Ander SpecZ

摘要

In this article we use the theories of relative algebraic geometry and of homotopical algebraic geometry (cf. [HAGII]) to construct some categories of schemes defined under SpecZ. We define the categories of N-schemes, F-1- schemes, S-schemes, S+-schemes and S-1-schemes, where (from an intuitive point of view) N is the semi-ring of natural numbers, F-1 is the field with one element, S is the ring spectra of integers, S+ is the semi-ring spectra of natural numbers and S I is the ring spectra with one element. These categories of schemes are linked together by base change functors, and all of them have a base change functor to the category of Z-schemes. We show that the linear group Gl(n) and the toric varieties call he defined as objects in these categories.