We will use the example in this section in the proof that the stack of polarized proper schemes is an algebraic stack. For this reason we will consider commutative graded algebras whose homogeneous parts are finite projective modules (sometimes called “locally finite”).

Example 92.7.1 (Graded algebras). Let $\mathcal{F}$ be the category defined as follows

1. an object is a pair $(A, P)$ consisting of an object $A$ of $\mathcal{C}_\Lambda$ and a graded $A$-algebra $P$ such that $P_ d$ is a finite projective $A$-module for all $d \geq 0$, and

2. a morphism $(f, g) : (B, Q) \to (A, P)$ consists of a morphism $f : B \to A$ in $\mathcal{C}_\Lambda$ together with a map $g : Q \to P$ which is $f$-linear and induces an isomorpism $Q \otimes _{B, f} A \cong P$.

The functor $p : \mathcal{F} \to \mathcal{C}_\Lambda$ sends $(A, P)$ to $A$ and $(f, g)$ to $f$. It is clear that $p$ is cofibred in groupoids. Given a graded $k$-algebra $P$ with $\dim _ k(P_ d) < \infty$ for all $d \geq 0$, let $x_0 = (k, P)$ be the corresponding object of $\mathcal{F}(k)$. We set

$\mathcal{D}\! \mathit{ef}_ P = \mathcal{F}_{x_0}$

Lemma 92.7.2. Example 92.7.1 satisfies the Rim-Schlessinger condition (RS). In particular, $\mathcal{D}\! \mathit{ef}_ P$ is a deformation category for any graded $k$-algebra $P$.

Proof. Let $A_1 \to A$ and $A_2 \to A$ be morphisms of $\mathcal{C}_\Lambda$. Assume $A_2 \to A$ is surjective. According to Formal Deformation Theory, Lemma 89.16.4 it suffices to show that the functor $\mathcal{F}(A_1 \times _ A A_2) \to \mathcal{F}(A_1) \times _{\mathcal{F}(A)} \mathcal{F}(A_2)$ is an equivalence of categories.

Consider an object

$((A_1, P_1), (A_2, P_2), (\text{id}_ A, \varphi ))$

of the category $\mathcal{F}(A_1) \times _{\mathcal{F}(A)} \mathcal{F}(A_2)$. Then we consider $P_1 \times _\varphi P_2$. Since $\varphi : P_1 \otimes _{A_1} A \to P_2 \otimes _{A_2} A$ is an isomorphism of graded algebras, we see that the graded pieces of $P_1 \times _\varphi P_2$ are finite projective $A_1 \times _ A A_2$-modules, see proof of Lemma 92.4.2. Thus $P_1 \times _\varphi P_2$ is an object of $\mathcal{F}(A_1 \times _ A A_2)$. This construction determines a quasi-inverse to our functor and the proof is complete. $\square$

Lemma 92.7.3. In Example 92.7.1 let $P$ be a graded $k$-algebra. Then

$T\mathcal{D}\! \mathit{ef}_ P \quad \text{and}\quad \text{Inf}(\mathcal{D}\! \mathit{ef}_ P) = \text{Der}_ k(P, P)$

are finite dimensional if $P$ is finitely generated over $k$.

Proof. We first deal with the infinitesimal automorphisms. Let $Q = P \otimes _ k k[\epsilon ]$. Then an element of $\text{Inf}(\mathcal{D}\! \mathit{ef}_ P)$ is given by an automorphism $\gamma = \text{id} + \epsilon \delta : Q \to Q$ as above where now $\delta : P \to P$. The fact that $\gamma$ is graded implies that $\delta$ is homogeneous of degree $0$. The fact that $\gamma$ is $k$-linear implies that $\delta$ is $k$-linear. The fact that $\gamma$ is multiplicative implies that $\delta$ is a $k$-derivation. Conversely, given a $k$-derivation $\delta : P \to P$ homogeneous of degree $0$, we obtain an automorphism $\gamma = \text{id} + \epsilon \delta$ as above. Thus we see that

$\text{Inf}(\mathcal{D}\! \mathit{ef}_ P) = \text{Der}_ k(P, P)$

as predicted in the lemma. Clearly, if $P$ is generated in degrees $P_ i$, $0 \leq i \leq N$, then $\delta$ is determined by the linear maps $\delta _ i : P_ i \to P_ i$ for $0 \leq i \leq N$ and we see that

$\dim _ k \text{Der}_ k(P, P) < \infty$

as desired.

To finish the proof of the lemma we show that there is a finite dimensional deformation space. To do this we choose a presentation

$k[X_1, \ldots , X_ n]/(F_1, \ldots , F_ m) \longrightarrow P$

of graded $k$-algebras where $\deg (X_ i) = d_ i$ and $F_ j$ is homogeneous of degree $e_ j$. Let $Q$ be any graded $k[\epsilon ]$-algebra finite free in each degree which comes with an isomorphsm $\alpha : Q/\epsilon Q \to P$ so that $(Q, \alpha )$ defines an element of $T\mathcal{D}\! \mathit{ef}_ P$. Choose a homogeneous element $q_ i \in Q$ of degree $d_ i$ mapping to the image of $X_ i$ in $P$. Then we obtain

$k[\epsilon ][X_1, \ldots , X_ n] \longrightarrow Q,\quad X_ i \longmapsto q_ i$

and since $P = Q/\epsilon Q$ this map is surjective by Nakayama's lemma. A small diagram chase shows we can choose homogeneous elements $F_{\epsilon , j} \in k[\epsilon ][X_1, \ldots , X_ n]$ of degree $e_ j$ mapping to zero in $Q$ and mapping to $F_ j$ in $k[X_1, \ldots , X_ n]$. Then

$k[\epsilon ][X_1, \ldots , X_ n]/(F_{\epsilon , 1}, \ldots , F_{\epsilon , m}) \longrightarrow Q$

is a presentation of $Q$ by flatness of $Q$ over $k[\epsilon ]$. Write

$F_{\epsilon , j} = F_ j + \epsilon G_ j$

There is some ambiguity in the vector $(G_1, \ldots , G_ m)$. First, using different choices of $F_{\epsilon , j}$ we can modify $G_ j$ by an arbitrary element of degree $e_ j$ in the kernel of $k[X_1, \ldots , X_ n] \to P$. Hence, instead of $(G_1, \ldots , G_ m)$, we remember the element

$(g_1, \ldots , g_ m) \in P_{e_1} \oplus \ldots \oplus P_{e_ m}$

where $g_ j$ is the image of $G_ j$ in $P_{e_ j}$. Moreover, if we change our choice of $q_ i$ into $q_ i + \epsilon p_ i$ with $p_ i$ of degree $d_ i$ then a computation (omitted) shows that $g_ j$ changes into

$g_ j^{new} = g_ j - \sum \nolimits _{i = 1}^ n p_ i \partial F_ j / \partial X_ i$

We conclude that the isomorphism class of $Q$ is determined by the image of the vector $(G_1, \ldots , G_ m)$ in the $k$-vector space

$W = \mathop{\mathrm{Coker}}(P_{d_1} \oplus \ldots \oplus P_{d_ n} \xrightarrow {(\frac{\partial F_ j}{\partial X_ i})} P_{e_1} \oplus \ldots \oplus P_{e_ m})$

In this way we see that we obtain an injection

$T\mathcal{D}\! \mathit{ef}_ P \longrightarrow W$

Since $W$ visibly has finite dimension, we conclude that the lemma is true. $\square$

In Example 92.7.1 if $P$ is a finitely generated graded $k$-algebra, then $\mathcal{D}\! \mathit{ef}_ P$ admits a presentation by a smooth prorepresentable groupoid in functors over $\mathcal{C}_\Lambda$ and a fortiori has a (minimal) versal formal object. This follows from Lemmas 92.7.2 and 92.7.3 and the general discussion in Section 92.3.

Lemma 92.7.4. In Example 92.7.1 assume $P$ is a finitely generated graded $k$-algebra. Assume $\Lambda$ is a complete local ring with residue field $k$ (the classical case). Then the functor

$F : \mathcal{C}_\Lambda \longrightarrow \textit{Sets},\quad A \longmapsto \mathop{\mathrm{Ob}}\nolimits (\mathcal{D}\! \mathit{ef}_ P(A))/\cong$

of isomorphism classes of objects has a hull.

Proof. This follows immediately from Lemmas 92.7.2 and 92.7.3 and Formal Deformation Theory, Lemma 89.16.6 and Remark 89.15.7. $\square$

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