Question: Since the given sequence is a short sequence of the following long exact sequence fi Mi-1 Mi fitl Mi+1 .. with N = Im(fi)

Since the given sequence is a short sequence of the following long

Since the given sequence is a short sequence of the following long exact sequence fi Mi-1 Mi fitl Mi+1 .. with N = 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() = 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() = Ker(g) but I do not know how. Could anyone clarify to me how to do this please?

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