Abstract. We develop homotopical algebraic geometry ([To-Ve 1, To-Ve 2]) in the special context where the base symmetric monoidal model category is that of spectra S, i.e. what might be called, after Waldhausen, brave new algebraic geometry. We discuss various model topologies on the model category of commutative algebras in S, and their associated theories of geometric S-stacks (a geometric S-stack being an analog of Artin notion of algebraic stack in Algebraic Geometry). Two examples of geometric S-stacks are given: a global moduli space of associative ring spectrum structures, and the stack of elliptic curves endowed with the sheaf of topological modular forms.

Key words: Sheaves, stacks, ring spectra, elliptic cohomology.

MSC-class: 55P43; 14A20; 18G55; 55U40; 18F10.

INTRODUCTION

Homotopical Algebraic Geometry is a kind of algebraic geometry where the affine objects are given by commutative ring-like objects in some homotopy theory (technically speaking, in a symmetric monoidal model category); these affine objects are then glued together according to an appropriate homotopical modification of a Grothendieck topology (a model topology, see [To-Ve 1, 4.3]). More generally, we allow ourselves to consider more exible objects like stacks, in order to deal with appropriate moduli problems. This theory is developed in full generality in [To-Ve 1, To-Ve 2] (see also [To-Ve 3]).