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

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 $T_{univ} = \mathop{\mathrm{Spec}}(R_{univ})$ which will be the spectrum of

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

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

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

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

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

$\sum b^ le_ l$ 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

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

$f_ l$ 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 $\mathop{\mathrm{Spec}}(R_{univ})$. $\square$

## Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like `$\pi$`

). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

## Comments (0)