Proposition 13

Let A be a ring, m a two-sided ideal of A contained in the Jacobson radical of A and E an A-module. Let A and E be given the m-adic filtrations (no. 1, Example 3). Suppose that one of the following conditions holds: (a) E is a finitely generated A-module and A is Hausdorff; (b) m is nilpotent.

For E to be a free A-module, it is necessary and sufficient that E/mE be a free (A/m)-module and that E satisfy property (GR) (no. 8).

If E is a free A-module and $(e_\lambda)$ a basis of E, $m^k E$ is the direct sum of the submodules $m^k e_\lambda$ of E for all $k > 0$ (Algebra, Chapter II, § 3, no. 7, Remark); then $m^k E / m^{k+1} E$ is identified with the direct sum of the $m^k e_\lambda / m^{k+1} e_\lambda$ (Algebra, Chapter 11, § 1, no. 6, Proposition 7). We deduce first (for $k = 0$) that the classes $1 \otimes e_\lambda$ of the $e_\lambda$ in $E / mE = (A / m) \otimes_A E$ form a basis of the (A/m)-module $E / mE$, since the canonical mapping

is bijective for all $k \geq 0$; hence $\gamma_E$ is bijective. Note that this part of the proof uses neither condition (a) nor condition (b).

Suppose conversely that the conditions of the statement hold and let $(x_i)_{i \in I}$ be a family of elements of E whose classes mod. $mE$ form a basis of the $(A/m)$-module $E/mE$; let L be the free A-module $A_s^{(I)}$, $(f_i)_{i \in I}$ its canonical basis and $u : L \to E$ the A-linear mapping such that $u(f_i) = x_i$ for all $i \in I$. The hypotheses already imply that $u$ is surjective (Chapter II, § 3, no. 2, Corollary 1 to Proposition 4) and it remains to prove that $u$ is injective. Now, each of the hypotheses (a) and (b) implies that A is Hausdorff and hence so is L with the m-adic filtration, since $m^kL = (m^k)^{(I)}$ (Algebra, Chapter II, § 3, no. 7, Remark) and $\operatorname{gr}(L)$ is identified with $\operatorname{gr}(A) \otimes_{A/m} (L/mL)$ from the first part of the proof; the homomorphism $u$ is compatible with the filtrations and it is possible to write $\operatorname{gr}(u) = \gamma_E \circ v$ where $v$ is the bijection of $\operatorname{gr}(L)$ onto

mapping the class off, mod. mM onto $1 \otimes \bar{x}_i$, where $\bar{x}_i$ is the class of $x_i$ mod. $mE$. The hypothesis then implies that $\operatorname{gr}(u)$ is injective and the conclusion follows with the aid of Corollary 1 to Theorem 1, no. 8.