Remark 1

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

Algebra, chapter VIII, § 13. Absolutely Semisimple Algebras

$*$ The morphisms $\partial^n: C^n(A,P)\rightarrow C^{n+1}(A,P)$ define a complex $(C(A,P), \partial )$ of K-modules (X, §2, n $^o1$ , p. 24). The mapping $\gamma_n$ therefore defines a homotopy from this complex to itself linking 0 to Id $_{C(A,P)}$ (X, §2, n $^o4$ , p. 32, définition $4$ ) $.*$

We keep the notation of No.6. For every integer $n\geqslant 0$ , we define a mapping $h_n: B_n\rightarrow B_{n+1}$ by the formula

h_n(x) =d^1_{n+2}(e\otimes x) =\sum_{i=1}^ra_i\otimes a'_ix

It is a homomorphism of $(A$ , A)-bimodules (formula (3)).

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

Nothing.

Cites

X, §2, n$^o1$, p. 24 not in this corpus: A X
X, §2, n$^o4$, p. 32 not in this corpus: A X