Under Spec Z
Abstract
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, F1-schemes, S-schemes, S+-schemes, and S1-schemes, where from a very intuitive point of view N is the semi-ring of natural numbers, F1 is the field with one element, S is the sphere ring spectrum, S+ is the semi-ring spectrum of natural numbers and S1 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 Gln and toric varieties can be defined as objects in certain of these categories.
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