Exercise 13, § 7

Let $A$ be a local ring and let $a$ and $b$ be two ideals of $A$ distinct from $A$. Prove that if $a \subset b$, one has $\overline{a} = \overline{b}$ if and only if the graded algebra gr_b(A) is integral over the graded algebra gr_a(A). Deduce that if $a = (x_1, ..., x_d) \subset b$ where $d = \dim(A)$, one has $\overline{a} = \overline{b}$ if and only if the initial forms $\xi_1, ..., \xi_d$ in gr_b(A) of the $x_i$ are of degree 1 and form a maximal secant sequence for gr_b(A).