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Resolution (algebra)
In mathematics, particularly in abstract algebra and homological algebra, a resolution (or left resolution; dually a coresolution or right resolution〔 uses ''coresolution'', though ''right resolution'' is more common, as in 〕) is an exact sequence of modules (or, more generally, of objects in an abelian category), which is used to describe the structure of a specific module or object of this category. In particular, projective and injective resolutions induce a quasi-isomorphism between the exact sequence and the module, which may be regarded as a weak equivalence, with the resolution having nicer properties as a space.〔, 〕
Generally, the objects in the sequence are restricted to have some property ''P'' (for example to be free). Thus one speaks of a ''P resolution'': for example, a flat resolution, a free resolution, an injective resolution, a projective resolution. The sequence is supposed to be infinite to the left (to the right for a coresolution). However, a finite resolution is one where only finitely many of the objects in the sequence are non-zero.
==Resolutions of modules==
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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