The Stacks project

Remark 24.23.5. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. A sheaf of graded sets on $\mathcal{C}$ is a sheaf of sets $\mathcal{S}$ endowed with a map $\deg : \mathcal{S} \to \underline{\mathbf{Z}}$ of sheaves of sets. Let us denote $\mathcal{O}[\mathcal{S}]$ the graded $\mathcal{O}$-module which is the free $\mathcal{O}$-module on the graded sheaf of sets $\mathcal{S}$. More precisely, the $n$th graded part of $\mathcal{O}[\mathcal{S}]$ is the sheafification of the rule

\[ U \longmapsto \bigoplus \nolimits _{s \in \mathcal{S}(U),\ \deg (s) = n} s \cdot \mathcal{O}(U) \]

With zero differential we also may consider this as a differential graded $\mathcal{O}$-module. Let $\mathcal{A}$ be a sheaf of graded $\mathcal{O}$-algebras Then we similarly define $\mathcal{A}[\mathcal{S}]$ to be the graded $\mathcal{A}$-module whose $n$th graded part is the sheafification of the rule

\[ U \longmapsto \bigoplus \nolimits _{s \in \mathcal{S}(U)} s \cdot \mathcal{A}^{n - \deg (s)}(U) \]

If $\mathcal{A}$ is a differential graded $\mathcal{O}$-algebra, the we turn this into a differential graded $\mathcal{O}$-module by setting $\text{d}(s) = 0$ for all $s \in \mathcal{S}(U)$ and sheafifying.

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