The Stacks project

\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

12.24 Injectives

Definition 12.24.1. 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 12.24.2. 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. $\square$

Lemma 12.24.3. 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 12.24.4. 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 12.24.5. 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

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

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

  3. 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 $A \mapsto (A \to J(A))$.


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