The Stacks project

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.

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