ueyama

Full-text: Open access


Let A=A(E,σ),A'=A(E',σ') be Noetherian Artin-Schelter regular geometric algebras with dimkA1=dimkA1'=n, and let ν,ν' be generalized Nakayama automorphisms of A,A'. In this paper, we study relationships between the conditions

(A) A is graded Morita equivalent to A', and

(B) A(E,νσn) is isomorphic to A(E',(ν')(σ')n) as graded algebras.

It is proved that if A,A' are “generic” 3-dimensional quadratic Artin-Schelter regular algebras, then (A) is equivalent to (B), and if A,A' are n-dimensional skew polynomial algebras, then (A) implies (B).

Article information

Kyoto J. Math., Volume 51, Number 2 (2011), 485-501.

First available in Project Euclid: 22 April 2011

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 16W50: Graded rings and modules 16D90: Module categories [See also 16Gxx, 16S90]; module theory in a category-theoretic context; Morita equivalence and duality 16S38: Rings arising from non-commutative algebraic geometry [See also 14A22] 16S37: Quadratic and Koszul algebras


Ueyama, Kenta. Graded Morita equivalences for generic Artin-Schelter regular algebras. Kyoto J. Math. 51 (2011), no. 2, 485--501. doi:10.1215/21562261-1214420.

Export citation


  • [1] M. Artin and W. Schelter, Graded algebras of global dimension 3, Adv. Math. 66 (1987), 171–216.
  • [2] M. Artin, J. Tate, and M. Van den Bergh, “Some algebras associated to automorphisms of elliptic curves” in The Grothendieck Festschrift, Vol. I, Progr. Math. 86, Birkhauser, Boston, 1990, 33–85.
  • [3] Y. Hattori, Noncommutative projective schemes of quantum affine coordinate rings which are birational but not isomorphic, preprint.
  • [4] P. Jørgensen, Local cohomology for non-commutative graded algebras, Comm. Algebra 25 (1997), 575–591.
  • [5] I. Mori, “Noncommutative projective schemes and point schemes” in Algebras, Rings and Their Representations, World Sci., Hackensack, N.J., 2006, 215–239.
  • [6] I. Mori, Co-point modules over Frobenius Koszul algebras, Comm. Algebra 36 (2008), 4659–4677.
  • [7] I. Mori, Asymmetry of Ext-groups, J. Algebra 322 (2009), 2235–2250.
  • [8] S. P. Smith, “Some finite-dimensional algebras related to elliptic curves” in Representation Theory of Algebras and Related Topics (Mexico City, 1994), CMS Conf. Proc. 19, Amer. Math. Soc., Providence, 1996, 315–348.
  • [9] M. Van den Bergh, Existence theorems for dualizing complexes over non-commutative graded and filtered rings, J. Algebra 195 (1997), 662–679.
  • [10] J. Vitoria, Equivalences for noncommutative projective spaces, preprint, arXiv:1001.4400v2 [math.RA]