Exercise 7, § 4

(a) Let A be a commutative ring and m an ideal of A such that A is Hausdorff and complete with the m-adic topology and $m/m^2$ is a finitely generated A-module. Show that the topology on $A' = A\{X_1, \ldots, X_r\}$ is the $m'$-adic topology, where $m' = mA'$ and that $m'/{m'}^2$ is a finitely generated $(A'/m')$-module (use Proposition 14 of § 2, no. 11). In particular, if A is Noetherian, so is $A'$.

(b) Let A be a commutative Noetherian ring and m an ideal of A such that A is Hausdorff and complete with the m-adic topology. Let $u : A \to B$ be a continuous homomorphism from A to a Hausdorff commutative topological ring B making B into an A-algebra. Show that the following conditions are equivalent:

(α) B is Noetherian, its topology is the mB-adic topology, B is complete and $B/mB$ is a finitely generated algebra over $A/m$.

(β) B is topologically A-isomorphic to $\lim \leftarrow B_n$ where $(B_n)_{n \geq 1}$ is an inverse system of discrete A-algebras such that the mappings $\phi_{nm} : B_n \to B_m$ for $m \geq n$ are surjective, the kernel of $\phi_{nm}$ is $m^{n+1}B_m$ and B, is a finitely generated algebra over $A/m$.

(γ) B is topologically A-isomorphic to a quotient of an algebra of the form $A\{X_1, \ldots, X_r\}$ by a closed ideal.

(To prove that (β) implies (γ), use Proposition 14 of § 2, no. 11 and Theorem 1 of § 2, no. 8.)