TITLE: Flatness and local freeness
QUESTION [35 upvotes]: The following statement is well-known:
Let $A$ be a commutative Noetherian ring and $M$ a finitely generated $A$-module. Then $M$ is flat if and only if $M_{\mathfrak{p}}$ is a free $A_{\mathfrak{p}}$-module for all $\mathfrak{p}$.
My question is: do we need the assumption that $A$ is Noetherian? I have a proof (from Matsumura) that doesn't require that assumption, but the fact that other references (e.g. Atiyah, Wikipedia) are including this assumption makes me rather uneasy.

REPLY [5 votes]: This is too long for a comment, so I'm forced to include it as an answer, even though it isn't one.  In defense of ashpool's question:  until reading Brian Conrad's excellent answer, I'd assumed that finite presentation (not just finite type) hypotheses were necessary (for the statement that flat modules over a local ring are free).  Reason 1:  there is a (slightly) easier proof in the finite presentation case, and when I proved it myself (something my advisor wisely told me to always do, rather than looking things up), I was led to the finite presentation argument (see 25.4.2 on the June 27 2011 version of the notes here --- it will be fixed in later versions), and didn't think to go further.  Reason 2:  When sources such as Mumford (see the 2nd edition of the Red Book, p. 215) state the result with finite presentation hypotheses, my prejudices are confirmed.  Then when I read Matsumura, I see what I expect to see, and not what is on the page.
Also, I liked Matt's interesting point:  you only need Noetherian hypotheses to get from "free stalks" to "locally free sheaf".