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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