Associated graded and flatness Let $M$ be a filtered module over a filtered algebra $A$, and suppose $gr(M)$ is flat over $gr(A)$, where $gr$ means the associated graded module and algebra, respectively.
What can one say in general about the flatness of $M$ over $A$, or with relevant assumptions (for instance in the above, we should assume both filtrations are complete to avoid dumb counterexamples)?  Are there good references for this sort of question?  I have played with the various definitions of flatness trying to find an obvious relationship, but I find flatness proofs confusing.
The particular examples I have in mind are comparing $U(\mathfrak{g})$-modules to $S(\mathfrak{g})$-modules, and $D(X)$-modules to $O(T^*X)$-modules for affine varieties $X$, if it helps.  I suspect the answer doesn't depend on any of the details though.
 A: Let me suppose, as in your examples, that we have a base field $k$.
It is well known that to check that a right $A$-module $M$ is flat it is enough to show that whenever $I\leq_\ell A$ is a left ideal, the map $M\otimes_AI\to M\otimes_A A$ induced by the inclusion $I\to A$ is injective. This condition can be rewritten: $M$ is flat iff for each left ideal $I\leq_\ell A$ we have $\mathrm{Tor}^A_1(M,A/I)=0$.
So now suppose $A$ and $M$ are (exhaustively, separatedly, increasingly from zero) filtered in such a way that $\mathrm{gr}M$ is a flat $\mathrm{gr}A$-module. 
Pick a left ideal $I\leq_\ell A$; notice that the filtration on $A$ induces a filtration on the quotient $A/I$. We can compute $\mathrm{Tor}^A_\bullet(M,A/I)$ as the homology of the homologically graded complex $$\cdots\to M\otimes_kA^{\otimes_kp}\otimes_kA/I\to M\otimes_kA^{\otimes_k(p-1)}\otimes_kA/I\to\cdots$$ with certain differentials whose formula does not fit in this margin, coming from the bar resolution. Now the filtrations on $M$, on $A$ and on $A/I$ all collaborate to provide a filtration of our complex. We've gotten ourselves a positively homologicaly graded with a canonically bounded below, increasing, exhaustive and separated filtration. The corresponding spectral sequence then converges, and its limit is $\mathrm{Tor}^A_\bullet(M,A/I)$. Its $E^0$ term is the complex
$$\cdots\to\mathrm{gr}M\otimes_k\mathrm{gr}A^{\otimes_kp}\otimes_k\mathrm{gr}(A/I)\to \mathrm{gr}M\otimes_k\mathrm{gr}A^{\otimes_k(p-1)}\otimes_k\mathrm{gr}(A/I)\to\cdots$$
with, again, the bar differential, and its homology, which is the $E^1$ page of the spectral sequence, is then precisely $\mathrm{Tor}^{\mathrm{gr}A}_\bullet(\mathrm{gr}M,\mathrm{gr}(A/I))$. Since we are assuming that $\mathrm{gr}M$ is $\mathrm{gr}A$-flat, this last $\mathrm{Tor}$ vanishes in positive degrees, so the limit of the spectral sequence also vanishes in positive degrees. In particular, $\mathrm{Tor}^A_1(M,A/I)=0$.
NB: As Victor observed above in a comment, Bjork's Rings of differential operators proves in its Proposition 3.12 that $\mathrm{w.dim}_AM\leq\mathrm{w.dim}_{\mathrm{gr}A}\mathrm{gr}M$ (here $\mathrm{w.dim}$ is the flat dimension) from which it follows at once that $M$ is flat as soon as $\mathrm{gr}M$ is; the argument given is essentialy the same one as mine. I am very suprised about not having found this result in McConnell and Robson's!
A: Dear David,
See Prop. 1.2 of Algebras of p-adic distributions and admissible representations by Peter Schneider and Jeremy Teitelbaum, for one such result:
Proposition 1.2. Suppose that $gr^{\bullet} R$ and $gr^{\bullet} A$ are left noetherian and that $gr^{\bullet} A$ as a right $gr^{\bullet}R$-module (via $gr^{\bullet}\phi$) is flat; then $A$ is flat as a right
$R$-module (via $\phi$).
There are some hypotheses which you can find in the preamble to section 1 of the paper,
as well as useful references to literature on this kind of question.  
