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Au-dessous de Specℤ

Published online by Cambridge University Press:  04 September 2008

Bertrand Toën and
Michel Vaquié
Bertrand Toën
Affiliation:
Laboratoire Emile Picard, UMR CNRS 5580 Université Paul Sabatier, Toulouse, France, toen@math.ups-tlse.fr.
Michel Vaquié
Affiliation:
Laboratoire Emile Picard, UMR CNRS 5580 Université Paul Sabatier, Toulouse, France, vaquie@math.ups-tlse.fr.
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Abstract

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, 1-schemes, -schemes, +-schemes and 1-schemes, where (from an intuitive point of view) ℕ is the semi-ring of natural numbers, 1 is the field with one element, is the ring spectra of integers, + is the semi-ring spectra of natural numbers and 1 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 Gln and the toric varieties can be defined as objects in these categories.

Keywords

Schemessymmetric monoidal categoriesGrothendieck topologieshomotopy theory

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Type
Research Article
Information
Journal of K-Theory , Volume 3 , Issue 3 , June 2009, pp. 437 - 500
Copyright
Copyright © ISOPP 2009

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