The Stacks project

Proof. Let $S = \mathop{\mathrm{Spec}}(A)$. Let $\mathcal{J}$ be an injective object of $\mathit{QCoh}(\mathcal{O}_ S)$. Since $\mathit{QCoh}(\mathcal{O}_ S)$ is equivalent to the category of $A$-modules we see that $\mathcal{J}$ is equal to $\widetilde{J}$ for some injective $A$-module $J$. By Dualizing Complexes, Proposition 47.5.9 we can write $J = \bigoplus E_\alpha$ with $E_\alpha$ indecomposable and therefore isomorphic to the injective hull of a reside field at a point. Thus (because finite disjoint unions of Artinian schemes are Artinian) we may assume that $J$ is the injective hull of $\kappa (\mathfrak p)$ for some prime $\mathfrak p$ of $A$. Then $J = \bigcup J[\mathfrak p^ n]$ where $J[\mathfrak p^ n]$ is the injective hull of $\kappa (\mathfrak p)$ over $A_\mathfrak /\mathfrak p^ nA_\mathfrak p$, see Dualizing Complexes, Lemma 47.7.3. Thus $\widetilde{J}$ is the colimit of the sheaves $(Z_ n \to X)_*\mathcal{G}_ n$ where $Z_ n = \mathop{\mathrm{Spec}}(A_\mathfrak p/\mathfrak p^ nA_\mathfrak p)$ and $\mathfrak G_ n$ the coherent sheaf associated to the finite $A_\mathfrak /\mathfrak p^ nA_\mathfrak p$-module $J[\mathfrak p^ n]$. Finiteness follows from Dualizing Complexes, Lemma 47.6.1. $\square$