Let \(f: X \to Y\) be an affine morphism of noetherian separated schemes (II, Ex. 5.17). Show that for any quasi-coherent sheaf \(\mcf\) on \(X\), there are natural isomorphisms for all \(i \geq 0\), \[ H^i(X, \mcf) \cong H^i(Y, f_* \mcf). \]
Hint: Use (II, 5.8).
Solution
1. For every open cover \(\mathfrak{V}\) of \(Y\), the Čech complexes \(C^\bullet(\mathfrak{V}, f_* \mcf)\) and \(C^\bullet(f^{-1}\mathfrak{V}, \mcf)\) are equal.
For every finite intersection \(V_J\) of members of \(\mathfrak{V}\), \(\Gamma(V_J, f_* \mcf) = \Gamma(f^{-1} V_J, \mcf)\) by definition of \(f_*\), with the same restriction maps.
2. If \(\mathfrak{V}\) is a finite open affine cover of \(Y\), then \(f^{-1}\mathfrak{V}\) is a finite open affine cover of \(X\).
\(f\) is affine, so the preimage of each affine open of \(Y\) is affine.
3. Q.E.D.
\(f_* \mcf\) is quasi-coherent by (II, 5.8), and \(X\) and \(Y\) are Noetherian and separated, so Čech cohomology for any open affine cover computes the cohomology of a quasi-coherent sheaf (III, 4.5). Choose a finite open affine cover \(\mathfrak{V}\) of \(Y\); by steps 1 and 2, \[H^i(Y, f_* \mcf) \cong \check{H}^i(\mathfrak{V}, f_* \mcf) = \check{H}^i(f^{-1}\mathfrak{V}, \mcf) \cong H^i(X, \mcf).\] The isomorphisms are natural in \(\mcf\), since a morphism of quasi-coherent sheaves induces compatible maps of both Čech complexes.