This is an elementary question which did [not get answered][1]  on math.stackexchange. I would like to know the answer for expository purposes.

I need either a reference or a counter-example to the following statement. Let $A$ be a noetherian ring which is Jacobson (i.e. every prime ideal $\mathfrak{p} \subset A$ is an intersection of maximal ideals). Suppose that $M$ is an $A$-module.

> Is it true that $M$ is flat if and only if
> $\operatorname{Tor}_1^A(M,A/\mathfrak{m}) = (0)$ for all *maximal
> ideals* $\mathfrak{m} \subset A$?

Note hat I have not assumed that $M$ be finitely generated. If $M$ is finitely generated then the answer is in the affirmative, as is well-known (and requires just the noetherian hypothesis). Note as well that under just the noetherian hypothesis, the same statement is true with *maximal* replaced by *prime*. See, for example, Lemma 2.1 of these [notes](http://math.berkeley.edu/~robin/math274root.pdf). I would be happy if the Jacobson property was superfluous here as well.


  [1]: http://math.stackexchange.com/questions/388818/flatness-over-jacobson-ring