ag.algebraic geometry - Flatness of modules via Tor

As far as I understand, this is false. Here is an example (familiar to $D$-module people):
$A=k[x,y]$; $M=k[a,b]$ on which $x$ (resp. $y$) acts as $frac{d}{da}$ (resp. $frac{d}{db}$).
Since the action of both $x$ and $y$ is locally nilpotent, $M$ is supported at the origin of
$Spec(A)$. Therefore, the only non-zero Tor's of the kind you consider are $Tor_i(M,k)$, where both $x$ and $y$ act on $k$ by zero. These Tor's are easy to compute (they amount to computing de Rham cohomology of affine plane with coordinates $a$ and $b$), and they are non-zero precisely when $i=2$. (Essentially, the calculation repeats the proof of Kashiwara's Lemma.)