Exercise 28, § 2

Let $A$ be a commutative ring with a linear topology and $E, F$ two topological $A$-modules which are linearly topologized. If $V$ (resp. $W$) runs through the set of open submodules of $E$ (resp. $F$), the submodules

of $E \otimes_A F$ form a fundamental system of neighbourhoods of 0 in $E \otimes_A F$ for a topology which is compatible with its module structure over the topological ring $A$ and is called the tensor product of the given topologies on $E$ and $F$. The Hausdorff completion $(E \otimes_A F)^{\wedge}$ of this $A$-module is an $\hat{A}$-module called the 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, $F$) consisting of submodules, $(E \otimes_A F)^{\wedge}$ is canonically isomorphic to the inverse limit of the inverse system of $A$-modules

deduce that $(E \otimes_A F)^{\prime\prime}$ is an $\hat{A}$-module canonically isomorphic to $(\hat{E} \otimes_A \hat{F}^{\wedge})$; it is also denoted by $\hat{E} \otimes_A \hat{F}$.

(b) Let $E', F'$ be two topological $A$-modules which are linearly topologized and $u : E \to E' \ v : F \to F'$ two continuous $A$-linear mappings; show that $u \otimes v : E \otimes F \to E' \otimes F'$ is continuous with the tensor product topologies on $E \otimes F$ and $E' \otimes F'$; we denote by $u \otimes v$ the continuous linear mapping $(E \otimes F)^{\wedge} \to (E' \otimes F')^{\wedge}$ corresponding to $u \otimes v$.

(c) Let $B, C$ be two commutative $A$-algebras with linear topologies such that the canonical mappings $A \to B, A \to C$ are continuous (to abbreviate, $B, C$ are called topological $A$-algebras). Show that $(B \otimes_A C)^{\wedge}$ has a canonical topological $A$-algebra structure called the completed tensor product of the algebras $B$ and $C$. Define canonical continuous representations

with the following property : for every complete Hausdorff commutative topological $A$-algebra $D$ and every ordered pair of continuous $A$-homomorphisms $u : B \to D, v : C \to D$, there exists a unique continuous $A$-homomorphism $w : (B \otimes_A C)^{\wedge} \to D$ such that $u = w \circ \rho$ and $v = w \circ \sigma$.

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