Bertrand Toën

Homotopical algebraic geometry. II. Geometric stacks and applications.

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 \cite{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.

Derived algebraic geometry

This text is a survey of derived algebraic geometry. It covers a variety of general notions and results from the subject with a view on the recent developments at the interface with deformation quantization.

Derived Hall algebras.

The purpose of this work is to define a derived Hall algebra

From Hag To Dag: Derived Moduli Stacks

\mathcal{DH}(T)

Brave new algebraic geometry and global derived moduli spaces of ring spectra.

, associated to any dg-category

Simplicial localization of monoidal structures, and a non-linear version of Deligne's conjecture

T

Schematic homotopy types and non-abelian Hodge theory

(under some finiteness conditions). Our main theorem states that

Simplicial methods for operads and algebraic geometry

\mathcal{DH}(T)

Algebraic and topological aspects of the schematization functor.

is associative and unital. It is shown that

On motives for Deligne-Mumford stacks

\mathcal{DH}(T)