The Stacks project

Lemma 44.2.3. Let $S$ be a scheme. Let $i : X \to Y$ be a closed immersion of schemes. If $\mathrm{Hilb}^ d_{Y/S}$ is representable by a scheme, so is $\mathrm{Hilb}^ d_{X/S}$ and the corresponding morphism of schemes $\underline{\mathrm{Hilb}}^ d_{X/S} \to \underline{\mathrm{Hilb}}^ d_{Y/S}$ is a closed immersion.

Proof. Let $T$ be a scheme over $S$ and let $Z \in \mathrm{Hilb}^ d_{Y/S}(T)$. Claim: there is a closed subscheme $T_ X \subset T$ such that a morphism of schemes $T' \to T$ factors through $T_ X$ if and only if $Z_{T'} \to Y_{T'}$ factors through $X_{T'}$. Applying this to a scheme $T_{univ}$ representing $\mathrm{Hilb}^ d_{Y/S}$ and the universal object1 $Z_{univ} \in \mathrm{Hilb}^ d_{Y/S}(T_{univ})$ we get a closed subscheme $T_{univ, X} \subset T_{univ}$ such that $Z_{univ, X} = Z_{univ} \times _{T_{univ}} T_{univ, X}$ is a closed subscheme of $X \times _ S T_{univ, X}$ and hence defines an element of $\mathrm{Hilb}^ d_{X/S}(T_{univ, X})$. A formal argument then shows that $T_{univ, X}$ is a scheme representing $\mathrm{Hilb}^ d_{X/S}$ with universal object $Z_{univ, X}$.

Proof of the claim. Consider $Z' = X_ T \times _{Y_ T} Z$. Given $T' \to T$ we see that $Z_{T'} \to Y_{T'}$ factors through $X_{T'}$ if and only if $Z'_{T'} \to Z_{T'}$ is an isomorphism. Thus the claim follows from the very general More on Flatness, Lemma 38.23.4. However, in this special case one can prove the statement directly as follows: first reduce to the case $T = \mathop{\mathrm{Spec}}(A)$ and $Z = \mathop{\mathrm{Spec}}(B)$. After shrinking $T$ further we may assume there is an isomorphism $\varphi : B \to A^{\oplus d}$ as $A$-modules. Then $Z' = \mathop{\mathrm{Spec}}(B/J)$ for some ideal $J \subset B$. Let $g_\beta \in J$ be a collection of generators and write $\varphi (g_\beta ) = (g_\beta ^1, \ldots , g_\beta ^ d)$. Then it is clear that $T_ X$ is given by $\mathop{\mathrm{Spec}}(A/(g_\beta ^ j))$. $\square$

[1] See Categories, Section 4.3

Comments (0)

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Your email address will not be published.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.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0B97. Beware of the difference between the letter 'O' and the digit '0'.