Lemma 50.16.3. Let $E = 0(P)$ be the exceptional divisor of the blowing up $b$. For any locally free $\mathcal{O}_ X$-module $\mathcal{E}$ and $0 \leq i \leq n - 1$ the map
is an isomorphism in $D(\mathcal{O}_ X)$.
Proof. By the projection formula it is enough to show this for $\mathcal{E} = \mathcal{O}_ X$, see Cohomology, Lemma 20.54.2. Since $X$ is affine it suffices to show that the maps
are isomorphisms and that $H^ j(X, \mathcal{O}_ L(iE)) = 0$ for $j > 0$ and $0 \leq i \leq n - 1$, see Cohomology of Schemes, Lemma 30.4.6. Since $\pi $ is affine, we can compute global sections and cohomology after taking $\pi _*$, see Cohomology of Schemes, Lemma 30.2.4. If $n = 1$, then $L \to X$ is an isomorphism and $i = 0$ hence the first statement holds. If $n > 1$, then we consider the composition
Since $H^0(X \setminus Z, \mathcal{O}_ X) = H^0(X, \mathcal{O}_ X)$ in this case as $Z$ has codimension $n \geq 2$ in $X$ (details omitted) we conclude the first statement holds. For the second, recall that $\mathcal{O}_ L(E) = \mathcal{O}_ L(-1)$, see Divisors, Lemma 31.32.4. Hence we have
as discussed in More on Morphisms, Section 37.51. Thus we conclude by the vanishing of the cohomology of twists of the structure sheaf on $P = \mathbf{P}^{n - 1}_ Z$ shown in Cohomology of Schemes, Lemma 30.8.1. $\square$
Your email address will not be published. Required fields are marked.
In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).
All contributions are licensed under the GNU Free Documentation License.
Comments (0)