Let $(R,\mathfrak{m})$ be a regular local ring with regular system of parameters $x_1, . . . , x_n$. Let $f : R \rightarrow S$ be a homomorphism of local rings, $f_i = f(x_i)$ for $i = 1, . . . , n$, and let $M$ be a finite $S$-module. Show that the following are equivalent:
$(a)$ $M$ is $R$-flat,
$(b)$ depth$_R(M) = n$, and
$(c)$ $f_1, . . . , f_n$ is an $M$-sequence.
We can say that $R$ is Cohen-Macaulay. Hence $x_1, . . . , x_n$ is an $R$-sequence. Now, $\mathfrak{m}$ is generated by $x_1, . . . , x_n$. Then $f_i = f(x_i)$ is also an $S$-sequence. If $M$ is $R$-flat, then $f_i$ will be an $M$-sequence. Hence we have established $(a) \implies (c)$. But I am unable to solve the other two equivalences. Any help will be appreciated. Thanks in advance!