The Stacks project

Lemma 44.2.5. Let $X \to S$ be a morphism of affine schemes. Let $d \geq 0$. Then $\mathrm{Hilb}^ d_{X/S}$ is representable.

Proof. Say $S = \mathop{\mathrm{Spec}}(R)$. Then we can choose a closed immersion of $X$ into the spectrum of $R[x_ i; i \in I]$ for some set $I$ (of sufficiently large cardinality. Hence by Lemma 44.2.3 we may assume that $X = \mathop{\mathrm{Spec}}(A)$ where $A = R[x_ i; i \in I]$. We will use Schemes, Lemma 26.15.4 to prove the lemma in this case.

Condition (1) of the lemma follows from Lemma 44.2.1.

For every subset $W \subset A$ of cardinality $d$ we will construct a subfunctor $F_ W$ of $\mathrm{Hilb}^ d_{X/S}$. (It would be enough to consider the case where $W$ consists of a collection of monomials in the $x_ i$ but we do not need this.) Namely, we will say that $Z \in \mathrm{Hilb}^ d_{X/S}(T)$ is in $F_ W(T)$ if and only if the $\mathcal{O}_ T$-linear map

\[ \bigoplus \nolimits _{f \in W} \mathcal{O}_ T \longrightarrow (Z \to T)_*\mathcal{O}_ Z,\quad (g_ f) \longmapsto \sum g_ f f|_ Z \]

is surjective (equivalently an isomorphism). Here for $f \in A$ and $Z \in \mathrm{Hilb}^ d_{X/S}(T)$ we denote $f|_ Z$ the pullback of $f$ by the morphism $Z \to X_ T \to X$.

Openness, i.e., condition (2)(b) of the lemma. This follows from Algebra, Lemma 10.79.4.

Covering, i.e., condition (2)(c) of the lemma. Since

\[ A \otimes _ R \mathcal{O}_ T = (X_ T \to T)_*\mathcal{O}_{X_ T} \to (Z \to T)_*\mathcal{O}_ Z \]

is surjective and since $(Z \to T)_*\mathcal{O}_ Z$ is finite locally free of rank $d$, for every point $t \in T$ we can find a finite subset $W \subset A$ of cardinality $d$ whose images form a basis of the $d$-dimensional $\kappa (t)$-vector space $((Z \to T)_*\mathcal{O}_ Z)_ t \otimes _{\mathcal{O}_{T, t}} \kappa (t)$. By Nakayama's lemma there is an open neighbourhood $V \subset T$ of $t$ such that $Z_ V \in F_ W(V)$.

Representable, i.e., condition (2)(a) of the lemma. Let $W \subset A$ have cardinality $d$. We claim that $F_ W$ is representable by an affine scheme over $R$. We will construct this affine scheme here, but we encourage the reader to think it through for themselves. Choose a numbering $f_1, \ldots , f_ d$ of the elements of $W$. We will construct a universal element $Z_{univ} = \mathop{\mathrm{Spec}}(B_{univ})$ of $F_ W$ over $T_{univ} = \mathop{\mathrm{Spec}}(R_{univ})$ which will be the spectrum of

\[ B_{univ} = R_{univ}[e_1, \ldots , e_ d]/ (e_ ke_ l - \sum c_{kl}^ m e_ m) \]

where the $e_ l$ will be the images of the $f_ l$ and where the closed immersion $Z_{univ} \to X_{T_{univ}}$ is given by the ring map

\[ A \otimes _ R R_{univ} \longrightarrow B_{univ} \]

mapping $1 \otimes 1$ to $\sum b^ le_ l$ and $x_ i$ to $\sum b_ i^ le_ l$. In fact, we claim that $F_ W$ is represented by the spectrum of the ring

\[ R_{univ} = R[c_{kl}^ m, b^ l, b_ i^ l]/\mathfrak a_{univ} \]

where the ideal $\mathfrak a_{univ}$ is generated by the following elements:

  1. multiplication on $B_{univ}$ is commutative, i.e., $c_{lk}^ m - c_{kl}^ m \in \mathfrak a_{univ}$,

  2. multiplication on $B_{univ}$ is associative, i.e., $c_{lk}^ m c_{m n}^ p - c_{lq}^ p c_{kn}^ q \in \mathfrak a_{univ}$,

  3. $\sum b^ le_ l$ is a multiplicative $1$ in $B_{univ}$, in other words, we should have $(\sum b^ le_ l)e_ k = e_ k$ for all $k$, which means $\sum b^ lc_{lk}^ m - \delta _{km} \in \mathfrak a_{univ}$ (Kronecker delta).

After dividing out by the ideal $\mathfrak a'_{univ}$ of the elements listed sofar we obtain a well defined ring map

\[ \Psi : A \otimes _ R R[c_{kl}^ m, b^ l, b_ i^ l]/\mathfrak a'_{univ} \longrightarrow \left(R[c_{kl}^ m, b^ l, b_ i^ l]/\mathfrak a'_{univ}\right) [e_1, \ldots , e_ d]/(e_ ke_ l - \sum c_{kl}^ m e_ m) \]

sending $1 \otimes 1$ to $\sum b^ le_ l$ and $x_ i \otimes 1$ to $\sum b_ i^ le_ l$. We need to add some more elements to our ideal because we need

  1. $f_ l$ to map to $e_ l$ in $B_{univ}$. Write $\Psi (f_ l) - e_ l = \sum h_ l^ me_ m$ with $h_ l^ m \in R[c_{kl}^ m, b^ l, b_ i^ l]/\mathfrak a'_{univ}$ then we need to set $h_ l^ m$ equal to zero.

Thus setting $\mathfrak a_{univ} \subset R[c_{kl}^ m, b^ l, b_ i^ l]$ equal to $\mathfrak a'_{univ} + $ ideal generated by lifts of $h_ l^ m$ to $R[c_{kl}^ m, b^ l, b_ i^ l]$, then it is clear that $F_ W$ is represented by $\mathop{\mathrm{Spec}}(R_{univ})$. $\square$

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