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.
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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