Finitely Generated and Flat imply Projective

Question

A result of S. Endo says that a finitely generated flat module over an integral domain $A$ is projective.

Do we have a counterexample to the above statement when $A$ is a reduced ring?

I feel the statement should be false for a reduced ring, but I cannot find a counter example. Any help/hint is appreciated.

Answer 1

See Exercise 7.24 of my commutative algebra notes, which by the way is based on an answer Georges Elencwajg gave on this site. It shows that if a commutative ring $R$ is absolutely flat and not Noetherian and $I$ is an ideal of $R$ which is not finitely generated, then $R/I$ is a finitely generated $R$-module which is flat and not projective. Since every quotient of an absolutely flat ring is reduced, this gives an example of what you want. Explicitly, you can take $R$ to be an infinite product of fields or any infinite Boolean ring.

Answer 2

Another example of a non-noetherian, reduced, absolutely flat ring $R$ is the ring of step functions on $[0,1)$, with discontinuities at the points $\dfrac k{2^n}$ $\;(0\le k<2^n)$, and right continuous at these points.

(Cf Bourbaki, Commutative Algebra, ch. II, §4, exercise 17 c)).