Higher direct image sheaf ($R^q{f}_{\ast }\mathcal{F}$)

For a continuous map $f:X\to Y$ and a sheaf $\mathcal{F}$ of abelian groups, the higher direct image sheaf $R^q{f}_{\ast }\mathcal{F}$ is the $q$th right derived functor of the direct image sheaf construction. An injective resolution $\mathcal{F}\to I^{\bullet }$ givesEquivalently, it is the sheafification of the presheaf $V\mapsto H^q(f^{-1}(V),\mathcal{F})$ on open subsets of $Y$. Sheafification matters: those higher cohomology groups need not already satisfy the sheaf gluing axiom as a presheaf.

At $q=0$ this is the ordinary direct image $f_{\ast }\mathcal{F}$. If $Y$ is a point, higher direct images are the ordinary sheaf cohomology groups of $X$, regarded as sheaves on that point. For the identity map, every higher direct image with $q>0$ vanishes because the identity direct-image functor is exact.

For a family of schemes, these sheaves organize cohomological information over the base. Their comparison with cohomology on individual fibers is the subject of cohomology and base change, whose hypotheses control when that comparison is valid.