The Stacks project

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

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

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 $T_{univ} = \mathop{\mathrm{Spec}}(R_{univ})$ as follows. First, we set

for some ideal $\mathfrak a_{univ}$ which we describe below. Here the indices $k, l, m, i$ are all ranging through $\{ 1, \ldots , d\} $. Then we set

as an $R_{univ}$-module with algebra structure given by the rule

The closed immersion $Z_{univ} \to X_{T_{univ}}$ is given by the $R_{univ}$-algebra map

mapping $1 \otimes 1$ to $\sum b^ le_ l$ and $x_ i$ to $\sum b_ i^ le_ l$ (for existence see below). As for the ideal $\mathfrak a_{univ}$ we have the following prescription:

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

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

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).

Since we've guaranteed that $B_{univ}$ is a commutative $R_{univ}$-algebra with $1$ given by $\sum b^ le_ l$ the homomorphism $\Psi $ exists. The final condition is that

1. $f_ l$ to map to $e_ l$ in $B_{univ}$. Write $\Psi (f_ l) - e_ l \equiv \sum h_ l^ me_ m$ for some $h_ l^ m \in R[c_{kl}^ m, b^ l, b_ i^ l]$ then we should have $h_ l^ m \in \mathfrak a_{univ}$.

With $\mathfrak a_{univ} \subset R[c_{kl}^ m, b^ l, b_ i^ l]$ generated by the elements listed in (1) – (5) it is clear that $F_ W$ is represented by $\mathop{\mathrm{Spec}}(R_{univ})$. $\square$