For \(f: X \to Y\), the functor \(f_*\) is left exact; set \(R^i f_* \mcf \da R^i(f_*)(\mcf)\). Equivalently, \(R^i f_* \mcf\) is the sheafification of \[ V \mapsto H^i\qty{\inverseof{f}(V), \restrictionof{\mcf}{\inverseof{f}(V)}} . \]
(Functoriality in the space, and affine morphisms)
Let \(f \colon X \to Y\) be a morphism of schemes.
For a sheaf of abelian groups \(\mcf\) on \(X\) there are natural maps \(H^i(Y, f_* \mcf) \to H^i(X, \mcf)\), and for a sheaf of \(\OO_Y\)-modules \(\mcg\) natural maps \(H^i(Y, \mcg) \to H^i(X, f^* \mcg)\).
If \(X\) and \(Y\) are quasicompact and separated, \(f\) is affine and \(\mcf\) is quasicoherent, then \(H^i(Y, f_* \mcf) \to H^i(X, \mcf)\) is an isomorphism for every \(i\) (P-AGH341AFFINEMORPH).
If \(X\) is Noetherian, \(Y = \Spec A\) is affine, and \(\mcf\) is quasicoherent, then \(R^i f_* \mcf \cong \widetilde{H^i(X,\mcf)}\) [@Har10a, Proposition III.8.5]. If \(f\) is an affine morphism and \(\mcf\) is quasicoherent, then \(R^{i} f_* \mcf = 0\) for \(i>0\) and \(H^i(X,\mcf) \cong H^i(Y, f_*\mcf)\). For a closed subscheme \(i\colon X\hookrightarrow\PP^n\) and a quasicoherent sheaf \(\mcf\) on \(X\), this gives \(H^i(X,\mcf)\cong H^i(\PP^n,i_*\mcf)\).
If \(Y\) is Noetherian, \(f\) is projective, and \(\mcf\) is coherent, then \(R^i f_* \mcf\) is coherent [@Har10a, Theorem III.8.8]; for \(Y=\Spec k\) this is the finite-dimensionality of \(H^i(X,\mcf)\) [@Har10a, Theorem III.5.2].
Dependencies
D-COHDERSheaf cohomology as the derived functor of global sectionsuses
Backlinks
T-COHLERAYThe Leray spectral sequenceusesT-COHFFThe theorem on formal functionsuses