Find a non-quasi-compact scheme s.t. all finitely generated + globally generated quasi-coherent modules are finitely globally generated

Question

Closely related to this question in MSE, but the difference is that we will set $X$ to be scheme and $\mathcal{F}$ to be quasi-coherent.

Let $X$ be a locally ringed space. We say an $\mathcal{O}_X$-module is finitely globally generated if it is globally generated by a finite subset of its global sections. Clearly (finitely globally generated) $\Rightarrow$ (finitely generated + globally generated). The reverse is false in general but true when $X$ is quasi-compact.

So we have [$X$ is quasi-compact] $\Rightarrow$ [$\forall \mathcal{F}\in \mathrm{Mod}(\mathcal{O}_X)$, (finitely globally generated) $\Leftrightarrow$ (finitely generated + globally generated)]. Now I am interested in if the above implication can be reversed. In the answer of this question in MSE, we have a counter-example (not scheme), i.e. we have shown that

[$\forall \mathcal{F}\in \mathrm{Mod}(\mathcal{O}_X)$, (finitely globally generated) $\Leftrightarrow$ (finitely generated + globally generated)]

$\not\Rightarrow$

[the locally ringed space $X$ is quasi-compact]

, so the case is solved for locally ringed space but not for the setting of schemes+quasi-coherent sheaves.

Now I want to show

[$\forall \mathcal{F}\in \mathrm{QS}(\mathcal{O}_X)$, (finitely globally generated) $\Leftrightarrow$ (finitely generated + globally generated)]

$\not\Rightarrow$

[the scheme $X$ is quasi-compact]

Equivalently I want to show there exists a non-quasi-compact scheme $X$ s.t. $\forall \mathcal{F}\in \mathrm{QC}(\mathcal{O}_X)$, (finitely generated + globally generated) $\Rightarrow$ (finitely globally generated) , since all finitely globally generated modules are automatically finitely generated and globally generated.

(I hope the newest edit makes my statement clear enough.)

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Here is a rough approach (not yet complete) by a friend of mine, generalizing the answer mentioned above:

1. Let $\omega_1$ be the first uncountable ordinal. Consider the abelian group $\Gamma=\prod_{\beta<\omega_1}\mathbb{Z}$ ($\bigoplus_{\beta<\omega_1}\mathbb{Z}$ also works) and equip it with the lexicographic order.

2. Fix a field $k$. Let $K=\mathrm{Frac}(k[\Gamma])$ and $v:k[\Gamma]\backslash\{0\}\to \Gamma$ be the canonical valuation map sending $a=\sum_{g\in \Gamma} a_g \cdot g$ to $\min\{g:a_g\neq0\}$. The valuation map extends to $K^*$. Let $A$ be the corresponding valuation ring. Let $Q:=v^{-1}(0)$, then $K=\{0\}\cup\bigcup_{g\in \Gamma} Q\cdot g$ and $A=\{0\}\cup\bigcup_{g\in \Gamma^{\geq 0}} Q\cdot g$.

3. It's known that the prime ideals of $A$ correspond 1-1 with prime ideals of $\Gamma$, see this tag in Stacks Project.

4. It can be shown that every prime ideal of $\Gamma$ (also viewed as in $A$) is exactly one of following: $$\{0\}=:P_{0}\subsetneq P_{1}\subsetneq \cdots \subsetneq P_\beta \subsetneq \cdots \subsetneq P_{\omega_1}$$ where $P_{\beta}:=\{a\in \Gamma^{>0}:\min\{\alpha:a_\alpha\neq 0\}<\beta \}$ for $\beta\in [1,\omega_1]$ and $P_0:=\emptyset$ in $\Gamma$ and $P_0:=\{0\}$ in $A$. Moreover, $P_{\omega_1}=\bigcup_{\beta<\omega_1} P_\beta$.

5. It follows that every proper open subset of $\mathrm{Spec} A$ is one of the following: $$\emptyset=U_{0}\subsetneq U_{1}\subsetneq \cdots \subsetneq U_\beta \subsetneq \cdots \subsetneq U_{\omega_1}$$ where $U_{\beta}:=V(P_\beta)^c=\{P_\alpha:\alpha<\beta\}$. We have $U_{\beta^+}$ is the minimal open neighborhood of $P_{\beta}$, so those opens must be distinguished open affines in $\mathrm{Spec}A$ and $\mathcal{O}_{\mathrm{Spec}A,P_{\beta}}=\mathcal{O}_{\mathrm{Spec}A}(U_{\beta^+})=A_{P_\beta}$. Moreover $U_{\alpha^+}$ is distinguished open affine in $U_{\beta^+}$ if $\alpha<\beta$. And $U_{\omega_1}=\bigcup_{\beta<\omega_1} U_\beta=:X$ is our non-quasi-compact candidate. Also $\mathcal{O}(X)=A$.

6. Then $X$ has a base $(U_{\beta^+})_{\beta<\omega_1}$ consisting all the non-empty open affines in $X$. The minimality of each $U_{\beta^+}$ of containing $P_{\beta}$ results that every presheaf on the base is also a sheaf. With the characterization of quasi-coherence on the distinguished affine base, any quasi-coherent module is determined by the data of a presheaf $F$ of modules on that base s.t. for each $\alpha<\alpha^\prime$, the induced map $F(U_{\alpha^{\prime +}})\otimes_{\mathcal{O}(U_{\alpha^{\prime +}})}\mathcal{O}(U_{\alpha^+})\to F(U_{\alpha^{+}})$ is an isormophism.

7. Let $\mathcal{F}\in \mathrm{QC}(\mathcal{O}_X)$ be finitely generated and globally generated. Then $\mathcal{F}_\beta:=\mathcal{F}_{P_\beta}$ is a finite $\mathcal{O}_\beta:=\mathcal{O}_{P_\beta}$-module. Define $n_\beta=\min\{n:\exists(\mathcal{O}_\beta^n\twoheadrightarrow \mathcal{F}_{\beta})\}$. Clearly $n_\beta\geq n_\alpha$ if $\beta>\alpha$. So we have a non-decreasing function $[0,\omega_1)\to \mathbb{N}$, by the Pressing Down lemma, there exists $\beta_0$ s.t. $\forall \beta\geq\beta_0,n_\beta=n_{\beta_0}=:m$. Globally generation implies that $\mathcal{F}(X)\otimes_A A_{P_\beta}\twoheadrightarrow \mathcal{F}_\beta$ is surjective for all $\beta$.

8. We know $\mathcal{F}(X)$ is the inverse limit of $(\mathcal{F}_\beta)_\beta$. I want to show that 1)$\mathcal{F}(X)\otimes_A \mathcal{O}_\beta \cong \mathcal{F}_\beta$ so $\mathcal{F}$ extends to $\mathrm{Spec} A$, 2) $\mathcal{F}(X)$ is a finite $A$-module (Note that $\mathcal{O}(X)=A$). It follows that $\mathcal{F}$ is globally finitely generated.

There could be mistakes in the above claims.

Answer 1

Edit. The argument below applies only to Lindelöf schemes.

Proposition. For every Lindelöf non-quasi-compact scheme $X$, there exists a quasi-coherent $\mathcal{O}_X$-module that is globally generated and that is locally finitely generated, but whose $\mathcal{O}_X(X)$-module of global sections is not finitely generated.

Proof. Since $X$ is not quasi-compact, there exist a strictly increasing sequence of open subsets whose union equals $X$, $$U = U_0 \subsetneq U_1 \subsetneq \dots \subsetneq U_n \subsetneq \dots \subsetneq X,\ \ \ n\in \mathbb{Z}_{\geq 0}.$$ For every $n\in \mathbb{Z}_{\geq 0}$, denote the closed complement of $U_n$ by $C_n$ with its reduced structure. Denote by $\mathcal{O}_{C_n}$ the pushforward to $X$ of the structure sheaf of $C_n$, considered as a $\mathcal{O}_X$-module. Let $\mathcal{F}$ denote the direct sum, $$\mathcal{F} := \bigoplus_{n=0}^\infty \mathcal{O}_{C_n},$$ considered as a quotient of the direct sum $\bigoplus_{n=0}^\infty \mathcal{O}_X$.

The restriction of $\mathcal{F}$ to each open subset $U_m$ is isomorphic to the finitely generated quasi-coherent sheaf $\bigoplus_{n=0}^m \mathcal{O}_{C_n}$, so it is locally finitely generated. Globally, $\mathcal{F}$ is a quotient of the direct sum $\bigoplus_{n=0}^\infty \mathcal{O}_X$. Yet $\mathcal{F}(X) = \bigoplus_{n=0}^\infty \mathcal{O}_{C_n}(C_n)$ is not finitely generated as an $\mathcal{O}_X(X)$-module. QED

Answer 2

Warning. This does not answer the question. See the comments.

Let $k$ be a field. For each $n\in\mathbb{N}$, let $X_n$ be a copy of $\mathrm{Spec}(k)$. Put $X:=\coprod_{n\in\mathbb{N}}X_n$. We view $\mathscr{O}_{X_n}$ as a quasi-coherent sheaf on $X$ supported on $X_n$. Now put $\mathscr{F}:=\bigoplus_{n\in\mathbb{N}}\mathscr{O}_{X_n}^{n+1}$.

Clearly $\mathscr{F}$ is finitely generated (this is a local condition), and $$\Gamma(X,\mathscr{F})\cong \prod_{n\in\mathbb{N}}\Gamma(X_n,\mathscr{O}_{X_n}^{n+1})\cong \prod_{n\in\mathbb{N}}k^{n+1}$$ so $\mathscr{F}$ is globally generated but not finitely so.