The Stacks project

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. $\square$