Exercise 27, § 11

Let B be an associative unital algebra over a commutative field K, and let $m : B \otimes_K B \to B$ be the K-linear mapping defining the multiplication in B. The tensor product $B^* \otimes_K B^*$ (where $B^*$ is the dual vector space of the vector space B) is canonically identified with a vector subspace of $(B \otimes_K B)^*$ (II, p. 110, prop. 16).

a) Now prove that, for an element $w \in B^*$, the following conditions are equivalent: $1^\circ \ ^t m(w) \in B^* \otimes B^* ; 2^\circ$ the set of the $x \cdot w$, where x runs through B, is a finite-dimensional subspace of $B^* ; 3^\circ$ the set of the $w \cdot x$, where x runs through B, is a finite-dimensional subspace of $B^* ; 4^\circ$ the set of the $x \cdot w \cdot y$, where x and y run through B, is contained in a finite-dimensional subspace of $B^*$.

b) Let $B' \subset B^*$ be the set of the $w \in B^*$ satisfying the equivalent conditions of a). Now prove that $^t m(B') \subset B' \otimes_K B'$. (For $w \in B'$, write $^t m(w) = \sum_i u_i \otimes v_i$, where for example the $u_i$ are linearly independent in $B^*$; now prove that one then has $v_i \in B'$ for every i by using the associativity of B, and reasoning by contradiction.) $B'$ is therefore the largest cogebra contained in $B^*$, for which $^t m$ is the coproduct. One says that $B'$ is the dual cogebra of the algebra B. When B is of finite dimension, $B' = B^*$.

c) Now prove that if B is a commutative field of infinite dimension over K, one has $B' = \{0\}$. (Note that the orthogonal of the set of the $x \cdot w$, for a $w \in B^*$, x running through B, is an ideal of B).

d) Let $B_1, B_2$ be two K-algebras, $f : B_1 \to B_2$ a K-homomorphism of algebras. Now prove that one has $^t f(B'_2) \subset B'_1$, and that $^t f$, restricted to $B'_2$, is a homomorphism of cogebras of $B'_2$ into $B'_1$.

e) If C is a coassociative and co-unital cogebra over K, the canonical injection C → C** of the vector space C into its bidual transforms C into a subspace of (C*)′, dual cogebra of the dual algebra C* of C, and is a homomorphism of cogebras of C into (C*)′.