The Stacks project

Proof. Lemma 28.26.11 is (1) $\Rightarrow $ (2). Lemmas 28.26.2 and 28.26.12 provide the implication (1) $\Leftarrow $ (2). The implications (2) $\Rightarrow $ (4) $\Rightarrow $ (3) are clear from Constructions, Section 27.8. Lemma 28.26.6 is (3) $\Rightarrow $ (1). Thus we see that the first 4 conditions are all equivalent.

Assume the equivalent conditions (1) – (4). Note that in particular $X$ is separated (as an open subscheme of the separated scheme $\text{Proj}(S)$). Let $\mathcal{F}$ be a quasi-coherent sheaf on $X$. Choose $s \in S_{+}$ homogeneous such that $X_ s$ is affine. We claim that any section $m \in \Gamma (X_ s, \mathcal{F})$ is in the image of one of the maps displayed in (5) above. This will imply (5) since these affines $X_ s$ cover $X$. Namely, by Lemma 28.17.2 we may write $m$ as the image of $m' \otimes s^{-n}$ for some $n \geq 1$, some $m' \in \Gamma (X, \mathcal{F} \otimes \mathcal{L}^{\otimes n})$. This proves the claim.

Clearly (5) $\Rightarrow $ (6). Let us assume (6) and prove $\mathcal{L}$ is ample. Pick $x \in X$. Let $U \subset X$ be an affine open which contains $x$. Set $Z = X \setminus U$. We may think of $Z$ as a reduced closed subscheme, see Schemes, Section 26.12. Let $\mathcal{I} \subset \mathcal{O}_ X$ be the quasi-coherent sheaf of ideals corresponding to the closed subscheme $Z$. By assumption (6), there exists an $n \geq 1$ and a section $s \in \Gamma (X, \mathcal{I} \otimes \mathcal{L}^{\otimes n})$ such that $s$ does not vanish at $x$ (more precisely such that $s \not\in \mathfrak m_ x \mathcal{I}_ x \otimes \mathcal{L}_ x^{\otimes n}$). We may think of $s$ as a section of $\mathcal{L}^{\otimes n}$. Since it clearly vanishes along $Z$ we see that $X_ s \subset U$. Hence $X_ s$ is affine, see Lemma 28.26.4. This proves that $\mathcal{L}$ is ample. At this point we have proved that (1) – (6) are equivalent.

Assume the equivalent conditions (1) – (6). In the following we will use the fact that the tensor product of two sheaves of modules which are globally generated is globally generated without further mention (see Modules, Lemma 17.4.3). By (1) we can find elements $s_ i \in S_{d_ i}$ with $d_ i \geq 1$ such that $X = \bigcup _{i = 1, \ldots , n} X_{s_ i}$. Set $d = d_1\ldots d_ n$. It follows that $\mathcal{L}^{\otimes d}$ is globally generated by

This means that if $\mathcal{L}^{\otimes j}$ is globally generated then so is $\mathcal{L}^{\otimes j + dn}$ for all $n \geq 0$. Fix a $j \in \{ 0, \ldots , d - 1\} $. For any point $x \in X$ there exists an $n \geq 1$ and a global section $s$ of $\mathcal{L}^{j + dn}$ which does not vanish at $x$, as follows from (5) applied to $\mathcal{F} = \mathcal{L}^{\otimes j}$ and ample invertible sheaf $\mathcal{L}^{\otimes d}$. Since $X$ is quasi-compact there we may find a finite list of integers $n_ i$ and global sections $s_ i$ of $\mathcal{L}^{\otimes j + dn_ i}$ which do not vanish at any point of $X$. Since $\mathcal{L}^{\otimes d}$ is globally generated this means that $\mathcal{L}^{\otimes j + dn}$ is globally generated where $n = \max \{ n_ i\} $. Since we proved this for every congruence class mod $d$ we conclude that there exists an $n_0 = n_0(\mathcal{L})$ such that $\mathcal{L}^{\otimes n}$ is globally generated for all $n \geq n_0$. At this point we see that if $\mathcal{F}$ is globally generated then so is $\mathcal{F} \otimes \mathcal{L}^{\otimes n}$ for all $n \geq n_0$.

We continue to assume the equivalent conditions (1) – (6). Let $\mathcal{F}$ be a quasi-coherent sheaf of $\mathcal{O}_ X$-modules of finite type. Denote $\mathcal{F}_ n \subset \mathcal{F}$ the image of the canonical map of (5). By construction $\mathcal{F}_ n \otimes \mathcal{L}^{\otimes n}$ is globally generated. By (5) we see $\mathcal{F}$ is the sum of the subsheaves $\mathcal{F}_ n$, $n \geq 1$. By Modules, Lemma 17.9.7 we see that $\mathcal{F} = \sum _{n = 1, \ldots , N} \mathcal{F}_ n$ for some $N \geq 1$. It follows that $\mathcal{F} \otimes \mathcal{L}^{\otimes n}$ is globally generated whenever $n \geq N + n_0(\mathcal{L})$ with $n_0(\mathcal{L})$ as above. We conclude that (1) – (6) implies (7).

Assume (7). Let $\mathcal{F}$ be a quasi-coherent sheaf of $\mathcal{O}_ X$-modules of finite type. By (7) there exists an integer $n \geq 1$ such that the canonical map

is surjective. Let $I$ be the set of finite subsets of $\Gamma (X, \mathcal{F} \otimes _{\mathcal{O}_ X} \mathcal{L}^{\otimes n})$ partially ordered by inclusion. Then $I$ is a directed partially ordered set. For $i = \{ s_1, \ldots , s_{r(i)}\} $ let $\mathcal{F}_ i \subset \mathcal{F}$ be the image of the map

which is multiplication by $s_ j$ on the $j$th factor. The surjectivity above implies that $\mathcal{F} = \mathop{\mathrm{colim}}\nolimits _{i \in I} \mathcal{F}_ i$. Hence Modules, Lemma 17.9.7 applies and we conclude that $\mathcal{F} = \mathcal{F}_ i$ for some $i$. Hence we have proved (8). In other words, (7) $\Rightarrow $ (8).

The implication (8) $\Rightarrow $ (9) is trivial.

Finally, assume (9). Let $\mathcal{I} \subset \mathcal{O}_ X$ be a quasi-coherent sheaf of ideals. By Lemma 28.22.3 (this is where we use the condition that $X$ be quasi-separated) we see that $\mathcal{I} = \mathop{\mathrm{colim}}\nolimits _\alpha I_\alpha $ with each $I_\alpha $ quasi-coherent of finite type. Since by assumption each of the $I_\alpha $ is a quotient of negative tensor powers of $\mathcal{L}$ we conclude the same for $\mathcal{I}$ (but of course without the finiteness or boundedness of the powers). Hence we conclude that (9) implies (6). This ends the proof of the proposition. $\square$