# Non-projective flat module over a local ring

Could you give me an example of a finitely generated module that is flat over a local ring but not projective?

For a non finitely generated I took $\mathbb{Q}$ over $\mathbb{Z}_p$, but I cannot find an example of a finitely generated one. Of course it should be a module over a local non-noetherian ring. I don't know a lot of local non-noetherian rings, the first that came to my mind was $k[x_1,x_2,\ldots]/(x_1,x_2^2,x_3^3,\ldots)$, since localization is a good way to find flat modules I wanted to localize this ring, but it has only one prime ideal (the maximal ideal); so I really don't know what to do. Any help?

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Power series rings in infinitely many indeterminates might be a better place to look for primes. – Kevin Carlson Oct 26 '12 at 4:29
mathoverflow.net/questions/32847/… – user26857 Oct 26 '12 at 12:37
See Manny Reyes' comment to the un-accepted answer in navigetor's link! That's the best way to think of it... – rschwieb Oct 26 '12 at 19:22

in that proposition $M$ is supposed to be finitely presented – Chris Oct 26 '12 at 18:31