Talk:Flat module

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[edit] Notherian local rings need not be perfect

"over a Noetherian local ring, flatness, freeness, and projectivity" are all equivalent.

I don't believe this. Q_p is a flat Z_p module, but not a free one.

I don't believe it either. I think the author must be thinking of finitely generated modules.

Dave Benson, Aberdeen. —Preceding unsigned comment added by 139.133.7.37 (talk) 22:36, 2 May 2009 (UTC)

Rings in which all flat modules are projective are called perfect rings. Commutative noetherian perfect rings are artinian (and perfect commutative integral domains are fields). I added "finitely generated" to the claim, but then it is true in much larger generality than noetherian local rings (well, even without finitely generated, there are non-noetherian perfect rings). I was not sure how to make the comment more interesting. JackSchmidt (talk) 23:41, 2 May 2009 (UTC)