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- In mathematics, especially homological algebra and other applications of abelian category theory, the short five lemma is a special case of the five lemma.It states that for the following commutative diagram (in any abelian category, or in the category of groups), if the rows are short exact sequences, and if g and h are isomorphisms, then f is an isomorphism as well. It follows immediately from the five lemma. The essence of the lemma can be summarized as follows: if you have a homomorphism f from an object B to an object B′, and this homomorphism induces an isomorphism from a subobject A of B to a subobject A′ of B′ and also an isomorphism from the factor object B/A to B′/A′, then f itself is an isomorphism. Note however that the existence of f (such that the diagram commutes) has to be assumed from the start; two objects B and B′ that simply have isomorphic sub- and factor objects need not themselves be isomorphic (for example, in the category of abelian groups, B could be the cyclic group of order four and B′ the Klein four-group). (en)
- 在同調代數中，短五引理是五引理的一個特例，它斷言：在任何阿貝爾範疇或群範疇中，若以下交換圖的橫行正合，而 皆為同構，則 也是同構。 此斷言是五引理的直接推論。 這個引理可以有如下詮釋：假設有態射 ，此態射在子對象及相應的商對象上誘導出的態射 皆為同構，則 本身也是同構。重點是必須先假設 的存在性。 (zh)
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- 在同調代數中，短五引理是五引理的一個特例，它斷言：在任何阿貝爾範疇或群範疇中，若以下交換圖的橫行正合，而 皆為同構，則 也是同構。 此斷言是五引理的直接推論。 這個引理可以有如下詮釋：假設有態射 ，此態射在子對象及相應的商對象上誘導出的態射 皆為同構，則 本身也是同構。重點是必須先假設 的存在性。 (zh)
- In mathematics, especially homological algebra and other applications of abelian category theory, the short five lemma is a special case of the five lemma.It states that for the following commutative diagram (in any abelian category, or in the category of groups), if the rows are short exact sequences, and if g and h are isomorphisms, then f is an isomorphism as well. It follows immediately from the five lemma. (en)
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- Short five lemma (en)
- 短五引理 (zh)
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