Definition:Injective Resolution

Definition

Let $\mathbf A$ be an abelian category.

Let $X$ be an object in $\mathbf A$.

An injective resolution of $X$ is a cochain complex $P := \family {d^i : I^i \to I^{i + 1} }_{i \mathop \in \Z}$ in $\mathbf A$, such that:

$(1): \quad \forall i < 0 : I^i = 0$

$(2): \quad I^i$ is injective for all $i \ge 0$

together with a morphism $\varepsilon : X \to I^0$, such that the cochain complex:

$\begin{xy}

\xymatrix{ \dots \ar[r] & 0 \ar[r] & 0 \ar[r] & X \ar[r]^{\varepsilon} & I^0 \ar[r]^{d^0} & I^1 \ar[r]^{d^1} & I^2 \ar[r]^{d^2} & \dots } \end{xy}$ is exact.

Also see

• Injective Resolution Exists Iff Enough Injectives