Serre global-generation criterion for ampleness

Assume first that $L$ is ample. By [F1] every point $x$ has an $n_x\ge 1$ and $s_x\in \mathrm{\Gamma }(X,L^{n_x})$ with $x\in X_{s_x}$ and $X_{s_x}$ affine; since $X$ is quasi-compact by [F5], finitely many points $x_1,\dots ,x_k$ have $X={\bigcup }_i{X}_{s_i}$, where $s_i=s_{x_i}$ and $X_{s_i}$ is affine.

The underlying space of $X$ is Noetherian: by [F5] choose a finite affine open cover $X=U_1\cup \cdots \cup U_r$ with $U_i=\mathrm{Spec}{A}_i$ and $A_i$ Noetherian; each $U_i$ is a Noetherian topological space by [F12], and a descending chain of closed subsets of $X$ has traces in the finitely many $U_i$ which stabilize by some index, whence the chain itself stabilizes because a closed subset of $X$ is determined by its traces on a finite open cover. Consequently $X$ is quasi-compact, and every open subset of $X$ is compact by [F13], so in particular the intersection of two affine opens is quasi-compact and $X$ is quasi-separated in the sense of [F15].

For an invertible ${\mathcal{O}}_X$-module $M$, the sheaf $M$ is globally generated if and only if for every $x\in X$ some global section $s\in \mathrm{\Gamma }(X,M)$ has $s(x)\mathrm{\ne }0$: by [F2] global generation means that the images of the global sections generate $M_x$, and by [F3] $M_x$ is free of rank one over ${\mathcal{O}}_{X,x}$, so a family generates $M_x$ exactly when one member is a unit; a germ in $M_x\cong {\mathcal{O}}_{X,x}$ is a unit exactly when its value in the fibre is nonzero.

Conversely, assume that for every coherent sheaf there is a bound beyond which all its twists by $L$ are globally generated. Let $x\in X$ and let $U$ be an affine open subscheme containing $x$; put $Z=X\setminus U$, a closed subset of $X$. Define $\mathcal{I}\subseteq {\mathcal{O}}_X$ by $\mathrm{\Gamma }(W,\mathcal{I})=\{f\in \mathrm{\Gamma }(W,{\mathcal{O}}_X):f_z\in {\mathfrak{m}}_z{\mathcal{O}}_{X,z}\text{ for every }z\in Z\cap W\}$; this is an ideal sheaf [F17], its sections vanish at every point of $Z$, and its restriction to $U$ is the unit ideal: for an affine chart $V\subseteq U$ one has $Z\cap V=\mathrm{\varnothing }$ and the defining condition is vacuous, so $\mathrm{\Gamma }(V,\mathcal{I})=\mathrm{\Gamma }(V,{\mathcal{O}}_X)$.

Choose the finite affine open cover $X=U_1\cup \cdots \cup U_r$ with $U_i=\mathrm{Spec}{A}_i$ and $A_i$ Noetherian from [F5]. For each $i$ the ideal $\mathrm{\Gamma }(U_i,\mathcal{I})=\{f\in A_i:f\in \mathfrak{p}\text{ for every }\mathfrak{p}\in Z\cap U_i\}$ is, by [F9], the radical defining ideal of the closed subset $Z\cap U_i$ (with value $A_i$ when $Z\cap U_i=\mathrm{\varnothing }$), hence is finitely generated because $A_i$ is Noetherian. For $D(f)\subseteq U_i$, both $\mathrm{\Gamma }(D(f),\mathcal{I})$ and $(\mathrm{\Gamma }(U_i,\mathcal{I}){)}_f$ are the radical defining ideal of $Z\cap D(f)$ in $A_f$: the first by definition, the second by [F8]. Thus $\mathcal{I}{\mathrm{\mid }}_{U_i}$ is the sheaf associated to the module $\mathrm{\Gamma }(U_i,\mathcal{I})$ by [F16], so $\mathcal{I}$ is quasi-coherent by the affine-cover criterion [F7] and of finite type because a finite generating set of each $\mathrm{\Gamma }(U_i,\mathcal{I})$ generates $\mathcal{I}$ on $U_i$; since $X$ is locally Noetherian, [F6] makes $\mathcal{I}$ coherent.

With $d=d_1\cdots d_k$, where $d_i=\mathrm{deg}{s}_i$, each power $s_i^{d\mathrm{/}{d}_i}\in \mathrm{\Gamma }(X,L^d)$ has $X_{s_i^{d\mathrm{/}{d}_i}}=X_{s_i}$; these loci cover $X$, so step 1.3 applies and $L^d$ is globally generated.

Let $M$ be an ample invertible sheaf on $X$ and let $G$ be a quasi-coherent ${\mathcal{O}}_X$-module. For $m\ge 1$ let $G_m\subseteq G$ be the image of the canonical map $\mathrm{\Gamma }(X,G\otimes M^m){\otimes }_{\mathbb{Z}}{M}^{-m}\to G$. Then $G={\sum }_{m\ge 1}{G}_m$. Indeed, applying step 1.1 to $M$ gives finitely many $t_j\in \mathrm{\Gamma }(X,M^{e_j})$, $e_j\ge 1$, with $X={\bigcup }_j{X}_{t_j}$ and each $X_{t_j}$ affine. For a section $u\in \mathrm{\Gamma }(X_{t_j},G)$ the extension lemma [F4] applied on the quasi-compact quasi-separated scheme $X$ of step 1.2 to $s=t_j$ yields $r\ge 0$ and $a\in \mathrm{\Gamma }(X,G\otimes M^{e_{j}r})$ whose image after division by $t_j^r$ is $u$. If $r\ge 1$, this places $u$ in $G_{e_{j}r}{\mathrm{\mid }}_{X_{t_j}}$. If $r=0$, multiply $a$ by $t_j$ to obtain a global section of $G\otimes M^{e_j}$ whose image after division by $t_j$ is still $u$, placing $u$ in $G_{e_j}{\mathrm{\mid }}_{X_{t_j}}$. Thus every local section of $G$ lies locally in ${\sum }_{m\ge 1}{G}_m$, and a subsheaf whose sections surject onto those of $G$ over a cover equals $G$.

Let $n\ge \max (n_0(\mathcal{I}),1)$ be such that $\mathcal{I}\otimes L^n$ is globally generated. By step 1.4 the stalk ${\mathcal{I}}_x$ is ${\mathcal{O}}_{X,x}$, so $(\mathcal{I}\otimes L^n{)}_x\cong (L^n{)}_x$ is free of rank one [F3]; global generation produces a global section $t$ whose germ at $x$ is not in ${\mathfrak{m}}_x(\mathcal{I}\otimes L^n{)}_x$, and viewing $t$ as a section $s\in \mathrm{\Gamma }(X,L^n)$ through this identification, $s(x)\mathrm{\ne }0$. For $z\in Z$ the stalk of $\mathcal{I}$ at $z$ is contained in ${\mathfrak{m}}_z$ by the description in step 1.5 and [F9], so $s$ vanishes at every point of $Z$ and therefore $X_s\subseteq X\setminus Z=U$. As $U$ is affine, $X_s=U\cap X_s$ is affine by [F10].

For every $j\ge 0$ and every $x\in X$ there are $m\ge 1$ and $a\in \mathrm{\Gamma }(X,L^{j+dm})$ with $a(x)\mathrm{\ne }0$. Indeed, $L^d$ is ample by [F11], so step 2.2 applied to $M=L^d$ and $G=L^j$ gives $L^j={\sum }_m{G}_m$; the stalk $(L^j{)}_x$ is free of rank one by [F3], so if every global section of every $L^{j+dm}$ vanished at $x$, then every summand $G_m$ would vanish at $x$ — its value at $x$ is spanned by products of values $g(x)a(x)$ — contradicting that the sum equals $L^j$.

Let $F$ be quasi-coherent of finite type, and define $F_m$ as in step 2.2 for $M=L$; put $J_N={\sum }_{m\le N}{F}_m$. Then $F={\sum }_m{F}_m={\bigcup }_N{J}_N$, and there is $N$ with $F=J_N$. To see this, use [F14] and quasi-compactness of $X$ to choose a finite cover of $X$ by opens $V_i$ on which $F$ is generated by finitely many sections $g_{il}\in F(V_i)$; since $(J_N{)}_x$ is increasing in $N$ with union $F_x$, each germ $(g_{il}{)}_x$ lies in some $(J_{N_{il}(x)}{)}_x$, and because only finitely many pairs occur at $x$, a single $N(x)$ works. A germ $(g_{il}{)}_x\in (J_{N(x)}{)}_x$ means that $g_{il}$ lies in $J_{N(x)}$ over some open neighbourhood of $x$; over such a neighbourhood contained in all relevant $V_i$ the sections $g_{il}$ lie in $J_{N(x)}$ and generate $F$, so $J_{N(x)}=F$ over it. These neighbourhoods cover the quasi-compact space $X$, so finitely many suffice and $N$ is their maximum.

Fix $j\ge 0$. By step 3.1 the loci $X_a$, for $a\in \mathrm{\Gamma }(X,L^{j+dm})$, $m\ge 1$, cover $X$; quasi-compactness of $X$ [F5] gives finitely many $a_1,\dots ,a_r$ with degrees $j+d{m}_1,\dots ,j+d{m}_r$ such that $X={\bigcup }_l{X}_{a_l}$. Put $n_j={\max }_l{m}_l$. Then $L^{j+d{n}_j}$ is globally generated: at any point $x$ choose $l$ with $a_l(x)\mathrm{\ne }0$ and, by step 2.1 and step 1.3, a section $u\in \mathrm{\Gamma }(X,L^d)$ with $u(x)\mathrm{\ne }0$; then $a_l\otimes u^{n_j-m_l}\in \mathrm{\Gamma }(X,L^{j+d{n}_j})$ has nonzero value at $x$, so step 1.3 applies.

Put $n_0(L)=d\cdot {\max }_{0\le j<d}{n}_j+(d-1)$. For every $n\ge n_0(L)$ the sheaf $L^n$ is globally generated: write $n=j+d{n}^{\mathrm{\prime }}$ with $0\le j<d$; then $n^{\mathrm{\prime }}\ge {\max }_i{n}_i\ge n_j$ and $L^n=L^{j+d{n}_j}\otimes (L^d{)}^{n^{\mathrm{\prime }}-n_j}$ is a tensor product of two globally generated invertible sheaves, which is globally generated because the tensor product of sections nonzero at a point is nonzero at that point, so step 1.3 applies.

For $m\ge N+n_0(L)$ the sheaf $F\otimes L^m$ is globally generated: by step 3.2 and right exactness of tensoring with the invertible sheaf $L^m$, one has $F\otimes L^m={\sum }_{m^{\mathrm{\prime }}\le N}(F_{m^{\mathrm{\prime }}}\otimes L^m)$; each summand is isomorphic to $(F_{m^{\mathrm{\prime }}}\otimes L^{m^{\mathrm{\prime }}})\otimes L^{m-m^{\mathrm{\prime }}}$, where $F_{m^{\mathrm{\prime }}}\otimes L^{m^{\mathrm{\prime }}}$ is globally generated by construction of $F_{m^{\mathrm{\prime }}}$ as the image of an evaluation map, and $L^{m-m^{\mathrm{\prime }}}$ is globally generated by step 5.1 because $m-m^{\mathrm{\prime }}\ge n_0(L)$. A tensor product of globally generated sheaves is globally generated, and a finite sum of globally generated subsheaves is globally generated because the evaluation map of the ambient sheaf maps onto each summand.

If $L$ is ample, then every coherent $F$ is quasi-coherent of finite type by [F6], so steps 5.1 and 6.1 provide a bound, depending on $F$, beyond which all twists $F\otimes L^n$ are globally generated. This proves the forward implication.

Steps 1.4, 1.5 and 2.3 show that the global-generation hypothesis provides, for every $x\in X$, an $n\ge 1$ and $s\in \mathrm{\Gamma }(X,L^n)$ with $x\in X_s$ and $X_s$ affine; since $X$ is quasi-compact by [F5], the definition [F1] makes $L$ ample. Together with step 7.1 this proves both implications. If $X=\mathrm{\varnothing }$, then $L$ is ample by [F1] vacuously, and the zero sheaf is globally generated, so the empty case contributes no exception. The recursive open-cover selection in [F13] uses [A1]; the other uses of choice are inherited from the cited sheaf, Proj and closed-subset interfaces, and the proof selects nothing beyond the choices recorded there. [A1, F1, F5, F13, step 1.4, step 1.5, step 2.3, step 7.1, cases: empty] \qed