Lemma 10.99.7 (Local criterion for flatness). Let R \to S be a local homomorphism of local Noetherian rings. Let \mathfrak m be the maximal ideal of R, and let \kappa = R/\mathfrak m. Let M be a finite S-module. If \text{Tor}_1^ R(\kappa , M) = 0, then M is flat over R.
Proof. Let I \subset R be an ideal. By Lemma 10.39.5 it suffices to show that I \otimes _ R M \to M is injective. By Remark 10.75.9 we see that this kernel is equal to \text{Tor}_1^ R(M, R/I). By Lemma 10.99.6 we see that J \otimes _ R M \to M is injective for all ideals of finite colength.
Choose n >> 0 and consider the following short exact sequence
0 \to I \cap \mathfrak m^ n \to I \oplus \mathfrak m^ n \to I + \mathfrak m^ n \to 0
This is a sub sequence of the short exact sequence 0 \to R \to R^{\oplus 2} \to R \to 0. Thus we get the diagram
\xymatrix{ (I\cap \mathfrak m^ n) \otimes _ R M \ar[r] \ar[d] & I \otimes _ R M \oplus \mathfrak m^ n \otimes _ R M \ar[r] \ar[d] & (I + \mathfrak m^ n) \otimes _ R M \ar[d] \\ M \ar[r] & M \oplus M \ar[r] & M }
Note that I + \mathfrak m^ n and \mathfrak m^ n are ideals of finite colength. Thus a diagram chase shows that \mathop{\mathrm{Ker}}((I \cap \mathfrak m^ n)\otimes _ R M \to M) \to \mathop{\mathrm{Ker}}(I \otimes _ R M \to M) is surjective. We conclude in particular that K = \mathop{\mathrm{Ker}}(I \otimes _ R M \to M) is contained in the image of (I \cap \mathfrak m^ n) \otimes _ R M in I \otimes _ R M. By Artin-Rees, Lemma 10.51.2 we see that K is contained in \mathfrak m^{n-c}(I \otimes _ R M) for some c > 0 and all n >> 0. Since I \otimes _ R M is a finite S-module (!) and since S is Noetherian, we see that this implies K = 0. Namely, the above implies K maps to zero in the \mathfrak mS-adic completion of I \otimes _ R M. But the map from S to its \mathfrak mS-adic completion is faithfully flat by Lemma 10.97.3. Hence K = 0, as desired. \square
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