From HAG to DAG: derived moduli spaces

Contributo su invito. Contiene risultati originali, non apparsi in precedenza. ABSTRACT -These are expanded notes of some talks given during the fall 2002, about homotopical algebraic geometry with special emphasis on its applications to derived algebraic geometry and derived deformation theory. We use the general framework developed in [HAG-I], and in particular the notions of model topology, model sites and stacks over them, in order to define various derived moduli functors and study their geometric properties. We start by defining the model category of D-stacks, with respect to an extension of the ´etale topology to the category of commutative differential graded algebras, and we show that its homotopy category contains interesting objects, such as schemes, algebraic stacks, higher algebraic stacks, dg-schemes, etc. We define the notion of geometric D-stacks and present some related geometric constructions (O-modules, perfect complexes, K-theory, derived tangent stacks, cotangent complexes, various notions of smoothness, etc.). Finally, we define and study the derived moduli problems classifying local systems on a topological space, vector bundles on a smooth projective variety, and A_{\infty}-categorical structures. We state geometricity and smoothness results for these examples. The proofs of the results presented in this paper will be mainly given in [HAG-II].