Projective Module - Projective Resolutions

Projective Resolutions

Given a module, M, a projective resolution of M is an infinite exact sequence of modules

· · · → P_n → · · · → P₂ → P₁ → P₀ → M → 0,

with all the P_i's projective. Every module possesses a projective resolution. In fact a free resolution (resolution by free modules) exists. The exact sequence of projective modules may sometimes be abbreviated to P(M) → M → 0 or P_• → M → 0. A classic example of a projective resolution is given by the Koszul complex of a regular sequence, which is a free resolution of the ideal generated by the sequence.

The length of a finite resolution is the subscript n such that P_n is nonzero and P_i=0 for i greater than n. If M admits a finite projective resolution, the minimal length among all finite projective resolutions of M is called its projective dimension and denoted pd(M). If M does not admit a finite projective resolution, then by convention the projective dimension is said to be infinite. As an example, consider a module M such that pd(M) = 0. In this situation, the exactness of the sequence 0 → P₀ → M → 0 indicates that the arrow in the center is an isomorphism, and hence M itself is projective.