Exercise 15, § 1

With the notation of the preceding exercise, suppose that the residue field and the field of fractions of the ring $V$ are isomorphic (this is the case, for example, when $V = k[[U]]$, the field $k$ being the field of fractions of a ring of formal power series in an infinite number of indeterminates with coefficients in a field). Let $\sigma$ be an isomorphism of $V[T]/m_1$ onto $V[T]/m_2$. Let C denote the subring of $V[T] = E$ formed by the elements of E whose classes modulo $m_1$ and $m_2$ correspond under $\sigma$. Now prove that C is noetherian and that $m_1 m_2 = m_1 \cap m_2$ is a maximal ideal of C. Now prove that E is the integral closure of C (note that $E = C + C.(\pi T - 1)$ and that $(\pi T - 1)^2 + (\pi T - 1)$ belongs to C).

Let A be the local ring $C_{m_1 m_2}$. Then A is integral and of dimension 2, hence catenary; the integral closure $B = E \otimes_C A$ of A is a noetherian semilocal ring possessing exactly two maximal ideals $n_1$ and $n_2$, and we have $\dim(B_{n_1}) = 1$, $\dim(B_{n_2}) = 2$.