Exercise 4, § 2

Soient A a local ring, $\mathfrak{p}$ a prime ideal of A. We denote by B the ring of fractions $A[Z][S^{-1}]$, where S is the set of elements $Z + a$ of $A[Z]$ with $a \in \mathfrak{m}_A - \mathfrak{p}$. We denote by z the element $Z/1$ of B. a) Prove that $B/(z)$ is isomorphic to $A_p$ and that $B/(z-1)$ is isomorphic to A. b) Suppose now that the ring A is integral and noetherian, and that $\mathfrak{p}$ is generated by a nonzero element p of A. Let C be the ring $B[T]/(z(pT-1))$; if A is a Macaulay ring, the same is true of C. Prove that Spec C has two irreducible components $X_1$ and $X_2$ which meet at a closed point x, such that $\operatorname{codim}(\{x\}, X_1) = \operatorname{codim}(\{x\}, X_2) = 1$, $\dim(X_1) = 2$ and $\dim(X_2) \geqslant \dim(A_p)$. For every integer n, give an example with $\dim(A_p) = n$.