Exercise 28, § 2

Let A be a commutative ring endowed with a linear topology, E, F two topological A-modules, linearly topologized. When V (resp. W) runs through the set of open submodules of E (resp. F), the submodules $\operatorname{Im}(V \otimes_A F) + \operatorname{Im}(E \otimes_A W)$ of $E \otimes_A F$ form a fundamental system of neighbourhoods of 0 in $E \otimes_A F$ for a topology compatible with its module structure over the topological ring A, and called tensor product of the given topologies on E and F. The separated completion $(E \otimes_A F)^\sim$ of this A-module is a $\hat{A}$-module called completed tensor product of E and F.

(a) Show that if $(V_\lambda)$ (resp. $(W_\mu)$) is a fundamental system of neighbourhoods of 0 in E (resp. F) formed by submodules, $(E \otimes_A F)^\sim$ is canonically isomorphic to the projective limit of the projective system of A-modules $(E/V_\lambda) \otimes_A (F/W_\mu)$; deduce that $(E \otimes_A F)^\sim$ is a $\hat{A}$-module canonically isomorphic to $(\hat{E} \otimes_{\hat{A}} \hat{F})^\sim$; it is also denoted by $\hat{E} \otimes_{\hat{A}} \hat{F}$.

(b) Let E', F' be two topological A-modules, linearly topologized, $u : E \to E', \ \varphi : F \to F'$ two continuous A-linear mappings; show that $u \otimes \varphi : E \otimes F \to E' \otimes F'$ is continuous for the tensor product topologies on $E \otimes F$ and $E' \otimes F'$; $u \hat{\otimes} \varphi$ denotes the continuous linear mapping $(E \otimes F)^{\sim} \to (E' \otimes F')^{\sim}$ corresponding to $u \otimes \varphi$.

(c) Let B, C be two commutative A-algebras endowed with linear topologies such that the canonical mappings $A \to B, A \to C$ are continuous (one says for short that B and C are A-\emph{topological algebras}). Show that $(B \otimes_A C)^{\sim}$ is canonically endowed with a structure of topological A-algebra, called \emph{completed tensor product} of the algebras B and C. Define canonical continuous representations $\rho : B \to (B \otimes_A C)^{\sim}, \ \sigma : C \to (B \otimes_A C)^{\sim}$ having the following property: for every commutative topological A-algebra D, separated and complete, and every pair of continuous A-homomorphisms $u : B \to D, \ \varphi : C \to D$, there exists one and only one continuous A-homomorphism $\omega : (B \otimes_A C)^{\sim} \to D$ such that $u = \omega \circ \rho$ and $\varphi = \omega \circ \sigma$.

(d) Let $\mathfrak{m}$ be an ideal of A such that the topology of A is the $\mathfrak{m}$-adic topology. If E and F are each endowed with the $\mathfrak{m}$-adic topology, show that on $E \otimes_A F$ the tensor product of the topologies of E and F is the $\mathfrak{m}$-adic topology. Deduce that $(E \otimes_A F)^\sim$ is isomorphic to the projective limits $\lim ((E/\mathfrak{m}^{n+1}E) \otimes_A F)$ and $\lim (E \otimes_A (F/\mathfrak{m}^{n+1}F))$.