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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
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Abstract

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
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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References

B.Barr, M. - Exact categories - Springer, Lecture notes in mathematics 236, pp1–120, 1971.Google Scholar
BQ.Borceux, F., Quinteiro, C. - A theory of enriched sheaves - Cahiers de topologie et géométrie différentielle catégorique volume XXXVII-2, 1996, pages 145–162.Google Scholar
D.Deitmar, A. - Schemes over F 1 in: Number Fields and Function - Progress in Mathematics 239 Geer, Gerard van der; Moonen, Ben J.J.; Schoof, Ren (Eds.) 2005. Fields - Two Parallel Worlds.Google Scholar
G.Gabriel, P. - Des Catégories abéliennes - Bulletin de la société mathématique de France 90 (1962) pp 323–448.Google Scholar
SGA4.Artin, M., Grothendieck, A. et Verdier, J.L. - Théorie des topos et cohomologie étale des schémas. Tome 1: Théorie des topos - Séminaire de Géométrie Algébrique du Bois-Marie 1963-1964 (SGA4) - Lecture notes in mathematics 269 - Springer-Verlag, Berlin-New York, 1972. xix+525pp.Google Scholar
H.Hovey, M. - Model Categories - Mathematical Surveys and Monographs 63 - American Mathematical Society, Providence, RI, 1999. xii+209pp.Google Scholar
Kap.Kaplansky, I. - An introduction to differential algebra - Hermann, Paris, 1957.Google Scholar
K.Kelly, G.M. - Basic concepts of enriched category theory - London Mathematical Society Lecture Note Series 64 - Cambridge University Press, Cambridge-New York, 1982. 285pp. Also available in Reprints in theory and applications of categories 10(2005).Google Scholar
DK.Day, B.J., Kelly, G.M. - Enriched functor categories - Springer Lecture Notes in Mathematics 106 - Springer-Verlag, Berlin and New York, 1969. pp179–191.Google Scholar
McL.Mac Lane, S. - Categories for the working mathematician - Graduate text in mathematics 5 - Springer-Verlag, New York-Berlin, 1971. ix+262pp.Google Scholar
TV.Toën, B., Vaquie, M. - Au dessous de Spec(ℤ) - J. K-Theory 3 (2009), 437–500.Google Scholar