Reduced cohomology (${\tilde{H}}^{\ast }(X;R)$)

For a based space, reduced cohomology removes the coefficient-ring summand coming from the basepoint. In nonnegative degrees it is the kernel of restriction $H^{\ast }(X;R)\to H^{\ast }(\{x_0\};R)$. It agrees with ordinary cohomology in positive degrees and admits the suspension isomorphism and exact sequences for based cofibrations.