## 24.12 Sheaves of differential graded algebras

This section is the analogue of Differential Graded Algebra, Section 22.3.

Definition 24.12.1. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. A *sheaf of differential graded $\mathcal{O}$-algebras* or a *sheaf of differential graded algebras* on $(\mathcal{C}, \mathcal{O})$ is a cochain complex $\mathcal{A}^\bullet $ of $\mathcal{O}$-modules endowed with $\mathcal{O}$-bilinear maps

\[ \mathcal{A}^ n \times \mathcal{A}^ m \to \mathcal{A}^{n + m},\quad (a, b) \longmapsto ab \]

called the multiplication maps with the following properties

multiplication is associative,

there is a global section $1$ of $\mathcal{A}^0$ which is a two-sided identity for multiplication,

for $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$, $a \in \mathcal{A}^ n(U)$, and $b \in \mathcal{A}^ m(U)$ we have

\[ \text{d}^{n + m}(ab) = \text{d}^ n(a)b + (-1)^ n a\text{d}^ m(b) \]

We often denote such a structure $(\mathcal{A}, \text{d})$. A *homomorphism of differential graded $\mathcal{O}$-algebras* from $(\mathcal{A}, \text{d})$ to $(\mathcal{B}, \text{d})$ is a map $f : \mathcal{A}^\bullet \to \mathcal{B}^\bullet $ of complexes of $\mathcal{O}$-modules compatible with the multiplication maps.

Given a differential graded $\mathcal{O}$-algebra $(\mathcal{A}, \text{d})$ and an object $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ we use the notation

\[ \mathcal{A}(U) = \Gamma (U, \mathcal{A}) = \bigoplus \nolimits _{n \in \mathbf{Z}} \mathcal{A}^ n(U) \]

This is a differential graded $\mathcal{O}(U)$-algebra.

As much as possible, we will think of a differential graded $\mathcal{O}$-algebra $(\mathcal{A}, \text{d})$ as a graded $\mathcal{O}$-algebra $\mathcal{A}$ endowed with the operator $\text{d} : \mathcal{A} \to \mathcal{A}$ of degree $1$ (where $\mathcal{A}$ is viewed as a graded $\mathcal{O}$-module) satisfying the Leibniz rule given in the definition.

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