Exercise 7, § 7

Let $A$ be a local ring, with maximal ideal $m$, $M$ a finitely generated $A$-module, $(x_1, ..., x_d)$ a sequence of elements of $m$ secant for the $A$-module $M$, and $x$ the ideal $x_1 A + \cdots + x_d A$.

a) Suppose the $A$-module $M$ is faithful. Then the quotient of $\operatorname{gr}_x(A)$ by its nilradical $n$ is isomorphic to $(A/m)[X_1, ..., X_d]$ and $(\operatorname{gr}_x(M))_n$ is a $(\operatorname{gr}_x(A))_n$-module of length $e_x(M)$.

b) In the general case, $\operatorname{gr}_x(A)$ possesses a unique minimal prime ideal $n$ and $(\operatorname{gr}_x(M))_n$ is a $\operatorname{gr}_x(A)$-module of length $e_x(M)$.