Editing Projective module (section)

== Projective resolutions ==
{{Main|Projective resolution}}
Given a module, ''M'', a '''projective [[resolution (algebra)|resolution]]''' of ''M'' is an infinite [[exact sequence]] of modules
:&sdot;&sdot;&sdot; → ''P''<sub>''n''</sub> → &sdot;&sdot;&sdot; → ''P''<sub>2</sub> → ''P''<sub>1</sub> → ''P''<sub>0</sub> → ''M'' → 0,
with all the ''P''<sub>''i''</sub>&thinsp;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 {{nowrap|''P''(''M'') → ''M'' → 0}} or {{nowrap|''P''<sub>•</sub> → ''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 (ring theory)|ideal]] generated by the sequence.

The ''length'' of a finite resolution is the index ''n'' such that ''P''<sub>''n''</sub> is [[zero module|nonzero]] and {{nowrap|1=''P''<sub>''i''</sub> = 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 {{nowrap|1=pd(''M'') = 0}}. In this situation, the exactness of the sequence 0 → ''P''<sub>0</sub> → ''M'' → 0 indicates that the arrow in the center is an isomorphism, and hence ''M'' itself is projective.