Lemma 10.99.11. Let R \to S be a ring map. Let I \subset R be an ideal. Let M be an S-module. Assume
R is a Noetherian ring,
S is a Noetherian ring,
M is a finite S-module, and
for each n \geq 1 the module M/I^ n M is flat over R/I^ n.
Then for every \mathfrak q \in V(IS) the localization M_{\mathfrak q} is flat over R. In particular, if S is local and IS is contained in its maximal ideal, then M is flat over R.
Proof. We are going to use Lemma 10.99.10. By assumption M/IM is flat over R/I. Hence it suffices to check that \text{Tor}_1^ R(M, R/I) is zero on localization at \mathfrak q. By Remark 10.75.9 this Tor group is equal to K = \mathop{\mathrm{Ker}}(I \otimes _ R M \to M). We know that the kernel of I/I^ n \otimes _{R/I^ n} M/I^ nM \to M/I^ nM is zero for all n \geq 1. Hence an element of K maps to zero in I/I^ n \otimes _{R/I^ n} M/I^ nM. Since
we conclude that K \subset I^{n - 1}(I \otimes _ R M) for all n \geq 1. By the Artin-Rees lemma, and more precisely Lemma 10.51.5 we conclude that K_{\mathfrak q} = 0, as desired. \square
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