Section 24.12 (0FRE): Sheaves of differential graded algebras

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

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.

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.
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.
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.

The Stacks project

24.12 Sheaves of differential graded algebras

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