## Derived categories of projective bundles

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- by L. Costa and R. M. Miró-Roig PDF
- Proc. Amer. Math. Soc.
**133**(2005), 2533-2537 Request permission

## Abstract:

The goal of this short note is to prove that any projective bundle $\mathbb {P}(\mathcal {E}) \rightarrow X$ has a tilting bundle whose summands are line bundles whenever the same holds for $X$.## References

- I. N. Bernšteĭn, I. M. Gel′fand, and S. I. Gel′fand,
*Algebraic vector bundles on $\textbf {P}^{n}$ and problems of linear algebra*, Funktsional. Anal. i Prilozhen.**12**(1978), no. 3, 66–67 (Russian). MR**509387** - A. I. Bondal,
*Representations of associative algebras and coherent sheaves*, Izv. Akad. Nauk SSSR Ser. Mat.**53**(1989), no. 1, 25–44 (Russian); English transl., Math. USSR-Izv.**34**(1990), no. 1, 23–42. MR**992977**, DOI 10.1070/IM1990v034n01ABEH000583 - Alexei Bondal and Dmitri Orlov,
*Reconstruction of a variety from the derived category and groups of autoequivalences*, Compositio Math.**125**(2001), no. 3, 327–344. MR**1818984**, DOI 10.1023/A:1002470302976 - Tom Bridgeland and Antony Maciocia,
*Complex surfaces with equivalent derived categories*, Math. Z.**236**(2001), no. 4, 677–697. MR**1827500**, DOI 10.1007/PL00004847 - Tom Bridgeland,
*Fourier-Mukai transforms for elliptic surfaces*, J. Reine Angew. Math.**498**(1998), 115–133. MR**1629929**, DOI 10.1515/crll.1998.046 - L. Costa, R.M. Miró-Roig,
*Tilting bundles on toric varieties*, Preprint, Univ. Barcelona (2003). - A. L. Gorodentsev and A. N. Rudakov,
*Exceptional vector bundles on projective spaces*, Duke Math. J.**54**(1987), no. 1, 115–130. MR**885779**, DOI 10.1215/S0012-7094-87-05409-3 - Robin Hartshorne,
*Algebraic geometry*, Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York-Heidelberg, 1977. MR**0463157** - A. King,
*Tilting bundles on some rational surfaces*, Preprint at http://www.maths.bath.ac.uk/masadk/papers/. - Maxim Kontsevich,
*Homological algebra of mirror symmetry*, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994) Birkhäuser, Basel, 1995, pp. 120–139. MR**1403918** - Antony Maciocia,
*Generalized Fourier-Mukai transforms*, J. Reine Angew. Math.**480**(1996), 197–211. MR**1420564**, DOI 10.1515/crll.1996.480.197 - Shigeru Mukai,
*Fourier functor and its application to the moduli of bundles on an abelian variety*, Algebraic geometry, Sendai, 1985, Adv. Stud. Pure Math., vol. 10, North-Holland, Amsterdam, 1987, pp. 515–550. MR**946249**, DOI 10.2969/aspm/01010515 - D. O. Orlov,
*Projective bundles, monoidal transformations, and derived categories of coherent sheaves*, Izv. Ross. Akad. Nauk Ser. Mat.**56**(1992), no. 4, 852–862 (Russian, with Russian summary); English transl., Russian Acad. Sci. Izv. Math.**41**(1993), no. 1, 133–141. MR**1208153**, DOI 10.1070/IM1993v041n01ABEH002182 *Helices and vector bundles*, London Mathematical Society Lecture Note Series, vol. 148, Cambridge University Press, Cambridge, 1990. Seminaire Rudakov; Translated from the Russian by A. D. King, P. Kobak and A. Maciocia. MR**1074776**

## Additional Information

**L. Costa**- Affiliation: Facultat de Matemàtiques, Departament d’Algebra i Geometria, Universitat de Barcelona, Gran Via de les Corts Catalanes 585, 08007 Barcelona, Spain
- Email: costa@ub.edu
**R. M. Miró-Roig**- Affiliation: Facultat de Matemàtiques, Departament d’Algebra i Geometria, Universitat de Barcelona, Gran Via de les Corts Catalanes 585, 08007 Barcelona, Spain
- MR Author ID: 125375
- ORCID: 0000-0003-1375-6547
- Email: miro@ub.edu
- Received by editor(s): July 15, 2003
- Received by editor(s) in revised form: May 17, 2004
- Published electronically: April 8, 2005
- Additional Notes: The first author was partially supported by MTM2004-00666

The second author was partially supported by MTM2004-00666 - Communicated by: Michael Stillman
- © Copyright 2005 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**133**(2005), 2533-2537 - MSC (2000): Primary 14F05; Secondary 14M25
- DOI: https://doi.org/10.1090/S0002-9939-05-07846-9
- MathSciNet review: 2146195