The homotopy theory of dg-categories and derived Morita theory
| dc.creator | Toen, B. | |
| dc.date | 2004-08-24 | |
| dc.date | 2006-10-19 | |
| dc.date.accessioned | 2026-07-07T06:38:45Z | |
| dc.date.available | 2026-07-07T06:38:45Z | |
| dc.description | The main purpose of this work is the study of the homotopy theory of dg-categories up to quasi-equivalences. Our main result provides a natural description of the mapping spaces between two dg-categories $C$ and $D$ in terms of the nerve of a certain category of $(C,D)$-bimodules. We also prove that the homotopy category $Ho(dg-Cat)$ is cartesian closed (i.e. possesses internal Hom's relative to the tensor product). We use these two results in order to prove a derived version of Morita theory, describing the morphisms between dg-categories of modules over two dg-categories $C$ and $D$ as the dg-category of $(C,D)$-bi-modules. Finally, we give three applications of our results. The first one expresses Hochschild cohomology as endomorphisms of the identity functor, as well as higher homotopy groups of the \emph{classifying space of dg-categories} (i.e. the nerve of the category of dg-categories and quasi-equivalences between them). The second application is the existence of a good theory of localization for dg-categories, defined in terms of a natural universal property. Our last application states that the dg-category of (continuous) morphisms between the dg-categories of quasi-coherent (resp. perfect) complexes on two schemes (resp. smooth and proper schemes) is quasi-equivalent to the dg-category of quasi-coherent complexes (resp. perfect) on their product. | |
| dc.description | 50 pages. Few mistakes corrected, and some references added. Thm. 8.15 is new. Minor corrections. Final version, to appear in Inventiones | |
| dc.identifier | https://arxiv.org/abs/math/0408337 | |
| dc.identifier | http://arxiv.org/abs/math/0408337 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/100848 | |
| dc.subject | Algebraic Geometry | |
| dc.subject | K-Theory and Homology | |
| dc.title | The homotopy theory of dg-categories and derived Morita theory | |
| dc.type | text |