Graded Morita equivalences for generic Artin-Schelter regular algebras. (English) Zbl 1228.16024

In the main result of this paper, Theorem 3.1, the author proves that the conditions
Two generic 3-dimensional Noetherian Artin-Schelter algebras \(A=\mathcal A(E,\sigma)\) and \(A'=\mathcal A(E',\sigma')\) are Morita equivalent, and
The corresponding graded algebras \(\mathcal A(E,\nu^*\sigma^n)\) and \(\mathcal A(E',(\nu')^*(\sigma')^n)\) are isomorphic,
are equivalent.
Moreover, if \(A\) and \(A'\) are \(n\)-dimensional skew polynomial algebras then (A) implies (B) (Theorem 4.1). The converse of Theorem 4.1 is not true, as it is proved in Example 4.2.


16S38 Rings arising from noncommutative algebraic geometry
16E65 Homological conditions on associative rings (generalizations of regular, Gorenstein, Cohen-Macaulay rings, etc.)
16W50 Graded rings and modules (associative rings and algebras)
16D90 Module categories in associative algebras
16S36 Ordinary and skew polynomial rings and semigroup rings
16S37 Quadratic and Koszul algebras
Full Text: DOI


[1] M. Artin and W. Schelter, Graded algebras of global dimension 3, Adv. Math. 66 (1987), 171-216. · Zbl 0633.16001 · doi:10.1016/0001-8708(87)90034-X
[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. · Zbl 0744.14024
[3] Y. Hattori, Noncommutative projective schemes of quantum affine coordinate rings which are birational but not isomorphic ,
[4] P. Jørgensen, Local cohomology for non-commutative graded algebras , Comm. Algebra 25 (1997), 575-591. · Zbl 0871.16021 · doi:10.1080/00927879708825875
[5] I. Mori, “Noncommutative projective schemes and point schemes” in Algebras, Rings and Their Representations , World Sci., Hackensack, N.J., 2006, 215-239. · Zbl 1110.14003 · doi:10.1142/9789812774552_0014
[6] I. Mori, Co-point modules over Frobenius Koszul algebras , Comm. Algebra 36 (2008), 4659-4677. · Zbl 1207.16032 · doi:10.1080/00927870802186847
[7] I. Mori, Asymmetry of Ext-groups , J. Algebra 322 (2009), 2235-2250. · Zbl 1173.13017 · doi:10.1016/j.jalgebra.2009.02.027
[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. · Zbl 0856.16009
[9] M. Van den Bergh, Existence theorems for dualizing complexes over non-commutative graded and filtered rings , J. Algebra 195 (1997), 662-679. · Zbl 0894.16020 · doi:10.1006/jabr.1997.7052
[10] J. Vitoria, Equivalences for noncommutative projective spaces , preprint, · Zbl 0473.15005
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.