Composition of Homomorphisms In An Exact Sequence

Theorem

Given an exact sequence

M0f1M1f2M2f3M3f4fnMn

fi+1fi=0 for any i{1,,n1}.

Proof

This follows directly from the definition of an exact sequence. For any xMi1, fi(x)im(fi)=ker(fi+1) and therefore fi+1(f(x))=0 by definition of the kernel.