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 Nschemes, F_1schemes, Sschemes, S_+schemes, and S_1schemes, where from a very intuitive point of view N is the semiring of natural numbers, F_1 is the field with one element, S is the sphere ring spectrum, S_+ is the semiring 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 Zschemes (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.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 September 2005
 DOI:
 10.48550/arXiv.math/0509684
 arXiv:
 arXiv:math/0509684
 Bibcode:
 2005math......9684T
 Keywords:

 Mathematics  Algebraic Geometry;
 Mathematics  Category Theory
 EPrint:
 45 pages, french. Several mistakes corrected.The central definition of scheme is slightly modified. Section 2.1 is new and contains more details about the fpqc topology