The Stacks project

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22.4 Differential graded modules

Just the definitions.

Definition 22.4.1. Let $R$ be a ring. Let $(A, \text{d})$ be a differential graded algebra over $R$. A (right) differential graded module $M$ over $A$ is a right $A$-module $M$ which has a grading $M = \bigoplus M^ n$ and a differential $\text{d}$ such that $M^ n A^ m \subset M^{n + m}$, such that $\text{d}(M^ n) \subset M^{n + 1}$, and such that

\[ \text{d}(ma) = \text{d}(m)a + (-1)^ n m\text{d}(a) \]

for $a \in A$ and $m \in M^ n$. A homomorphism of differential graded modules $f : M \to N$ is an $A$-module map compatible with gradings and differentials. The category of (right) differential graded $A$-modules is denoted $\text{Mod}_{(A, \text{d})}$.

Note that we can think of $M$ as a cochain complex $M^\bullet $ of (right) $R$-modules. Namely, for $r \in R$ we have $\text{d}(r) = 0$ and $r$ maps to a degree $0$ element of $A$, hence $\text{d}(mr) = \text{d}(m)r$.

We can define left differential graded $A$-modules in exactly the same manner. If $M$ is a left $A$-module, then we can think of $M$ as a right $A^{opp}$-module with multiplication $\cdot _{opp}$ defined by the rule

\[ m \cdot _{opp} a = (-1)^{\deg (a)\deg (m)} a m \]

for $a$ and $m$ homogeneous. The category of left differential graded $A$-modules is equivalent to the category of right differential graded $A^{opp}$-modules. We prefer to work with right modules (essentially because of what happens in Example 22.19.8), but the reader is free to switch to left modules if (s)he so desires.

Lemma 22.4.2. Let $(A, d)$ be a differential graded algebra. The category $\text{Mod}_{(A, \text{d})}$ is abelian and has arbitrary limits and colimits.

Proof. Kernels and cokernels commute with taking underlying $A$-modules. Similarly for direct sums and colimits. In other words, these operations in $\text{Mod}_{(A, \text{d})}$ commute with the forgetful functor to the category of $A$-modules. This is not the case for products and limits. Namely, if $N_ i$, $i \in I$ is a family of differential graded $A$-modules, then the product $\prod N_ i$ in $\text{Mod}_{(A, \text{d})}$ is given by setting $(\prod N_ i)^ n = \prod N_ i^ n$ and $\prod N_ i = \bigoplus _ n (\prod N_ i)^ n$. Thus we see that the product does commute with the forgetful functor to the category of graded $A$-modules. A category with products and equalizers has limits, see Categories, Lemma 4.14.10. $\square$

Thus, if $(A, \text{d})$ is a differential graded algebra over $R$, then there is an exact functor

\[ \text{Mod}_{(A, \text{d})} \longrightarrow \text{Comp}(R) \]

of abelian categories. For a differential graded module $M$ the cohomology groups $H^ n(M)$ are defined as the cohomology of the corresponding complex of $R$-modules. Therefore, a short exact sequence $0 \to K \to L \to M \to 0$ of differential graded modules gives rise to a long exact sequence

22.4.2.1
\begin{equation} \label{dga-equation-les} H^ n(K) \to H^ n(L) \to H^ n(M) \to H^{n + 1}(K) \end{equation}

of cohomology modules, see Homology, Lemma 12.12.12.

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

Definition 22.4.3. Let $(A, \text{d})$ be a differential graded algebra. Let $M$ be a differential graded module. For any $k \in \mathbf{Z}$ we define the $k$-shifted module $M[k]$ as follows

  1. as $A$-module $M[k] = M$,

  2. $M[k]^ n = M^{n + k}$,

  3. $\text{d}_{M[k]} = (-1)^ k\text{d}_ M$.

For a morphism $f : M \to N$ of differential graded $A$-modules we let $f[k] : M[k] \to N[k]$ be the map equal to $f$ on underlying $A$-modules. This defines a functor $[k] : \text{Mod}_{(A, \text{d})} \to \text{Mod}_{(A, \text{d})}$.

The remarks in Homology, Section 12.13 apply. In particular, we will identify the cohomology groups of all shifts $M[k]$ without the intervention of signs.

At this point we have enough structure to talk about triangles, see Derived Categories, Definition 13.3.1. In fact, our next goal is to develop enough theory to be able to state and prove that the homotopy category of differential graded modules is a triangulated category. First we define the homotopy category.


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