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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.

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