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24.13 Sheaves of differential graded modules

This section is the analogue of Differential Graded Algebra, Section 22.4.

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 $\textit{Mod}(\mathcal{A}, \text{d})$.

We can define left differential graded modules in exactly the same manner but our default in the chapter will be right modules.

Given a differential graded $\mathcal{A}$-module $\mathcal{M}$ and an object $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ we use the notation

\[ \mathcal{M}(U) = \Gamma (U, \mathcal{M}) = \bigoplus \nolimits _{n \in \mathbf{Z}} \mathcal{M}^ n(U) \]

This is a (right) differential graded $\mathcal{A}(U)$-module.

Lemma 24.13.2. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $(\mathcal{A}, \text{d})$ be a differential graded $\mathcal{O}$-algebra. The category $\textit{Mod}(\mathcal{A}, \text{d})$ is an abelian category with the following properties

  1. $\textit{Mod}(\mathcal{A}, \text{d})$ has arbitrary direct sums,

  2. $\textit{Mod}(\mathcal{A}, \text{d})$ has arbitrary colimits,

  3. filtered colimit in $\textit{Mod}(\mathcal{A}, \text{d})$ are exact,

  4. $\textit{Mod}(\mathcal{A}, \text{d})$ has arbitrary products,

  5. $\textit{Mod}(\mathcal{A}, \text{d})$ has arbitrary limits.

The forgetful functor

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

sending a differential graded $\mathcal{A}$-module to its underlying graded module commutes with all limits and colimits.

Proof. Let us denote $F : \textit{Mod}(\mathcal{A}, \text{d}) \to \textit{Mod}(\mathcal{A})$ the functor in the statement of the lemma. Observe that the category $\textit{Mod}(\mathcal{A})$ has properties (1) – (5), see Lemma 24.4.2.

Consider a homomorphism $f : \mathcal{M} \to \mathcal{N}$ of graded $\mathcal{A}$-modules. The kernel and cokernel of $f$ as maps of graded $\mathcal{A}$-modules are additionally endowed with differentials as in Definition 24.13.1. Hence these are also the kernel and cokernel in $\textit{Mod}(\mathcal{A}, \text{d})$. Thus $\textit{Mod}(\mathcal{A}, \text{d})$ is an abelian category and taking kernels and cokernels commutes with $F$.

To prove the existence of limits and colimits it is sufficient to prove the existence of products and direct sums, see Categories, Lemmas 4.14.11 and 4.14.12. The same lemmas show that proving the commutation of limits and colimits with $F$ follows if $F$ commutes with direct sums and products.

Let $\mathcal{M}_ t$, $t \in T$ be a set of differential graded $\mathcal{A}$-modules. Then we can consider the direct sum $\bigoplus \mathcal{M}_ t$ as a graded $\mathcal{A}$-module. Since the direct sum of graded modules is done termwise, it is clear that $\bigoplus \mathcal{M}_ t$ comes endowed with a differential. The reader easily verifies that this is a direct sum in $\textit{Mod}(\mathcal{A}, \text{d})$. Similarly for products.

Observe that $F$ is an exact functor and that a complex $\mathcal{M}_1 \to \mathcal{M}_2 \to \mathcal{M}_3$ of $\textit{Mod}(\mathcal{A}, \text{d})$ is exact if and only if $F(\mathcal{M}_1) \to F(\mathcal{M}_2) \to F(\mathcal{M}_3)$ is exact in $\textit{Mod}(\mathcal{A})$. Hence we conclude that (3) holds as filtered colimits are exact in $\textit{Mod}(\mathcal{A})$. $\square$

Combining Lemmas 24.13.2 and 24.4.2 we find that there is an exact and faithful functor

\[ \textit{Mod}(\mathcal{A}, \text{d}) \longrightarrow \text{Comp}(\mathcal{O}) \]

of abelian categories. For a differential graded $\mathcal{A}$-module $\mathcal{M}$ the cohomology $\mathcal{O}$-modules, denoted $H^ i(\mathcal{M})$, are defined as the cohomology of the complex of $\mathcal{O}$-modules corresponding to $\mathcal{M}$. Therefore, a short exact sequence $0 \to \mathcal{K} \to \mathcal{L} \to \mathcal{M} \to 0$ of differential graded $\mathcal{A}$-modules gives rise to a long exact sequence
\begin{equation} \label{sdga-equation-les} H^ n(\mathcal{K}) \to H^ n(\mathcal{L}) \to H^ n(\mathcal{M}) \to H^{n + 1}(\mathcal{K}) \end{equation}

of cohomology modules, see Homology, Lemma 12.13.12.

Moreover, from now on we borrow all the terminology used for complexes of modules. For example, we say that a differential graded $\mathcal{A}$-module $\mathcal{M}$ is acyclic if $H^ k(\mathcal{M}) = 0$ for all $k \in \mathbf{Z}$. We say that a homomorphism $\mathcal{M} \to \mathcal{N}$ of differential graded $\mathcal{A}$-modules is a quasi-isomorphism if it induces isomorphisms $H^ k(\mathcal{M}) \to H^ k(\mathcal{N})$ for all $k \in \mathbf{Z}$. And so on and so forth.

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