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

· · · → E_n → · · · → E₂ → E₁ → E₀ → M → 0,

with all the E_i modules over R. The resolution is said to be finite if the sequence of E_i 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) → M → 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_•(x).

See also

• Resolution (mathematics)

References

• Iain T. Adamson (1972). Elementary rings and modules. Oliver and Boyd. ISBN 0-05-002192-3. 
• Serge Lang (1993). Algebra, 3rd ed.. Addison-Wesley. ISBN 0-201-55540-9.