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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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3  
Sure: every module carries the discrete topology, and also the indiscrete topology. But this is not a particularly useful observation. – Qiaochu Yuan Mar 25 '12 at 21:11
    
the Krull-toplogy, see en.wikipedia.org/wiki/Completion_(ring_theory – Blah Mar 25 '12 at 21:11
    
@QiaochuYuan surely, but any kind of interesting topology, I mean, not the trivial one. – Jr. Mar 25 '12 at 21:36
    
@Blah dead link – Jr. Mar 25 '12 at 21:36
1  
All you have to do is add ')' at the end of the link: en.wikipedia.org/wiki/Completion_(ring_theory) – William Mar 25 '12 at 21:47

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