A chain complex of projective modules (or projective objects in an abelian category) together with a quasi‑isomorphism (or augmentation) from the complex to a given module M that is exact except at degree zero; used to compute derived functors such as Ext and Tor.

Principle

Replace a module by a projective complex that is homologically equivalent so that right-derived functors can be computed by applying a functor to the resolution and taking homology; projective modules allow exactness of Hom(-,–) computations at appropriate places.

Demonstration

For the module Z/n over Z, one can construct a projective (even free) resolution ... → Z → Z → Z/n → 0 where the boundary maps are multiplication by n and appropriate inclusions; applying Hom_Z(–, A) to this resolution computes Ext^i_Z(Z/n, A).

Misapplication

Using a non-projective complex and expecting Hom or tensor to preserve exactness as if it were projective, or assuming that a resolution must be finite-length in categories that do not guarantee finite global dimension; such mistakes lead to incorrect derived functor computations.

Consequence

A projective resolution yields an explicit method to compute derived functors: applying a left-exact functor to the resolution and taking homology recovers right-derived functors, so Ext and Tor groups become computable from chosen resolutions.

Reversal

Dual notion: injective resolution. Where projective resolutions resolve objects from the left using projectives and are used to compute right-derived functors of left-exact functors, injective resolutions resolve from the right and compute left-derived functors of right-exact functors.

Boundary

Requires the ambient abelian category to have enough projectives for arbitrary modules to admit projective resolutions; in categories lacking enough projectives other techniques (e.g., projective generators, flat resolutions, or using derived categories with different models) are necessary.

Semantic Tension

Tension between free and projective resolutions: free resolutions are projective but sometimes larger; between minimal resolutions (when they exist) and arbitrary projective resolutions which may be redundant but computationally convenient.

Synthesis

A projective resolution is a projective chain complex quasi-isomorphic to a module M that replaces M by a homologically tractable object; it is the standard device in homological algebra for computing derived functors, converting local exactness properties into computable homology groups.