  We use techniques from relative algebraic geometry and homotopical algebraic
geometry in order to construct several categories of schemes defined "under
Spec Z". We define this way the categories of N-schemes, F_1-schemes,
S-schemes, S_+-schemes, and S_1-schemes, where from a very intuitive point of
view N is the semi-ring of natural numbers, F_1 is the field with one element,
S is the sphere ring spectrum, S_+ is the semi-ring spectrum of natural numbers
and S_1 is the ring spectrum with one element. These categories of schemes are
related by several base change functors, and they all possess a base change
functor to Z-schemes (in the usual sense). Finally, we show how the linear
group Gl_n and toric varieties can be defined as objects in certain of these
categories.
