Example

Tag 067O, permanent. Cite it as [Bourbaki, Tag 067O].

Algebra, chapter III, § 11. Cogebras, products of multilinear forms, inner products and duality

(4) It is immediately verified that the cogebra $A$ (no. 1, Example (1)) the cogebra $A^{(X)}$ (no. 1, Example 4) and the cogebra $\mathscr{T}(M)$ (no. 1, Example 5) are coassociative. If $B$ is an associative $A$ -algebra which is a finitely generated projective $A$ -module, the cogebra $B^*$ (no. 1, Example 3) is coassociative: for the commutativity of diagram (12) then follows by transposition from that of the diagram which expresses the associativity of $B$ (§ 1, no. 3). Conversely, the same argument and the canonical identification of the A-module B with its bidual (II, § 2, no. 7, Corollary 4 to Proposition 13) show that if the cogebras B* is coassociative, the algebra B is associative. Finally, the cogebras S(M) and $\Lambda(M)$ (no. 1, Examples 6 and 7) are coassociative; this follows from the commutativity of the diagram

\begin{array}{ccc} M & \xrightarrow{\Delta} & M \times M \\ \downarrow & & \downarrow 1_M \times \Delta \\ M \times M & \xrightarrow{\Delta \times 1_M} & M \times M \times M \end{array}

the functorial properties of S(M) (§ 6, no. 2) and $\Lambda(M)$ (§ 7, no. 2), which give the corresponding commutative diagrams

\left\{ \begin{array}{ccc} S(M) & \xrightarrow{S(\Delta)} & S(M \times M) \\ \downarrow S(\Delta) & & \downarrow S(\Delta_M \times 1) \\ S(M \times M) & \xrightarrow{S(\Delta \times 1_M)} & S(M \times M \times M) \end{array} \right.

\left\{ \begin{array}{ccc} \Lambda(M) & \xrightarrow{\Lambda(\Delta)} & \Lambda(M \times M) \\ \downarrow \Lambda(\Delta) & & \downarrow \Lambda(1_M \times \Delta) \\ \Lambda(M \times M) & \xrightarrow{\Lambda(\Delta \times 1_M)} & \Lambda(M \times M \times M) \end{array} \right.

and the existence and functoriality of canonical isomorphisms for symmetric and exterior algebras of a direct sum (§ 6, no. 6 and § 7, no. 7).

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

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Example, § 11

Example

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