Does Ample imply direct image algebra finitely generated?

Question

Let $S$ be a Noetherian scheme, $f: X \to S$ be a projective morphism and $\mathcal{L}$ an $f$-ample invertible $\mathcal{O}_X$-module. Is the $\mathcal{O}_S$-algebra $\bigoplus_{m ∈ \mathbb{Z}_{\geq 0}} f_*(\mathcal{L}^{\otimes m})$ finitely generated over $\mathcal{O}_S$?

There are many technical conditions in Stacks Project 01VJ, but none mention finite generation of the algebra above.

Answer 1

I answer because I’m not satisfied by the arguments in the link (they’re elliptic).

Edit: my argument itself was lacking. That’ll teach me to be a critic. It’s corrected now.

Clearly, we can assume that $S$ is affine and the spectrum of the Noetherian ring $A$.

Let $\iota: X \rightarrow \mathbb{P}^n_S$ be a closed immersion such that $\iota^{\star}O(1)=\mathcal{L}^{\otimes d}$ for some $d \geq 1$.

Let $\mathcal{K}$ be the kernel of $\mathcal{O}_{\mathbb{P}^n_A} \rightarrow \iota_{\star} \mathcal{O}_X$.

Then, for every $l \geq 1$, we have an exact sequence $0 \rightarrow \mathcal{K}(l) \rightarrow O(l) \rightarrow \iota_{\star} \mathcal{L}^{\otimes dl} \rightarrow 0$.

Therefore, for all $l\geq L$ for some $L > 0$, $H^1(\mathbb{P}^n_A, \mathcal{K}(l))=0$, hence $H^0(\mathbb{P}^n_A,O(l)) \rightarrow H^0(X, \mathcal{L}^{\otimes dl})$ is onto.

Since $H^0(\mathbb{P}^n_A, O(1))^{\otimes l} \rightarrow H^0(\mathbb{P}^n_A, O(l))$ is onto, it follows that $\bigoplus_{m \geq 0}{H^0(X,\mathcal{L}^{\otimes dm})}$ is generated in degree $\leq L$. Since every $f_{\star} \mathcal{L}^{\otimes l}$ is coherent, $G_d:=\bigoplus_{m \geq 0}{H^0(X,\mathcal{L}^{\otimes dm})}$ is finitely generated over $A$.

Now, let $\mathcal{F}$ be any coherent sheaf on $X$. Then there is an exact sequence $0 \rightarrow \mathcal{A}\rightarrow (\mathcal{L}^{\otimes -ds})^{\oplus r} \rightarrow \mathcal{F} \rightarrow 0$ for some $s \geq 0$.

Since, for all large enough $l \geq 0$, $H^1(X,\mathcal{A}\otimes \mathcal{L}^{\otimes dl})=0$, for all large enough $l \geq 0$, $H^0(X,\mathcal{L}^{\otimes d(l-s)})^{\oplus r} \rightarrow H^0(X,\mathcal{F} \otimes \mathcal{L}^{\otimes dl})$ is onto.

It follows that $\bigoplus_{l \geq 0}{H^0(X,\mathcal{F} \otimes \mathcal{L}^{\otimes dl})}$ is a finitely generated module over $G_d$.

Apply the above with $\mathcal{F}=\mathcal{L}^{\otimes k}$ for $0 \leq k < d$: it follows that $G_1$ is a finite $G_d$-algebra, while $G_d$ is finitely generated over $A$, which concludes.

Answer 2

Yes.

Since $S$ is quasi-compact, it suffices to prove the case where $S$ is a Noetherian affine scheme $\operatorname{Spec} R$ and $\mathcal{L}$ is ample. Since $X$ is quasicompact, taking global sections commutes with taking a direct sum, see Lemma 01AI. We have $$ \begin{align} H^0(S, \bigoplus_{m \in \mathbb{Z}_{\geq 0}} f_* (\mathcal{L}^{\otimes m})) & = \bigoplus_{m \in \mathbb{Z}_{\geq 0}} H^0(S, f_* (\mathcal{L}^{\otimes m}))\\ & = \bigoplus_{m \in \mathbb{Z}_{\geq 0}} H^0(X, \mathcal{L}^{\otimes m}). \end{align} $$ Since $S$ is affine and $\bigoplus_{m \in \mathbb{Z}_{\geq 0}} f_* (\mathcal{L}^{\otimes m})$ is quasi-coherent, it suffices to prove that $\bigoplus_{m \in \mathbb{Z}_{\geq 0}} H^0(X, \mathcal{L}^{\otimes m})$ is a finitely generated $R$-algebra. This is Lemma 0B5T(1).