Exercise 27, § 2

(a) Let A be a commutative ring with a linear topology. For every multiplicative subset S of A, let $A\{S^{-1}\}$ denote the Hausdorff completion of the ring $S^{-1}A$ with the topology for which a fundamental system of neighbourhoods of 0 consists of the ideals $S^{-1}U_\lambda$, where $(U_\lambda)$ is a fundamental system of neighbourhoods of 0 in A consisting of ideals of A. Show that $A\{S^{-1}\}$ is canonically isomorphic to the inverse limit of the rings $S_\lambda^{-1}(A/U_\lambda)$ where $S_\lambda$ is the canonical image of S in $A/U_\lambda$. If $\hat{A}$ is the Hausdorff completion of A, $A\{S^{-1}\}$ is canonically isomorphic to $\hat{A}\{S'-1\}$, where $S'$ is the canonical image of S in $\hat{A}$.

(b) For $A\{S^{-1}\}$ to be reduced to 0, it is necessary and sufficient that, in A, 0 belong to the closure of S. Deduce an example where A is Hausdorff and complete but $S^{-1}A$ is not so with the topology defined in (a).

(c) For every continuous homomorphism $u$ of A to a complete Hausdorff linearly topologized ring B such that $u(S)$ consists of invertible elements of B, show that $u = u' \circ j$, where $j : A \to A\{S^{-1}\}$ is the canonical mapping and $u'$ is continuous; moreover $u'$ is determined uniquely.

(d) Let $S_1, S_2$ be two multiplicative subsets of A and let $S'_2$ be the canonical image of $S_2$ in $A\{S^{-1}\}$; define a canonical isomorphism of $A\{(S_1S_2)^{-1}\}$ onto $A\{S_1^{-1}\}\{{S'_2}^{-1}\}$.

(e) Let $a$ be an open ideal of A and let $a\{S^{-1}\}$ be the Hausdorff completion of $S^{-1}a$ with respect to the topology induced by that on $S^{-1}A$; show that $a\{S^{-1}\}$ is canonically identified with an open ideal of $A\{S^{-1}\}$ and that the discrete ring $A\{S^{-1}\}/a\{S^{-1}\}$ is isomorphic to $S^{-1}(A/a)$. Conversely, if $a'$ is an open ideal of $A\{S^{-1}\}$, its inverse image a in A is an open ideal such that $a' = a\{S^{-1}\}$. In particular, the mapping $p \mapsto p\{S^{-1}\}$ is an increasing bijection of the set of open prime ideals of A not meeting S onto the set of open prime ideals of $A\{S^{-1}\}$.

(f) Let $p$ be an open prime ideal of A and let $S = A - p$. Show that $A\{S^{-1}\}$ is a local ring whose residue field is isomorphic to the field of fractions of $A/p$.

(g) For every element $f \in A$, let $A_{,,n}$ denote the ring $A\{S_f^{-1}\}$, where $S_f$ is the multiplicative set of thef'' ($n \geqslant 0$). If $f$ runs through a multiplicative subset S of A, the $A_{,,n}$ form a direct system of rings whose direct limit is denoted by $A_{\{S\}}$ (without any topology); define a canonical homomorphism

If $S = A - p$, where $p$ is an open prime ideal of A, show that $A_{\{S\}}$ is a local ring, the canonical homomorphism $A_{\{S\}} \to A\{S^{-1}\}$ is local and the residue fields of $A_{,,n}$ and $A\{S^{-1}\}$ are canonically isomorphic (cf. Chapter II, § 3, Exercise 16).

(h) Suppose that the topology on $A$ is the $m$-adic topology for some ideal $m$ of $A$, that $A$ is Hausdorff and complete and that $m/m^2$ is a finitely generated $(A/m)$-module. Show that, if $m' = m\{S^{-1}\}$, the topology on $A' = A\{S^{-1}\}$ is the $m'$-adic topology, $m' = mA'$ and $m'/{m'}^2$ is a finitely generated $(A'/m')$-module (use Proposition 14 of no. 10). If $A$ is Noetherian, so is $A'$.