Exercise 25, § 11

Let C be a cogebra over a ring A. A sub-A-module S of C is called a coideal if c(S) ⊂ Im(S ⊗ C) + Im(C ⊗ S). If C is counital, with counit γ, a coideal S is called conull if γ(x) = 0 for x ∈ S.

(a) A right or left coideal is a coideal. Every sum of coideals is a coideal. (b) If S is a coideal of C, the orthogonal S^0 is a subalgebra of the dual algebra C*. If C is counital and S is conull, S^0 is unital. (c) Suppose that A is a field and C a counital cogebra. Show that if E' is a unital subalgebra of C* with the same unit element as C*, the orthogonal E'^0 in C is a conull coideal. (d) If S is a coideal of C, show that there exists on C/S one and only one cogebra structure such that the canonical mapping p: C → C/S is a cogebra morphism. If C is counital and S is conull, C/S is counital. (e) For every cogebra morphism f: C → C', S = Ker(f) is a coideal of C, S' = Im(f) a subcogebra of C' and, in the canonical factorization

of f, g is a cogebra isomorphism.

(26) Let C be a coassociative counital cogebra over a commutative field K.

(a) For every vector subspace E of C which is a left C*-module under the law (u, x) ↦ u ↼ x, the left annihilator of this C*-module is a two-sided ideal J' of the algebra C*. Show that if E is finite-dimensional over K, C*/J' is finite-dimensional. (b) Deduce from (a) that for every element x ∈ C, the subcogebra of C generated by x is finite-dimensional. (Note that the C*-module E generated by x is finite-dimensional and that, if J' is its left annihilator, x ∈ J'^0, the orthogonal of J' in C.)

(27) Let B be an associative unital algebra over a commutative field K and let m: B ⊗_K B → B be the K-linear mapping defining the multiplication on B. The tensor product B* ⊗_K B* (where B* is the dual vector space of the vector space B) is canonically identified with a vector subspace of (B ⊗_K B)* (II, § 7, no. 7, Proposition 16).

(a) Show that for an element w ∈ B* the following conditions are equivalent: (1) ^t m(w) ∈ B* ⊗ B*; (2) the set of x ↼ w, where x runs through B, is a finite-dimensional subspace of B*; (3) the set of w ↼ x, where x runs through B, is a finite-dimensional subspace of B*; (4) the set of x ↼ w ↼ 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 $w \in B^*$ satisfying the equivalent conditions in (a). Show 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 $B^*$; show that then $v_i \in B'$ for all $i$ using the associativity of $B$ and arguing by reductio ad absurdum. ) $B'$ is therefore the largest cogebra contained in $B^*$, for which $^t m$ is the coproduct. $B'$ is called the dual cogebra of the algebra $B$. When $B$ is finite-dimensional, $B' = B^*$.

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

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

(e) If $C$ is a coassociative counital cogebr*a over $K$, the canonical injection $C \to C^{**}$ of the vector space $C$ into its bidual maps $C$ onto a subspace of $(C^*)'$, the dual cogebr*a of the algebra $C^*$ dual to $C$, and is a cogebr*a homomorphism of $C$ into $(C^*)'$.