Theorem 3

Let A be a commutative Noetherian ring, m an ideal of A and E, F, G threefinitely generated A-modules. Then:

(i) If $E \xrightarrow{u} F \xrightarrow{v} G$ is an exact sequence of A-linear mappings, the sequence $\hat{E} \xrightarrow{\hat{u}} \hat{F} \xrightarrow{\hat{v}} \hat{G}$ obtained by passing to the Hausdorff completions (with respect to the m-adic topologies) is exact.

(ii) The canonical A-linear mapping $\alpha_E : \hat{A} \otimes_A E \to \hat{E}$ is bijective.

(iii) The A-module $\hat{A}$ is flat.

We have seen that $u$ and $v$ are strict morphisms of topological groups (no. 2, Corollary to Theorem 2). Assertion (i) then follows from § 2, no. 12, Lemma 2. Assertion (ii) is obvious when $E = A$ and the case where E is a free finitely generated A-module can be immediately reduced to that. In the general case, E admits a finite presentation

(Chapter I, § 2, no. 8, Lemma 8). We derive a commutative diagram

The first row is exact (Chapter I, § 2, no. 1, Lemma 1) and so is the second by (i). We already know that $\alpha_E$ is surjective (§ 2, no. 12, Proposition 16); on the other hand, as $\alpha_L$ and $\alpha_{L'}$ are bijective and $1 \otimes v$ is surjective, $\alpha_E$ is injective by virtue of Chapter I, § 1, no. 4, Corollary 2 to Proposition 2; this shows (ii).

Then it follows from (i) and (ii) that, if $a$ is an ideal of A (necessarily finitely generated), the canonical mapping $\hat{A} \otimes_A a \to \hat{A}$ is injective, being the composition of $\hat{a} \to \hat{A}$ and $\alpha_E$, which proves that $\hat{A}$ is a flat $A$-module (Chapter I, § 2, no. 3, Proposition 1).

Under the conditions of Theorem 3 $\hat{A} \otimes_A E$ is often identified with $E$ by means of the canonical mapping $\alpha_E$. If $u : E \to F$ is a homomorphism of finitely generated $A$-modules, $\hat{u} : E \to \hat{F}$ is then identified with $1 @ u$ by virtue of the commutativity of diagram (4).