The Five Lemma is a result in the field of homological algebra, particularly in the context of derived categories and spectral sequences. It provides a criterion for when a five-term exact sequence of chain complexes splits. This lemma is commonly used in the study of abelian categories and the derived functor theory.
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In a commutative diagram of two exact five-term sequences, if the first and fourth vertical maps are epimorphisms and the second and fifth are monomorphisms, then an isomorphism in the middle follows; in particular, four surrounding isomorphisms force the fifth.