An exact chain complex I• of injective objects together with a monomorphism M → I0 embedding the given object or module M so that M → I• is a quasi-isomorphism in positive degrees; used to compute right-derived functors by applying a left-exact functor to I• and taking homology.

Principle

Replace an object by a homologically equivalent complex built from injectives so that applying left-exact functors preserves enough exactness to measure the failure of exactness via homology groups.

Demonstration

For an R-module M over a ring with enough injectives, choose an injective resolution 0 → M → I0 → I1 → I2 → ···; applying HomR(−,N) yields a complex whose cohomology in degree n is Ext^n_R(M,N).

Misapplication

Using a non-exact embedding or a complex of objects that are not injective and then interpreting the resulting homology as the correct right-derived functors; or attempting to use an injective resolution to compute left-derived functors without dualizing.

Consequence

Correctly constructed injective resolutions give well-defined right-derived functors (R^iF), long exact sequences in cohomology, and invariance up to homotopy or quasi-isomorphism; they enable explicit calculations of Ext and sheaf cohomology in suitable categories.

Reversal

A projective resolution is the dual construction: it resolves objects by projectives and is used to compute left-derived functors such as Tor, rather than right-derived functors.

Boundary

Exists and is useful in abelian categories with enough injectives; in categories lacking enough injectives one must pass to derived categories, use injective model structures, or replace with other resolutions (e.g., Čech or Godement for sheaves). The resolution is not canonical but is unique up to homotopy quasi-isomorphism.

Semantic Tension

Tension arises between minimal or canonical injective resolutions (when they exist) and arbitrary injective resolutions; also between injective and projective approaches — both compute derived functors but on opposite sides (right vs left) and with different existence hypotheses.

Synthesis

An injective resolution is a homologically equivalent embedding of an object into a complex of injectives that converts the problem of measuring a left-exact functor's failure of exactness into computing cohomology groups that define right-derived functors.