Proposition 6

Tag 00FS, permanent. Cite it as [Bourbaki, Tag 00FS].

Algebra, VIII, p. 241, § 13. Absolutely Semisimple Algebras

Let K be a commutative ring and A a K-algebra. Let $e=\sum^r_{i=1}a_i\otimes a'_i$ be an element of $B = A\otimes_KA^o$ satisfying the conditions in item (ii) of Proposition 5 of VIII, p. 236. For any integer $n\geqslant 1$ and any element $f$ of $C^n(A,P)$ , we denote by $\gamma^n(f)$ the element of $C^{n-1}(A,P)$ defined by the formula

\gamma^n(f)(x_1, . . . , x_{n-1}) =\sum_{i=1}^ra_if(a'_i, x_1. . . , x_{n-1}) \tag{15}

We then have

\partial^{n-1}(\gamma^n(f)) +\gamma^{n+1}(\partial^n(f)) =f \tag{16}

for every integer $n\geqslant 1$ and every $f\in C^n(A,P)$ .

Bourbaki sets the proof directly after the statement, and the corpus does not separate them, so what is above is the statement and its proof together.

Cited by

Lemma 5, § 13
Theorem 3, § 13

Cites

Proposition 5 of VIII, p. 236