Since the given sequence is a short sequence of the following long exact sequence fi ...

 

Transcribed Image Text:

Since the given sequence is a short sequence of the following long exact sequence fi → Mi-1 → Mi fitl → Mi+1 → ….. with N₂ = Im(fi) = Ker(fi+1) for each i, so I will fix i first. Now to show that the given short exact sequence is exact, I have to show that f is injective, g is surjective and Im(ƒ) = Ker(g) where f and g are given below: 9 0→ Ni →→ Mi → Ni+1 → 0 1- I do not know if writing the maps f and g in terms of the maps f; and fi+1 will help me or not in the proof of that f is injective and g is surjective and if so, how will this help? 2- Also, I want to use the first isomorphism theorem to prove that Im(ƒ) = Ker(g) but I do not know how. Could anyone clarify to me how to do this please? Since the given sequence is a short sequence of the following long exact sequence fi → Mi-1 → Mi fitl → Mi+1 → ….. with N₂ = Im(fi) = Ker(fi+1) for each i, so I will fix i first. Now to show that the given short exact sequence is exact, I have to show that f is injective, g is surjective and Im(ƒ) = Ker(g) where f and g are given below: 9 0→ Ni →→ Mi → Ni+1 → 0 1- I do not know if writing the maps f and g in terms of the maps f; and fi+1 will help me or not in the proof of that f is injective and g is surjective and if so, how will this help? 2- Also, I want to use the first isomorphism theorem to prove that Im(ƒ) = Ker(g) but I do not know how. Could anyone clarify to me how to do this please?

Answer rating: 100% (QA)

1 Writing the maps f and g in terms of the maps f i and il will help you to bet... View the full answer