The Stacks project

12.28 Projectives

Definition 12.28.1. Let $\mathcal{A}$ be an abelian category. An object $P \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{A})$ is called projective if for every surjection $A \rightarrow B$ and every morphism $P \to B$ there exists a morphism $P \to A$ making the following diagram commute

\[ \xymatrix{ A \ar[r] & B \\ P \ar@{-->}[u] \ar[ru] & } \]

Here is the obligatory characterization of projective objects.

Lemma 12.28.2. Let $\mathcal{A}$ be an abelian category. Let $P$ be an object of $\mathcal{A}$. The following are equivalent:

  1. The object $P$ is projective.

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

  3. Any short exact sequence

    \[ 0 \to A \to B \to P \to 0 \]

    in $\mathcal{A}$ is split.

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

Proof. Omitted. $\square$

Lemma 12.28.3. Let $\mathcal{A}$ be an abelian category. Suppose $P_\omega $, $\omega \in \Omega $ is a set of projective objects of $\mathcal{A}$. If $\coprod _{\omega \in \Omega } P_\omega $ exists then it is projective.

Proof. Omitted. $\square$

Definition 12.28.4. Let $\mathcal{A}$ be an abelian category. We say $\mathcal{A}$ has enough projectives if every object $A$ has an surjective morphism $P \to A$ from an projective object $P$ onto it.

Definition 12.28.5. Let $\mathcal{A}$ be an abelian category. We say that $\mathcal{A}$ has functorial projective surjections if there exists a functor

\[ P : \mathcal{A} \longrightarrow \text{Arrows}(\mathcal{A}) \]

such that

  1. $t \circ J = \text{id}_\mathcal {A}$,

  2. for any object $A \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{A})$ the morphism $P(A)$ is surjective, and

  3. for any object $A \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{A})$ the object $s(P(A))$ is an projective object of $\mathcal{A}$.

We will denote such a functor by $A \mapsto (P(A) \to A)$.

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