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?
Best Answer
Over a (commutative) local ring (non necessarily noetherian), any finitely generated flat module is free (Matsumura, Commutative Algebra, Prop. 3.G, p. 21), hence projective.