Dokument: Vector Bundles as Generators on Schemes and Stacks
| Titel: | Vector Bundles as Generators on Schemes and Stacks | |||||||
| URL für Lesezeichen: | https://docserv.uni-duesseldorf.de/servlets/DocumentServlet?id=15680 | |||||||
| URN (NBN): | urn:nbn:de:hbz:061-20100720-102019-5 | |||||||
| Kollektion: | Dissertationen | |||||||
| Sprache: | Englisch | |||||||
| Dokumententyp: | Wissenschaftliche Abschlussarbeiten » Dissertation | |||||||
| Medientyp: | Text | |||||||
| Autor: | Gross, Philipp [Autor] | |||||||
| Dateien: |
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| Beitragende: | Prof. Dr. Reich, Holger [Gutachter] Prof. Dr. Schröer, Stefan [Gutachter] | |||||||
| Dewey Dezimal-Klassifikation: | 500 Naturwissenschaften und Mathematik » 510 Mathematik | |||||||
| Beschreibung: | We investigate the resolution property of quasicompact and quasiseparated schemes, or more generally of algebraic stacks with pointwise affine stabilizer groups.
Such a space has the resolution property if every quasicoherent sheaf of finite type admits a surjection from a locally free sheaf of finite rank. As our first main result we verify the resolution property for a large class of surfaces in the first part of the present work. Namely, we show that all two-dimensional schemes that are proper over a noetherian ring satisfy the resolution property. This class includes many singular, non-normal, non-reduced and non-quasiprojective surfaces. The case of normal separated algebraic surfaces was settled by Schröer and Vezzosi (Stefan Schröer and Gabriele Vezzosi, Existence of vector bundles and global resolutions for singular surfaces, Compos. Math. 140 (2004), no. 3, 717--728) and we generalize their methods of gluing local resolutions to the non-normal and non-reduced case, using the pinching techniques of Ferrand (Daniel Ferrand, Conducteur, descente et pincement, Bull. Soc. Math. France 131 (2003), no.4, 553--585.) in combination with deformation theory of vector bundles. In the second part of the present work our main result states that for a large class of algebraic stacks the resolution property is equivalent to a stronger form: There exists a single locally free sheaf E such that the collection of sheaves, obtained by taking appropriate locally free subsheaves of direct sums, tensor products and duals of E, is sufficiently large in order to resolve arbitrary quasicoherent sheaves of finite type. Next, we interpret this geometrically: A sheaf has this property if and only if its associated frame bundle has quasiaffine total space. This yields a natural generalization of the concept of ample line bundles on separated schemes to vector bundles of higher rank on arbitrary quasicompact algebraic stacks with affine diagonal. As an immediate consequence of this result we infer a generalization of Totaro's Theorem to non-normal stacks which says that an algebraic stack with affine stabilizer groups has the resolution property if and only if it is a quotient of a quasiaffine scheme by an action of some general linear group (Burt Totaro, The resolution property for schemes and stacks, J. Reine Angew. Math. 577 (2004), 1--22.) | |||||||
| Lizenz: |  Urheberrechtsschutz | |||||||
| Fachbereich / Einrichtung: | Mathematisch- Naturwissenschaftliche Fakultät » WE Mathematik » Algebraische Geometrie | |||||||
| Dokument erstellt am: | 20.07.2010 | |||||||
| Dateien geändert am: | 19.07.2010 | |||||||
| Promotionsantrag am: | 26.05.2010 | |||||||
| Datum der Promotion: | 01.07.2010 |
