Editing Flat module (section)

=== Faithfully flat local homomorphisms ===
Here is one characterization of a faithfully flat homomorphism for a not-necessarily-flat homomorphism. Given an injective local homomorphism <math>(R, \mathfrak m) \hookrightarrow (S, \mathfrak n)</math> such that <math>\mathfrak{m} S</math> is an <math>\mathfrak{n}</math>-[[primary ideal]], the homomorphism <math>S \to B</math> is faithfully flat if and only if the [[theorem of transition]] holds for it; that is, for each <math>\mathfrak m</math>-primary ideal <math>\mathfrak q</math> of <math>R</math>, <math>\operatorname{length}_S (S/ \mathfrak q S) = \operatorname{length}_S (S/ \mathfrak{m} S) \operatorname{length}_R(R/\mathfrak q).</math>{{sfn|Matsumura|1986|loc=Ch. 8, Exercise 22.1|ps=none}}