71.15 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 71.15.1. 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
\mathcal{L} is ample on X/R (Definition 71.14.1),
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 69.16.9.
\square
Lemma 71.15.2. 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 69.11.2. 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 69.22 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 69.12.8. 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 69.8.3 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
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
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 69.22.5) 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 69.20.3 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 69.22.4 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 71.15.3. (For a more general version see Descent on Spaces, Lemma 74.13.2). 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 71.14.1).
Proof.
Pick d_0 as in Lemma 71.15.2 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 69.23.2 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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