Tilting Bundles on Rational Surfaces and Quasi-Hereditary Algebras
Annales de l'Institut Fourier, Volume 64 (2014) no. 2, pp. 625-644.

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 q2 vanishing, then X also admits a tilting sheaf (tilting bundle, or tilting complex, respectively) obtained as a universal extension of this exceptional sequence.

Nous construisons un faisceau basculant sur toute surface projective rationnelle lisse. Pour ce faire, nous partons d’une suite exceptionnelle complète de fibrés en droites auxquelles nous appliquons des extensions universelles. De plus, il est possible de choisir ce faisceau basculant de telle sorte que son algèbre d’endomorphismes est quasi-héréditaire.

DOI: 10.5802/aif.2860
Classification: 14J26, 16G20, 18E30
Keywords: tilting bundle, rational surface, quasi-hereditary algebra
Mot clés : fibrés basculant, surfaces rationnelles, algèbres quasi-héréditaires
Hille, Lutz 1; Perling, Markus 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)
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Hille, Lutz; Perling, Markus. Tilting Bundles on Rational Surfaces and Quasi-Hereditary Algebras. Annales de l'Institut Fourier, Volume 64 (2014) no. 2, pp. 625-644. doi : 10.5802/aif.2860. https://aif.centre-mersenne.org/articles/10.5802/aif.2860/

[1] Barth, W.; Peters, C.; Van de Ven, A. Compact complex surfaces, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 4, Springer-Verlag, Berlin, 1984, pp. x+304 | DOI | MR | Zbl

[2] Beĭlinson, A. A. Coherent sheaves on P n and problems in linear algebra, Funktsional. Anal. i Prilozhen., Volume 12 (1978) no. 3, pp. 68-69 | DOI | MR | Zbl

[3] Bondal, A. I. Representations of associative algebras and coherent sheaves, Izv. Akad. Nauk SSSR Ser. Mat., Volume 53 (1989) no. 1, pp. 25-44 | MR | Zbl

[4] Bondal, A. I.; van den Bergh, M. Generators and representability of functors in commutative and noncommutative geometry, Mosc. Math. J., Volume 3 (2003) no. 1, p. 1-36, 258 | MR | Zbl

[5] Buchweitz, Ragnar-Olaf; Hille, Lutz Hochschild (co-)homology of schemes with tilting object, Trans. Amer. Math. Soc., Volume 365 (2013) no. 6, pp. 2823-2844 | DOI | MR | Zbl

[6] Dlab, Vlastimil; Ringel, Claus Michael The module theoretical approach to quasi-hereditary algebras, Representations of algebras and related topics (Kyoto, 1990) (London Math. Soc. Lecture Note Ser.), Volume 168, Cambridge Univ. Press, Cambridge, 1992, pp. 200-224 | MR | Zbl

[7] Fulton, William Introduction to toric varieties, Annals of Mathematics Studies, 131, Princeton University Press, Princeton, NJ, 1993, pp. xii+157 (The William H. Roever Lectures in Geometry) | MR | Zbl

[8] Hille, L. Exceptional sequences of line bundles on toric varieties, Mathematisches Institut, Georg-August-Universität Göttingen: Seminars 2003/2004, Universitätsdrucke Göttingen, Göttingen, 2004, pp. 175-190 | MR | Zbl

[9] Hille, Lutz; Perling, Markus Exceptional sequences of invertible sheaves on rational surfaces, Compos. Math., Volume 147 (2011) no. 4, pp. 1230-1280 | DOI | MR | Zbl

[10] Kapranov, M. M. On the derived categories of coherent sheaves on some homogeneous spaces, Invent. Math., Volume 92 (1988) no. 3, pp. 479-508 | DOI | MR | Zbl

[11] Kawamata, Yujiro Derived categories of toric varieties, Michigan Math. J., Volume 54 (2006) no. 3, pp. 517-535 | DOI | MR | Zbl

[12] Oda, Tadao Convex bodies and algebraic geometry, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 15, Springer-Verlag, Berlin, 1988, pp. viii+212 (An introduction to the theory of toric varieties, Translated from the Japanese) | MR | Zbl

[13] Orlov, D. O. Projective bundles, monoidal transformations, and derived categories of coherent sheaves, Izv. Ross. Akad. Nauk Ser. Mat., Volume 56 (1992) no. 4, pp. 852-862 | DOI | MR | Zbl

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