Algebraic Geometry over model categories (a general approach to derived algebraic geometry)

Details Algebraic Geometry over model categories (a general approach to derived algebraic geometry) Algebraic Geometry over model categories (a general approach to derived algebraic geometry)

2001-10-10

arxiv.org/abs/math/0110109v1

Mathematics

Algebraic Geometry

LaTeX,xy-pic, 51 pages

Scientific paper

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 DG-schemes as geometric stacks over the model category of complexes and the second one is a definition of etale K-theory 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.

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