Exercise 18, § 3

Let $A$ be a local ring with maximal ideal $m$; an $A$-module $M$ is called quasi-finite if $M/mM$ is a vector space of finite rank over the residue field $k = A/m$. In particular, if $A$ is an integral domain, the field of fractions $K$ of $A$ is a quasi-finite $A$-module.

(a) Show that if $A$ is Noetherian and $M$ is a quasi-finite $A$-module, its Hausdorff completion $\hat{M}$ with respect to the $m$-adic topology is a finitely generated $A$-module (use §2, no. 11, Corollary 2 to Proposition 14). In particular, if $A$ is complete and $M$ Hausdorff with the $m$-adic topology, $M$ is a finitely generated $A$-module.

(b) Let $B$ be another local ring, $n$ its maximal ideal, $\phi : A \to B$ a local homomorphism and $M$ a finitely generated $B$-module. Show that, if $B$ is Noetherian and $M$ is a quasi-finite $A$-module, then the $m$-adic and $n$-adic topologies on $M$ are identical. (Note that $M/mM$ is a $B$-module of finite length and deduce that, if $b$ is the annihilator of the $B$-module $M/mM$, then $V(b) = \{n\}$ in $\operatorname{Spec}(B)$ and therefore $V(mB + b) = \{n\}$. Conclude using Chapter II, §4, no. 3, Corollary 2 to Proposition 11.)

(c) Under the hypotheses of (b), show that, if $M \neq 0$, $B/b$ is a quasi-finite $A$-module (note that $M \neq nM$ and $M/nM$ is a vector space of finite rank over $k$; deduce that $B/n$ is of finite rank over $k$).