The Stacks project

Proof. We check conditions (1) and (2) of Categories, Definition 4.35.1.

Condition (1). Let $(X \to S, \mathcal{L})$ be an object of $\mathcal{P}\! \mathit{olarized}$ and let $(X' \to S') \to (X \to S)$ be a morphism of $\mathcal{S}\! \mathit{paces}'_{fp, flat, proper}$. Then we let $\mathcal{L}'$ be the pullback of $\mathcal{L}$ to $X'$. Observe that $X, S, S'$ are schemes, hence $X'$ is a scheme as well (as the fibre product of schemes). Then $\mathcal{L}'$ is ample on $X'/S'$ by Morphisms, Lemma 29.37.9. In this way we obtain a morphism $(X' \to S', \mathcal{L}') \to (X \to S, \mathcal{L})$ lying over $(X' \to S') \to (X \to S)$.

Condition (2). Consider morphisms $(f, g, \varphi ) : (X' \to S', \mathcal{L}') \to (X \to S, \mathcal{L})$ and $(a, b, \psi ) : (Y \to T, \mathcal{N}) \to (X \to S, \mathcal{L})$ of $\mathcal{P}\! \mathit{olarized}$. Given a morphism $(k, h) : (Y \to T) \to (X' \to S')$ of $\mathcal{S}\! \mathit{paces}'_{fp, flat, proper}$ with $(f, g) \circ (k, h) = (a, b)$ we have to show there is a unique morphism $(k, h, \chi ) : (Y \to T, \mathcal{N}) \to (X' \to S', \mathcal{L}')$ of $\mathcal{P}\! \mathit{olarized}$ such that $(f, g, \varphi ) \circ (k, h, \chi ) = (a, b, \psi )$. We can just take

This proves condition (2). A composition of functors defining fibred categories defines a fibred category, see Categories, Lemma 4.33.12. This we see that $\mathcal{P}\! \mathit{olarized}$ is fibred in groupoids over $\mathit{Sch}_{fppf}$ (strictly speaking we should check the fibre categories are groupoids and apply Categories, Lemma 4.35.2). $\square$

Proof. We prove $\mathcal{P}\! \mathit{olarized}$ is a stack in groupoids over $\mathcal{S}\! \mathit{paces}'_{fp, flat, proper}$ by checking conditions (1), (2), and (3) of Stacks, Definition 8.5.1. We have already seen (1) in Lemma 99.14.2.

A covering of $\mathcal{S}\! \mathit{paces}'_{fp, flat, proper}$ comes about in the following manner: Let $X \to S$ be an object of $\mathcal{S}\! \mathit{paces}'_{fp, flat, proper}$. Suppose that $\{ S_ i \to S\} _{i \in I}$ is a covering of $\mathit{Sch}_{fppf}$. Set $X_ i = S_ i \times _ S X$. Then $\{ (X_ i \to S_ i) \to (X \to S)\} _{i \in I}$ is a covering of $\mathcal{S}\! \mathit{paces}'_{fp, flat, proper}$ and every covering of $\mathcal{S}\! \mathit{paces}'_{fp, flat, proper}$ is isomorphic to one of these. Set $S_{ij} = S_ i \times _ S S_ j$ and $X_{ij} = S_{ij} \times _ S X$ so that $(X_{ij} \to S_{ij}) = (X_ i \to S_ i) \times _{(X \to S)} (X_ j \to S_ j)$. Next, suppose that $\mathcal{L}, \mathcal{N}$ are ample invertible sheaves on $X/S$ so that $(X \to S, \mathcal{L})$ and $(X \to S, \mathcal{N})$ are two objects of $\mathcal{P}\! \mathit{olarized}$ over the object $(X \to S)$. To check descent for morphisms, we assume we have morphisms $(\text{id}, \text{id}, \varphi _ i)$ from $(X_ i \to S_ i, \mathcal{L}|_{X_ i})$ to $(X_ i \to S_ i, \mathcal{N}|_{X_ i})$ whose base changes to morphisms from $(X_{ij} \to S_{ij}, \mathcal{L}|_{X_{ij}})$ to $(X_{ij} \to S_{ij}, \mathcal{N}|_{X_{ij}})$ agree. Then $\varphi _ i : \mathcal{L}|_{X_ i} \to \mathcal{N}|_{X_ i}$ are isomorphisms of invertible modules over $X_ i$ such that $\varphi _ i$ and $\varphi _ j$ restrict to the same isomorphisms over $X_{ij}$. By descent for quasi-coherent sheaves (Descent on Spaces, Proposition 74.4.1) we obtain a unique isomorphism $\varphi : \mathcal{L} \to \mathcal{N}$ whose restriction to $X_ i$ recovers $\varphi _ i$.

Decent for objects is proved in exactly the same manner. Namely, suppose that $\{ (X_ i \to S_ i) \to (X \to S)\} _{i \in I}$ is a covering of $\mathcal{S}\! \mathit{paces}'_{fp, flat, proper}$ as above. Suppose we have objects $(X_ i \to S_ i, \mathcal{L}_ i)$ of $\mathcal{P}\! \mathit{olarized}$ lying over $(X_ i \to S_ i)$ and a descent datum

satisfying the obvious cocycle condition over $(X_{ijk} \to S_{ijk})$ for every triple of indices. Then by descent for quasi-coherent sheaves (Descent on Spaces, Proposition 74.4.1) we obtain a unique invertible $\mathcal{O}_ X$-module $\mathcal{L}$ and isomorphisms $\mathcal{L}|_{X_ i} \to \mathcal{L}_ i$ recovering the descent datum $\varphi _{ij}$. To show that $(X \to S, \mathcal{L})$ is an object of $\mathcal{P}\! \mathit{olarized}$ we have to prove that $\mathcal{L}$ is ample. This follows from Descent on Spaces, Lemma 74.13.1.

Since we already have seen that $\mathcal{S}\! \mathit{paces}'_{fp, flat, proper}$ is a stack in groupoids over $\mathit{Sch}_{fppf}$ (Lemma 99.13.3) it now follows formally that $\mathcal{P}\! \mathit{olarized}$ is a stack in groupoids over $\mathit{Sch}_{fppf}$. See Stacks, Lemma 8.10.6. $\square$

Proof. Omitted. Hints: Argue exactly as in the proof of Lemma 99.13.4 and use Descent on Spaces, Proposition 74.4.1 to descent the invertible sheaf in the construction of the quasi-inverse functor. The relative ampleness property descends by Descent on Spaces, Lemma 74.13.1. $\square$

Proof. By Lemmas 99.13.3 and 99.14.3 the statement makes sense. To prove it, we choose a scheme $S$ and an object $\xi = (X \to S)$ of $\mathcal{S}\! \mathit{paces}'_{fp, flat, proper}$ over $S$. We have to show that

is an algebraic stack over $S$. Observe that an object of $\mathcal{X}$ is given by a pair $(T/S, \mathcal{L})$ where $T$ is a scheme over $S$ and $\mathcal{L}$ is an invertible $\mathcal{O}_{X_ T}$-module which is ample on $X_ T/T$. Morphisms are defined in the obvious manner. In particular, we see immediately that we have an inclusion

of categories over $(\mathit{Sch}/S)_{fppf}$, inducing equality on morphism sets. Since $\mathcal{P}\! \mathit{ic}_{X/S}$ is an algebraic stack by Proposition 99.10.2 it suffices to show that the inclusion above is representable by open immersions. This is exactly the content of Descent on Spaces, Lemma 74.13.2. $\square$

Proof. This is a formal consequence of Lemmas 99.14.6 and 99.13.2. See Criteria for Representability, Lemma 97.8.4. $\square$

Proof. Let $I$ be a directed set and let $(A_ i, \varphi _{ii'})$ be a system of rings over $I$. Set $S = \mathop{\mathrm{Spec}}(A)$ and $S_ i = \mathop{\mathrm{Spec}}(A_ i)$. We have to show that on fibre categories we have

We know that the category of schemes of finite presentation over $S$ is the colimit of the category of schemes of finite presentation over $S_ i$, see Limits, Lemma 32.10.1. Moreover, given $X_ i \to S_ i$ of finite presentation, with limit $X \to S$, then the category of invertible $\mathcal{O}_ X$-modules $\mathcal{L}$ is the colimit of the categories of invertible $\mathcal{O}_{X_ i}$-modules $\mathcal{L}_ i$, see Limits, Lemma 32.10.2 and 32.10.3. If $X \to S$ is proper and flat, then for sufficiently large $i$ the morphism $X_ i \to S_ i$ is proper and flat too, see Limits, Lemmas 32.13.1 and 32.8.7. Finally, if $\mathcal{L}$ is ample on $X$ then $\mathcal{L}_ i$ is ample on $X_ i$ for $i$ sufficiently large, see Limits, Lemma 32.4.15. Putting everything together finishes the proof. $\square$

Proof. By More on Morphisms, Lemma 37.14.6 there is an equivalence

where $\textit{flat-lfp}_ S$ signifies the category of schemes flat and locally of finite presentation over $S$. Let $X'/S'$ on the left hand side correspond to the triple $(X/S, Y'/T', \varphi )$ on the right hand side. Set $Y = T \times _{T'} Y'$ which is isomorphic with $T \times _ S X$ via $\varphi $. Then More on Morphisms, Lemma 37.14.5 shows that we have an equivalence

where $\textit{QCoh-flat}_{X/S}$ signifies the category of quasi-coherent $\mathcal{O}_ X$-modules flat over $S$. Since $X \to S$, $Y \to T$, $X' \to S'$, $Y' \to T'$ are flat, this will in particular apply to invertible modules to give an equivalence of categories

where $\textit{Pic}(X)$ signifies the category of invertible $\mathcal{O}_ X$-modules. There is a small point here: one has to show that if an object $\mathcal{F}'$ of $\textit{QCoh-flat}_{X'/S'}$ pulls back to invertible modules on $X$ and $Y'$, then $\mathcal{F}'$ is an invertible $\mathcal{O}_{X'}$-module. It follows from the cited lemma that $\mathcal{F}'$ is an $\mathcal{O}_{X'}$-module of finite presentation. By More on Morphisms, Lemma 37.16.7 it suffices to check the restriction of $\mathcal{F}'$ to fibres of $X' \to S'$ is invertible. But the fibres of $X' \to S'$ are the same as the fibres of $X \to S$ and hence these restrictions are invertible.

Having said the above we obtain an equivalence of categories if we drop the assumption (for the category of objects over $S$) that $X \to S$ be proper and the assumption that $\mathcal{L}$ be ample. Now it is clear that if $X' \to S'$ is proper, then $X \to S$ and $Y' \to T'$ are proper (Morphisms, Lemma 29.41.5). Conversely, if $X \to S$ and $Y' \to T'$ are proper, then $X' \to S'$ is proper by More on Morphisms, Lemma 37.3.3. Similarly, if $\mathcal{L}'$ is ample on $X'/S'$, then $\mathcal{L}'|_ X$ is ample on $X/S$ and $\mathcal{L}'|_{Y'}$ is ample on $Y'/T'$ (Morphisms, Lemma 29.37.9). Finally, if $\mathcal{L}'|_ X$ is ample on $X/S$ and $\mathcal{L}'|_{Y'}$ is ample on $Y'/T'$, then $\mathcal{L}'$ is ample on $X'/S'$ by More on Morphisms, Lemma 37.3.2. $\square$

Proof. The discussion in Artin's Axioms, Section 98.8 only applies to fields of finite type over the base scheme $\mathop{\mathrm{Spec}}(\mathbf{Z})$. Our stack satisfies (RS*) by Lemma 99.14.9 and we may apply Artin's Axioms, Lemma 98.21.2 to get the vector spaces $T_ x(k)$ and $\text{Inf}_ x(k)$ mentioned in (2). Moreover, in the finite type case these spaces agree with the ones mentioned in part (1) by Artin's Axioms, Remark 98.21.7. With this out of the way we can start the proof.

One proof is to use an argument as in the proof of Lemma 99.13.8; this would require us to develop a deformation theory for pairs consisting of a scheme and a quasi-coherent module. Another proof would be the use the result from Lemma 99.13.8, the algebraicity of $\mathcal{P}\! \mathit{olarized}\to \mathcal{S}\! \mathit{paces}'_{fp, flat, proper}$, and a computation of the deformation space of an invertible module. However, what we will do instead is to translate the question into a deformation question on graded $k$-algebras and deduce the result that way.

Let $\mathcal{C}_ k$ be the category of Artinian local $k$-algebras $A$ with residue field $k$. We get a predeformation category $p : \mathcal{F} \to \mathcal{C}_ k$ from our object $x$ of $\mathcal{X}$ over $k$, see Artin's Axioms, Section 98.3. Thus $\mathcal{F}(A)$ is the category of triples $(X_ A, \mathcal{L}_ A, \alpha )$, where $(X_ A, \mathcal{L}_ A)$ is an object of $\mathcal{P}\! \mathit{olarized}$ over $A$ and $\alpha $ is an isomorphism $(X_ A, \mathcal{L}_ A) \times _{\mathop{\mathrm{Spec}}(A)} \mathop{\mathrm{Spec}}(k) \cong (X, \mathcal{L})$. On the other hand, let $q : \mathcal{G} \to \mathcal{C}_ k$ be the category cofibred in groupoids defined in Deformation Problems, Example 93.7.1. Choose $d_0 \gg 0$ (we'll see below how large). Let $P$ be the graded $k$-algebra

Then $y = (k, P)$ is an object of $\mathcal{G}(k)$. Let $\mathcal{G}_ y$ be the predeformation category of Formal Deformation Theory, Remark 90.6.4. Given $(X_ A, \mathcal{F}_ A, \alpha )$ as above we set

The isomorphism $\alpha $ induces a map $\beta : Q \to P$. By deformation theory of projective schemes (More on Morphisms, Lemma 37.10.6) we obtain a $1$-morphism

of categories cofibred in groupoids over $\mathcal{C}_ k$. In fact, this functor is an equivalence with quasi-inverse given by $Q \mapsto \underline{\text{Proj}}_ A(Q)$. Namely, the scheme $X_ A = \underline{\text{Proj}}_ A(Q)$ is flat over $A$ by Divisors, Lemma 31.30.6. Set $\mathcal{L}_ A = \mathcal{O}_{X_ A}(1)$; this is flat over $A$ by the same lemma. We get an isomorphism $(X_ A, \mathcal{L}_ A) \times _{\mathop{\mathrm{Spec}}(A)} \mathop{\mathrm{Spec}}(k) = (X, \mathcal{L})$ from $\beta $. Then we can deduce all the desired properties of the pair $(X_ A, \mathcal{L}_ A)$ from the corresponding properties of $(X, \mathcal{L})$ using the techniques in More on Morphisms, Sections 37.3 and 37.10. Some details omitted.

In conclusion, we see that $T\mathcal{F} = T\mathcal{G}_ y = T_ y\mathcal{G}$ and $\text{Inf}(\mathcal{F}) = \text{Inf}_ y(\mathcal{G})$. These vector spaces are finite dimensional by Deformation Problems, Lemma 93.7.3 and the proof is complete. $\square$

Proof. Choose $d_0$ for $X_1 \to S_1$ and $\mathcal{L}_1$ as in More on Morphisms, Lemma 37.10.6. For any $n \geq 1$ set

By the lemma each $A_ n$ is a finitely presented graded $R_ n$-algebra whose homogeneous parts $(A_ n)_ d$ are finite projective $R_ n$-modules such that $X_ n = \text{Proj}(A_ n)$ and $\mathcal{L}_ n = \mathcal{O}_{\text{Proj}(A_ n)}(1)$. The lemma also guarantees that the maps

induce isomorphisms $A_ n = A_ m \otimes _{R_ m} R_ n$ for $n \leq m$. We set

By More on Algebra, Lemma 15.13.4 we see that $B_ d$ is a finite projective $R$-module and that $B \otimes _ R R_ n = A_ n$. Thus the scheme

is flat over $S$ and $\mathcal{L}$ is a quasi-coherent $\mathcal{O}_ X$-module flat over $S$, see Divisors, Lemma 31.30.6. Because formation of Proj commutes with base change (Constructions, Lemma 27.11.6) we obtain canonical isomorphisms

compatible with the transition maps of the system. Thus we may think of $X_1 \subset X$ as a closed subscheme. Below we will show that $B$ is of finite presentation over $R$. By Divisors, Lemmas 31.30.4 and 31.30.7 this implies that $X \to S$ is of finite presentation and proper and that $\mathcal{L} = \mathcal{O}_ X(1)$ is of finite presentation as an $\mathcal{O}_ X$-module. Since the restriction of $\mathcal{L}$ to the base change $X_1 \to S_1$ is invertible, we see from More on Morphisms, Lemma 37.16.8 that $\mathcal{L}$ is invertible on an open neighbourhood of $X_1$ in $X$. Since $X \to S$ is closed and since $\mathop{\mathrm{Ker}}(R \to R_1)$ is contained in the Jacobson radical (More on Algebra, Lemma 15.11.3) we see that any open neighbourhood of $X_1$ in $X$ is equal to $X$. Thus $\mathcal{L}$ is invertible. Finally, the set of points in $S$ where $\mathcal{L}$ is ample on the fibre is open in $S$ (More on Morphisms, Lemma 37.50.3) and contains $S_1$ hence equals $S$. Thus $X \to S$ and $\mathcal{L}$ have all the properties required of them in the statement of the lemma.

We prove the claim above. Choose a presentation $A_1 = R_1[X_1, \ldots , X_ s]/(F_1, \ldots , F_ t)$ where $X_ i$ are variables having degrees $d_ i$ and $F_ j$ are homogeneous polynomials in $X_ i$ of degree $e_ j$. Then we can choose a map

lifting the map $R_1[X_1, \ldots , X_ s] \to A_1$. Since each $B_ d$ is finite projective over $R$ we conclude from Nakayama's lemma (Algebra, Lemma 10.20.1 using again that $\mathop{\mathrm{Ker}}(R \to R_1)$ is contained in the Jacobson radical of $R$) that $\Psi $ is surjective. Since $- \otimes _ R R_1$ is right exact we can find $G_1, \ldots , G_ t \in \mathop{\mathrm{Ker}}(\Psi )$ mapping to $F_1, \ldots , F_ t$ in $R_1[X_1, \ldots , X_ s]$. Observe that $\mathop{\mathrm{Ker}}(\Psi )_ d$ is a finite projective $R$-module for all $d \geq 0$ as the kernel of the surjection $R[X_1, \ldots , X_ s]_ d \to B_ d$ of finite projective $R$-modules. We conclude from Nakayama's lemma once more that $\mathop{\mathrm{Ker}}(\Psi )$ is generated by $G_1, \ldots , G_ t$. $\square$

Proof. For definitions of the notions in the lemma, please see Artin's Axioms, Section 98.9. From the definitions we see the lemma follows immediately from the more general Lemma 99.14.11. $\square$

Proof. This follows from Artin's Axioms, Lemma 98.20.3 and Lemmas 99.14.7, 99.14.9, 99.14.8, and 99.14.11. For the “usual” proof of this fact, please see the discussion in the remark following this proof. $\square$

Proof. The absolute case follows from Artin's Axioms, Lemma 98.17.1 and Lemmas 99.14.7, 99.14.9, 99.14.8, 99.14.12, and 99.14.13. The case over $B$ follows from this, the description of $B\textit{-Polarized}$ as a $2$-fibre product in Remark 99.14.5, and the fact that algebraic stacks have $2$-fibre products, see Algebraic Stacks, Lemma 94.14.3. $\square$