I need either a reference or a counter-example to the following statement. Let $A$ be a noetherian Jacobson ring (i.e. a noetherian ring where 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 $\mathfrak{m} \subset A$?

Let me point out that I have not assume that $M$ be finitely generated.

Note that under the noetherian hypothesis, the same statement is true with maximal replaced by prime. See, for example, Lemma 2.1 of these notes.


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