70.15 Relative ampleness and cohomology

This section contains some results related to the results in Cohomology of Schemes, Sections 30.21 and 30.17.

The following lemma is just an example of what we can do.

Lemma 70.15.1. Let $R$ be a Noetherian ring. Let $X$ be an algebraic space over $R$ such that the structure morphism $f : X \to \mathop{\mathrm{Spec}}(R)$ is proper. Let $\mathcal{L}$ be an invertible $\mathcal{O}_ X$-module. The following are equivalent

1. $\mathcal{L}$ is ample on $X/R$ (Definition 70.14.1),

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 $p > 0$.

Proof. The implication (1) $\Rightarrow$ (2) follows from Cohomology of Schemes, Lemma 30.16.1 because assumption (1) implies that $X$ is a scheme. The implication (2) $\Rightarrow$ (1) is Cohomology of Spaces, Lemma 68.16.9. $\square$

Lemma 70.15.2. Let $Y$ be a Noetherian scheme. Let $X$ be an algebraic space over $Y$ such that the structure morphism $f : X \to Y$ is proper. Let $\mathcal{L}$ be an invertible $\mathcal{O}_ X$-module. Let $\mathcal{F}$ be a coherent $\mathcal{O}_ X$-module. Let $y \in Y$ be a point such that $X_ y$ is a scheme and $\mathcal{L}_ y$ is ample on $X_ y$. Then there exists a $d_0$ such that for all $d \geq d_0$ we have

$R^ pf_*(\mathcal{F} \otimes _{\mathcal{O}_ X} \mathcal{L}^{\otimes d})_ y = 0 \text{ for }p > 0$

and the map

$f_*(\mathcal{F} \otimes _{\mathcal{O}_ X} \mathcal{L}^{\otimes d})_ y \longrightarrow H^0(X_ y, \mathcal{F}_ y \otimes _{\mathcal{O}_{X_ y}} \mathcal{L}_ y^{\otimes d})$

is surjective.

Proof. Note that $\mathcal{O}_{Y, y}$ is a Noetherian local ring. Consider the canonical morphism $c : \mathop{\mathrm{Spec}}(\mathcal{O}_{Y, y}) \to Y$, see Schemes, Equation (26.13.1.1). This is a flat morphism as it identifies local rings. Denote momentarily $f' : X' \to \mathop{\mathrm{Spec}}(\mathcal{O}_{Y, y})$ the base change of $f$ to this local ring. We see that $c^*R^ pf_*\mathcal{F} = R^ pf'_*\mathcal{F}'$ by Cohomology of Spaces, Lemma 68.11.2. Moreover, the fibres $X_ y$ and $X'_ y$ are identified. Hence we may assume that $Y = \mathop{\mathrm{Spec}}(A)$ is the spectrum of a Noetherian local ring $(A, \mathfrak m, \kappa )$ and $y \in Y$ corresponds to $\mathfrak m$. In this case $R^ pf_*(\mathcal{F} \otimes _{\mathcal{O}_ X} \mathcal{L}^{\otimes d})_ y = H^ p(X, \mathcal{F} \otimes _{\mathcal{O}_ X} \mathcal{L}^{\otimes d})$ for all $p \geq 0$. Denote $f_ y : X_ y \to \mathop{\mathrm{Spec}}(\kappa )$ the projection.

Let $B = \text{Gr}_\mathfrak m(A) = \bigoplus _{n \geq 0} \mathfrak m^ n/\mathfrak m^{n + 1}$. Consider the sheaf $\mathcal{B} = f_ y^*\widetilde{B}$ of quasi-coherent graded $\mathcal{O}_{X_ y}$-algebras. We will use notation as in Cohomology of Spaces, Section 68.22 with $I$ replaced by $\mathfrak m$. Since $X_ y$ is the closed subspace of $X$ cut out by $\mathfrak m\mathcal{O}_ X$ we may think of $\mathfrak m^ n\mathcal{F}/\mathfrak m^{n + 1}\mathcal{F}$ as a coherent $\mathcal{O}_{X_ y}$-module, see Cohomology of Spaces, Lemma 68.12.8. Then $\bigoplus _{n \geq 0} \mathfrak m^ n\mathcal{F}/\mathfrak m^{n + 1}\mathcal{F}$ is a quasi-coherent graded $\mathcal{B}$-module of finite type because it is generated in degree zero over $\mathcal{B}$ abd because the degree zero part is $\mathcal{F}_ y = \mathcal{F}/\mathfrak m \mathcal{F}$ which is a coherent $\mathcal{O}_{X_ y}$-module. Hence by Cohomology of Schemes, Lemma 30.19.3 part (2) there exists a $d_0$ such that

$H^ p(X_ y, \mathfrak m^ n \mathcal{F}/ \mathfrak m^{n + 1}\mathcal{F} \otimes _{\mathcal{O}_{X_ y}} \mathcal{L}_ y^{\otimes d}) = 0$

for all $p > 0$, $d \geq d_0$, and $n \geq 0$. By Cohomology of Spaces, Lemma 68.8.3 this is the same as the statement that $H^ p(X, \mathfrak m^ n \mathcal{F}/ \mathfrak m^{n + 1}\mathcal{F} \otimes _{\mathcal{O}_ X} \mathcal{L}^{\otimes d}) = 0$ for all $p > 0$, $d \geq d_0$, and $n \geq 0$.

Consider the short exact sequences

$0 \to \mathfrak m^ n\mathcal{F}/\mathfrak m^{n + 1} \mathcal{F} \to \mathcal{F}/\mathfrak m^{n + 1} \mathcal{F} \to \mathcal{F}/\mathfrak m^ n \mathcal{F} \to 0$

of coherent $\mathcal{O}_ X$-modules. Tensoring with $\mathcal{L}^{\otimes d}$ is an exact functor and we obtain short exact sequences

$0 \to \mathfrak m^ n\mathcal{F}/\mathfrak m^{n + 1} \mathcal{F} \otimes _{\mathcal{O}_ X} \mathcal{L}^{\otimes d} \to \mathcal{F}/\mathfrak m^{n + 1} \mathcal{F} \otimes _{\mathcal{O}_ X} \mathcal{L}^{\otimes d} \to \mathcal{F}/\mathfrak m^ n \mathcal{F} \otimes _{\mathcal{O}_ X} \mathcal{L}^{\otimes d} \to 0$

Using the long exact cohomology sequence and the vanishing above we conclude (using induction) that

1. $H^ p(X, \mathcal{F}/\mathfrak m^ n \mathcal{F} \otimes _{\mathcal{O}_ X} \mathcal{L}^{\otimes d}) = 0$ for all $p > 0$, $d \geq d_0$, and $n \geq 0$, and

2. $H^0(X, \mathcal{F}/\mathfrak m^ n \mathcal{F} \otimes _{\mathcal{O}_ X} \mathcal{L}^{\otimes d}) \to H^0(X_ y, \mathcal{F}_ y \otimes _{\mathcal{O}_{X_ y}} \mathcal{L}_ y^{\otimes d})$ is surjective for all $d \geq d_0$ and $n \geq 1$.

By the theorem on formal functions (Cohomology of Spaces, Theorem 68.22.5) we find that the $\mathfrak m$-adic completion of $H^ p(X, \mathcal{F} \otimes _{\mathcal{O}_ X} \mathcal{L}^{\otimes d})$ is zero for all $d \geq d_0$ and $p > 0$. Since $H^ p(X, \mathcal{F} \otimes _{\mathcal{O}_ X} \mathcal{L}^{\otimes d})$ is a finite $A$-module by Cohomology of Spaces, Lemma 68.20.3 it follows from Nakayama's lemma (Algebra, Lemma 10.20.1) that $H^ p(X, \mathcal{F} \otimes _{\mathcal{O}_ X} \mathcal{L}^{\otimes d})$ is zero for all $d \geq d_0$ and $p > 0$. For $p = 0$ we deduce from Cohomology of Spaces, Lemma 68.22.4 part (3) that $H^0(X, \mathcal{F} \otimes _{\mathcal{O}_ X} \mathcal{L}^{\otimes d}) \to H^0(X_ y, \mathcal{F}_ y \otimes _{\mathcal{O}_{X_ y}} \mathcal{L}_ y^{\otimes d})$ is surjective, which gives the final statement of the lemma. $\square$

Lemma 70.15.3. (For a more general version see Descent on Spaces, Lemma 73.13.2). Let $Y$ be a Noetherian scheme. Let $X$ be an algebraic space over $Y$ such that the structure morphism $f : X \to Y$ is proper. Let $\mathcal{L}$ be an invertible $\mathcal{O}_ X$-module. Let $y \in Y$ be a point such that $X_ y$ is a scheme and $\mathcal{L}_ y$ is ample on $X_ y$. Then there is an open neighbourhood $V \subset Y$ of $y$ such that $\mathcal{L}|_{f^{-1}(V)}$ is ample on $f^{-1}(V)/V$ (as in Definition 70.14.1).

Proof. Pick $d_0$ as in Lemma 70.15.2 for $\mathcal{F} = \mathcal{O}_ X$. Pick $d \geq d_0$ so that we can find $r \geq 0$ and sections $s_{y, 0}, \ldots , s_{y, r} \in H^0(X_ y, \mathcal{L}_ y^{\otimes d})$ which define a closed immersion

$\varphi _ y = \varphi _{\mathcal{L}_ y^{\otimes d}, (s_{y, 0}, \ldots , s_{y, r})} : X_ y \to \mathbf{P}^ r_{\kappa (y)}.$

This is possible by Morphisms, Lemma 29.39.4 but we also use Morphisms, Lemma 29.41.7 to see that $\varphi _ y$ is a closed immersion and Constructions, Section 27.13 for the description of morphisms into projective space in terms of invertible sheaves and sections. By our choice of $d_0$, after replacing $Y$ by an open neighbourhood of $y$, we can choose $s_0, \ldots , s_ r \in H^0(X, \mathcal{L}^{\otimes d})$ mapping to $s_{y, 0}, \ldots , s_{y, r}$. Let $X_{s_ i} \subset X$ be the open subspace where $s_ i$ is a generator of $\mathcal{L}^{\otimes d}$. Since the $s_{y, i}$ generate $\mathcal{L}_ y^{\otimes d}$ we see that $|X_ y| \subset U = \bigcup |X_{s_ i}|$. Since $X \to Y$ is closed, we see that there is an open neighbourhood $y \in V \subset Y$ such that $|f|^{-1}(V) \subset U$. After replacing $Y$ by $V$ we may assume that the $s_ i$ generate $\mathcal{L}^{\otimes d}$. Thus we obtain a morphism

$\varphi = \varphi _{\mathcal{L}^{\otimes d}, (s_0, \ldots , s_ r)} : X \longrightarrow \mathbf{P}^ r_ Y$

with $\mathcal{L}^{\otimes d} \cong \varphi ^*\mathcal{O}_{\mathbf{P}^ r_ Y}(1)$ whose base change to $y$ gives $\varphi _ y$ (strictly speaking we need to write out a proof that the construction of morphisms into projective space given in Constructions, Section 27.13 also works to describe morphisms of algebraic spaces into projective space; we omit the details).

We will finish the proof by a sleight of hand; the “correct” proof proceeds by directly showing that $\varphi$ is a closed immersion after base changing to an open neighbourhood of $y$. Namely, by Cohomology of Spaces, Lemma 68.23.2 we see that $\varphi$ is a finite over an open neighbourhood of the fibre $\mathbf{P}^ r_{\kappa (y)}$ of $\mathbf{P}^ r_ Y \to Y$ above $y$. Using that $\mathbf{P}^ r_ Y \to Y$ is closed, after shrinking $Y$ we may assume that $\varphi$ is finite. In particular $X$ is a scheme. Then $\mathcal{L}^{\otimes d} \cong \varphi ^*\mathcal{O}_{\mathbf{P}^ r_ Y}(1)$ is ample by the very general Morphisms, Lemma 29.37.7. $\square$

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