Localization of ringed spaces

Details Localization of ringed spaces Localization of ringed spaces

2011-03-10

arxiv.org/abs/1103.2139v1

Mathematics

Algebraic Geometry

Scientific paper

Let $X$ be a ringed space together with the data $M$ of a set $M_x$ of prime ideals of $\O_{X,x}$ for each point $x \in X$. We introduce the localization of $(X,M)$, which is a locally ringed space $Y$ and a map of ringed spaces $Y \to X$ 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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