Misc,
Homotopical Algebraic Geometry II: geometric stacks and applications
(2004)cite http://arxiv.org/abs/math/0404373arxiv:math.AG/0404373Comment: 228 pages; final version to appear in Memoirs of the AMS.
Abstract
This is the second part of a series of papers devoted to develop Homotopical
Algebraic Geometry. We start by defining and studying generalizations of
standard notions of linear and commutative algebra in an abstract monoidal
model category, such as derivations, etale and smooth maps, flat and projective
modules, etc. We then use the theory of stacks over model categories introduced
in hagI in order to define a general notion of geometric stack over a
base symmetric monoidal model category C, and prove that this notion satisfies
the expected properties. The rest of the paper consists in specializing C to
several different contexts. First of all, when C=k-Mod is the category of
modules over a ring k, with the trivial model structure, we show that our
notion gives back the algebraic n-stacks of C. Simpson. Then we set C=sk-Mod,
the model category of simplicial k-modules, and obtain this way a notion of
geometric derived stacks which are the main geometric objects of Derived
Algebraic Geometry. We give several examples of derived version of classical
moduli stacks, as for example the derived stack of local systems on a space, of
algebra structures over an operad, of flat bundles on a projective complex
manifold, etc. Finally, we present the cases where C=(k) is the model category
of unbounded complexes of modules over a char 0 ring k, and C=Sp^\Sigma the
model category of symmetric spectra. In these two contexts, called respectively
Complicial and Brave New Algebraic Geometry, we give some examples of geometric
stacks such as the stack of associative dg-algebras, the stack of
dg-categories, and a geometric stack constructed using topological modular
forms.
