Remark

Tag 0604, permanent. Cite it as [Bourbaki, Tag 0604].

Algebra, chapter III, § 3. Graded algebras

Definition 1 can also be interpreted by saying that $E$ is a graded $A$ -module and that the $A$ -linear mapping

m : E \otimes_A E \to E

defining the multiplication on $E$ (§ 1, no. 3), is homogeneous of degree 0 when $E \otimes_A E$ is given its graduation of type $\Delta$ (II, § 11, no. 5).

To define a graded $A$ -algebra structure of type $\Delta$ on the graded ring $A$ , with $E$ as underlying graded $A$ -module, therefore amounts to defining for each ordered pair $(\lambda, \mu)$ of elements of $\Delta$ a $\mathbf{Z}$ -bilinear mapping

m_{\lambda \mu} : E_\lambda \times E_\mu \to E_{\lambda + \mu}

such that for every triple of indices $(\lambda, \mu, \nu)$ and for $\alpha \in A_\lambda,\ x \in E_\mu,\ y \in E_\nu$ ,

\alpha . m_{\mu \nu}(x, y) = m_{\lambda + \mu, \nu}(\alpha x, y) = m_{\mu, \lambda + \nu}(x, \alpha y).

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

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