Exercise 3, § 5

Let A be a Noetherian integral domain, K its field of fractions, $\mathscr{T}_u$ the topology $\mathscr{T}_u(A)$ and $\mathscr{T}_K$ the corresponding topology on K (Exercise 1).

(a) If A is a Zariski ring with Jacobson radical $r \neq 0$ and A is complete with the r-adic topology, show that A is complete with the topology $\mathscr{T}_u$ (General Topology, Chapter III, § 3, no. 5, Corollary 2 to Proposition 9).

(b) Suppose that $\mathscr{T}_K$ is not discrete and is defined by a valuation $v$ on K. Show that $v$ is a discrete valuation and that A is an open subring of the ring V of $v$; obtain the converse. (Using the hypothesis, which implies A $\neq K$, and Proposition 1 of § 4, no. 1, we may assume that A $\subset V$. Using the fact that A is open under $\mathscr{T}_K$, show that V is a finitely generated A-module and conclude using § 3, no. 5, Proposition 9. For the converse, observe that A is a local ring of which the maximal ideal m and (0) are the only prime ideals and in which every ideal $a \neq 0$ is therefore m-primary). The integral closure of A is then V. Give an example where A $\neq V$ (take V to be a ring of formal power series $k[[T]]$, where $k$ is a field).

(c) Deduce an example of a non-discrete complete topological field whose topology is not locally retrobounded (use (b) and Exercise 2).