The Stacks project

24.20 Shift functors on sheaves of differential graded modules

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{A}$ be a sheaf of differential graded algebras on $(\mathcal{C}, \mathcal{O})$. Let $\mathcal{M}$ be a differential graded $\mathcal{A}$-module. Let $k \in \mathbf{Z}$. We define the $k$th shift of $\mathcal{M}$, denoted $\mathcal{M}[k]$, as follows

1. as a graded $\mathcal{A}$-module we let $\mathcal{M}[k]$ be as defined in Section 24.11,

2. the differential $d_{\mathcal{M}[k]} : (\mathcal{M}[k])^ n \to (\mathcal{M}[k])^{n + 1}$ is defined to be $(-1)^ k\text{d}_\mathcal {M} : \mathcal{M}^{n + k} \to \mathcal{M}^{n + k + 1}$.

For a homomorphism $f : \mathcal{L} \to \mathcal{M}$ of $\mathcal{A}$-modules homogeneous of degree $n$, we let $f[k] : \mathcal{L}[k] \to \mathcal{M}[k]$ be given by the same component maps as $f$. Then $f[k]$ is a homogeneous $\mathcal{A}$-module map of degree $n$. This gives a map

compatible with differentials (it follows from the fact that the signs of the differentials of $\mathcal{L}$ and $\mathcal{M}$ are changed by the same amount). These choices are compatible with the choice in Differential Graded Algebra, Definition 22.4.3. It is clear that we have defined a functor

of differential graded categories and that we have $[k + l] = [k] \circ [l]$.

We claim that the isomorphism

defined in Section 24.11 on underlying graded modules is compatible with the differentials. To see this, suppose we have a right $\mathcal{A}$-module map $f : \mathcal{L} \to \mathcal{M}[k]$ homogeneous of degree $n$; this is an element of degree $n$ of the LHS. Denote $f' : \mathcal{L} \to \mathcal{M}$ the homogeneous $\mathcal{A}$-module map of degree $n + k$ with the same component maps as $f$. By our conventions, this is the corresponding element of degree $n$ of the RHS. By definition of the differential of LHS we obtain

and for the differential on the RHS we obtain

These maps have the same component maps and the proof is complete.