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

Bertrand Toen (1) , Gabriele Vezzosi (2)

1 LEP - Laboratoire Émile Picard
2 DMA - Dipartimento di Matematica Applicata [Firenze]

We develop homotopical algebraic geometry (see math.AG/0207028) in the special context where the base symmetric monoidal model category is the category S of spectra, 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, the associated theories of geometric S-stacks (a geometric S-stack being an analog of Artin notion of algebraic stack in Algebraic Geometry), and finally show how to define global moduli spaces of associative ring spectra structures and a moduli space related to topological modular forms as geometric S-stacks. 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.

Domaines: Topologie algébrique [math.AT]

Bertrand Toen, Gabriele Vezzosi. Brave new algebraic geometry and global derived moduli spaces of ring spectra.. Geometry, Applications, and Higher Chromatic Analogues, Cambridge University Press, pp.325-359, 2007, London Mathematical Society Note Series, 9780521700405. ⟨10.1017/CBO9780511721489.018⟩. ⟨hal-00772967⟩