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In mathematics, especially homological algebra and other applications of abelian category theory, the five lemma is an important and widely used lemma about commutative diagrams. The five lemma is valid ''not only'' for abelian categories but also works in the category of groups, for example. The five lemma can be thought of as a combination of two other theorems, the four lemmas, which are dual to each other. ==Statements== Consider the following commutative diagram in any abelian category (such as the category of abelian groups or the category of vector spaces over a given field) or in the category of groups. file:5 lemma.svg The five lemma states that, if the rows are exact, ''m'' and ''p'' are isomorphisms, ''l'' is an epimorphism, and ''q'' is a monomorphism, then ''n'' is also an isomorphism. The two four-lemmas state: (1) If the rows in the commutative diagram file:4 lemma right.svg are exact and ''m'' and ''p'' are epimorphisms and ''q'' is a monomorphism, then ''n'' is an epimorphism. (2) If the rows in the commutative diagram file:4 lemma left.svg are exact and ''m'' and ''p'' are monomorphisms and ''l'' is an epimorphism, then ''n'' is a monomorphism. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「five lemma」の詳細全文を読む スポンサード リンク
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