The Stacks project

96.13 The finite Hilbert stack of a point

Let $d \geq 1$ be an integer. In Examples of Stacks, Definition 94.18.2 we defined a stack in groupoids $\mathcal{H}_ d$. In this section we prove that $\mathcal{H}_ d$ is an algebraic stack. We will throughout assume that $S = \mathop{\mathrm{Spec}}(\mathbf{Z})$. The general case will follow from this by base change. Recall that the fibre category of $\mathcal{H}_ d$ over a scheme $T$ is the category of finite locally free morphisms $\pi : Z \to T$ of degree $d$. Instead of classifying these directly we first study the quasi-coherent sheaves of algebras $\pi _*\mathcal{O}_ Z$.

Let $R$ be a ring. Let us temporarily make the following definition: A free $d$-dimensional algebra over $R$ is given by a commutative $R$-algebra structure $m$ on $R^{\oplus d}$ such that $e_1 = (1, 0, \ldots , 0)$ is a unit1. We think of $m$ as an $R$-linear map

\[ m : R^{\oplus d} \otimes _ R R^{\oplus d} \longrightarrow R^{\oplus d} \]

such that $m(e_1, x) = m(x, e_1) = x$ and such that $m$ defines a commutative and associative ring structure. If we write $m(e_ i, e_ j) = \sum a_{ij}^ ke_ k$ then we see this boils down to the conditions

\[ \left\{ \begin{matrix} \sum _ l a_{ij}^ la_{lk}^ m = \sum _ l a_{il}^ ma_{jk}^ l & \forall i, j, k, m \\ a_{ij}^ k = a_{ji}^ k & \forall i, j, k \\ a_{i1}^ j = \delta _{ij} & \forall i, j \end{matrix} \right. \]

where $\delta _{ij}$ is the Kronecker $\delta $-function. OK, so let's define

\[ R_{univ} = \mathbf{Z}[a_{ij}^ k]/J \]

where the ideal $J$ is the ideal generated by the relations displayed above. Denote

\[ m_{univ} : R_{univ}^{\oplus d} \otimes _{R_{univ}} R_{univ}^{\oplus d} \longrightarrow R_{univ}^{\oplus d} \]

the free $d$-dimensional algebra $m$ over $R_{univ}$ whose structure constants are the classes of $a_{ij}^ k$ modulo $J$. Then it is clear that given any free $d$-dimensional algebra $m$ over a ring $R$ there exists a unique $\mathbf{Z}$-algebra homomorphism $\psi : R_{univ} \to R$ such that $\psi _*m_{univ} = m$ (this means that $m$ is what you get by applying the base change functor $- \otimes _{R_{univ}} R$ to $m_{univ}$). In other words, setting $X = \mathop{\mathrm{Spec}}(R_{univ})$ we obtain a canonical identification

\[ X(T) = \{ \text{free }d\text{-dimensional algebras }m\text{ over }R\} \]

for varying $T = \mathop{\mathrm{Spec}}(R)$. By Zariski localization we obtain the following seemingly more general identification

96.13.0.1
\begin{equation} \label{criteria-equation-objects} X(T) = \{ \text{free }d\text{-dimensional algebras } m\text{ over }\Gamma (T, \mathcal{O}_ T)\} \end{equation}

for any scheme $T$.

Next we talk a little bit about isomorphisms of free $d$-dimensional $R$-algebras. Namely, suppose that $m$, $m'$ are two free $d$-dimensional algebras over a ring $R$. An isomorphism from $m$ to $m'$ is given by an invertible $R$-linear map

\[ \varphi : R^{\oplus d} \longrightarrow R^{\oplus d} \]

such that $\varphi (e_1) = e_1$ and such that

\[ m \circ \varphi \otimes \varphi = \varphi \circ m'. \]

Note that we can compose these so that the collection of free $d$-dimensional algebras over $R$ becomes a category. In this way we obtain a functor

96.13.0.2
\begin{equation} \label{criteria-equation-FAd} FA_ d : \mathit{Sch}_{fppf}^{opp} \longrightarrow \textit{Groupoids} \end{equation}

from the category of schemes to groupoids: to a scheme $T$ we associate the set of free $d$-dimensional algebras over $\Gamma (T, \mathcal{O}_ T)$ endowed with the structure of a category using the notion of isomorphisms just defined.

The above suggests we consider the functor $G$ in groups which associates to any scheme $T$ the group

\[ G(T) = \{ g \in \text{GL}_ d(\Gamma (T, \mathcal{O}_ T)) \mid g(e_1) = e_1\} \]

It is clear that $G \subset \text{GL}_ d$ (see Groupoids, Example 39.5.4) is the closed subgroup scheme cut out by the equations $x_{11} = 1$ and $x_{i1} = 0$ for $i > 1$. Hence $G$ is a smooth affine group scheme over $\mathop{\mathrm{Spec}}(\mathbf{Z})$. Consider the action

\[ a : G \times _{\mathop{\mathrm{Spec}}(\mathbf{Z})} X \longrightarrow X \]

which associates to a $T$-valued point $(g, m)$ with $T = \mathop{\mathrm{Spec}}(R)$ on the left hand side the free $d$-dimensional algebra over $R$ given by

\[ a(g, m) = g^{-1} \circ m \circ g \otimes g. \]

Note that this means that $g$ defines an isomorphism $m \to a(g, m)$ of $d$-dimensional free $R$-algebras. We omit the verification that $a$ indeed defines an action of the group scheme $G$ on the scheme $X$.

Lemma 96.13.1. The functor in groupoids $FA_ d$ defined in (96.13.0.2) is isomorphic (!) to the functor in groupoids which associates to a scheme $T$ the category with

  1. set of objects is $X(T)$,

  2. set of morphisms is $G(T) \times X(T)$,

  3. $s : G(T) \times X(T) \to X(T)$ is the projection map,

  4. $t : G(T) \times X(T) \to X(T)$ is $a(T)$, and

  5. composition $G(T) \times X(T) \times _{s, X(T), t} G(T) \times X(T) \to G(T) \times X(T)$ is given by $((g, m), (g', m')) \mapsto (gg', m')$.

Proof. We have seen the rule on objects in (96.13.0.1). We have also seen above that $g \in G(T)$ can be viewed as a morphism from $m$ to $a(g, m)$ for any free $d$-dimensional algebra $m$. Conversely, any morphism $m \to m'$ is given by an invertible linear map $\varphi $ which corresponds to an element $g \in G(T)$ such that $m' = a(g, m)$. $\square$

In fact the groupoid $(X, G \times X, s, t, c)$ described in the lemma above is the groupoid associated to the action $a : G \times X \to X$ as defined in Groupoids, Lemma 39.16.1. Since $G$ is smooth over $\mathop{\mathrm{Spec}}(\mathbf{Z})$ we see that the two morphisms $s, t : G \times X \to X$ are smooth: by symmetry it suffices to prove that one of them is, and $s$ is the base change of $G \to \mathop{\mathrm{Spec}}(\mathbf{Z})$. Hence $(G \times X, X, s, t, c)$ is a smooth groupoid scheme, and the quotient stack $[X/G]$ is an algebraic stack by Algebraic Stacks, Theorem 93.17.3.

Proposition 96.13.2. The stack $\mathcal{H}_ d$ is equivalent to the quotient stack $[X/G]$ described above. In particular $\mathcal{H}_ d$ is an algebraic stack.

Proof. Note that by Groupoids in Spaces, Definition 77.20.1 the quotient stack $[X/G]$ is the stackification of the category fibred in groupoids associated to the “presheaf in groupoids” which associates to a scheme $T$ the groupoid

\[ (X(T), G(T) \times X(T), s, t, c). \]

Since this “presheaf in groupoids” is isomorphic to $FA_ d$ by Lemma 96.13.1 it suffices to prove that the $\mathcal{H}_ d$ is the stackification of (the category fibred in groupoids associated to the “presheaf in groupoids”) $FA_ d$. To do this we first define a functor

\[ \mathop{\mathrm{Spec}}: FA_ d \longrightarrow \mathcal{H}_ d \]

Recall that the fibre category of $\mathcal{H}_ d$ over a scheme $T$ is the category of finite locally free morphisms $Z \to T$ of degree $d$. Thus given a scheme $T$ and a free $d$-dimensional $\Gamma (T, \mathcal{O}_ T)$-algebra $m$ we may assign to this the object

\[ Z = \underline{\mathop{\mathrm{Spec}}}_ T(\mathcal{A}) \]

of $\mathcal{H}_{d, T}$ where $\mathcal{A} = \mathcal{O}_ T^{\oplus d}$ endowed with a $\mathcal{O}_ T$-algebra structure via $m$. Moreover, if $m'$ is a second such free $d$-dimensional $\Gamma (T, \mathcal{O}_ T)$-algebra and if $\varphi : m \to m'$ is an isomorphism of these, then the induced $\mathcal{O}_ T$-linear map $\varphi : \mathcal{O}_ T^{\oplus d} \to \mathcal{O}_ T^{\oplus d}$ induces an isomorphism

\[ \varphi : \mathcal{A}' \longrightarrow \mathcal{A} \]

of quasi-coherent $\mathcal{O}_ T$-algebras. Hence

\[ \underline{\mathop{\mathrm{Spec}}}_ T(\varphi ) : \underline{\mathop{\mathrm{Spec}}}_ T(\mathcal{A}) \longrightarrow \underline{\mathop{\mathrm{Spec}}}_ T(\mathcal{A}') \]

is a morphism in the fibre category $\mathcal{H}_{d, T}$. We omit the verification that this construction is compatible with base change so we get indeed a functor $\mathop{\mathrm{Spec}}: FA_ d \to \mathcal{H}_ d$ as claimed above.

To show that $\mathop{\mathrm{Spec}}: FA_ d \to \mathcal{H}_ d$ induces an equivalence between the stackification of $FA_ d$ and $\mathcal{H}_ d$ it suffices to check that

  1. $\mathit{Isom}(m, m') = \mathit{Isom}(\mathop{\mathrm{Spec}}(m), \mathop{\mathrm{Spec}}(m'))$ for any $m, m' \in FA_ d(T)$.

  2. for any scheme $T$ and any object $Z \to T$ of $\mathcal{H}_{d, T}$ there exists a covering $\{ T_ i \to T\} $ such that $Z|_{T_ i}$ is isomorphic to $\mathop{\mathrm{Spec}}(m)$ for some $m \in FA_ d(T_ i)$, and

see Stacks, Lemma 8.9.1. The first statement follows from the observation that any isomorphism

\[ \underline{\mathop{\mathrm{Spec}}}_ T(\mathcal{A}) \longrightarrow \underline{\mathop{\mathrm{Spec}}}_ T(\mathcal{A}') \]

is necessarily given by a global invertible matrix $g$ when $\mathcal{A} = \mathcal{A}' = \mathcal{O}_ T^{\oplus d}$ as modules. To prove the second statement let $\pi : Z \to T$ be a finite locally free morphism of degree $d$. Then $\mathcal{A}$ is a locally free sheaf $\mathcal{O}_ T$-modules of rank $d$. Consider the element $1 \in \Gamma (T, \mathcal{A})$. This element is nonzero in $\mathcal{A} \otimes _{\mathcal{O}_{T, t}} \kappa (t)$ for every $t \in T$ since the scheme $Z_ t = \mathop{\mathrm{Spec}}(\mathcal{A} \otimes _{\mathcal{O}_{T, t}} \kappa (t))$ is nonempty being of degree $d > 0$ over $\kappa (t)$. Thus $1 : \mathcal{O}_ T \to \mathcal{A}$ can locally be used as the first basis element (for example you can use Algebra, Lemma 10.79.4 parts (1) and (2) to see this). Thus, after localizing on $T$ we may assume that there exists an isomorphism $\varphi : \mathcal{A} \to \mathcal{O}_ T^{\oplus d}$ such that $1 \in \Gamma (\mathcal{A})$ corresponds to the first basis element. In this situation the multiplication map $\mathcal{A} \otimes _{\mathcal{O}_ T} \mathcal{A} \to \mathcal{A}$ translates via $\varphi $ into a free $d$-dimensional algebra $m$ over $\Gamma (T, \mathcal{O}_ T)$. This finishes the proof. $\square$

[1] It may be better to think of this as a pair consisting of a multiplication map $m : R^{\oplus d} \otimes _ R R^{\oplus d} \to R^{\oplus d}$ and a ring map $\psi : R \to R^{\oplus d}$ satisfying a bunch of axioms.

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