Corollary 1

Let $\rho : A \to B$ be a local homomorphism of noetherian local rings.

a) One has

with equality if $\rho$ makes $B$ into a flat $A$-module.

b) Suppose that $\rho$ makes $B$ into a flat $A$-module, and let $S$ be a subset of $\mathfrak{m}_A$. Then $S$ is secant for $A$ if and only if $\rho(S)$ is secant for $B$.

Assertion a) is the particular case $M = A, N = B$ of Prop. 7. Let us prove b). Under the hypotheses made, we have

with $\overline{B} = B \otimes_A \kappa_A = B/\rho(\mathfrak{m}_A).B$. Since $\rho$ is injective (I, § 3, No. 5, Prop. 9), we have

Finally $B' = B/\rho(S).B$ is a flat module over $A' = A/SA$, whence

since $B'/\rho(\mathfrak{m}_{A'}).B'$ is isomorphic to $\overline{B}$. Our assertion follows at once from formulas (17), (18) and (19).