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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

   

 

Non-commutative $ \mathbb{P}^1$-bundles over commutative schemes


Author: M. Van den Bergh
Journal: Trans. Amer. Math. Soc. 364 (2012), 6279-6313
MSC (2010): Primary 18E15; Secondary 14D15
Published electronically: July 11, 2012
MathSciNet review: 2958936
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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.


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  • 1. M. Artin and J. Zhang, Noncommutative projective schemes, Adv. in Math. 109 (1994), no. 2, 228-287. MR 1304753 (96a:14004)
  • 2. -, Abstract Hilbert schemes. I., Algebr. Represent. Theory 4 (2001), no. 4, 305-394 (English). MR 1863391 (2002h:16046)
  • 3. M. Artin, J. Tate, and M. Van den Bergh, Some algebras associated to automorphisms of elliptic curves, The Grothendieck Festschrift, vol. 1, Birkhäuser, 1990, pp. 33-85. MR 1086882 (92e:14002)
  • 4. M. Artin and M. Van den Bergh, Twisted homogeneous coordinate rings, J. Algebra 188 (1990), 249-271. MR 1067406 (91k:14003)
  • 5. D. Ben-Zvi and T. Nevins, Perverse bundles and Calogero-Moser spaces, Compos. Math. 144 (2008), no. 6, 1403-1428. MR 2474315 (2009i:14012)
  • 6. A. Bondal and M. Kapranov, Representable functors, Serre functors, and reconstructions, Izv. Akad. Nauk SSSR Ser. Mat. 53 (1989), no. 6, 1183-1205, 1337. MR 1039961 (91b:14013)
  • 7. A. Bondal and A. Polishchuk, Homological properties of associative algebras: the method of helices, Russian Acad. Sci. Izv. Math 42 (1994), 219-260. MR 1230966 (94m:16011)
  • 8. D. Chan and A. Nyman, Non-commutative Mori contractions and $ \mathbb{P}^1$-bundles, arXiv:0904.1717.
  • 9. O. De Deken and W. Lowen, Abelian and derived deformations in the presence of Z-generating geometric helices, J. Noncommut. Geom. 5 (2011), no. 4, 477-505. MR 2838522
  • 10. R. Hartshorne, Algebraic geometry, Springer-Verlag, 1977. MR 0463157 (57:3116)
  • 11. J. P. Jouanolou, Systèmes projectifs $ J$-adiques, Cohomologie $ l$-adique et fonctions $ L$, SGA5 (Berlin), Lecture Notes in Mathematics, vol. 589, Springer Verlag, Berlin, 1977. MR 0491704 (58:10907)
  • 12. J. Kollár and S. Mori, Birational geometry of algebraic varieties, Cambridge University Press, Cambridge, 1998. MR 1658959 (2000b:14018)
  • 13. M. Kontsevich, Course notes from the E.N.S., 1998.
  • 14. W. Lowen and M. Van den Bergh, Deformation theory of abelian categories, Trans. Amer. Math. Soc. 358 (2006), no. 12, 5441-5483. MR 2238922 (2008b:18016)
  • 15. Wendy Lowen, Obstruction theory for objects in abelian and derived categories, Comm. Algebra 33 (2005), no. 9, 3195–3223. MR 2175388 (2006k:18016), 10.1081/AGB-200066155
  • 16. I. Mori, Intersection theory over quantum ruled surfaces, J. Pure Appl. Algebra 211 (2007), no. 1, 25-41. MR 2333760 (2008i:14001)
  • 17. -, Quantum ruled surfaces defined by quivers, in preparation.
  • 18. C. Nastacescu and F. Van Oystaeyen, Graded ring theory, North-Holland, Amsterdam, 1982. MR 676974 (84i:16002)
  • 19. A. Nyman, The Eilenberg-Watts theorem over schemes, arXiv:0902.4886.
  • 20. -, Points on quantum projectivizations, Mem. Amer. Math. Soc. 167 (2004), no. 795, vi+142. MR 2026268 (2005b:14004)
  • 21. -, Serre finiteness and Serre vanishing for non-commutative $ \Bbb P^1$-bundles, J. Algebra 278 (2004), no. 1, 32-42. MR 2068065 (2005f:14003)
  • 22. -, Serre duality for non-commutative $ {\Bbb P}^1$-bundles, Trans. Amer. Math. Soc. 357 (2005), no. 4, 1349-1416 (electronic). MR 2115370 (2006c:14002)
  • 23. D. Patrick, Non-commutative symmetric algebras, Ph.D. thesis, MIT, 1997.
  • 24. -, Noncommutative symmetric algebras of two-sided vector spaces, J. Algebra 233 (2000), no. 1, 16-36. MR 1793588 (2001i:16042)
  • 25. A. Polishchuk, Noncommutative proj and coherent algebras, Math. Res. Lett. 12 (2005), no. 1, 63-74. MR 2122731 (2005k:14003)
  • 26. A. L. Rosenberg, Non-commutative algebraic geometry and representations of quantized algebras, Mathematics and Its Applications, vol. 330, Kluwer Academic Publishers, Dordrecht, 1995. MR 1347919 (97b:14004)
  • 27. S. Sierra, G-algebras, twistings, and equivalences of graded categories, arXiv:math/0608791. MR 2776790 (2012d:16128)
  • 28. M. Van den Bergh, Non-commutative quadrics, arXiv:0807.375.
  • 29. -, Notes on formal deformations of abelian categories, arXiv:1002.0259.
  • 30. -, A translation principle for Sklyanin algebras, J. Algebra 184 (1996), 435-490. MR 1409223 (98a:16047)
  • 31. -, Blowing up of non-commutative smooth surfaces, Mem. Amer. Math. Soc. 154 (2001), no. 734, x+140. MR 1846352 (2002k:16057)

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Additional Information

M. Van den Bergh
Affiliation: Department of Mathematics, Universiteit Hasselt, 3590 Diepenbeek, Belgium
Email: michel.vandenbergh@uhasselt.be

DOI: http://dx.doi.org/10.1090/S0002-9947-2012-05469-9
Keywords: Non-commutative geometry, Hirzebruch surfaces, deformations
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
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.