Proposition 12

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

For a graded K-linear mapping $f : A \to E$ of degree $\delta$ to be an $\varepsilon$ -derivation, it is necessary and sufficient that the mapping $x \mapsto (x, f(x))$ of A into $A \oplus E(\delta)$ be a graded K-algebra homomorphism.

Using the fact that for $x$ homogeneous in $A$ and $y \in A$

(xy, f(xy)) = (x, f(x)).(y, f(y)),

we obtain, taking account of (28), the equivalent relation

f(xy) = f(x).y + \varepsilon_{\delta, \deg(x)} x.f(y),

whence the proposition.

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

Cited by

Proposition 14, § 10

Cites

Nothing.