Non-commutative $\mathbb {P}^1$-bundles over commutative schemes
HTML articles powered by AMS MathViewer
- by M. Van den Bergh PDF
- Trans. Amer. Math. Soc. 364 (2012), 6279-6313 Request permission
Abstract:
In this paper we develop the theory of non-commutative $\mathbb {P}^1$-bundles over commutative (smooth) schemes. Such non-commutative $\mathbb {P}^1$-bundles occur in the theory of $D$-modules but our definition is more general. We can show that every non-commutative deformation of a Hirzebruch surface is given by a non-commutative $\mathbb {P}^1$-bundle over $\mathbb {P}^1$ in our sense.References
- M. Artin and J. J. Zhang, Noncommutative projective schemes, Adv. Math. 109 (1994), no. 2, 228–287. MR 1304753, DOI 10.1006/aima.1994.1087
- M. Artin and J. J. Zhang, Abstract Hilbert schemes, Algebr. Represent. Theory 4 (2001), no. 4, 305–394. MR 1863391, DOI 10.1023/A:1012006112261
- M. Artin, J. Tate, and M. Van den Bergh, Some algebras associated to automorphisms of elliptic curves, The Grothendieck Festschrift, Vol. I, Progr. Math., vol. 86, Birkhäuser Boston, Boston, MA, 1990, pp. 33–85. MR 1086882
- M. Artin and M. Van den Bergh, Twisted homogeneous coordinate rings, J. Algebra 133 (1990), no. 2, 249–271. MR 1067406, DOI 10.1016/0021-8693(90)90269-T
- David Ben-Zvi and Thomas Nevins, Perverse bundles and Calogero-Moser spaces, Compos. Math. 144 (2008), no. 6, 1403–1428. MR 2474315, DOI 10.1112/S0010437X0800359X
- A. I. Bondal and M. M. Kapranov, Representable functors, Serre functors, and reconstructions, Izv. Akad. Nauk SSSR Ser. Mat. 53 (1989), no. 6, 1183–1205, 1337 (Russian); English transl., Math. USSR-Izv. 35 (1990), no. 3, 519–541. MR 1039961, DOI 10.1070/IM1990v035n03ABEH000716
- A. I. Bondal and A. E. Polishchuk, Homological properties of associative algebras: the method of helices, Izv. Ross. Akad. Nauk Ser. Mat. 57 (1993), no. 2, 3–50 (Russian, with Russian summary); English transl., Russian Acad. Sci. Izv. Math. 42 (1994), no. 2, 219–260. MR 1230966, DOI 10.1070/IM1994v042n02ABEH001536
- D. Chan and A. Nyman, Non-commutative Mori contractions and $\mathbb {P}^1$-bundles, arXiv:0904.1717.
- Olivier De Deken and Wendy Lowen, Abelian and derived deformations in the presence of $\Bbb Z$-generating geometric helices, J. Noncommut. Geom. 5 (2011), no. 4, 477–505. MR 2838522, DOI 10.4171/JNCG/83
- Robin Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York-Heidelberg, 1977. MR 0463157
- Cohomologie $l$-adique et fonctions $L$, Lecture Notes in Mathematics, Vol. 589, Springer-Verlag, Berlin-New York, 1977 (French). Séminaire de Géometrie Algébrique du Bois-Marie 1965–1966 (SGA 5); Edité par Luc Illusie. MR 0491704
- János Kollár and Shigefumi Mori, Birational geometry of algebraic varieties, Cambridge Tracts in Mathematics, vol. 134, Cambridge University Press, Cambridge, 1998. With the collaboration of C. H. Clemens and A. Corti; Translated from the 1998 Japanese original. MR 1658959, DOI 10.1017/CBO9780511662560
- M. Kontsevich, Course notes from the E.N.S., 1998.
- Wendy Lowen and Michel Van den Bergh, Deformation theory of abelian categories, Trans. Amer. Math. Soc. 358 (2006), no. 12, 5441–5483. MR 2238922, DOI 10.1090/S0002-9947-06-03871-2
- Wendy Lowen, Obstruction theory for objects in abelian and derived categories, Comm. Algebra 33 (2005), no. 9, 3195–3223. MR 2175388, DOI 10.1081/AGB-200066155
- Izuru Mori, Intersection theory over quantum ruled surfaces, J. Pure Appl. Algebra 211 (2007), no. 1, 25–41. MR 2333760, DOI 10.1016/j.jpaa.2006.12.006
- —, Quantum ruled surfaces defined by quivers, in preparation.
- C. Năstăsescu and F. van Oystaeyen, Graded ring theory, North-Holland Mathematical Library, vol. 28, North-Holland Publishing Co., Amsterdam-New York, 1982. MR 676974
- A. Nyman, The Eilenberg-Watts theorem over schemes, arXiv:0902.4886.
- Adam Nyman, Points on quantum projectivizations, Mem. Amer. Math. Soc. 167 (2004), no. 795, vi+142. MR 2026268, DOI 10.1090/memo/0795
- Adam Nyman, Serre finiteness and Serre vanishing for non-commutative $\Bbb P^1$-bundles, J. Algebra 278 (2004), no. 1, 32–42. MR 2068065, DOI 10.1016/j.jalgebra.2004.04.003
- Adam Nyman, Serre duality for non-commutative ${\Bbb P}^1$-bundles, Trans. Amer. Math. Soc. 357 (2005), no. 4, 1349–1416. MR 2115370, DOI 10.1090/S0002-9947-04-03523-8
- D. Patrick, Non-commutative symmetric algebras, Ph.D. thesis, MIT, 1997.
- David Patrick, Noncommutative symmetric algebras of two-sided vector spaces, J. Algebra 233 (2000), no. 1, 16–36. MR 1793588, DOI 10.1006/jabr.2000.8445
- A. Polishchuk, Noncommutative proj and coherent algebras, Math. Res. Lett. 12 (2005), no. 1, 63–74. MR 2122731, DOI 10.4310/MRL.2005.v12.n1.a7
- Alexander L. Rosenberg, Noncommutative algebraic geometry and representations of quantized algebras, Mathematics and its Applications, vol. 330, Kluwer Academic Publishers Group, Dordrecht, 1995. MR 1347919, DOI 10.1007/978-94-015-8430-2
- Susan J. Sierra, $G$-algebras, twistings, and equivalences of graded categories, Algebr. Represent. Theory 14 (2011), no. 2, 377–390. MR 2776790, DOI 10.1007/s10468-009-9193-y
- M. Van den Bergh, Non-commutative quadrics, arXiv:0807.375.
- —, Notes on formal deformations of abelian categories, arXiv:1002.0259.
- Michel Van den Bergh, A translation principle for the four-dimensional Sklyanin algebras, J. Algebra 184 (1996), no. 2, 435–490. MR 1409223, DOI 10.1006/jabr.1996.0269
- Michel Van den Bergh, Blowing up of non-commutative smooth surfaces, Mem. Amer. Math. Soc. 154 (2001), no. 734, x+140. MR 1846352, DOI 10.1090/memo/0734
Additional Information
- M. Van den Bergh
- Affiliation: Department of Mathematics, Universiteit Hasselt, 3590 Diepenbeek, Belgium
- MR Author ID: 176980
- Email: michel.vandenbergh@uhasselt.be
- Received by editor(s): February 15, 2010
- Received by editor(s) in revised form: September 20, 2010
- Published electronically: July 11, 2012
- Additional Notes: The author is a senior researcher at the FWO
- © Copyright 2012
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 364 (2012), 6279-6313
- MSC (2010): Primary 18E15; Secondary 14D15
- DOI: https://doi.org/10.1090/S0002-9947-2012-05469-9
- MathSciNet review: 2958936