The Stacks project

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

\[ \mathcal{E} \longrightarrow Rb_*(b^*\mathcal{E} \otimes _{\mathcal{O}_ L} \mathcal{O}_ L(iE)) \]

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

\[ H^0(X, \mathcal{O}_ X) \to H^0(L, \mathcal{O}_ L) \to H^0(L, \mathcal{O}_ L(iE)) \]

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

\[ H^0(X, \mathcal{O}_ X) \to H^0(L, \mathcal{O}_ L) \to H^0(L, \mathcal{O}_ L(iE)) \to H^0(L \setminus E, \mathcal{O}_ L) = H^0(X \setminus Z, \mathcal{O}_ X) \]

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

\[ \pi _*\mathcal{O}_ L(iE) = \pi _*\mathcal{O}_ L(-i) = \bigoplus \nolimits _{k \geq -i} \mathcal{O}_ P(k) \]

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. $\square$

Comments (0)

Post a comment

Your email address will not be published.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).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0G5H. Beware of the difference between the letter 'O' and the digit '0'.