The Stacks project

Proof. We will prove this by induction on $d$. The case $d = 0$ holds because in that case $X$ is the spectrum of a finite dimensional $k$-algebra $A$ (Varieties, Lemma 33.20.2) and every coherent sheaf $\mathcal{F}$ corresponds to a finite $A$-module $M = H^0(X, \mathcal{F})$ which has $\dim _ k M < \infty $.

Assume $d > 0$ and the result has been shown for separated schemes of finite type of dimension $< d$. The scheme $X$ is Noetherian. Consider the property $\mathcal{P}$ of coherent sheaves on $X$ defined by the rule

We are going to use the result of Cohomology of Schemes, Lemma 30.12.4 to prove that $\mathcal{P}$ holds for every coherent sheaf on $X$.

Let

be a short exact sequence of coherent sheaves on $X$. Consider the long exact sequence of cohomology

Thus if $\mathcal{P}$ holds for $\mathcal{F}_1$ and $\mathcal{F}_2$, then it holds for $\mathcal{F}$.

Let $Z \subset X$ be an integral closed subscheme. Let $\mathcal{I}$ be a coherent sheaf of ideals on $Z$. To finish the proof we have to show that $H^ d(X, i_*\mathcal{I}) = H^ d(Z, \mathcal{I})$ is finite dimensional. If $\dim (Z) < d$, then the result holds because the cohomology group will be zero (Cohomology, Proposition 20.20.7). In this way we reduce to the situation discussed in the following paragraph.

Assume $X$ is a variety of dimension $d$ and $\mathcal{F} = \mathcal{I}$ is a coherent ideal sheaf. In this case we have a short exact sequence

where $i : Z \to X$ is the closed subscheme defined by $\mathcal{I}$. By induction hypothesis we see that $H^{d - 1}(Z, \mathcal{O}_ Z) = H^{d - 1}(X, i_*\mathcal{O}_ Z)$ is finite dimensional. Thus we see that it suffices to prove the result for the structure sheaf.

We can apply Chow's lemma (Cohomology of Schemes, Lemma 30.18.1) to the morphism $X \to \mathop{\mathrm{Spec}}(k)$. Thus we get a diagram

as in the statement of Chow's lemma. Also, let $U \subset X$ be the dense open subscheme such that $\pi ^{-1}(U) \to U$ is an isomorphism. We may assume $X'$ is a variety as well, see Cohomology of Schemes, Remark 30.18.2. The morphism $i' = (i, \pi ) : X' \to \mathbf{P}^ n_ X$ is a closed immersion (loc. cit.). Hence

is $\pi $-relatively ample (for example by Morphisms, Lemma 29.39.7). Hence by Cohomology of Schemes, Lemma 30.16.2 there exists an $n \geq 0$ such that $R^ p\pi _*\mathcal{L}^{\otimes n} = 0$ for all $p > 0$. Set $\mathcal{G} = \pi _*\mathcal{L}^{\otimes n}$. Choose any nonzero global section $s$ of $\mathcal{L}^{\otimes n}$. Since $\mathcal{G} = \pi _*\mathcal{L}^{\otimes n}$, the section $s$ corresponds to section of $\mathcal{G}$, i.e., a map $\mathcal{O}_ X \to \mathcal{G}$. Since $s|_ U \not= 0$ as $X'$ is a variety and $\mathcal{L}$ invertible, we see that $\mathcal{O}_ X|_ U \to \mathcal{G}|_ U$ is nonzero. As $\mathcal{G}|_ U = \mathcal{L}^{\otimes n}|_{\pi ^{-1}(U)}$ is invertible we conclude that we have a short exact sequence

where $\mathcal{Q}$ is coherent and supported on a proper closed subscheme of $X$. Arguing as before using our induction hypothesis, we see that it suffices to prove $\dim H^ d(X, \mathcal{G}) < \infty $.

By the Leray spectral sequence (Cohomology, Lemma 20.13.6) we see that $H^ d(X, \mathcal{G}) = H^ d(X', \mathcal{L}^{\otimes n})$. Let $\overline{X}' \subset \mathbf{P}^ n_ k$ be the closure of $X'$. Then $\overline{X}'$ is a projective variety of dimension $d$ over $k$ and $X' \subset \overline{X}'$ is a dense open. The invertible sheaf $\mathcal{L}$ is the restriction of $\mathcal{O}_{\overline{X}'}(n)$ to $X$. By Cohomology, Proposition 20.22.4 the map

is surjective. Since the cohomology group on the left has finite dimension by Cohomology of Schemes, Lemma 30.14.1 the proof is complete. $\square$