= Higher direct image sheaf
{title2=$R^qf_*\mathcal F$}
For a continuous map $f:X\to Y$ and a sheaf $\mathcal F$ of abelian groups, the higher direct image sheaf $R^qf_*\mathcal F$ is the $q$th right derived functor of the <direct image sheaf> construction. An injective resolution $\mathcal F\to I^\bullet$ gives
$$
R^qf_*\mathcal F=\mathcal H^q(f_*I^\bullet).
$$
Equivalently, 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 $f_*\mathcal F$. 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.
Back to article page
