The Stacks project

Lemma 37.10.6 (Deformations of projective schemes). Let $f : X \to S$ be a morphism of schemes which is proper, flat, and of finite presentation. Let $\mathcal{L}$ be $f$-ample. Assume $S$ is quasi-compact. There exists a $d_0 \geq 0$ such that for every cartesian diagram

\[ \vcenter { \xymatrix{ X \ar[r]_{i'} \ar[d]_ f & X' \ar[d]^{f'} \\ S \ar[r]^ i & S' } } \quad \text{and}\quad \begin{matrix} \text{invertible }\mathcal{O}_{X'}\text{-module} \\ \mathcal{L}'\text{ with }\mathcal{L} \cong (i')^*\mathcal{L}' \end{matrix} \]

where $S \subset S'$ is a thickening and $f'$ is proper, flat, of finite presentation we have

  1. $R^ p(f')_*(\mathcal{L}')^{\otimes d} = 0$ for all $p > 0$ and $d \geq d_0$,

  2. $\mathcal{A}'_ d = (f')_*(\mathcal{L}')^{\otimes d}$ is finite locally free for $d \geq d_0$,

  3. $\mathcal{A}' = \mathcal{O}_{S'} \oplus \bigoplus _{d \geq d_0} \mathcal{A}'_ d$ is a quasi-coherent $\mathcal{O}_{S'}$-algebra of finite presentation,

  4. there is a canonical isomorphism $r' : X' \to \underline{\text{Proj}}_{S'}(\mathcal{A}')$, and

  5. there is a canonical isomorphism $\theta ' : (r')^*\mathcal{O}_{\underline{\text{Proj}}_{S'}(\mathcal{A}')}(1) \to \mathcal{L}'$.

The construction of $\mathcal{A}'$, $r'$, $\theta '$ is functorial in the data $(X', S', i, i', f', \mathcal{L}')$.

Proof. We first describe the maps $r'$ and $\theta '$. Observe that $\mathcal{L}'$ is $f'$-ample, see Lemma 37.3.2. There is a canonical map of quasi-coherent graded $\mathcal{O}_{S'}$-algebras $\mathcal{A}' \to \bigoplus _{d \geq 0} (f')_*(\mathcal{L}')^{\otimes d}$ which is an isomorphism in degrees $\geq d_0$. Hence this induces an isomorphism on relative Proj compatible with the Serre twists of the structure sheaf, see Constructions, Lemma 27.18.4. Hence we get the morphism $r'$ by Morphisms, Lemma 29.37.4 (which in turn appeals to the construction given in Constructions, Lemma 27.19.1) and it is an isomorphism by Morphisms, Lemma 29.43.17. We get the map $\theta '$ from Constructions, Lemma 27.19.1. By Properties, Lemma 28.28.2 we find that $\theta '$ is an isomorphism (this also uses that the morphism $r'$ over affine opens of $S'$ is the same as the morphism from Properties, Lemma 28.26.9 as is explained in the proof of Morphisms, Lemma 29.43.17).

Assuming the vanishing and local freeness stated in parts (1) and (2), the functoriality of the construction can be seen as follows. Suppose that $h : T \to S'$ is a morphism of schemes, denote $f_ T : X'_ T \to T$ the base change of $f'$ and $\mathcal{L}_ T$ the pullback of $\mathcal{L}$ to $X'_ T$. By cohomology and base change (as formulated in Derived Categories of Schemes, Lemma 36.22.5 for example) we have the corresponding vanishing over $T$ and moreover $h^*\mathcal{A}'_ d = f_{T, *}\mathcal{L}_ T^{\otimes d}$ (and thus the local freeness of pushforwards as well as the finite generation of the corresponding graded $\mathcal{O}_ T$-algebra $\mathcal{A}_ T$). Hence the morphism $r_ T : X_ T \to \underline{\text{Proj}}_ T(\bigoplus f_{T, *}\mathcal{L}_ T^{\otimes d})$ is simply the base change of $r'$ to $T$ and the pullback of $\theta '$ is the map $\theta _ T$.

Having said all of the above, we see that it suffices to prove (1), (2), and (3). Pick $d_0$ such that $R^ pf_*\mathcal{L}^{\otimes d} = 0$ for all $d \geq d_0$ and $p > 0$, see Cohomology of Schemes, Lemma 30.16.1. We claim that $d_0$ works.

By cohomology and base change (Derived Categories of Schemes, Lemma 36.30.4) we see that $E'_ d = Rf'_*(\mathcal{L}')^{\otimes d}$ is a perfect object of $D(\mathcal{O}_{S'})$ and its formation commutes with arbitrary base change. In particular, $E_ d = Li^*E'_ d = Rf_*\mathcal{L}^{\otimes d}$. By Derived Categories of Schemes, Lemma 36.32.4 we see that for $d \geq d_0$ the complex $E_ d$ is isomorphic to the finite locally free $\mathcal{O}_ S$-module $f_*\mathcal{L}^{\otimes d}$ placed in cohomological degree $0$. Then by Derived Categories of Schemes, Lemma 36.31.3 we conclude that $E'_ d$ is isomorphic to a finite locally free module placed in cohomological degree $0$. Of course this means that $E'_ d = \mathcal{A}'_ d[0]$, that $R^ pf'_*(\mathcal{L}')^{\otimes d} = 0$ for $p > 0$, and that $\mathcal{A}'_ d$ is finite locally free. This proves (1) and (2).

The last thing we have to show is finite presentation of $\mathcal{A}'$ as a sheaf of $\mathcal{O}_{S'}$-algebras (this notion was introduced in Properties, Section 28.22). Let $U' = \mathop{\mathrm{Spec}}(R') \subset S'$ be an affine open. Then $A' = \mathcal{A}'(U')$ is a graded $R'$-algebra whose graded parts are finite projective $R'$-modules. We have to show that $A'$ is a finitely presented $R'$-algebra. We will prove this by reduction to the Noetherian case. Namely, we can find a finite type $\mathbf{Z}$-subalgebra $R'_0 \subset R'$ and a pair1 $(X'_0, \mathcal{L}'_0)$ over $R'_0$ whose base change is $(X'_{U'}, \mathcal{L}'|_{X'_{U'}})$, see Limits, Lemmas 32.10.2, 32.10.3, 32.13.1, 32.8.7, and 32.4.15. Cohomology of Schemes, Lemma 30.16.1 implies $A'_0 = \bigoplus _{d \geq 0} H^0(X'_0, (\mathcal{L}'_0)^{\otimes d})$ is a finitely generated graded $R'_0$-algebra and implies there exists a $d'_0$ such that $H^ p(X'_0, (\mathcal{L}'_0)^{\otimes d}) = 0$, $p > 0$ for $d \geq d'_0$. By the arguments given above applied to $X'_0 \to \mathop{\mathrm{Spec}}(R'_0)$ and $\mathcal{L}'_0$ we see that $(A'_0)_ d$ is a finite projective $R'_0$-module and that

\[ A'_ d = \mathcal{A}'_ d(U') = H^0(X'_{U'}, (\mathcal{L}')^{\otimes d}|_{X'_{U'}}) = H^0(X'_0, (\mathcal{L}'_0)^{\otimes d}) \otimes _{R'_0} R' = (A'_0)_ d \otimes _{R'_0} R' \]

for $d \geq d'_0$. Now a small twist in the argument is that we don't know that we can choose $d'_0$ equal to $d_0$2. To get around this we use the following sequence of arguments to finish the proof:

  1. The algebra $B = R'_0 \oplus \bigoplus _{d \geq \max (d_0, d'_0)} (A'_0)_ d$ is an $R'_0$-algebra of finite type: apply the Artin-Tate lemma to $B \subset A'_0$, see Algebra, Lemma 10.51.7.

  2. As $R'_0$ is Noetherian we see that $B$ is an $R'_0$-algebra of finite presentation.

  3. By right exactness of tensor product we see that $B \otimes _{R'_0} R'$ is an $R'$-algebra of finite presentation.

  4. By the displayed equalities this exactly says that $C = R' \oplus \bigoplus _{d \geq \max (d_0, d'_0)} A'_ d$ is an $R'$-algebra of finite presentation.

  5. The quotient $A'/C$ is the direct sum of the finite projective $R'$-modules $A'_ d$, $d_0 \leq d \leq \max (d_0, d'_0)$, hence finitely presented as $R'$-module.

  6. The quotient $A'/C$ is finitely presented as a $C$-module by Algebra, Lemma 10.6.4.

  7. Thus $A'$ is finitely presented as a $C$-module by Algebra, Lemma 10.5.3.

  8. By Algebra, Lemma 10.7.4 this implies $A'$ is finitely presented as a $C$-algebra.

  9. Finally, by Algebra, Lemma 10.6.2 applied to $R' \to C \to A'$ this implies $A'$ is finitely presented as an $R'$-algebra.

This finishes the proof. $\square$

[1] With the same properties as those enjoyed by $X' \to S'$ and $\mathcal{L}'$, i.e., $X'_0 \to \mathop{\mathrm{Spec}}(R'_0)$ is flat and proper and $\mathcal{L}'_0$ is ample.
[2] Actually, one can reduce to this case by doing more limit arguments.

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