Toen and Vezzosi, file Toen web publ del.pdf
Toen and Vaquie: Under Spec Z. Some notes: Idea: Relative alg geom. Think of commutative monoids in a symm monoidal cat C as models for affine schemes relative to C. If there is a reasonable symmetric monoidal functor from C to Z-modules, get a base change functor, and a notion of scheme under Spec(Z). Homotopical version of this requires C to have a model structure. Now have flat and Zariski topology. Can make sense of schemes: a functor with a Zariski covering. Stuff about toric varieties and GL. Brave new AG over the sphere spectrum, and the spectrum with one element. For the sphere spectrum, we start with the symmetric monoidal model cat of Gamma-spaces (very special??), whose homotopy cat is equivalent to the homotopy cat of connective symmetric spectra. The corresponding model cat of commutative monoids is a model for the homotopy theory of brave new comm rings. For the spectrum with one element, we start with C being simplicial sets.
nLab page on Brave new algebraic geometry