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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