Exercise 24, § 7

Let A be a unital associative algebra and M the set of its finite codimensional two-sided ideals.

a) Let $U^*$ be the vector space dual of U. If $\theta \in U^*$, show the equivalence of the following properties:

(i) there exists $\mathfrak{m}\in M$ such that $\theta (\mathfrak{m}) = 0$;

(ii) there exist two finite families $(\theta '_i)$ and $(\theta ''_i)$ of elements of $U^*$ such that

$\theta (xy) =\sum_i\theta '_i(x)\theta ''_i(y)$ for all $x, y\in U$.

The elements $\theta$ with these properties form a subspace $U'$ of $U^*$, which coincides with that denoted by $B'$ in Algebra, Chap. III, §11, Exerc. 27. There exists a unique coalgebra structure on $U'$ whose coproduct $c: U'\rightarrow U'\otimes U'$ is given by

where $\theta '_i, \theta ''_i\in U'$ are such that $\theta (xy) =\sum_i\theta '_i(x)\theta ''_i(y)$ for all $x, y\in U$, cf. (ii).

The coalgebra $U'$ is the union of the increasing filtration by the subspaces $(U/\mathfrak{m})^*,\mathfrak{m}\in M$, which are finite dimensional. The dual of $U'$ can be identified with the algebra $\widehat{U} =$ lim$_{\leftarrow -}U/\mathfrak{m}$; if $\widehat{U}$ is given the projective limit of the discrete topologies on the $U/\mathfrak{m},\mathfrak{m}\in M$, the continuous linear forms on $\widehat{U}$ are given by the elements of $U'$.

b) Let E be a finite dimensional left U-module. Its annihilator $\mathfrak{m}_E$ belongs to M; the composite $\widehat{U}\rightarrow U/\mathfrak{m}_E\rightarrow$ End(E) gives E the structure of a left $\widehat{U}$-module. If F is a finite dimensional left U-module, a linear map $f: E\rightarrow F$ is a U-homomorphism if and only if it is a $\widehat{U}$-homomorphism. If $a\in E, b\in E^*$, the linear form $\theta_{a,b}:x \rightarrow \langle xa, b\rangle$ belongs to $U'$, and

$\langle x, \theta_{a,b}\rangle =\langle xa, b\rangle$ for all $x\in \widehat{U}$.

The $\theta_{a,b}$ (for varying $E, a, b$) generate the $k$-vector space $U'$. c) Let $X_U$ be the set of isomorphism classes of finite dimensional left U-modules. For all $E\in X_U$, let $u_E$ be a $k$-linear endomorphism of E; assume that

$f\circ u_E=u_F\circ f$ for all $E,F\in X_U$ and $f\in$ Hom$_U(E,F)$.

Show that there exists a unique element $x\in \widehat{U}$ such that $x_E=u_E$ for all $E\in X_U$. (Reduce to the case in which U is finite dimensional.)