Proposition 6

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

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

Let E be a graded cogebra over A. In order that, for every commutative A-algebra B, with the trivial graduation, the graded A-algebra of type $\mathbf{Z} \mathrm{Homgr}_A(E, B)$ (no. 2) be anticommutative (§ 4, no. 9, Definition 7), it is necessary and sufficient that, if $\sigma_g$ denotes the automorphism of the A-module $E \otimes_A E$ such that

\sigma_g(x_p \otimes x_q) = (-1)^{pq} x_q \otimes x_p

for $x_p \in E_p, x_q \in E_q$ , where p and q are arbitrary elements of $\mathbf{N}$ , the diagram

\begin{array}{ccc} E & & \\ & \searrow^c & \swarrow^c \\ E \otimes_A E & \xrightarrow{\sigma_g} & E \otimes_A E \end{array}

be commutative.

The proof is analogous to that of Proposition 2 of no. 2.

When the graded cogebra E satisfies the condition of Proposition 6, it is said to be anticocommutative.

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

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

Proposition 6

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