Tor and Flatness of finite type modules in local rings

Question

We know that a finitely presented module over a local ring $(R,m)$ is free if and only if it is flat if and only if $$\text{Tor}_1(M, R/m)=0$$ We know that a finitely generated module is flat over the local ring $(R,m)$ if and only if it is free. Do we still have the equivalence with $\text{Tor}_1(M, R/m)=0$? If not, does one know a counterexample (obviously where $m$ is not of finite type)?

Answer 1

Let $R$ be a valuation ring with maximal ideal $\mathfrak m$ satisfying $\mathfrak m=\mathfrak m^2$. Set $M=R/\mathfrak m$. Then $\mathrm{Tor}_1(M,R/\mathfrak m)=0$; see here. On the other side $M$ is not torsion-free, hence not flat.