Lemma 12.25.2. Let $\mathcal{A}$ be an abelian category. Let $P$ be an object of $\mathcal{A}$. The following are equivalent:

The object $P$ is projective.

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

Any short exact sequence

\[ 0 \to A \to B \to P \to 0 \]in $\mathcal{A}$ is split.

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

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