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Localization of Ringed Spaces

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Author(s): William D. Gillam

Journal: Advances in Pure Mathematics
ISSN 2160-0368

Volume: 01;
Issue: 05;
Start page: 250;
Date: 2011;
Original page

Keywords: Localization | Fibered Product | Spec | Relative Scheme

ABSTRACT
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.
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