Exercise 16, § 3

Let $A$ be a Commutative Noetherian ring, $m$ an ideal of $A$ and $E$ a finitely generated $A$-module with the $m$-adic topology. For every infinite set $I$, let $E$, denote the set of families $(x_i)_{i \in I}$ of elements of $E$ such that $\lim x_i = 0$ with respect to the filter of complements of finite subsets of $I$; it is a submodule of the product $A$-module $E^I$.

(a) Show that, if $0 \to E' \to E \to E'' \to 0$ is an exact sequence of finitely generated $A$-modules, the corresponding sequence $0 \to E'_1 \to E_1 \to E''_1 \to 0$ is exact.

(b) Define a canonical $A$-homomorphism $A, \otimes_A E \to E_1$ for every $A$-module $E$ and show that it is an isomorphism (first verify this when $E$ is free and finitely generated and then use (a)).

(c) Deduce from (b) that $A$, is a faithfully flat $A$-module.