The Dixmier-Moeglin equivalence for twisted homogeneous coordinate rings. (English) Zbl 1219.16030
Suppose that \(k\) is an uncountable algebraically closed field of characteristic zero. Let \(X\) be a projective variety of dimension at most 2 with an automorphism \(\sigma\) and with a \(\sigma\)-ample invertible sheaf \(L\). Then the twisted homogeneous coordinate \(k\)-algebra \(B(X,L,\sigma)\) satisfies Dixmier-Moeglin equivalence. If \(S\) is a commutative affine \(k\)-algebra with an automorphism \(\sigma\) and \(\dim S\leqslant 2\), then the algebras \(S[t;\sigma]\), \(S[t^\pm;\sigma]\) satisfy Dixmier-Moeglin equivalence.
Reviewer: Vyacheslav A. Artamonov (Moskva)
MSC:
| 16S38 | Rings arising from noncommutative algebraic geometry |
| 14A22 | Noncommutative algebraic geometry |
| 16S35 | Twisted and skew group rings, crossed products |
| 16S60 | Associative rings of functions, subdirect products, sheaves of rings |
References:
| [1] | M. Artin and J. T. Stafford, Noncommutative graded domains with quadratic growth, Inventiones Mathematicae 122 (1995), 231–276. · Zbl 0849.16022 · doi:10.1007/BF01231444 |
| [2] | M. Artin, J. Tate and M. Van den Bergh, Some algebras associated to automorphisms of elliptic curves, in The Grothendieck Festschrift, Vol. I, Birkhäuser Boston, Boston, MA, 1990, pp. 33–85. · Zbl 0744.14024 |
| [3] | M. Artin and M. Van den Bergh, Twisted homogeneous coordinate rings, Journal of Algebra 133 (1990), 249–271. · Zbl 0717.14001 · doi:10.1016/0021-8693(90)90269-T |
| [4] | J. P. Bell, Noetherian algebras over algebraically closed fields, Journal of Algebra 310 (2007), 148–155. · Zbl 1117.16010 · doi:10.1016/j.jalgebra.2006.08.026 |
| [5] | G. M. Bergman, Zero-divisors in tensor products, in Noncommutative Ring Theory (Internat. Conf., Kent State Univ., Kent, Ohio, 1975), Lecture Notes in Mathematics 545, Springer, Berlin, 1976, pp. 32–82. |
| [6] | K. A. Brown and K. R. Goodearl, Lectures on Algebraic Quantum Groups, Advanced Courses in Mathematics, CRM Barcelona, Birkhäuser Verlag, Basel, 2002. · Zbl 1027.17010 |
| [7] | D. A. Cox, The homogeneous coordinate ring of a toric variety, Journal of Algebraic Geometry 4 (1995), 17–50. · Zbl 0846.14032 |
| [8] | J. Diller and C. Favre, Dynamics of bimeromorphic maps of surfaces, American Journal of Mathematics 123 (2001), 1135–1169. · Zbl 1112.37308 · doi:10.1353/ajm.2001.0038 |
| [9] | I. Dolgachev, Infinite Coxeter groups and automorphisms of algebraic surfaces, in The Lefschetz Centennial Conference, Part I (Mexico City, 1984), Contemporory Mathematics, Vol. 58, American Mathematical Society, Providence, RI, 1986, pp. 91–106. |
| [10] | S. Friedland and J. Milnor, Dynamical properties of plane polynomial automorphisms, Ergodic Theory and Dynamical Systems 9 (1989), 67–99. · Zbl 0651.58027 · doi:10.1017/S014338570000482X |
| [11] | K. R. Goodearl and J. T. Stafford, The graded version of Goldie’s theorem, in Algebra and its Applications (Athens, OH, 1999), Contemporary Mathematics, Vol. 259, American Mathematical Society, Providence, RI, 2000, pp. 237–240. · Zbl 0985.16031 |
| [12] | R. Hartshorne, Algebraic Geometry, Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York, 1977. · Zbl 0367.14001 |
| [13] | M. Hanamura, On the birational automorphism groups of algebraic varieties, Compositio Mathematica 63 (1987), 123–142. · Zbl 0655.14007 |
| [14] | R. S. Irving, Primitive ideals of certain Noetherian algebras, Mathematische Zeitschrift 169 (1979), 77–92. · doi:10.1007/BF01214914 |
| [15] | D. A. Jordan, Primitivity in skew Laurent polynomial rings and related rings, Mathematische Zeitschrift 213 (1993), 353–371. · Zbl 0797.16037 · doi:10.1007/BF03025725 |
| [16] | D. S. Keeler, Criteria for {\(\sigma\)}-ampleness, Journal of the American Mathematical Society 13 (2000), 517–532. · Zbl 0952.14002 · doi:10.1090/S0894-0347-00-00334-9 |
| [17] | S. Lang, Abelian Varieties, Springer-Verlag, New York, 1983, (Reprint of the 1959 original). · Zbl 0516.14031 |
| [18] | S. Lang, Fundamentals of Diophantine Geometry, Springer-Verlag, New York, 1983. · Zbl 0528.14013 |
| [19] | R. Lazarsfeld, Positivity in Algebraic Geometry. I, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, Vol. 48, Springer-Verlag, Berlin, 2004, Classical setting: line bundles and linear series. |
| [20] | A. Leroy and J. Matczuk, Primitivity of skew polynomial and skew Laurent polynomial rings, Communications in Algebra 24 (1996), 2271–2284. · Zbl 0863.16020 · doi:10.1080/00927879608825699 |
| [21] | M. Lorenz, Primitive ideals of group algebras of supersoluble groups, Mathematische Annalen 225 (1977), 115–122. · Zbl 0341.16003 · doi:10.1007/BF01351715 |
| [22] | H. Matsumura, Commutative Ring Theory, Cambridge Studies in Advanced Mathematics, Vol. 8, Cambridge University Press, Cambridge, 1986, (Translated from the Japanese by M. Reid). · Zbl 0603.13001 |
| [23] | J. C. McConnell and J. C. Robson, Noncommutative Noetherian Rings, revised edn., American Mathematical Society, Providence, RI, 2001. · Zbl 0980.16019 |
| [24] | D. Mumford, Abelian Varieties, Tata Institute of Fundamental Research Studies in Mathematics, No. 5, Tata Institute of Fundamental Research, Bombay, 1970. · Zbl 0223.14022 |
| [25] | C. Năstăsescu and F. van Oystaeyen, Graded Ring Theory, North-Holland Publ. Co., Amsterdam, 1982. |
| [26] | D. Rogalski, GK-dimension of birationally commutative surfaces, Transactions of the American Mathematical Society 361 (2009), 5921–5945. · Zbl 1181.14005 · doi:10.1090/S0002-9947-09-04885-5 |
| [27] | D. Rogalski and J. T. Stafford, A class of noncommutative projective surfaces, to appear in Proceedings of the London Mathematical Society, arXiv:math/0612657. · Zbl 1173.14005 |
| [28] | D. Rogalski and J. T. Stafford, Naïve noncommutative blowups at zero-dimensional schemes, Journal of Algebra 318 (2007), 794–833. · Zbl 1141.14001 · doi:10.1016/j.jalgebra.2007.02.017 |
| [29] | D. Rogalski and J. J. Zhang, Canonical maps to twisted rings, Mathematische Zeitschrift 259 (2008), 433–455. · Zbl 1170.16021 · doi:10.1007/s00209-006-0964-4 |
| [30] | L. H. Rowen and L. Small, Primitive ideals of algebras, Communications in Algebra 25 (1997), 3853–3857. · Zbl 0904.16003 · doi:10.1080/00927879708826091 |
| [31] | Théorie des intersections et théor‘eme de Riemann-Roch (French), Séminaire de Géométrie Algébrique du Bois-Marie 1966–1967 (SGA 6), Dirigé par P. Berthelot, A. Grothendieck et L. Illusie, avec la collaboration de D. Ferrand, J. P. Jouanolou, O. Jussila, S. Kleiman, M. Raynaud et J. P. Serre, Lecture Notes in Mathematics 225, Springer-Verlag, Berlin-New York, 1971. |
| [32] | S. J. Sierra, Geometric idealizers, Transactions of the American Mathematical Society, to appear. |
| [33] | L. W. Small, Prime ideals in Noetherian PI-rings, American Mathematical Society. Bulletin 79 (1973), 421–422. · Zbl 0278.16012 · doi:10.1090/S0002-9904-1973-13196-9 |
| [34] | T. A. Springer, Linear Algebraic Groups, second edn., Progress in Mathematics, Vol. 9, Birkhäuser Boston Inc., Boston, MA, 1998. · Zbl 0927.20024 |
| [35] | D. Q. Zhang, Dynamics of automorphisms on projective complex manifolds, Journal of Differential Geometry 82 (2009), 691–722. · Zbl 1187.14061 |
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.
