Exercise 1, § 6

Let A be a ring, $\mathfrak{p}$ a graded prime ideal of A[T] and $\mathfrak{p}_0$ the prime ideal $\mathfrak{p} \cap A$ of A. (a) If $T \in \mathfrak{p}$, then $\mathfrak{p} = \mathfrak{p}_0 + T.A[T]$. (b) If $T \notin \mathfrak{p}$, then $\mathfrak{p} = \mathfrak{p}_0 . A[T]$. (c) Deduce that, in case (a) one has $\mathrm{htgr}(\mathfrak{p}) = \mathrm{ht}(\mathfrak{p}_0) + 1$, and in case (b), one has $\mathrm{htgr}(\mathfrak{p}) = \mathrm{ht}(\mathfrak{p}_0)$. (d) Now prove that $\dim_{\mathrm{gr}}(A[T]) = \dim(A) + 1$. (e) Deduce an example of a graded ring B such that $\dim(B) \neq \dim_{\mathrm{gr}}(B)$ (p. 84, exerc. 7).