Proposition 6

Tag 066N, permanent. Cite it as [Bourbaki, Tag 066N].

*Suppose that $\mathbf{A}$ is a unital associative graded $\mathbf{K}$ -algebra and $\mathbf{E}$ a graded $(\mathbf{A}, \mathbf{A})$ -bimodule. If $d : \mathbf{A} \to \mathbf{E}$ is an $\varepsilon$ -derivation of degree $\delta$ , then, for every finite sequence $(x_i)_{1 \leq i \leq n}$ of homogeneous elements of $\mathbf{A}$ , of respective degrees $\xi_i$ ( $1 \leq i \leq n$ ),

d(x_1 x_2 \ldots x_n) = \sum_{i=1}^n \varepsilon_{\delta, \xi_1 + \cdots + \xi_{i-1}} x_1 \cdots x_{i-1} (dx_i) x_{i+1} \cdots x_n.

Formula (21) is trivial for

n = 0

and is proved by induction on

n

, taking account of (4) (no. 2).

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

Cited by

Proposition 4, § 1

Cites

Nothing.