Algebraic Geometry over model categories (a general approach to derived algebraic geometry)
Abstract
For a (semi)model category M, we define a notion of a ''homotopy'' Grothendieck topology on M, as well as its associated model category of stacks. We use this to define a notion of geometric stack over a symmetric monoidal base model category; geometric stacks are the fundamental objects to "do algebraic geometry over model categories". We give two examples of applications of this formalism. The first one is the interpretation of DGschemes as geometric stacks over the model category of complexes and the second one is a definition of etale Ktheory of E_{\infty}ring spectra. This first version is very preliminary and might be considered as a detailed research announcement. Some proofs, more details and more examples will be added in a forthcoming version.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 October 2001
 arXiv:
 arXiv:math/0110109
 Bibcode:
 2001math.....10109T
 Keywords:

 Mathematics  Algebraic Geometry;
 Mathematics  Algebraic Topology;
 14A20;
 18G55;
 55P43;
 55U40;
 18F10
 EPrint:
 LaTeX,xypic, 51 pages