Relative homology

Suppose K₀ ⊆ K is a subcomplex.
If ∂ : C_*(K) → C_*(K) and
∂' : C_*(K)/C_*(K₀) → C_*(K)/C_*(K₀)
are boundary maps, then
Im ∂' = ( Im ∂ + C_*(K₀) ) / C_*(K₀),
and ker ∂' = ∂^-1(C_*(K₀) / C_*(K₀).
In particular, if two *-chains are homologous modulo ∂, then they are also homologous modulo ∂'.