Let $X$ be a ringed space (update: with a non-zero-ringed structure sheaf). It can be shown that $\forall\mathcal{F}\in \mathrm{Mod}(\mathcal{O}_X)$,
$\mathcal{F}$ is globally generated by a finite subset of its global sections (latter to be called finitely globally generated) $\Rightarrow$ $\mathcal{F}$ is finitely generated (also called finite type in some textbooks) and globally generated.
The reverse (i.e. $\forall \mathcal{F}$, finitely genreated+globally generated $\Rightarrow$ finitely globally generated) is false. But the reverse is true if $X$ is quasi-compact (around every point, find an open neighborhood with a finite set of generators consisting of global sections).
Now we have ($X$ is quasi-compact) $\Rightarrow$ ($\forall \mathcal{F},$finitely genreated+globally generated $\Rightarrow$ finitely globally generated)
Now I am asking if $X$ being quasi-compact is necessary, i.e. if the above implication can be reversed. So I want to
- either show: $\exists X$ non-quasicompact, $\forall\mathcal{F}\in \mathrm{Mod}(\mathcal{O}_X)$, finitely generated + globally generated $\Rightarrow$ finitely globally generated.
- or show the negation: $\forall X$ non-quasicompact, $\exists\mathcal{F}\in \mathrm{Mod}(\mathcal{O}_X)$ s.t. $\mathcal{F}$ is finitely generated + globally generated, but not finitely globally generated.
Feel free to set $X$ to be locally ringed space or scheme and $\mathcal{F}$ to be quasi-coherent.
I personally believe the first one is true (the second one is too strong and is not proved in any literature I read).
My approach 1: We may find a scenario s.t. for certain open (maybe affine) $U\subset V$, we can always extend the surjection $\mathcal{O}_U^n\twoheadrightarrow \mathcal{F}|_U$ to $\mathcal{O}_V^n\twoheadrightarrow \mathcal{F}|_V$. Then we use this property to construct a non-quasicompact scheme by $X=(\coprod_i V_i)/\sim$ identifying $U\subset V_i$ for each $i$.
Update 1: My approach is a dead end. For any proper open $U\subset X$ with complement closed $Z$, let $i:Z\hookrightarrow X$ be the closed immersion, then $i_* \mathcal{O}_Z$ is finitely generated quasi-coherent and globally generated (by the unit section) but $(i^* \mathcal{O}_Z)|_U=0$ so we can not lift $\mathcal{O}_U^0\twoheadrightarrow (i^* \mathcal{O}_Z)|_U$.
My approach 2: Using Nakayama lemma we could extend surjection $(\frac{R}{I})^n\to \frac{M}{IM}$ to $R^n\to M$. Now consider a ring $R$ with Jacobson radical $I$, let $V=\mathrm{Spec}R$ and $Z=V(I)$. Consider the scheme $X=(\coprod_{i\in \mathbb{N}} V_i)/\sim$ identifying $Z\subset V_i=V$ for each $i$ (at this point we assume this scheme exist). For each finitely generated and globally generated quasi-coherent sheaf $\mathcal{F}$ on $X$, it can be shown that $\mathcal{F}|_Z$ stays finitely generated and globally generated over the closed subscheme $Z$, hence finitely globally generated say by $n$ sections. Then $\mathcal{F}|_{V_i}$ is also finitely globally generated by $n$ sections that agree on $Z$ (each $V_i$ is closed in $X$). Two problems remain, (1) Deduce that $\mathcal{F}$ is finitely globally generated by $n$ sections. (2) Show $X$ is non-quasi-compact.
Best Answer
A friend of mine told me an example proving (1) in the case of locally ringed space.
Let $\omega_1$ be the first uncoutable ordinal. Let $X=[0,\omega_1)$ be the set of all ordinals smaller than $\omega_1$. Note that any non-decreasing function $f:[0,\omega_1)\to \mathbb{N}$ is eventually constant (also known as the Pressing Down lemma).
For each $\beta <\omega_1$, let $U_\beta =[0,\beta]\subset X$. Give $X$ the topology generated by the base $(U_\beta)_{\beta<\omega_1}$. Clearly $U\subset X$ is open iff $U=U_\beta$ for some $\beta<\omega_1$. The open cover $X=\bigcup_\beta U_\beta$ makes sure $X$ is non-quasi-compact.
Fix a field $k$ (e.g. $\mathbb{Q}$). Define $\mathcal{O}_X$ by $\mathcal{O}_X(U)=k$. Let $\mathcal{F}\in \mathrm{Mod}(\mathcal{O}_X)$ be finitely generated and globally generated.
[finitely generated] implies that for each $\beta\in X$, then $\mathcal{F}(U_\beta)=\mathcal{F}_\beta=k^{n_\beta}$ for some $n_\beta\geq 0$.
[globally generated] implies that for each pair $\alpha<\beta$ in $X$, $\mathcal{F}_\alpha$ is generated by the images of the composition map $\mathcal{F}(X)\to \mathcal{F}(U_\beta)\to\mathcal{F}(U_\alpha)=\mathcal{F}_\alpha$. Since they are all $k$-linear, it means $\mathcal{F}(X)\to \mathcal{F}(U_\beta)\to\mathcal{F}(U_\alpha)=\mathcal{F}_\alpha$ is surjective and so is $\mathcal{F}(U_\beta)\to\mathcal{F}(U_\alpha)=\mathcal{F}_\alpha$. By rank-nullity, we have $n_\beta\geq n_\alpha$. Define $f:X\to \mathbb{N}$ by $f(\beta)=n_\beta$, then $f$ is non-decreasing and thus eventually constant, say for all $\beta\geq \beta_0,n_\beta=m$.
Then for all $\beta>\alpha\geq \beta_0$, the map $k^{n_\beta}\twoheadrightarrow k^{n_\alpha}$ is surjective and thus bijective. So $\mathcal{F}(U_\beta)\cong \mathcal{F}(U_\alpha)$ via the restriction map. By sheaf property we can see that $\mathcal{F}(X)\cong k^m$. Let $e_1,\dots ,e_m$ be a system of generators of $k^m$, it is easy to see that $\mathcal{O}_X^m\to \mathcal{F}$ associated to $e_1,\dots,e_m$ is surjective. So $\mathcal{F}$ is finitely globally generated.