# Is there a non-projective flat module over a local ring?

Is there a non-projective flat module over a local ring? Here I assume the ring is commutative with unit.

$\mathbb{Q}$ is flat over $\mathbb{Z}_p$, but not projective.
• Well done! So is $k((x))$ over $k[[x]]$... – Bugs Bunny Jul 21 '10 at 21:39
• So is the $\mathfrak{m}$-adic completion of any non-Artinian local ring. – Graham Leuschke Jul 21 '10 at 22:25
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.