Exercise 16, § 7

Let $A$ be a local noetherian ring and $x = (x_1, ..., x_d)$ an ideal generated by a maximal secant sequence for $A$. Now prove that one has the inequality $e_x(A) \leqslant \mathrm{long}(A/x)$ and that equality occurs only if the homomorphism $(A/x)[T_1, ..., T_d] \xrightarrow{\varphi} \mathrm{gr}_x(A)$ defined by $\varphi(T_i) = \xi_i$, where $\xi_i$ is the class of $x_i$ modulo $x^2$, is an isomorphism (which means that $(x_1, ..., x_d)$ is completely secant).