The Stacks project

30.17 Ample invertible sheaves and cohomology

Here is a criterion for ampleness on proper schemes over affine bases in terms of vanishing of cohomology after twisting.

reference

Lemma 30.17.1. Let $R$ be a Noetherian ring. Let $f : X \to \mathop{\mathrm{Spec}}(R)$ be a proper morphism. Let $\mathcal{L}$ be an invertible $\mathcal{O}_ X$-module. The following are equivalent

  1. $\mathcal{L}$ is ample on $X$ (this is equivalent to many other things, see Properties, Proposition 28.26.13 and Morphisms, Lemma 29.39.4),

  2. for every coherent $\mathcal{O}_ X$-module $\mathcal{F}$ there exists an $n_0 \geq 0$ such that $H^ p(X, \mathcal{F} \otimes \mathcal{L}^{\otimes n}) = 0$ for all $n \geq n_0$ and $p > 0$, and

  3. for every quasi-coherent sheaf of ideals $\mathcal{I} \subset \mathcal{O}_ X$, there exists an $n \geq 1$ such that $H^1(X, \mathcal{I} \otimes \mathcal{L}^{\otimes n}) = 0$.

Proof. The implication (1) $\Rightarrow $ (2) follows from Lemma 30.16.1. The implication (2) $\Rightarrow $ (3) is trivial. The implication (3) $\Rightarrow $ (1) is Lemma 30.3.3. $\square$

Lemma 30.17.2. Let $R$ be a Noetherian ring. Let $f : Y \to X$ be a morphism of schemes proper over $R$. Let $\mathcal{L}$ be an invertible $\mathcal{O}_ X$-module. Assume $f$ is finite and surjective. Then $\mathcal{L}$ is ample if and only if $f^*\mathcal{L}$ is ample.

Proof. The pullback of an ample invertible sheaf by a quasi-affine morphism is ample, see Morphisms, Lemma 29.37.7. This proves one of the implications as a finite morphism is affine by definition.

Assume that $f^*\mathcal{L}$ is ample. Let $P$ be the following property on coherent $\mathcal{O}_ X$-modules $\mathcal{F}$: there exists an $n_0$ such that $H^ p(X, \mathcal{F} \otimes \mathcal{L}^{\otimes n}) = 0$ for all $n \geq n_0$ and $p > 0$. We will prove that $P$ holds for any coherent $\mathcal{O}_ X$-module $\mathcal{F}$, which implies $\mathcal{L}$ is ample by Lemma 30.17.1. We are going to apply Lemma 30.12.8. Thus we have to verify (1), (2) and (3) of that lemma for $P$. Property (1) follows from the long exact cohomology sequence associated to a short exact sequence of sheaves and the fact that tensoring with an invertible sheaf is an exact functor. Property (2) follows since $H^ p(X, -)$ is an additive functor. To see (3) let $Z \subset X$ be an integral closed subscheme with generic point $\xi $. Let $\mathcal{F}$ be a coherent sheaf on $Y$ such that the support of $f_*\mathcal{F}$ is equal to $Z$ and $(f_*\mathcal{F})_\xi $ is annihilated by $\mathfrak m_\xi $, see Lemma 30.13.1. We claim that taking $\mathcal{G} = f_*\mathcal{F}$ works. We only have to verify part (3)(c) of Lemma 30.12.8. Hence assume that $\mathcal{J} \subset \mathcal{O}_ X$ is a quasi-coherent sheaf of ideals such that $\mathcal{J}_\xi = \mathcal{O}_{X, \xi }$. A finite morphism is affine hence by Lemma 30.13.2 we see that $\mathcal{J}\mathcal{G} = f_*(f^{-1}\mathcal{J}\mathcal{F})$. Also, as pointed out in the proof of Lemma 30.13.2 the sheaf $f^{-1}\mathcal{J}\mathcal{F}$ is a coherent $\mathcal{O}_ Y$-module. As $\mathcal{L}$ is ample we see from Lemma 30.17.1 that there exists an $n_0$ such that

\[ H^ p(Y, f^{-1}\mathcal{J}\mathcal{F} \otimes _{\mathcal{O}_ Y} f^*\mathcal{L}^{\otimes n}) = 0, \]

for $n \geq n_0$ and $p > 0$. Since $f$ is finite, hence affine, we see that

\begin{align*} H^ p(X, \mathcal{J}\mathcal{G} \otimes _{\mathcal{O}_ X} \mathcal{L}^{\otimes n}) & = H^ p(X, f_*(f^{-1}\mathcal{J}\mathcal{F}) \otimes _{\mathcal{O}_ X} \mathcal{L}^{\otimes n}) \\ & = H^ p(X, f_*(f^{-1}\mathcal{J}\mathcal{F} \otimes _{\mathcal{O}_ Y} f^*\mathcal{L}^{\otimes n})) \\ & = H^ p(Y, f^{-1}\mathcal{J}\mathcal{F} \otimes _{\mathcal{O}_ Y} f^*\mathcal{L}^{\otimes n}) = 0 \end{align*}

Here we have used the projection formula (Cohomology, Lemma 20.51.2) and Lemma 30.2.4. Hence the quasi-coherent subsheaf $\mathcal{G}' = \mathcal{J}\mathcal{G}$ satisfies $P$. This verifies property (3)(c) of Lemma 30.12.8 as desired. $\square$

Cohomology is functorial. In particular, given a ringed space $X$, an invertible $\mathcal{O}_ X$-module $\mathcal{L}$, a section $s \in \Gamma (X, \mathcal{L})$ we get maps

\[ H^ p(X, \mathcal{F}) \longrightarrow H^ p(X, \mathcal{F} \otimes _{\mathcal{O}_ X} \mathcal{L}), \quad \xi \longmapsto s\xi \]

induced by the map $\mathcal{F} \to \mathcal{F} \otimes _{\mathcal{O}_ X} \mathcal{L}$ which is multiplication by $s$. We set $\Gamma _*(X, \mathcal{L}) = \bigoplus _{n \geq 0} \Gamma (X, \mathcal{L}^{\otimes n})$ as a graded ring, see Modules, Definition 17.24.7. Given a sheaf of $\mathcal{O}_ X$-modules $\mathcal{F}$ and an integer $p \geq 0$ we set

\[ H^ p_*(X, \mathcal{L}, \mathcal{F}) = \bigoplus \nolimits _{n \in \mathbf{Z}} H^ p(X, \mathcal{F} \otimes _{\mathcal{O}_ X} \mathcal{L}^{\otimes n}) \]

This is a graded $\Gamma _*(X, \mathcal{L})$-module by the multiplication defined above. Warning: the notation $H^ p_*(X, \mathcal{L}, \mathcal{F})$ is nonstandard.

Lemma 30.17.3. Let $X$ be a scheme. Let $\mathcal{L}$ be an invertible sheaf on $X$. Let $s \in \Gamma (X, \mathcal{L})$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module. If $X$ is quasi-compact and quasi-separated, the canonical map

\[ H^ p_*(X, \mathcal{L}, \mathcal{F})_{(s)} \longrightarrow H^ p(X_ s, \mathcal{F}) \]

which maps $\xi /s^ n$ to $s^{-n}\xi $ is an isomorphism.

Proof. Note that for $p = 0$ this is Properties, Lemma 28.17.2. We will prove the statement using the induction principle (Lemma 30.4.1) where for $U \subset X$ quasi-compact open we let $P(U)$ be the property: for all $p \geq 0$ the map

\[ H^ p_*(U, \mathcal{L}, \mathcal{F})_{(s)} \longrightarrow H^ p(U_ s, \mathcal{F}) \]

is an isomorphism.

If $U$ is affine, then both sides of the arrow displayed above are zero for $p > 0$ by Lemma 30.2.2 and Properties, Lemma 28.26.4 and the statement is true. If $P$ is true for $U$, $V$, and $U \cap V$, then we can use the Mayer-Vietoris sequences (Cohomology, Lemma 20.8.2) to obtain a map of long exact sequences

\[ \xymatrix{ H^{p - 1}_*(U \cap V, \mathcal{L}, \mathcal{F})_{(s)} \ar[r] \ar[d] & H^ p_*(U \cup V, \mathcal{L}, \mathcal{F})_{(s)} \ar[r] \ar[d] & H^ p_*(U, \mathcal{L}, \mathcal{F})_{(s)} \oplus H^ p_*(V, \mathcal{L}, \mathcal{F})_{(s)} \ar[d] \\ H^{p - 1}(U_ s \cap V_ s, \mathcal{F}) \ar[r]& H^ p(U_ s \cup V_ s, \mathcal{F}) \ar[r] & H^ p(U_ s, \mathcal{F}) \oplus H^ p(V_ s, \mathcal{F}) } \]

(only a snippet shown). Observe that $U_ s \cap V_ s = (U \cap V)_ s$ and that $U_ s \cup V_ s = (U \cup V)_ s$. Thus the left and right vertical maps are isomorphisms (as well as one more to the right and one more to the left which are not shown in the diagram). We conclude that $P(U \cup V)$ holds by the 5-lemma (Homology, Lemma 12.5.20). This finishes the proof. $\square$

Lemma 30.17.4. Let $X$ be a scheme. Let $\mathcal{L}$ be an invertible $\mathcal{O}_ X$-module. Let $s \in \Gamma (X, \mathcal{L})$ be a section. Assume that

  1. $X$ is quasi-compact and quasi-separated, and

  2. $X_ s$ is affine.

Then for every quasi-coherent $\mathcal{O}_ X$-module $\mathcal{F}$ and every $p > 0$ and all $\xi \in H^ p(X, \mathcal{F})$ there exists an $n \geq 0$ such that $s^ n\xi = 0$ in $H^ p(X, \mathcal{F} \otimes _{\mathcal{O}_ X} \mathcal{L}^{\otimes n})$.

Proof. Recall that $H^ p(X_ s, \mathcal{G})$ is zero for every quasi-coherent module $\mathcal{G}$ by Lemma 30.2.2. Hence the lemma follows from Lemma 30.17.3. $\square$

For a more general version of the following lemma see Limits, Lemma 32.11.4.

Lemma 30.17.5. Let $i : Z \to X$ be a closed immersion of Noetherian schemes inducing a homeomorphism of underlying topological spaces. Let $\mathcal{L}$ be an invertible sheaf on $X$. Then $i^*\mathcal{L}$ is ample on $Z$, if and only if $\mathcal{L}$ is ample on $X$.

Proof. If $\mathcal{L}$ is ample, then $i^*\mathcal{L}$ is ample for example by Morphisms, Lemma 29.37.7. Assume $i^*\mathcal{L}$ is ample. We have to show that $\mathcal{L}$ is ample on $X$. Let $\mathcal{I} \subset \mathcal{O}_ X$ be the coherent sheaf of ideals cutting out the closed subscheme $Z$. Since $i(Z) = X$ set theoretically we see that $\mathcal{I}^ n = 0$ for some $n$ by Lemma 30.10.2. Consider the sequence

\[ X = Z_ n \supset Z_{n - 1} \supset Z_{n - 2} \supset \ldots \supset Z_1 = Z \]

of closed subschemes cut out by $0 = \mathcal{I}^ n \subset \mathcal{I}^{n - 1} \subset \ldots \subset \mathcal{I}$. Then each of the closed immersions $Z_ i \to Z_{i - 1}$ is defined by a coherent sheaf of ideals of square zero. In this way we reduce to the case that $\mathcal{I}^2 = 0$.

Consider the short exact sequence

\[ 0 \to \mathcal{I} \to \mathcal{O}_ X \to i_*\mathcal{O}_ Z \to 0 \]

of quasi-coherent $\mathcal{O}_ X$-modules. Tensoring with $\mathcal{L}^{\otimes n}$ we obtain short exact sequences

30.17.5.1
\begin{equation} \label{coherent-equation-ses} 0 \to \mathcal{I} \otimes _{\mathcal{O}_ X} \mathcal{L}^{\otimes n} \to \mathcal{L}^{\otimes n} \to i_*i^*\mathcal{L}^{\otimes n} \to 0 \end{equation}

As $\mathcal{I}^2 = 0$, we can use Morphisms, Lemma 29.4.1 to think of $\mathcal{I}$ as a quasi-coherent $\mathcal{O}_ Z$-module and then $\mathcal{I} \otimes _{\mathcal{O}_ X} \mathcal{L}^{\otimes n} = \mathcal{I} \otimes _{\mathcal{O}_ Z} i^*\mathcal{L}^{\otimes n}$ with obvious abuse of notation. Moreover, the cohomology of this sheaf over $Z$ is canonically the same as the cohomology of this sheaf over $X$ (as $i$ is a homeomorphism).

Let $x \in X$ be a point and denote $z \in Z$ the corresponding point. Because $i^*\mathcal{L}$ is ample there exists an $n$ and a section $s \in \Gamma (Z, i^*\mathcal{L}^{\otimes n})$ with $z \in Z_ s$ and with $Z_ s$ affine. The obstruction to lifting $s$ to a section of $\mathcal{L}^{\otimes n}$ over $X$ is the boundary

\[ \xi = \partial s \in H^1(X, \mathcal{I} \otimes _{\mathcal{O}_ X} \mathcal{L}^{\otimes n}) = H^1(Z, \mathcal{I} \otimes _{\mathcal{O}_ Z} i^*\mathcal{L}^{\otimes n}) \]

coming from the short exact sequence of sheaves (30.17.5.1). If we replace $s$ by $s^{e + 1}$ then $\xi $ is replaced by $\partial (s^{e + 1}) = (e + 1) s^ e \xi $ in $H^1(Z, \mathcal{I} \otimes _{\mathcal{O}_ Z} i^*\mathcal{L}^{\otimes (e + 1)n})$ because the boundary map for

\[ 0 \to \bigoplus \nolimits _{m \geq 0} \mathcal{I} \otimes _{\mathcal{O}_ X} \mathcal{L}^{\otimes m} \to \bigoplus \nolimits _{m \geq 0} \mathcal{L}^{\otimes m} \to \bigoplus \nolimits _{m \geq 0} i_*i^*\mathcal{L}^{\otimes m} \to 0 \]

is a derivation by Cohomology, Lemma 20.25.5. By Lemma 30.17.4 we see that $s^ e \xi $ is zero for $e$ large enough. Hence, after replacing $s$ by a power, we can assume $s$ is the image of a section $s' \in \Gamma (X, \mathcal{L}^{\otimes n})$. Then $X_{s'}$ is an open subscheme and $Z_ s \to X_{s'}$ is a surjective closed immersion of Noetherian schemes with $Z_ s$ affine. Hence $X_ s$ is affine by Lemma 30.13.3 and we conclude that $\mathcal{L}$ is ample. $\square$

For a more general version of the following lemma see Limits, Lemma 32.11.5.

Lemma 30.17.6. Let $i : Z \to X$ be a closed immersion of Noetherian schemes inducing a homeomorphism of underlying topological spaces. Then $X$ is quasi-affine if and only if $Z$ is quasi-affine.

Proof. Recall that a scheme is quasi-affine if and only if the structure sheaf is ample, see Properties, Lemma 28.27.1. Hence if $Z$ is quasi-affine, then $\mathcal{O}_ Z$ is ample, hence $\mathcal{O}_ X$ is ample by Lemma 30.17.5, hence $X$ is quasi-affine. A proof of the converse, which can also be seen in an elementary way, is gotten by reading the argument just given backwards. $\square$

Lemma 30.17.7. Let $X$ be a scheme. Let $\mathcal{L}$ be an ample invertible $\mathcal{O}_ X$-module. Let $n_0$ be an integer. If $H^ p(X, \mathcal{L}^{\otimes -n}) = 0$ for $n \geq n_0$ and $p > 0$, then $X$ is affine.

Proof. We claim $H^ p(X, \mathcal{F}) = 0$ for every quasi-coherent $\mathcal{O}_ X$-module and $p > 0$. Since $X$ is quasi-compact by Properties, Definition 28.26.1 the claim finishes the proof by Lemma 30.3.1. The scheme $X$ is separated by Properties, Lemma 28.26.8. Say $X$ is covered by $e + 1$ affine opens. Then $H^ p(X, \mathcal{F}) = 0$ for $p > e$, see Lemma 30.4.2. Thus we may use descending induction on $p$ to prove the claim. Writing $\mathcal{F}$ as a filtered colimit of finite type quasi-coherent modules (Properties, Lemma 28.22.3) and using Cohomology, Lemma 20.19.1 we may assume $\mathcal{F}$ is of finite type. Then we can choose $n > n_0$ such that $\mathcal{F} \otimes _{\mathcal{O}_ X} \mathcal{L}^{\otimes n}$ is globally generated, see Properties, Proposition 28.26.13. This means there is a short exact sequence

\[ 0 \to \mathcal{F}' \to \bigoplus \nolimits _{i \in I} \mathcal{L}^{\otimes -n} \to \mathcal{F} \to 0 \]

for some set $I$ (in fact we can choose $I$ finite). By induction hypothesis we have $H^{p + 1}(X, \mathcal{F}') = 0$ and by assumption (combined with the already used commutation of cohomology with colimits) we have $H^ p(X, \bigoplus _{i \in I} \mathcal{L}^{\otimes -n}) = 0$. From the long exact cohomology sequence we conclude that $H^ p(X, \mathcal{F}) = 0$ as desired. $\square$

Lemma 30.17.8. Let $X$ be a quasi-affine scheme. If $H^ p(X, \mathcal{O}_ X) = 0$ for $p > 0$, then $X$ is affine.

Proof. Since $\mathcal{O}_ X$ is ample by Properties, Lemma 28.27.1 this follows from Lemma 30.17.7. $\square$


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