The Stacks project

Lemma 30.19.3. Let $A$ be a Noetherian ring. Let $B$ be a finitely generated graded $A$-algebra. Let $f : X \to \mathop{\mathrm{Spec}}(A)$ be a proper morphism. Set $\mathcal{B} = f^*\widetilde B$. Let $\mathcal{F}$ be a quasi-coherent graded $\mathcal{B}$-module of finite type.

  1. For every $p \geq 0$ the graded $B$-module $H^ p(X, \mathcal{F})$ is a finite $B$-module.

  2. If $\mathcal{L}$ is an ample invertible $\mathcal{O}_ X$-module, then there exists an integer $d_0$ such that $H^ p(X, \mathcal{F} \otimes \mathcal{L}^{\otimes d}) = 0$ for all $p > 0$ and $d \geq d_0$.

Proof. To prove this we consider the fibre product diagram

\[ \xymatrix{ X' = \mathop{\mathrm{Spec}}(B) \times _{\mathop{\mathrm{Spec}}(A)} X \ar[r]_-\pi \ar[d]_{f'} & X \ar[d]^ f \\ \mathop{\mathrm{Spec}}(B) \ar[r] & \mathop{\mathrm{Spec}}(A) } \]

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

\[ H^ p(X, \mathcal{F}) = H^ p(X', \mathcal{F}') \]

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. $\square$

Comments (2)

Comment #8106 by Laurent Moret-Bailly on

The gradings on and are completely irrelevant (and not even mentioned in the proof), so why not forget about them? The same comment applies to 69.20.4.

There are also:

  • 2 comment(s) on Section 30.19: Higher direct images of coherent sheaves

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