III.8: Higher Direct Images of Sheaves
III.8.1. #to_work
Let \(f: X \rightarrow Y\) be a continuous map of topological spaces. Let \(\mathcal{F}\) be a sheaf of abelian groups on \(X\), and assume that \(R^i f_*(\mathcal{F})=0\) for all \(i>0\). Show that there are natural isomorphisms, for each \(i \geqslant 0\), 1
\begin{align*} H^i(X, \mathcal{F}) \cong H^i\left(Y, f_* \mathcal{F}\right) . \end{align*}
III.8.2. #to_work
Let \(f: X \rightarrow Y\) be an affine morphism of schemes (II, Ex. 5.17) with \(X\) noetherian, and let \(\mathcal{F}\) be a quasi-coherent sheaf on \(X\). Show that the hypotheses of (Ex. 8.1) are satisfied, and hence that \(H^i(X, \mathcal{F}) \cong H^i\left(Y, f_* \mathcal{F}\right)\) for each \(i \geqslant 0\).
III.8.3. The Projection Formula. #to_work
Let \(f: X \rightarrow Y\) be a morphism of ringed spaces, let \(\mathcal{F}\) be an \(\mathcal{O}_X\)-module, and let \(\mathcal{E}\) be a locally free \(\mathcal{O}_Y\)-module of finite rank. Prove the projection formula (cf. (II, Ex. 5.1)) \begin{align*} R^i f_*\left(\mathcal{F} \otimes f^* \mathcal{E}\right) \cong R^i f_*(\mathcal{F}) \otimes \mathcal{E} . \end{align*}
III.8.4. #to_work
Let \(Y\) be a noetherian scheme, and let \(\mathcal{E}\) be a locally free \(\mathcal{O}_Y\)-module of rank \(n+1\), \(n \geqslant 1\). Let \(X=\mathbf{P}(\mathcal{E})\) (II, \(\S\) ), with the invertible sheaf \(\mathcal{O}_X(1)\) and the projection morphism \(\pi: X \rightarrow Y\).
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Then
- \(\pi_*(\mathcal{O}(l)) \cong S^l(\mathcal{E})\) for \(l \geqslant 0, \pi_*(\mathcal{O}(l))=0\) for \(l<0\) (II, 7.11);
- \(R^i \pi_*(\mathcal{O}(l))=0\) for \(0<i<n\) and all \(l \in \mathbf{Z}\); and
- \(R^n \pi_*(\mathcal{O}(l))=0\) for \(l>-n-1\).
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Show there is a natural exact sequence \begin{align*} 0 \rightarrow \Omega_{X / Y} \rightarrow\left(\pi^* \mathcal{E}\right)(-1) \rightarrow \mathcal{O} \rightarrow 0, \end{align*} cf. (II, 8.13), and conclude that the relative canonical sheaf \(\omega_{X / Y}=\wedge^n \Omega_{X / Y}\) is isomorphic to \(\left(\pi^* \wedge^{n+1} \mathcal{E}\right)(-n-1)\). Show furthermore that there is a natural isomorphism \(R^n \pi_*\left(\omega_{X / Y}\right) \cong \mathcal{O}_Y\) (cf. (7.1.1)).
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Now show, for any \(l \in \mathbf{Z}\), that \begin{align*} R^n \pi_*(\mathcal{O}(l)) \cong \pi_*(\mathcal{O}(-l-n-1)) {}^{ \vee }\otimes\left(\wedge^{n+1} \mathcal{E}\right) {}^{ \vee }. \end{align*}
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Show that \(p_a(X)=(-1)^n p_a(Y)\) (use (Ex. 8.1)) and \(p_g(X)=0\) (use (II, 8.11)).
- In particular, if \(Y\) is a nonsingular projective curve of genus \(g\), and \(\mathcal{E}\) a locally free sheaf of rank 2 , then \(X\) is a projective surface with \(p_a=-g, p_g=0\), and irregularity \(g\) (7.12.3). This kind of surface is called a geometrically ruled surface (V, §2).