The Stacks project

97.13 The finite Hilbert stack of a point

Let $d \geq 1$ be an integer. In Examples of Stacks, Definition 95.18.2 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

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

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

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

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

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

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

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

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

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

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

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

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 $X$.

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 94.17.3.