Lemma 12.24.2. Let $\mathcal{A}$ be an abelian category. Let $I$ be an object of $\mathcal{A}$. The following are equivalent:

1. The object $I$ is injective.

2. The functor $B \mapsto \mathop{\mathrm{Hom}}\nolimits _\mathcal {A}(B, I)$ is exact.

3. Any short exact sequence

$0 \to I \to A \to B \to 0$

in $\mathcal{A}$ is split.

4. We have $\mathop{\mathrm{Ext}}\nolimits _\mathcal {A}(B, I) = 0$ for all $B \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{A})$.

Proof. Omitted. $\square$

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