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Relative Zariski Open Objects

Published online by Cambridge University Press:  31 January 2012

Florian Marty
Florian Marty*
Affiliation:
Laboratoire Émile Picard, Université Paul Sabatier, 118 route de Narbonne, 31062 Toulouse Cedex 04, Francefmarty9@ac-toulouse.fr

Abstract

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In [TV], Bertrand Toën and Michel Vaquié define a scheme theory for a closed monoidal category (,⊗, 1) One of the key ingredients of this theory is the definition of a Zariski topology on the category of commutative monoidal objects in . The purpose of this article is to prove that under some hypotheses, Zariski open subobjects of affine schemes can be classified almost as in the usual case of rings (ℤ-mod,⊗,ℤ). The main result states that for any commutative monoidal object A in , the locale of Zariski open subobjects of the affine scheme Spec(A) is associated to a topological space whose points are prime ideals of A and whose open subsets are defined by the same formula as in rings. As a consequence, we can compare the notions of scheme over in [D] and in [TV].

Keywords

Categoryenriched categorysievesrelative geometry

Information

Type
Research Article
Information
Journal of K-Theory , Volume 10 , Issue 1 , August 2012, pp. 9 - 39
Copyright
Copyright © ISOPP 2011

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