 Ghost morphism in a derived category

A ghost is a morphism inducing zero on every cohomology sheaf, but it need not itself be zero. For the equator $j:S^1\hookrightarrow S^2$ and complement $i:U\hookrightarrow S^2$, the connecting morphism $j_{\ast }{\mathbb{Q}}_{S^1}\to i_{!}{\mathbb{Q}}_U[1]$ is a ghost. Its source and target have cohomology sheaves in different degrees, but its induced map $H^1(S^1)\to H_c^2(U)$ is nonzero, proving that the derived morphism is nonzero.