Lemma 24.11.1. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{A}$ be a graded $\mathcal{O}$-algebra. The category $\textit{Mod}(\mathcal{A})$ is a Grothendieck abelian category.
24.11 Shift functors on sheaves of graded modules
We urge the reader to skip this section. It turns out that sheaves of graded modules over a graded algebra are an example of the phenomenon discussed in Differential Graded Algebra, Remark 22.25.7.
Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{A}$ be a sheaf of graded algebras on $(\mathcal{C}, \mathcal{O})$. Let $\mathcal{M}$ be a graded $\mathcal{A}$-module. Let $k \in \mathbf{Z}$. We define the $k$th shift of $\mathcal{M}$, denoted $\mathcal{M}[k]$, to be the graded $\mathcal{A}$-module whose $n$th part is given by
\[ (\mathcal{M}[k])^ n = \mathcal{M}^{n + k} \]
is the $(n + k)$th part of $\mathcal{M}$. As multiplication maps
\[ (\mathcal{M}[k])^ n \times \mathcal{A}^ m \longrightarrow (\mathcal{M}[k])^{n + m} \]
we simply use the multiplication maps
\[ \mathcal{M}^{n + k} \times \mathcal{A}^ m \longrightarrow \mathcal{M}^{n + m + k} \]
of $\mathcal{M}$. It is clear that we have defined a functor $[k]$, that we have $[k + l] = [k] \circ [l]$, and that we have
\[ \mathop{\mathrm{Hom}}\nolimits _{\textit{Mod}^{gr}(\mathcal{A})}(\mathcal{L}, \mathcal{M}[k]) = \mathop{\mathrm{Hom}}\nolimits _{\textit{Mod}^{gr}(\mathcal{A})}(\mathcal{L}, \mathcal{M})[k] \]
(without the intervention of signs) functorially in $\mathcal{M}$ and $\mathcal{L}$. Thus we see indeed that the graded category of graded $\mathcal{A}$-modules can be recovered from the ordinary category of graded $\mathcal{A}$-modules and the shift functors as discussed in Differential Graded Algebra, Remark 22.25.7.
Proof. By Lemma 24.4.2 and the definition of a Grothendieck abelian category (Injectives, Definition 19.10.1) it suffices to show that $\textit{Mod}(\mathcal{A})$ has a generator. We claim that
\[ \mathcal{G} = \bigoplus \nolimits _{k, U} j_{U!}\mathcal{A}_ U[k] \]
is a generator where the sum is over all objects $U$ of $\mathcal{C}$ and $k \in \mathbf{Z}$. Indeed, given a graded $\mathcal{A}$-module $\mathcal{M}$ if there are no nonzero maps from $\mathcal{G}$ to $\mathcal{M}$, then we see that for all $k$ and $U$ we have
\[ \mathop{\mathrm{Hom}}\nolimits _{\textit{Mod}(\mathcal{A})}(j_{U!}\mathcal{A}_ U[k], \mathcal{M}) = \mathop{\mathrm{Hom}}\nolimits _{\textit{Mod}(\mathcal{A}_ U)}(\mathcal{A}_ U[k], \mathcal{M}|_ U) = \Gamma (U, \mathcal{M}^{-k}) \]
is equal to zero. Hence $\mathcal{M}$ is zero. $\square$
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