Exercise 6, § 1

a) Let $A$ be a (commutative) noetherian ring, $m$ an ideal of $A$ contained in the radical of $A$, and $gr(A)$ the associated graded ring of the $m$-adic filtration of $A$. Associate to each ideal of $A$ an ideal of $gr(A)$, and deduce that $Kdev(A) \leq Kdev(gr(A))$ (cf. exerc. 4, c)). Consequently, one has $\dim(A) \leq \dim(gr(A))$ (exerc. 5, c)). b) We now suppose that $m = xA$, where $x$ belongs to the radical of $A$. Now prove that $gr(A)$ is isomorphic to a quotient of the graded algebra $(A/m)[T]$; now prove that the ordered set $F$ of graded ideals of $(A/m)[T]$ is isomorphic to the set of increasing sequences of ideals of $A/m$; deduce (exerc. 4, d)) that one has $dev(F) = Kdev(A/(x)) + 1$, and hence $\dim(A) \leq \dim(A/m) + 1$.