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 20.52.2. 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 37.50. 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. $\square$


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