Resolution (algebra)

In mathematics, particularly in abstract algebra and homological algebra, a resolution is a sequence which is used to describe the structure of a module.

If the modules involved in the sequence have a property P then one speaks of a P resolution: for example, a flat resolution, a free resolution, an injective resolution, a projective resolution and so on.

Definition
Given a module M over a ring R, a resolution of M is an exact sequence (possibly infinite) of modules
 * &middot; &middot; &middot; &rarr; En &rarr; &middot; &middot; &middot; &rarr; E2 &rarr; E1 &rarr; E0 &rarr; M &rarr; 0,

with all the Ei modules over R. The resolution is said to be finite if the sequence of Ei is zero from some point onwards.

Properties
Every module possesses a free resolution: that is, a resolution by free modules. A fortiori, every module admits a projective resolution. Such an exact sequence may sometimes be seen written as an exact sequence P(M) &rarr; M &rarr; 0. The minimal length of a finite projective resolution of a module M is called its projective dimension and denoted pd(M). If M does not admit a finite projective resolution then the projective dimension is infinite.

Examples
A classic example of a projective resolution is given by the Koszul complex K&bull;(x).