The Stacks project

24.14 The differential graded category of modules

This section is the analogue of Differential Graded Algebra, Example 22.26.8. For our conventions on differential graded categories, please see Differential Graded Algebra, Section 22.26.

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $(\mathcal{A}, \text{d})$ be a sheaf of differential graded algebras on $(\mathcal{C}, \mathcal{O})$. We will construct a differential graded category

\[ \textit{Mod}^{dg}(\mathcal{A}, \text{d}) \]

over $R = \Gamma (\mathcal{C}, \mathcal{O})$ whose associated category of complexes is the category of differential graded $\mathcal{A}$-modules:

\[ \textit{Mod}(\mathcal{A}, \text{d}) = \text{Comp}(\textit{Mod}^{dg}(\mathcal{A}, \text{d})) \]

As objects of $\textit{Mod}^{dg}(\mathcal{A}, \text{d})$ we take right differential graded $\mathcal{A}$-modules, see Section 24.13. Given differential graded $\mathcal{A}$-modules $\mathcal{L}$ and $\mathcal{M}$ we set

\[ \mathop{\mathrm{Hom}}\nolimits _{\textit{Mod}^{dg}(\mathcal{A}, \text{d})}(\mathcal{L}, \mathcal{M}) = \mathop{\mathrm{Hom}}\nolimits _{\textit{Mod}^{gr}(\mathcal{A})}(\mathcal{L}, \mathcal{M}) = \bigoplus \nolimits _{n \in \mathbf{Z}} \mathop{\mathrm{Hom}}\nolimits ^ n(\mathcal{L}, \mathcal{M}) \]

as a graded $R$-module, see Section 24.5. In other words, the $n$th graded piece $\mathop{\mathrm{Hom}}\nolimits ^ n(\mathcal{L}, \mathcal{M})$ is the $R$-module of right $\mathcal{A}$-module maps homogeneous of degree $n$. For an element $f \in \mathop{\mathrm{Hom}}\nolimits ^ n(\mathcal{L}, \mathcal{M})$ we set

\[ \text{d}(f) = \text{d}_\mathcal {M} \circ f - (-1)^ n f \circ \text{d}_\mathcal {L} \]

To make sense of this we think of $\text{d}_\mathcal {M}$ and $\text{d}_\mathcal {L}$ as graded $\mathcal{O}$-module maps and we use composition of graded $\mathcal{O}$-module maps. It is clear that $\text{d}(f)$ is homogeneous of degree $n + 1$ as a graded $\mathcal{O}$-module map, and it is $\mathcal{A}$-linear because for homogeneous local sections $x$ and $a$ of $\mathcal{M}$ and $\mathcal{A}$ we have

\begin{align*} \text{d}(f)(xa) & = \text{d}_\mathcal {M}(f(x) a) - (-1)^ n f (\text{d}_\mathcal {L}(xa)) \\ & = \text{d}_\mathcal {M}(f(x)) a + (-1)^{\deg (x) + n} f(x) \text{d}(a) - (-1)^ n f(\text{d}_\mathcal {L}(x)) a - (-1)^{n + \deg (x)} f(x) \text{d}(a) \\ & = \text{d}(f)(x) a \end{align*}

as desired (observe that this calculation would not work without the sign in the definition of our differential on $\mathop{\mathrm{Hom}}\nolimits $).

For differential graded $\mathcal{A}$-modules $\mathcal{K}$, $\mathcal{L}$, $\mathcal{M}$ we have already defined the composition

\[ \mathop{\mathrm{Hom}}\nolimits ^ m(\mathcal{L}, \mathcal{M}) \times \mathop{\mathrm{Hom}}\nolimits ^ n(\mathcal{K}, \mathcal{L}) \longrightarrow \mathop{\mathrm{Hom}}\nolimits ^{n + m}(\mathcal{K}, \mathcal{M}) \]

in Section 24.5 by the usual composition of maps of sheaves. This defines a map of differential graded modules

\[ \mathop{\mathrm{Hom}}\nolimits _{\textit{Mod}^{dg}(\mathcal{A}, \text{d})}(\mathcal{L}, \mathcal{M}) \otimes _ R \mathop{\mathrm{Hom}}\nolimits _{\textit{Mod}^{dg}(\mathcal{A}, \text{d})}(\mathcal{K}, \mathcal{L}) \longrightarrow \mathop{\mathrm{Hom}}\nolimits _{\textit{Mod}^{dg}(\mathcal{A}, \text{d})}(\mathcal{K}, \mathcal{M}) \]

as required in Differential Graded Algebra, Definition 22.26.1 because

\begin{align*} \text{d}(g \circ f) & = \text{d}_\mathcal {M} \circ g \circ f - (-1)^{n + m} g \circ f \circ \text{d}_\mathcal {K} \\ & = \left(\text{d}_\mathcal {M} \circ g - (-1)^ m g \circ \text{d}_ L\right) \circ f + (-1)^ m g \circ \left(\text{d}_\mathcal {L} \circ f - (-1)^ n f \circ \text{d}_\mathcal {K}\right) \\ & = \text{d}(g) \circ f + (-1)^ m g \circ \text{d}(f) \end{align*}

if $f$ has degree $n$ and $g$ has degree $m$ as desired.


Comments (0)


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 0FRL. Beware of the difference between the letter 'O' and the digit '0'.