Resolution (algebra)

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This editable Main Article is under development and subject to a disclaimer.

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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.

## Contents

* 1 Definition
* 2 Properties
* 3 Examples
* 4 See also
* 5 References

## Definition[edit]

Given a module M over a ring R, a resolution of M is an exact sequence (possibly infinite) of modules

* * * -> En -> * * * -> E2 -> E1 -> E0 -> M -> 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[edit]

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[edit]

A classic example of a projective resolution is given by the Koszul complex K•(x).

## See also[edit]

* Resolution (mathematics)

## References[edit]

* 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.