Definition 24.13.1. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $(\mathcal{A}, \text{d})$ be a sheaf of differential graded algebras on $(\mathcal{C}, \mathcal{O})$. A (right) differential graded $\mathcal{A}$-module or (right) differential graded module over $\mathcal{A}$ is a cochain complex $\mathcal{M}^\bullet$ endowed with $\mathcal{O}$-bilinear maps

$\mathcal{M}^ n \times \mathcal{A}^ m \to \mathcal{M}^{n + m},\quad (x, a) \longmapsto xa$

called the multiplication maps with the following properties

1. multiplication satisfies $(xa)a' = x(aa')$,

2. the identity section $1$ of $\mathcal{A}^0$ acts as the identity on $\mathcal{M}^ n$ for all $n$,

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

$\text{d}^{n + m}(xa) = \text{d}^ n(x)a + (-1)^ n x\text{d}^ m(a)$

We often say “let $\mathcal{M}$ be a differential graded $\mathcal{A}$-module” to indicate this situation. A homomorphism of differential graded $\mathcal{A}$-modules from $\mathcal{M}$ to $\mathcal{N}$ is a map $f : \mathcal{M}^\bullet \to \mathcal{N}^\bullet$ of complexes of $\mathcal{O}$-modules compatible with the multiplication maps. The category of (right) differential graded $\mathcal{A}$-modules is denoted $\textit{Mod}(\mathcal{A}, \text{d})$.

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0FRI. Beware of the difference between the letter 'O' and the digit '0'.