Tilting Bundles on Rational Surfaces and Quasi-Hereditary Algebras

Lutz Hille[1]; Markus Perling[2]

  • [1] Universität Münster Mathematisches Institut Einsteinstr. 62 D–48149 Münster (Germany)
  • [2] Ruhr-Universität Bochum Fakultät für Mathematik Universitätsstrasse 150 D–44780 Bochum (Germany)

Annales de l’institut Fourier (2014)

  • Volume: 64, Issue: 2, page 625-644
  • ISSN: 0373-0956

Abstract

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Let X be any rational surface. We construct a tilting bundle T on X . Moreover, we can choose T in such way that its endomorphism algebra is quasi-hereditary. In particular, the bounded derived category of coherent sheaves on X is equivalent to the bounded derived category of finitely generated modules over a finite dimensional quasi-hereditary algebra A . The construction starts with a full exceptional sequence of line bundles on X and uses universal extensions. If X is any smooth projective variety with a full exceptional sequence of coherent sheaves (or vector bundles, or even complexes of coherent sheaves) with all groups Ext q for q ≥ 2 vanishing, then X also admits a tilting sheaf (tilting bundle, or tilting complex, respectively) obtained as a universal extension of this exceptional sequence.

How to cite

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Hille, Lutz, and Perling, Markus. "Tilting Bundles on Rational Surfaces and Quasi-Hereditary Algebras." Annales de l’institut Fourier 64.2 (2014): 625-644. <http://eudml.org/doc/275505>.

@article{Hille2014,
abstract = {Let $X$ be any rational surface. We construct a tilting bundle $T$ on $X$. Moreover, we can choose $T$ in such way that its endomorphism algebra is quasi-hereditary. In particular, the bounded derived category of coherent sheaves on $X$ is equivalent to the bounded derived category of finitely generated modules over a finite dimensional quasi-hereditary algebra $A$. The construction starts with a full exceptional sequence of line bundles on $X$ and uses universal extensions. If $X$ is any smooth projective variety with a full exceptional sequence of coherent sheaves (or vector bundles, or even complexes of coherent sheaves) with all groups $\rm \{Ext\}^q$ for $q \ge 2$ vanishing, then $X$ also admits a tilting sheaf (tilting bundle, or tilting complex, respectively) obtained as a universal extension of this exceptional sequence.},
affiliation = {Universität Münster Mathematisches Institut Einsteinstr. 62 D–48149 Münster (Germany); Ruhr-Universität Bochum Fakultät für Mathematik Universitätsstrasse 150 D–44780 Bochum (Germany)},
author = {Hille, Lutz, Perling, Markus},
journal = {Annales de l’institut Fourier},
keywords = {tilting bundle; rational surface; quasi-hereditary algebra},
language = {eng},
number = {2},
pages = {625-644},
publisher = {Association des Annales de l’institut Fourier},
title = {Tilting Bundles on Rational Surfaces and Quasi-Hereditary Algebras},
url = {http://eudml.org/doc/275505},
volume = {64},
year = {2014},
}

TY - JOUR
AU - Hille, Lutz
AU - Perling, Markus
TI - Tilting Bundles on Rational Surfaces and Quasi-Hereditary Algebras
JO - Annales de l’institut Fourier
PY - 2014
PB - Association des Annales de l’institut Fourier
VL - 64
IS - 2
SP - 625
EP - 644
AB - Let $X$ be any rational surface. We construct a tilting bundle $T$ on $X$. Moreover, we can choose $T$ in such way that its endomorphism algebra is quasi-hereditary. In particular, the bounded derived category of coherent sheaves on $X$ is equivalent to the bounded derived category of finitely generated modules over a finite dimensional quasi-hereditary algebra $A$. The construction starts with a full exceptional sequence of line bundles on $X$ and uses universal extensions. If $X$ is any smooth projective variety with a full exceptional sequence of coherent sheaves (or vector bundles, or even complexes of coherent sheaves) with all groups $\rm {Ext}^q$ for $q \ge 2$ vanishing, then $X$ also admits a tilting sheaf (tilting bundle, or tilting complex, respectively) obtained as a universal extension of this exceptional sequence.
LA - eng
KW - tilting bundle; rational surface; quasi-hereditary algebra
UR - http://eudml.org/doc/275505
ER -

References

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  1. W. Barth, C. Peters, A. Van de Ven, Compact complex surfaces, 4 (1984), Springer-Verlag, Berlin Zbl1036.14016MR749574
  2. A. A. Beĭlinson, Coherent sheaves on P n and problems in linear algebra, Funktsional. Anal. i Prilozhen. 12 (1978), 68-69 Zbl0402.14006MR509388
  3. A. I. Bondal, Representations of associative algebras and coherent sheaves, Izv. Akad. Nauk SSSR Ser. Mat. 53 (1989), 25-44 Zbl0692.18002MR992977
  4. A. I. Bondal, M. van den Bergh, Generators and representability of functors in commutative and noncommutative geometry, Mosc. Math. J. 3 (2003), 1-36, 258 Zbl1135.18302MR1996800
  5. Ragnar-Olaf Buchweitz, Lutz Hille, Hochschild (co-)homology of schemes with tilting object, Trans. Amer. Math. Soc. 365 (2013), 2823-2844 Zbl1274.14018MR3034449
  6. Vlastimil Dlab, Claus Michael Ringel, The module theoretical approach to quasi-hereditary algebras, Representations of algebras and related topics (Kyoto, 1990) 168 (1992), 200-224, Cambridge Univ. Press, Cambridge Zbl0793.16006MR1211481
  7. William Fulton, Introduction to toric varieties, 131 (1993), Princeton University Press, Princeton, NJ Zbl0813.14039MR1234037
  8. L. Hille, Exceptional sequences of line bundles on toric varieties, Mathematisches Institut, Georg-August-Universität Göttingen: Seminars 2003/2004 (2004), 175-190, Universitätsdrucke Göttingen, Göttingen Zbl1098.14524MR2181579
  9. Lutz Hille, Markus Perling, Exceptional sequences of invertible sheaves on rational surfaces, Compos. Math. 147 (2011), 1230-1280 Zbl1237.14043MR2822868
  10. M. M. Kapranov, On the derived categories of coherent sheaves on some homogeneous spaces, Invent. Math. 92 (1988), 479-508 Zbl0651.18008MR939472
  11. Yujiro Kawamata, Derived categories of toric varieties, Michigan Math. J. 54 (2006), 517-535 Zbl1159.14026MR2280493
  12. Tadao Oda, Convex bodies and algebraic geometry, 15 (1988), Springer-Verlag, Berlin Zbl0628.52002MR922894
  13. D. O. Orlov, Projective bundles, monoidal transformations, and derived categories of coherent sheaves, Izv. Ross. Akad. Nauk Ser. Mat. 56 (1992), 852-862 Zbl0798.14007MR1208153

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