I am self studying the book "Commutative Ring Theory" by H. Matsumura. The main theorem of section 22 is the [theorem 22.3](https://books.google.com/books?id=yJwNrABugDEC&pg=PA174), which characterizes flatness of a module $M$ over any ring $A$. The (part of the) theorem states :


Let $A$ be a ring, $I$ an ideal of $A$ and $M$ an $A$-module. Then the following are equivalent


(1) $Tor\_1^A(N,M)=0$ for every $A/I$ module $N$;


(2) $M/IM$ is flat over $A/I$ and $I\otimes M =IM$;


(3) $M/IM$ is flat over $A/I$ and $Tor\_1^A(A/I,M)=0$.


I have been able to show the above three are equivalent. Furthermore, the theorem also states if one of the above three holds, then one must have : 


(4) $M/IM$ is flat over $A/I$ and 


$\gamma\_n$ : $I^n/I^{(n+1)} \otimes M/IM \rightarrow I^nM/I^{(n+1)}M$ is an isomorphism for all $n\geq 0$.


While proving this, the author of the book uses the result that, if $I\otimes M =IM$ then one must have $I^n\otimes M =I^nM$ for all $n\geq 2$, which I am unable to show. Any help would be appreciated.



The local flatness criterion
edit: I apologize, the original answer was nonsense (I mixed different Tor functors in a silly way). 


The following works, but uses the equivalent condition (1) of the question, and does not show $I\otimes M = IM \implies I^n \otimes M = I^n M$ unconditionally.


Consider the exact sequence:


$$ I^2\otimes M \rightarrow I \otimes M \rightarrow I/I^2 \otimes M \rightarrow 0.$$


The rightmost module is an $A/I$-module, so $\mathrm{Tor}\_1(M, I/I^2)=0$. Hence the first map is injective. Also, $I\otimes M = IM$ and $I/I^2 \otimes M = I \otimes A/I \otimes M$ which by associativity and commutativity of tensor is $IM \otimes A/I = IM/I^2M$. So we have an exact sequence:


$$ 0\rightarrow I^2\otimes M \rightarrow IM \rightarrow IM/I^2M \rightarrow 0,$$
and $I^2 \otimes M$ is the kernel of the second map, which is $I^2 M$.


It seems this process can be continued: consider the exact sequence


$$ I^3\otimes M \rightarrow I^2 \otimes M \rightarrow I^2/I^3 \otimes M \rightarrow 0,$$


where again $I^2 / I^3$ is an $A/I$-module and the last two terms are $I^2 M$ and $I^2 M / I^3 M$ respectively; by the vanishing of $\mathrm{Tor}\_1(M,N)$ for $N$ an $A/I$-module, the first arrow is an injection and it equals the kernel of the second map, i.e. it is $I^3 M$.


