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As I understand it, many of the ideas that were introduced into algebraic geometry in the mid 20th century by french mathematicians were done by transporting over ideas from the theory of manifolds (for example defining things locally). Instead of there being an analogy between these 2 areas, is it possible to do more and use a theory that generalizes both of them? (Perhaps like the analogy between function fields and number fields, and how they can be studied simultaneously with valuations as described in that paper by Artin and Whapples).

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what about topology as a generalization? –  Eric O. Korman Mar 4 '11 at 23:16
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Varieties and manifolds are both locally ringed spaces (en.wikipedia.org/wiki/Ringed_space), but I am not sure if there is much one can say in this generality. –  Qiaochu Yuan Mar 4 '11 at 23:26
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What about diffeological spaces (ncatlab.org/nlab/show/diffeological+space)? Along the lines of locally ringed spaces, there are ringed toposes, or toposes, or sheaves (thought of as spaces). There are lots of generalizations, and lots have been said about all of them (including locally ringed spaces). –  Matt Mar 5 '11 at 1:18
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Echoing Matt's comment here: I think that when you mention mid-20th century french mathematicians introducing the idea of working locally on a variety you are speaking in a roundabout way about the étale topology or its many variants. If so then the generalization that you are looking for is probably the notion of a topos. This is not an easy subject to learn but Sheaves in Geometry and Logic is a nice place to start (although you will find no algebraic geometry in that book). –  Dan Petersen Mar 5 '11 at 8:55

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