TITLE: Is there a non-projective flat module over a local ring?
QUESTION [5 upvotes]: Is there a non-projective flat module over a local ring?
Here I assume the ring is commutative with unit.

REPLY [6 votes]: It is related to Bass' theorem. Flat modules are projective iff the ring is perfect. $p$-adic integers or formal power series are examples of local rings which are not perfect and have nonprojective flat modules.