Short exact sequence of cochain complexes ($0\to A^{\ast }\to B^{\ast }\to C^{\ast }\to 0$)

A degreewise short exact sequence of cochain complexes induces a long exact sequence from a coefficient sequence in the coefficient case, and a long exact cohomology sequence generally. Its degree-one connecting homomorphism lifts a quotient cocycle, takes its coboundary, and identifies that result in the subcomplex.