For every p, the kernel of d_p :C_p->C_(p - 1) is called the group of cycles, Z_p = {c element C_p :d(c) = 0}. The letter Z is short for the German word for cycle, "Zyklus." The image d(C_(p + 1)) is contained in the group of cycles because d°d = 0, and is called the group of boundaries, B_p = {c element C_p :(exists b element C_(p + 1) :d(b) = c)}. The quotients H_p = Z_p/B_p are the homology groups of the chain. Given a short exact sequence of chain complexes 0->A_*->B_*->C_*->0, there is a long exact sequence in homology.

chain complex | chain equivalence | chain homomorphism | chain homotopy | homology | snake lemma