习题 28, § 2

令$A$为交换环，并具有线性拓扑和$E, F$两个线性拓扑的$A$模块。如果$V$（或$W$）穿过$E$（或$F$）的开子模的集合，子模

of $E \otimes_A F$ 形式 a fundamental 系统 of 邻域 of 0 in $E \otimes_A F$ for 拓扑 which is compatible with its 模结构 over the topological 环 $A$ and is called 张量积 of the given topologies on $E$ 和 $F$。 The Hausdorff completion $(E \otimes_A F)^{\wedge}$ of this $A$-module is an $\hat{A}$-module called the completed 张量 乘积 of $E$ and $F$.

(a) Show that, if $(V_\lambda)$ (resp. $(W_\mu)$) is a fundamental 系统 of 邻域 of 0 in $E$ (resp, $F$) 由……组成 of 子模, $(E \otimes_A F)^{\wedge}$ is canonically isomorphic to $A$ 模块中的逆系逆极限

推出 $(E \otimes_A F)^{\prime\prime}$ 是与$(\hat{E} \otimes_A \hat{F}^{\wedge})$ 规范同构的 $\hat{A}$ 模块；也用$\hat{E} \otimes_A \hat{F}$表示。

(b) 令$E', F'$为两个线性拓扑的$A$拓扑模块，以及$u : E \to E' \ v : F \to F'$两个连续的$A$线性映射；证明$u \otimes v : E \otimes F \to E' \otimes F'$与$E \otimes F$和$E' \otimes F'$上的张量积拓扑连续；我们用$u \otimes v$表示连续的线性映射$(E \otimes F)^{\wedge} \to (E' \otimes F')^{\wedge}$相应的到$u \otimes v$。

(c) Let $B, C$ be two commutative $A$-algebras with linear topologies such that 典范映射 $A \to B, A \to C$ are continuous (to abbreviate, $B, C$ are called topological $A$-algebras).证明$(B \otimes_A C)^{\wedge}$有一个规范拓扑$A$-代数结构，称为代数$B$和$C$的完备张量积。 定义规范连续表示

with the following 性质 : for every complete Hausdorff commutative topological $A$-algebra $D$ and every 有序对 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 $u = w \circ \rho$ 和 $v = w \circ \sigma$。

(d) 令$m$为$A$的理想，使得$A$上的拓扑为$m$进数拓扑。如果$E$和$F$各自给出$m$-adic 拓扑，则证明$E$和$F$上的拓扑的$E \otimes_A F$张量积是$m$-adic拓扑。推出$(E \otimes_A F)^*$与逆极限$\lim((E/m^{n+1}) \otimes_A F)$和$\lim(E \otimes_A (F/m^n + 1F))$同构的。