Localization of Ringed Spaces

DOI: 10.4236/apm.2011.15045   PDF   HTML     4,798 Downloads   8,637 Views   Citations


Let X be a ringed space together with the data Μ of a set Μ of prime ideals of ΟΧx for each point x∈Χ . We introduce the localization of (X,M') , which is a locally ringed space Y and a map of ringed spaces YΧ enjoying a universal property similar to the localization of a ring at a prime ideal. We use this to prove that the category of locally ringed spaces has all inverse limits, to compare them to the inverse limit in ringed spaces, and to construct a very general Spec functor. We conclude with a discussion of relative schemes.

Share and Cite:

W. Gillam, "Localization of Ringed Spaces," Advances in Pure Mathematics, Vol. 1 No. 5, 2011, pp. 250-263. doi: 10.4236/apm.2011.15045.

Conflicts of Interest

The authors declare no conflicts of interest.


[1] M. Hakim, “Topos Annelés et Schémas Relatifs. Ergebnisse der Mathematik und ihrer Grenzgebiete,” Springer- Verlag, Berlin, 1972.
[2] R. Hartshorne, “Algebraic Geometry,” Springer-Verlag Berlin, 1977.
[3] H. Becker, Faserprodukt in LRS. http://www.unibonn.de/~habecker/Faserprodukt―in―LRS.pdf.
[4] L. Illusie, “Complexe Cotangent et Deformations I. L.N.M. 239,” Springer-Verlag, Berlin, 1971.
[5] A. Grothendieck and J. Dieudonné, “éléments de Géométrie Al-gébrique,” Springer, Berlin, 1960.
[6] J. Giraud, “Cohomolo-gie non Abélienne,” Springer, Berlin, 1971.
[7] A. Vistoli, “Notes on Grothendieck Topologies, Fibered Categories, and Descent Theory,” Citeseer, Princeton 2004.

comments powered by Disqus

Copyright © 2020 by authors and Scientific Research Publishing Inc.

Creative Commons License

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.