# 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

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topHille, 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

top- W. Barth, C. Peters, A. Van de Ven, Compact complex surfaces, 4 (1984), Springer-Verlag, Berlin Zbl1036.14016MR749574
- A. A. Beĭlinson, Coherent sheaves on ${\mathbf{P}}^{n}$ and problems in linear algebra, Funktsional. Anal. i Prilozhen. 12 (1978), 68-69 Zbl0402.14006MR509388
- A. I. Bondal, Representations of associative algebras and coherent sheaves, Izv. Akad. Nauk SSSR Ser. Mat. 53 (1989), 25-44 Zbl0692.18002MR992977
- 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
- Ragnar-Olaf Buchweitz, Lutz Hille, Hochschild (co-)homology of schemes with tilting object, Trans. Amer. Math. Soc. 365 (2013), 2823-2844 Zbl1274.14018MR3034449
- 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
- William Fulton, Introduction to toric varieties, 131 (1993), Princeton University Press, Princeton, NJ Zbl0813.14039MR1234037
- 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
- Lutz Hille, Markus Perling, Exceptional sequences of invertible sheaves on rational surfaces, Compos. Math. 147 (2011), 1230-1280 Zbl1237.14043MR2822868
- M. M. Kapranov, On the derived categories of coherent sheaves on some homogeneous spaces, Invent. Math. 92 (1988), 479-508 Zbl0651.18008MR939472
- Yujiro Kawamata, Derived categories of toric varieties, Michigan Math. J. 54 (2006), 517-535 Zbl1159.14026MR2280493
- Tadao Oda, Convex bodies and algebraic geometry, 15 (1988), Springer-Verlag, Berlin Zbl0628.52002MR922894
- 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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