Last time on Wild Spectral Sequences, we conquered the snake lemma using a spectral sequence argument. This time, we meet a new beast: the five lemma. The objective is the usual: prove the five lemma using spectral sequences.

Recall that the five lemma states that given a diagram

in an abelian category, if the rows are exact and $ a,b,d,e$ are isomorphisms, then so is $ c$. Actually, the hypotheses are too strong. It suffices to have $ b,d$ isomorphisms, $ a$ an epimorphism and $ e$ a monomorphism. One can deduce this via J. Leicht’s “strong four lemma” (which we might try and prove via a spectral sequence too) or just by using the regular diagram-chasing proof of the five lemma.

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