User:TakuyaMurata/Local criterion flatness

In algebra, the local criterion flatness states:
 * Let R be a local noetherian ring, S a local noetherian R-algebra with $$\mathfrak{m}_R S \subset \mathfrak{m}_S$$, and M a finitely generated S-module. Then M is flat over R if and only if $$\operatorname{Tor}_1^R(M, k) = 0.$$

For the proof, see.