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 Y→Χ 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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