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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

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

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

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

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

We claim that the isomorphism

\[ \mathop{\mathrm{Hom}}\nolimits _{\textit{Mod}^{dg}(\mathcal{A}, \text{d})}(\mathcal{L}, \mathcal{M}[k]) = \mathop{\mathrm{Hom}}\nolimits _{\textit{Mod}^{dg}(\mathcal{A}, \text{d})}(\mathcal{L}, \mathcal{M})[k] \]

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

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

and for the differential on the RHS we obtain

\[ \text{d}_{RHS}(f') = (-1)^ k\left( \text{d}_\mathcal {M} \circ f' - (-1)^{n + k} f' \circ \text{d}_\mathcal {L} \right) = (-1)^ k\text{d}_\mathcal {M} \circ f' - (-1)^ n f' \circ \text{d}_\mathcal {L} \]

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

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