Unformatted text preview: Rmodule D , the sequence → Hom R ( N,D ) g *→ Hom R ( M,D ) f *→ Hom R ( L,D ) is an exact sequence of abelian groups. (4) The Five Lemma states that given a diagram of abelian groups A 1 / f 1 ± A 2 / f 2 ± A 3 / f 3 ± A 4 / f 4 ± A 5 f 5 ± A 1 / A 2 / A 3 / A 4 / A 5 where the rows are exact, and f 1 ,f 2 ,f 4 and f 5 are isomorphisms, f 3 is an isomorphism as well. (a) Prove the Five Lemma. (b) Consider the following eight hypotheses: f i is injective, for i = 1 , 2 , 4 , 5 , f i is surjective, for i = 1 , 2 , 4 , 5 . Which of these hypothese suﬃce to prove that f 3 is injective? Which suﬃce to prove that f 3 is surjective?...
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 Fall '08
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 Algebra, Vector Space, Ring, Dr. Matthew Macauley

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