Projective and injective resolutions

For a left module $M$, choose a projective module $P_0$ surjecting onto it, then a projective module $P_1$ surjecting onto the kernel, and continue. The resulting augmented chain sequence is exact. Every module over a unital ring permits such a construction with free modules. An abelian category needs enough projectives for this argument. The unaugmented chain complex has zeroth homology $M$ and no positive homology; including the augmentation makes the entire displayed sequence exact.

To use our cochain convention, place $P_i$ in degree $-i$. Thus a module resolution is bounded above by degree zero, even when infinitely many negative degrees are needed. A finite resolution is an additional property, not a consequence of finite generation. Nor does a finite presentation guarantee that all higher kernels disappear. In computations, identify the kernel at each stage before declaring a sequence a resolution; simply repeating a matrix with square zero proves only that it is a complex.

An injective resolution starts with an embedding $N\to I^0$. Embed its cokernel into an injective $I^1$, then repeat. This produces an exact augmented cochain sequence with nonnegative degrees. Left-module categories have enough injectives; in an arbitrary abelian category this must be assumed. Projective and injective resolutions point in opposite directions because one repeatedly covers kernels and the other repeatedly embeds cokernels. Their augmentations therefore have opposite orientations as maps of complexes.

The original object maps into an injective resolution, whereas a projective resolution maps onto the original object. Each map is a quasi-isomorphism when the object is considered in degree zero. A functor applied to a resolution can destroy its exactness; that destruction is precisely the information a derived functor will measure. Applying an exact functor would preserve the exact augmented sequence and produce no positive-degree defect. Resolution machinery is useful because common tensor and Hom functors are only one-sided exact.

A module map lifts to a map between projective resolutions by lifting degree zero and then lifting successive maps on kernels. Two such lifts are homotopic through an inductive construction. Applying any additive functor preserves the homotopy identity, so the induced homology maps agree. In particular, two projective resolutions of the same module are homotopy equivalent through comparison maps lifting its identity. The dual statement holds for injective resolutions and extensions across the successive inclusions.

These comparison arguments justify independence, but not careless use of arbitrary unbounded complexes of projectives or injectives. The standard cochain argument uses projectives bounded above or injectives bounded below, allowing induction to start. Unbounded derived constructions require stronger replacement properties. For module-level derived functors the ordinary resolutions already have the needed direction of boundedness, even if their total length is infinite. Later examples will exploit infinite resolutions to detect a genuine failure of finite homological dimension.