Exercise 20, § 7

Let $A$ be a noetherian ring, $M$ a finitely generated $A$-module, and $q_1, ..., q_k$ ideals of $A$ such that $M/q_i \cdot M$ is of finite length for every i. Suppose that $M \neq 0$ and put $d = \dim_A(M)$. (a) Show that $M/q_1^{n_1} ... q_k^{n_k} \cdot M$ is of finite length for every k-tuple of positive integers $(n_1, ..., n_k)$. (b) Consider the graded ring H of type $\mathbf{N}^k$ such that

and the graded H-module N such that show that N is a finitely generated graded H-module generated by $N_{0,...,0}$ and a finite number of elements whose degrees are vectors of the canonical basis of $\mathbf{Z}^k$. Deduce that there exists a polynomial $Q(T) \in \mathbf{Z}[T_1, ..., T_k]$ and integers $s_1 \geqslant 1, ..., s_k \geqslant 1$ such that one has (Proceed by induction on d.)

(c) Deduce, by extending to $Z((T_1, ..., T_k))$ the results of § 6, No. 1, the existence of integers $c(\alpha_1, ..., \alpha_k)$ for the sequences $(\alpha_1, ..., \alpha_k)$ such that $|\alpha| = \sum_{i=1}^k \alpha_i = d$, such that one has

for $(v_1, ..., v_k)$ in $\mathbf{N}^k$.

(d) Suppose now that A is local, with infinite residue field $\kappa$. Show, using the existence of superficial elements for the graded H-module N, that one has

where $q_\alpha$ runs through the set of ideals of A generated by $\alpha_1$ elements of $q_1$, $\alpha_2$ elements of $q_2$, ..., $\alpha_k$ elements of $q_k$. One sometimes denotes the integer $c(\alpha_1, ..., \alpha_k)$ by $e(q_1^{[\alpha_1]} + \cdots + q_k^{[\alpha_k]}; M)$, which is abbreviated to $e(q_1^{[\alpha_1]} + \cdots + q_k^{[\alpha_k]})$ when $M = A$.

(e) Prove the equality

(f) Show that one has :

where $\overline{q_i}$ denotes the integral closure in A of the ideal $q_i$ (cf. exerc. 19).