Proposition 15

Let $A$ be a filtered ring and $E$ a filtered $A$-module, the respective filtrations $(A_n), (E_n)$ of $A$ and $E$ being exhaustive. Then $(\hat{A}_n)$ is a filtration compatible with the ring structure on $\hat{A}$ and $(\hat{E}_n)$ a filtration compatible with the module structure on $\hat{E}$ over the filtered ring $\hat{A}$; moreover these filtrations are exhaustive and define respectively the topologies on $\hat{A}$ and $\hat{E}$. Finally, the canonical mappings $\operatorname{gr}(A) \to \operatorname{gr}(\hat{A})$ and $\operatorname{gr}(E) \to \operatorname{gr}(\hat{E})$ of graded $\mathbf{Z}$-modules are respectively a graded ring isomorphism and a graded $\operatorname{gr}(A)$-module isomorphism.

In what follows, for every uniform space $X$, $j_X$ will denote the canonical mapping from $X$ to its Hausdorff completion $\hat{X}$ and $X_0 = j_X(X)$ the uniform subspace of $\hat{X}$, which is the Hausdorff space associated with $X$. Recall that the topology on $X$ is the inverse image under $j_X$ of that on $X_0$ (General Topology,

Chapter II, § 3, no. 7, Proposition 12). Recall also that, for every uniformly continuous mapping $f : X \to Y$, $\hat{f}$ denotes the uniformly continuous mapping from $\hat{X}$ to $\hat{Y}$ such that $\hat{f} \circ j_X = j_Y \circ f$ (loc. cit., Proposition 15); if $X$ is a uniform subspace of $Y$ and $\mathscr{E}$ the canonical injection, $\hat{X}$ is identified with a uniform subspace of $\hat{Y}$ and $\hat{f}$ is the canonical injection of $X$ into $\hat{Y}$ (loc. cit., no. 9. Corollary 1 to Proposition 18).