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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

  1. multiplication is associative,

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

  3. 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.

Remark 24.12.2. Let $(f, f^\sharp ) : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}_\mathcal {C}) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}_\mathcal {D})$ be a morphism of ringed topoi.

  1. Let $(\mathcal{A}, \text{d})$ be a differential graded $\mathcal{O}_\mathcal {C}$-algebra. The pushforward will be the differential graded $\mathcal{O}_\mathcal {D}$-algebra $(f_*\mathcal{A}, \text{d})$ where $f_*\mathcal{A}$ is as in Remark 24.3.2 and $\text{d} = f_*\text{d}$ as maps $f_*\mathcal{A}^ n \to f_*\mathcal{A}^{n + 1}$. We omit the verification that the Leibniz rule is satisfied.

  2. Let $\mathcal{B}$ be a differential graded $\mathcal{O}_\mathcal {D}$-algebra. The pullback will be the differential graded $\mathcal{O}_\mathcal {C}$-algebra $(f^*\mathcal{B}, \text{d})$ where $f^*\mathcal{B}$ is as in Remark 24.3.2 and $\text{d} = f^*\text{d}$ as maps $f^*\mathcal{B}^ n \to f^*\mathcal{B}^{n + 1}$. We omit the verification that the Leibniz rule is satisfied.

  3. The set of homomorphisms $f^*\mathcal{B} \to \mathcal{A}$ of differential graded $\mathcal{O}_\mathcal {C}$-algebras is in $1$-to-$1$ correspondence with the set of homomorphisms $\mathcal{B} \to f_*\mathcal{A}$ of differential graded $\mathcal{O}_\mathcal {D}$-algebras.

Part (3) follows immediately from the usual adjunction between $f^*$ and $f_*$ on sheaves of modules.

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