Is there any kind of topology, natural or unnatural, that modules do have? Is there any geometric interpretation for flat modules? Is "exactness" of a sequence, any kind of geometric condition? Thanks.
Do modules have any topology?
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$\begingroup$
general-topology
commutative-algebra
modules
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3Sure: every module carries the discrete topology, and also the indiscrete topology. But this is not a particularly useful observation. – 2012-03-25
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0the Krull-toplogy, see http://en.wikipedia.org/wiki/Completion_(ring_theory – 2012-03-25
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0@QiaochuYuan surely, but any kind of interesting topology, I mean, not the trivial one. – 2012-03-25
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0@Blah dead link – 2012-03-25
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1All you have to do is add ')' at the end of the link: http://en.wikipedia.org/wiki/Completion_(ring_theory) – 2012-03-25
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0@Jr.: If your base is noetherian and your module is finitely generated, flat modules are the same thing as locally free modules, so you can think of a flat module as being something like a vector bundle. But exactness is very much algebraic. – 2012-03-25
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0You can put a Zariski topology on the set of prime submodules of $M$ if $M$ is a multiplication module. – 2012-04-08