Let \(f: X \to Y\) be an affine morphism of schemes (II, Ex. 5.17) with \(X\) noetherian, and let \(\mcf\) be a quasi-coherent sheaf on \(X\). Show that the hypotheses of (Ex. 8.1) are satisfied, and hence that \[ H^i(X, \mcf) \cong H^i(Y, f_* \mcf) \] for each \(i \geq 0\).
Solution
1. If \(V\subseteq Y\) is affine open, then \(f^{-1}(V)\) is a noetherian affine scheme and \(\mcf|_{f^{-1}(V)}\) is quasi-coherent.
Because \(f\) is an affine morphism, the inverse image of every affine open subset of \(Y\) is affine. Thus \[ f^{-1}(V)=\Spec B \] for some ring \(B\). It is an open subscheme of the noetherian scheme \(X\), hence is noetherian. Quasi-coherence is local and is preserved by restriction to an open subscheme, so \[ \mcf|_{f^{-1}(V)} \] is quasi-coherent.
2. For every affine open \(V\subseteq Y\) and every \(i>0\),\[ H^i\bigl(f^{-1}(V),\mcf|_{f^{-1}(V)}\bigr)=0. \]
By step 1, \(f^{-1}(V)\) is a noetherian affine scheme and the restricted sheaf is quasi-coherent. The affine vanishing theorem, Hartshorne III.3.5, therefore gives \[ H^i\bigl(f^{-1}(V),\mcf|_{f^{-1}(V)}\bigr)=0 \] for every \(i>0\).
3. The higher direct images of \(\mcf\) vanish:\[ \boxed{R^if_*\mcf=0\qquad(i>0)}. \]
Proposition III.8.1 identifies \(R^if_*\mcf\) with the sheaf associated to the presheaf \[ V\longmapsto H^i\bigl(f^{-1}(V),\mcf|_{f^{-1}(V)}\bigr). \] Affine open subsets form a basis for the topology of the scheme \(Y\). By step 2 this presheaf is zero on every member of that basis when \(i>0\). Consequently all its stalks vanish, and its associated sheaf is zero. Thus \(R^if_*\mcf=0\) for \(i>0\).
4. For every \(i\ge0\) there is a natural isomorphism\[ \boxed{H^i(X,\mcf)\cong H^i(Y,f_*\mcf)}. \]
Step 3 verifies exactly the vanishing hypothesis of Exercise III.8.1. Applying that exercise gives the natural isomorphisms \[ H^i(X,\mcf)\cong H^i(Y,f_*\mcf) \] in every degree \(i\ge0\).
5. Q.E.D.
Step 3 verifies the required higher-direct-image hypothesis, and step 4 gives the requested cohomology isomorphisms.