Relative cohomology ($H^{\ast }(X,A;R)$)

Relative cohomology is the cohomology of cochains on $X$ vanishing on chains in $A$. A pair has a natural long exact sequence in cohomology, and excision identifies relative groups for suitable neighborhood pairs.

For an inclusion $A\subseteq X$, the restriction cochain map fits into an exact triangle with the relative cochain complex. Its cohomology gives $\cdots \to H^i(X,A)\to H^i(X)\to H^i(A)\stackrel{\delta }{\to }{H}^{i+1}(X,A)\to \cdots$. The same construction with derived sections of a sheaf complex gives relative hypercohomology and is natural under restrictions of pairs.