Flatness ascends up a inverse system Let $R$ be a ring with $f \in R$ a nonzero divisor. Then, the claim is that specifying an f-adically complete and f-torsion free $R$-algebra $S$ with $R/f\to S/f$ flat is the same as specifying a projective system $S_n$ of flat $R/f^n$ algebras such that $S_n/f^{n-1} \cong S_{n-1}$.
So suppose $S$ satisfies the conditions. Then, we should define $S_n = S/f^n$ but I am not sure why this is a flat $R/f^n$ module. Does flatness ascend in this way along nilpotent ideals in general?
 A: We will first see a result called local criterion of Flatness.
Theorem : Let A be a ring, $I$ and ideal of $A$ and $M$ an $A$-module. Assume that $I$ is a nilpotent ideal. Then the following are equivalent
(1) $M$ is $A$-flat.
(2) $M \otimes A/I$ is flat over $A/I$ and $I \otimes_A M \xrightarrow{\sim} IM$ by the natural map.
We will use use this theorem with $A = R/(f)^n, I = (\overline{f}), M = S/(f)^n$, where $\overline{f}$ is the image of $f$ in $R/(f)^n$. We want the result stated in (1) above and so we verify (2). The first part is built into the hypothesis i.e. $S/(f)^n \otimes_{R/(f)^n} R/(f) \cong S/(f)$ is flat over $R/(f)$. To verify the second part of (2), we proceed as follows
$$(\overline{f}) \otimes_{R/(f)^n} S/(f)^n \cong (f) \otimes_{R} (R/(f)^n \otimes_R S) \cong ((f) \otimes_R S) \otimes_R R/(f)^n \cong (f)S \otimes_{R} R/(f)^n = (\overline{f})(S/(f)^nS)$$
I wrote $(f)S$ in the last line to indicate that this ideal is in $S$ and not that this is the ideal in $S$ generated by $f$. The first and second isomorphism can be seen to be true from simple properties about tensor products. The third isomorphism follows from the fact that $S$ is $f$-torsion free. Actually $S$ is $f$-torsion free implies that the map $(f) \otimes_R S \rightarrow (f).S$ is injective, but the maps is obviously surjective. Hence the isomorphism.
There are several other criteria for flatness of which I have mentioned only two.The theorem can be found in Commutative Algebra, 2nd edition by H.Matsumura,  Chapter 8 Theorem 49. It says in the loc. cit. that theorems of such kind are due to Bourbaki. So that could be another reference but I haven't checked them.
