Exercise 9, § 4

Let $\rho : A \to B$ be a local homomorphism of local noetherian rings, $N$ a finite type Macaulay $B$-module. Suppose that the ring $A$ is regular and that we have $\dim_B(N) = \dim(A) + \dim_{B/m_A B}(N/m_A N)$. Prove that the $A$-module $N$ is flat (reason by induction on $\dim(A)$; if $x \in m_A - m_A^2$, one will show that $x$ is $N$-regular and that the homomorphism $A/xA \to B/xB$ and the $(B/xB)$-module $N/xN$ satisfy the same hypotheses as $\rho$ and $N$.