Resolution (algebra): Difference between revisions
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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 (mathematics)|module]]. | In [[mathematics]], particularly in [[abstract algebra]] and [[homological algebra]], a '''resolution''' is a sequence which is used to describe the structure of a [[module (mathematics)|module]]. | ||
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* {{cite book | author= Iain T. Adamson | title=Elementary rings and modules | series=University Mathematical Texts | publisher=Oliver and Boyd | year=1972 | isbn=0-05-002192-3 }} | * {{cite book | author= Iain T. Adamson | title=Elementary rings and modules | series=University Mathematical Texts | publisher=Oliver and Boyd | year=1972 | isbn=0-05-002192-3 }} | ||
* {{cite book | author=Serge Lang | authorlink=Serge Lang | title=Algebra | edition=3rd ed. | publisher=[[Addison-Wesley]] | year=1993 | isbn=0-201-55540-9 }} | * {{cite book | author=Serge Lang | authorlink=Serge Lang | title=Algebra | edition=3rd ed. | publisher=[[Addison-Wesley]] | year=1993 | isbn=0-201-55540-9 }} | ||
Revision as of 13:49, 28 October 2008
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
- · · · → 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
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
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.