An injective resolution of an object M in an abelian category is an exact complex 0→M→I^0→I^1→… with each I^i injective; such a coresolution is used to compute right-derived functors, e.g., Ext^i(−,M) or the derived functors R^iΓ for a left-exact functor Γ.

Principle

Injective objects absorb extensions: embedding M into an injective hull and iterating yields a coresolution whose Hom into other objects computes right-derived functors because Hom(−,I^•) is exact. The organizing idea is to replace M by a tractable complex of coflabby objects that eliminate Ext-obstructions when applying left-exact covariant functors.

Demonstration

In the category of abelian groups, every divisible group (e.g., the rationals Q) is injective; for a group A one constructs an injective resolution by embedding A into an injective group I^0 (for instance an injective envelope), then taking the cokernel and embedding it into I^1, and so on. Applying Hom(−,N) to the resolution yields a complex whose cohomology computes Ext^*(A,N).

Misapplication

Using noninjective modules in place of injective terms destroys exactness needed for computing derived functors and leads to incorrect Ext or cohomology groups. Another common mistake is assuming injective resolutions exist in every abelian category without verifying 'enough injectives'; some categories require different techniques (e.g., derived categories, model structures) when injectives are absent.

Consequence

When injective resolutions exist they provide the standard method to compute right-derived functors, to define derived categories concretely, and to study local cohomology and duality theories (Matlis duality, Grothendieck duality) where injective coresolutions expose extension and support phenomena.

Reversal

The dual notion is a projective resolution used to compute left-derived functors; comparing the two highlights that left versus right derived computations require projective versus injective replacements, and that in many practical categories one chooses the side (projective or injective) for which there are enough nice objects.

Boundary

Requires an abelian category with enough injectives to guarantee existence of injective resolutions; in categories lacking enough injectives one must use alternatives (K‑injective complexes, Brown representability, or model categorical replacements). Even when they exist, injective resolutions can be large or nonconstructive, making explicit computation difficult in practice.

Semantic Tension

Competes with projective resolutions and other replacement techniques (K‑injectives, injective model structures): injective resolutions are canonical when available but may be unwieldy, while derived category methods allow more flexible but abstract replacements; confusion occurs when one treats existence, computability, and functoriality as interchangeable.

Synthesis

An injective resolution is the canonical coresolution of an object by injective objects whose Hom complexes compute right-derived functors; it is the dual tool to projective resolutions, indispensable where injectives exist for calculating Ext, local cohomology, and duality phenomena, though alternatives are required in categories lacking enough injectives.