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 exact, and if g and h are isomorphisms, then f is an isomorphism as well.
The fivelemma can be thought of as a combination of two other theorems, the four lemmas, which are dual to each other.
The fivelemma 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 fivelemma is often applied to long exact sequences: when computing homology or cohomology of a given object, one typically employs a simpler subobject whose homology/cohomology is known, and arrives at a long exact sequence which involves the unknown homology groups of the original object.
The fivelemma (proved below) is a generalization of the shortfivelemma, and this step of the proof will have to be modified.
Short exact sequences can be clumped together into equivalence classes.
The shortfivelemma can be generalized to five modules, top and bottom, with 5 homomorphisms connecting corresponding modules and forming a commutative diagram.
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