## 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 $i \geqslant 0$,[^degen_leray]

\[
H^i(X, \mathcal{F}) \cong H^i\left(Y, f_* \mathcal{F}\right) .
\]

[^degen_leray]: 
This is a degenerate case of the Leray spectral sequence-see Godement $[1, II, 4.17.1]$.

### 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))
\[
R^i f_*\left(\mathcal{F} \otimes f^* \mathcal{E}\right) \cong R^i f_*(\mathcal{F}) \otimes \mathcal{E} .
\]

### 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 $\pi: X \rightarrow Y$.

a. 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 $l>-n-1$.

b. Show there is a natural exact sequence
\[
0 \rightarrow \Omega_{X / Y} \rightarrow\left(\pi^* \mathcal{E}\right)(-1) \rightarrow \mathcal{O} \rightarrow 0,
\]
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)).

c. Now show, for any $l \in \mathbf{Z}$, that
\[
R^n \pi_*(\mathcal{O}(l)) \cong \pi_*(\mathcal{O}(-l-n-1))\dual \otimes\left(\wedge^{n+1} \mathcal{E}\right)\dual .
\]

d. Show that $p_a(X)=(-1)^n p_a(Y)$ (use (Ex. 8.1)) and $p_g(X)=0$ (use (II, 8.11)).

e. 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).