Proposition 5

Suppose A semi-local, and let $\rho : A \to B$ be a ring homomorphism making B a finitely generated A-module. Let N be a nonzero finitely generated B-module, and let q be an ideal of A contained in the radical of A, such that $N/qN$ is of finite length. Among the maximal ideals of B (finite in number, by IV, § 2, No. 5, cor. 3 to prop. 9), let us denote by $m_1, ..., m_r$ those for which one has $\dim_{B_{m_i}}(N_{m_i}) = \dim_B(N)$. Put $B_i = B_{m_i}$ and $q_i = qB_i$ for $1 \leq i \leq r$. Then one has the equalities

The first equality follows from § 2, No. 3, th. 1, c); the second follows from Remark 4 of No. 1 (note that $m_i$ belongs to $V(qB)$ for $1 \leq i \leq r$ since $\rho^{-1}(m_i) \supset q$ by V, § 2, No. 1, prop. 1). Let us prove the third equality. Let E be a B-module of finite length; one has

$m$ ranging over the set of maximal ideals of B; this is in fact evident when E is one of the $B/m$, and the general case follows from this, since E possesses a composition series whose quotients are isomorphic to $B/m$. Applying this formula to the B-modules $N/q^{n+1}N$, one deduces the required equality by definition of the multiplicities.