Proposition 11

Let $\rho : A \to B$ be a local homomorphism of noetherian local rings, $M$ a finitely generated A-module and $N$ a finitely generated B-module; suppose that the A-module $N$ is flat.

a) Let $(x_1, \ldots, x_r)$ be a sequence of elements of $m_A$ which is regular for the A-module $M$, and $(y_1, \ldots, y_s)$ a sequence of elements of $m_B$ which is regular for the B-module $\kappa_A \otimes_A N$; then the sequence $(y_1, \ldots, y_s, \rho(x_1), \ldots, \rho(x_r))$ of elements of $m_B$ is regular for the B-module $M \otimes_A N$.

b) One has the equality

Let $x$ denote the ideal of A generated by the sequence $x$ and $y$ the ideal of B generated by $y$. By Prop. 10, the sequence $y$ is $M \otimes_A N$-regular and $N/yN$ is flat over A, so that the sequence $\rho(x) = (\rho(x_1), \ldots, \rho(x_r))$ is regular for $M \otimes_A (N/yN) = (M \otimes_A N)/y(M \otimes_A N)$. This proves a).

To prove b), we may suppose that M and N are nonzero. By Nakayama's lemma, $\kappa_A \otimes_A N$ is likewise nonzero, so that $\operatorname{prof}_A(M)$ and $\operatorname{prof}_B(\kappa_A \otimes_A N)$ are finite (No. 4, Cor. 2 of Th. 2). Take maximal regular sequences $x$ and $y$; then one has $r = \operatorname{prof}_A(M)$, $s = \operatorname{prof}_B(\kappa_A \otimes_A N)$, and there exists an injective a-linear mapping $u : \kappa_A \to M/xM$ and an injective b-linear mapping $v : \kappa_B \to \kappa_A \otimes_A (N/yN)$ (No. 4, Cor. 1 of Th. 2). Since $N/yN$ is flat over A, the b-linear mapping $(u \otimes 1_{N/yN}) \circ v$ of $\kappa_B$ into $(M/xM) \otimes_A (N/yN) = (M \otimes_A N)/(\rho(x) + y)(M \otimes_A N)$ is injective. This implies the equality $\operatorname{prof}_B(M \otimes_A N) = r + s$ (loc. cit.).