Exercise 27, § 2

a) Let $A$ be a commutative ring endowed with a linear topology. For every multiplicative subset $S$ of $A$, denote by $A \left\{ S^{-1} \right\}$ the separated completion of the ring $S^{-1}A$ endowed with the topology whose fundamental system of neighbourhoods of $0$ is formed by the ideals $S^{-1}U_\lambda$, where $(U_\lambda)$ is a fundamental system of neighbourhoods of $0$ in $A$ formed by ideals of $A$. Now prove that $A \left\{ S^{-1} \right\}$ is canonically isomorphic to the projective 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 separated completion of $A$, $A \left\{ S^{-1} \right\}$ is canonically isomorphic to $\hat{A} \left\{ {S'}^{-1} \right\}$, where $S'$ is the canonical image of $S$ in $\hat{A}$.

b) In order that $A \left\{ S^{-1} \right\}$ be reduced to $0$, it is necessary and sufficient that, in $A$, $0$ be adherent to $S$. Deduce an example where $A$ is separated and complete, but where this is not so for $S^{-1}A$ for the topology defined in a).

c) For every continuous homomorphism of $A$ into a linearly topologized ring $B$, separated and complete, such that $u(S)$ is formed by invertible elements in $B$, now prove that $u = u' \circ j$, where $j : A \to A \left\{ S^{-1} \right\}$ is the canonical mapping, and $u'$ is continuous; moreover $u'$ is determined in a unique way.

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 separated completion of $S^{-1}a$ for the topology induced by that of $S^{-1}A$; now prove that $a\{S^{-1}\}$ identifies canonically 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 $p \to p\{S^{-1}\}$ is an increasing bijection from 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 in A, and let $S = A - p$. Now prove 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 $f \in A$, we denote by $A\{f\}$ the ring $A\{S_f^{-1}\}$, where $S_f$ is the multiplicative set of the $f^n(n \geq 0)$. When $f$ runs through a multiplicative subset S of A, the $A\{f\}$ form an inductive filtered system of rings, whose inductive limit (not provided with a topology) is denoted by $A\{s\}$; define a canonical homomorphism

When $S = A - p$, where $p$ is a open prime ideal of A, now prove that $A\{s\}$ is a local ring, that the canonical homomorphism $A\{s\} \to A\{S^{-1}\}$ is local and that the residue fields of $A\{s\}$ and $A\{S^{-1}\}$ are canonically isomorphic (cf. chap. II, § 3, exerc. 16).

h) Suppose that the topology of A is the $m$-adic topology for an ideal $m$ of A, that A is separated and complete, and that $m/m^2$ is a $(A/m)$-module of type finite. Prove that if $m' = m\{S^{-1}\}$, the topology of $A' = A\{S^{-1}\}$ is the $m'$-adic topology, one has $m' = mA'$ and $m'/{m'}^2$ is a $(A'/m')$-module of type finite (use prop. 14 of No. 10). If A is noetherian, the same is true of $A'$.