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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.

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    Sure: 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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    the Krull-toplogy, see http://en.wikipedia.org/wiki/Completion_(ring_theory2012-03-25
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    @QiaochuYuan surely, but any kind of interesting topology, I mean, not the trivial one.2012-03-25
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    @Blah dead link2012-03-25
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    All 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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    @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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    You can put a Zariski topology on the set of prime submodules of $M$ if $M$ is a multiplication module.2012-04-08

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