Lemma 12.27.2. Let \mathcal{A} be an abelian category. Let I be an object of \mathcal{A}. The following are equivalent:
The object I is injective.
The functor B \mapsto \mathop{\mathrm{Hom}}\nolimits _\mathcal {A}(B, I) is exact.
Any short exact sequence
0 \to I \to A \to B \to 0in \mathcal{A} is split.
We have \mathop{\mathrm{Ext}}\nolimits _\mathcal {A}(B, I) = 0 for all B \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{A}).
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