Proposition 6

Suppose $A$ semi-local and reduced; let $q$ be an ideal of definition of $A$; let $A'$ be the total ring of fractions of $A$, and let $B$ be a finite $A$-subalgebra of $A'$. Then $B$ is semi-local and $qB$ is an ideal of definition of it. Suppose that, for every maximal ideal m of B such that dim(Bm) = dim(B), one has [B/m : A/(A ∩ m)] = 1. Then one has $e_q^A(A) = e_{q_B}^B(B)$.

By IV, § 2, No. 5, cor. 3 of prop. 9, B is semi-local with ideal of definition qB. One has $e_{q_B}^B(B) = e_q^A(B)$ by the corollary to Proposition 5. Since A′ is identified with $\prod_p A_p$ where p runs through the set of minimal prime ideals of A (IV, § 2, No. 5, prop. 10), the canonical mapping $A_p \to A_p \otimes_A B$ is bijective for every minimal prime ideal p of A. It then follows from Lemmas 1 and 2 that $e_q^A(B) = e_q^A(A)$, whence the proposition.