The Stacks project

Lemma 24.25.1. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{A}$ be a sheaf of graded algebras on $(\mathcal{C}, \mathcal{O})$. There exists a set $T$ and for each $t \in T$ an injective map $\mathcal{N}_ t \to \mathcal{N}'_ t$ of graded $\mathcal{A}$-modules such that an object $\mathcal{I}$ of $\textit{Mod}(\mathcal{A})$ is injective if and only if for every solid diagram

\[ \xymatrix{ \mathcal{N}_ t \ar[r] \ar[d] & \mathcal{I} \\ \mathcal{N}'_ t \ar@{..>}[ru] } \]

a dotted arrow exists in $\textit{Mod}(\mathcal{A})$ making the diagram commute.

Proof. This is true in any Grothendieck abelian category, see Injectives, Lemma 19.11.6. By Lemma 24.11.1 the category $\textit{Mod}(\mathcal{A})$ is a Grothendieck abelian category. $\square$

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