Tilting Bundles on Rational Surfaces and Quasi-Hereditary Algebras

Hille, Lutz; Perling, Markus

Annales de l'Institut Fourier, Tome 64 (2014) no. 2, pp. 625-644.

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.

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 ${\mathrm{Ext}}^{\mathrm{q}}$ 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.

Classification : 14J26, 16G20, 18E30
Keywords: tilting bundle, rational surface, quasi-hereditary algebra
Mot clés : fibrés basculant, surfaces rationnelles, algèbres quasi-héréditaires

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