The Stacks project

Proof. To prove this we consider the fibre product diagram

Note that $f'$ is a proper morphism, see Morphisms, Lemma 29.41.5. Also, $B$ is a finitely generated $A$-algebra, and hence Noetherian (Algebra, Lemma 10.31.1). This implies that $X'$ is a Noetherian scheme (Morphisms, Lemma 29.15.6). Note that $X'$ is the relative spectrum of the quasi-coherent $\mathcal{O}_ X$-algebra $\mathcal{B}$ by Constructions, Lemma 27.4.6. Since $\mathcal{F}$ is a quasi-coherent $\mathcal{B}$-module we see that there is a unique quasi-coherent $\mathcal{O}_{X'}$-module $\mathcal{F}'$ such that $\pi _*\mathcal{F}' = \mathcal{F}$, see Morphisms, Lemma 29.11.6 Since $\mathcal{F}$ is finite type as a $\mathcal{B}$-module we conclude that $\mathcal{F}'$ is a finite type $\mathcal{O}_{X'}$-module (details omitted). In other words, $\mathcal{F}'$ is a coherent $\mathcal{O}_{X'}$-module (Lemma 30.9.1). Since the morphism $\pi : X' \to X$ is affine we have

by Lemma 30.2.4. Thus (1) follows from Lemma 30.19.2. Given $\mathcal{L}$ as in (2) we set $\mathcal{L}' = \pi ^*\mathcal{L}$. Note that $\mathcal{L}'$ is ample on $X'$ by Morphisms, Lemma 29.37.7. By the projection formula (Cohomology, Lemma 20.54.2) we have $\pi _*(\mathcal{F}' \otimes \mathcal{L}') = \mathcal{F} \otimes \mathcal{L}$. Thus part (2) follows by the same reasoning as above from Lemma 30.16.2. $\square$