Remark 50.10.1 (de Rham complex of a graded ring). Let $G$ be an abelian monoid written additively with neutral element $0$. Let $R \to A$ be a ring map and assume $A$ comes with a grading $A = \bigoplus _{g \in G} A_ g$ by $R$-modules such that $R$ maps into $A_0$ and $A_ g \cdot A_{g'} \subset A_{g + g'}$. Then the module of differentials comes with a grading

$\Omega _{A/R} = \bigoplus \nolimits _{g \in G} \Omega _{A/R, g}$

where $\Omega _{A/R, g}$ is the $R$-submodule of $\Omega _{A/R}$ generated by $a_0 \text{d}a_1$ with $a_ i \in A_{g_ i}$ such that $g = g_0 + g_1$. Similarly, we obtain

$\Omega ^ p_{A/R} = \bigoplus \nolimits _{g \in G} \Omega ^ p_{A/R, g}$

where $\Omega ^ p_{A/R, g}$ is the $R$-submodule of $\Omega ^ p_{A/R}$ generated by $a_0 \text{d}a_1 \wedge \ldots \wedge \text{d}a_ p$ with $a_ i \in A_{g_ i}$ such that $g = g_0 + g_1 + \ldots + g_ p$. Of course the differentials preserve the grading and the wedge product is compatible with the gradings in the obvious manner.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).