Is there a non-projective flat module over a local ring? Here I assume the ring is commutative with unit.
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2 Answers
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$\mathbb{Q}$ is flat over $\mathbb{Z}_p$, but not projective.
answered Jul 21, 2010 at 21:32
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2$\begingroup$ Well done! So is $k((x))$ over $k[[x]]$... $\endgroup$ Jul 21, 2010 at 21:39
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4$\begingroup$ So is the $\mathfrak{m}$-adic completion of any non-Artinian local ring. $\endgroup$ Jul 21, 2010 at 22:25
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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.
answered Jul 21, 2010 at 21:36
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5$\begingroup$ But we can also make a slightly positive statement. Every local ring is semiperfect, and every finitely generated flat module over a semiperfect ring is projective. Moreover, since projective modules over a local ring are free, we see that every finitely generated flat module over a local ring is free. $\endgroup$ Jul 21, 2010 at 21:54
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1$\begingroup$ For those, who -- like me -- didn't know what a perfect ring was, see: en.wikipedia.org/wiki/Perfect_ring. $\endgroup$ Jul 22, 2010 at 3:26