projective resolution

A projective resolution replaces a module by an approximation made entirely of well-behaved building blocks. Projective modules are the modules over which maps lift and out of which exact sequences split — they are the algebraic analogue of free, flexible material. The idea is to express any module as the end of a long exact chain of projectives, trading a possibly complicated object for a complex whose terms we understand completely.

Formally, a projective resolution of an R-module M is an exact sequence ... -> P_2 -> P_1 -> P_0 -> M -> 0 in which each P_i is projective. Such a resolution always exists because every module is a quotient of a free (hence projective) module; one builds it step by step, covering M by P_0, then covering the kernel by P_1, and so on. Discarding M, the truncated complex ... -> P_1 -> P_0 -> 0 has homology M in degree 0 and zero elsewhere, so it is a stand-in for M.

Projective resolutions are not unique, but any two are chain homotopy equivalent, so any functor applied to them and then passed to homology gives a well-defined answer. This is precisely how left derived functors — Tor among them — are computed: resolve, apply the functor, take homology. The minimal number of nonzero terms needed measures the projective dimension of M, a fundamental invariant.