Section 12.27 (0134): Injectives

12.27 Injectives

Definition 12.27.1. Let $\mathcal{A}$ be an abelian category. An object $J \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{A})$ is called injective if for every injection $A \hookrightarrow B$ and every morphism $A \to J$ there exists a morphism $B \to J$ making the following diagram commute

\[ \xymatrix{ A \ar[r] \ar[d] & B \ar@{-->}[ld] \\ J & } \]

Here is the obligatory characterization of injective objects.

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

\[ 0 \to I \to A \to B \to 0 \]

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

Proof. Omitted. $\square$

Lemma 12.27.3. Let $\mathcal{A}$ be an abelian category. Suppose $I_\omega $, $\omega \in \Omega $ is a set of injective objects of $\mathcal{A}$. If $\prod _{\omega \in \Omega } I_\omega $ exists then it is injective.

Proof. Omitted. $\square$

Definition 12.27.4. Let $\mathcal{A}$ be an abelian category. We say $\mathcal{A}$ has enough injectives if every object $A$ has an injective morphism $A \to J$ into an injective object $J$.

Definition 12.27.5. Let $\mathcal{A}$ be an abelian category. We say that $\mathcal{A}$ has functorial injective embeddings if there exists a functor

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

such that

$s \circ J = \text{id}_\mathcal {A}$,
for any object $A \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{A})$ the morphism $J(A)$ is injective, and
for any object $A \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{A})$ the object $t(J(A))$ is an injective object of $\mathcal{A}$.

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

Comments (3)

Comment #4832 by Weixiao Lu on December 14, 2019 at 02:15

Any cofiltered limit of injective objects is injective.

Comment #4835 by Johan on December 16, 2019 at 09:22

@#4832. This is not true. To give an example, working dually, it suffices to show that there is an abelian category and a non-projective object which is a filtered colimit of projective modules. An example is $\mathbf{Q}$ as the usual filtered colimit of copies of $\mathbf{Z}$ in the category of abelian groups.

Comment #8523 by Haohao Liu on June 26, 2023 at 06:51

To Weixiao Lu: In a locally noetherian category, every inductive limit of injective objects is an injective object (Cor. 1, p.358 of Des Catégories Abéliennes by Gabriel).

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12.27 Injectives

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