The Stacks project

94.18 Finite Hilbert stacks

We formulate this in somewhat greater generality than is perhaps strictly needed. Fix a $1$-morphism

\[ F : \mathcal{X} \longrightarrow \mathcal{Y} \]

of stacks in groupoids over $(\mathit{Sch}/S)_{fppf}$. For each integer $d \geq 1$ consider a category $\mathcal{H}_ d(\mathcal{X}/\mathcal{Y})$ defined as follows:

  1. An object $(U, Z, y, x, \alpha )$ where $U, Z$ are objects of in $(\mathit{Sch}/S)_{fppf}$ and $Z$ is a finite locally free of degree $d$ over $U$, where $y \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{Y}_ U)$, $x \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{X}_ Z)$ and $\alpha : y|_ Z \to F(x)$ is an isomorphism1.

  2. A morphism $(U, Z, y, x, \alpha ) \to (U', Z', y', x', \alpha ')$ is given by a morphism of schemes $f : U \to U'$, a morphism of schemes $g : Z \to Z'$ which induces an isomorphism $Z \to Z' \times _ U U'$, and isomorphisms $b : y \to f^*y'$, $a : x \to g^*x'$ inducing a commutative diagram

    \[ \xymatrix{ y|_ Z \ar[rr]_\alpha \ar[d]_{b|_ Z} & & F(x) \ar[d]^{F(a)} \\ f^*y'|_ Z \ar[rr]^{\alpha '} & & F(g^*x') \\ } \]

It is clear from the definitions that there is a canonical forgetful functor

\[ p : \mathcal{H}_ d(\mathcal{X}/\mathcal{Y}) \longrightarrow (\mathit{Sch}/S)_{fppf} \]

which assigns to the quintuple $(U, Z, y, x, \alpha )$ the scheme $U$ and to the morphism $(f, g, b, a) : (U, Z, y, x, \alpha ) \to (U', Z', y', x', \alpha ')$ the morphism $f : U \to U'$.

Lemma 94.18.1. The category $\mathcal{H}_ d(\mathcal{X}/\mathcal{Y})$ endowed with the functor $p$ above defines a stack in groupoids over $(\mathit{Sch}/S)_{fppf}$.

Proof. As usual, the hardest part is to show descent for objects. To see this let $\{ U_ i \to U\} $ be a covering of $(\mathit{Sch}/S)_{fppf}$. Let $\xi _ i = (U_ i, Z_ i, y_ i, x_ i, \alpha _ i)$ be an object of $\mathcal{H}_ d(\mathcal{X}/\mathcal{Y})$ lying over $U_ i$, and let $\varphi _{ij} : \text{pr}_0^*\xi _ i \to \text{pr}_1^*\xi _ j$ be a descent datum. First, observe that $\varphi _{ij}$ induces a descent datum $(Z_ i/U_ i, \varphi _{ij})$ which is effective by Descent, Lemma 35.37.1 This produces a scheme $Z/U$ which is finite locally free of degree $d$ by Descent, Lemma 35.23.30. From now on we identify $Z_ i$ with $Z \times _ U U_ i$. Next, the objects $y_ i$ in the fibre categories $\mathcal{Y}_{U_ i}$ descend to an object $y$ in $\mathcal{Y}_ U$ because $\mathcal{Y}$ is a stack in groupoids. Similarly the objects $x_ i$ in the fibre categories $\mathcal{X}_{Z_ i}$ descend to an object $x$ in $\mathcal{X}_ Z$ because $\mathcal{X}$ is a stack in groupoids. Finally, the given isomorphisms

\[ \alpha _ i : (y|_ Z)_{Z_ i} = y_ i|_{Z_ i} \longrightarrow F(x_ i) = F(x|_{Z_ i}) \]

glue to a morphism $\alpha : y|_ Z \to F(x)$ as the $\mathcal{Y}$ is a stack and hence $\mathit{Isom}_\mathcal {Y}(y|_ Z, F(x))$ is a sheaf. Details omitted. $\square$

Definition 94.18.2. We will denote $\mathcal{H}_ d(\mathcal{X}/\mathcal{Y})$ the degree $d$ finite Hilbert stack of $\mathcal{X}$ over $\mathcal{Y}$ constructed above. If $\mathcal{Y} = S$ we write $\mathcal{H}_ d(\mathcal{X}) = \mathcal{H}_ d(\mathcal{X}/\mathcal{Y})$. If $\mathcal{X} = \mathcal{Y} = S$ we denote it $\mathcal{H}_ d$.

Note that given $F : \mathcal{X} \to \mathcal{Y}$ as above we have the following natural $1$-morphisms of stacks in groupoids over $(\mathit{Sch}/S)_{fppf}$:
\begin{equation} \label{examples-stacks-equation-diagram-hilbert-d-stack} \vcenter { \xymatrix{ \mathcal{H}_ d(\mathcal{X}) \ar[rd] & \mathcal{H}_ d(\mathcal{X}/\mathcal{Y}) \ar[d] \ar[l] \ar[r] & \mathcal{Y} \\ & \mathcal{H}_ d } } \end{equation}

Each of the arrows is given by a "forgetful functor".

Lemma 94.18.3. The $1$-morphism $\mathcal{H}_ d(\mathcal{X}/\mathcal{Y}) \to \mathcal{H}_ d(\mathcal{X})$ is faithful.

Proof. To check that $\mathcal{H}_ d(\mathcal{X}/\mathcal{Y}) \to \mathcal{H}_ d(\mathcal{X})$ is faithful it suffices to prove that it is faithful on fibre categories. Suppose that $\xi = (U, Z, y, x, \alpha )$ and $\xi ' = (U, Z', y', x', \alpha ')$ are two objects of $\mathcal{H}_ d(\mathcal{X}/\mathcal{Y})$ over the scheme $U$. Let $(g, b, a), (g', b', a') : \xi \to \xi '$ be two morphisms in the fibre category of $\mathcal{H}_ d(\mathcal{X}/\mathcal{Y})$ over $U$. The image of these morphisms in $\mathcal{H}_ d(\mathcal{X})$ agree if and only if $g = g'$ and $a = a'$. Then the commutative diagram

\[ \xymatrix{ y|_ Z \ar[rr]_\alpha \ar[d]_{b|_ Z, \ b'|_ Z} & & F(x) \ar[d]^{F(a) = F(a')} \\ y'|_ Z \ar[rr]^-{\alpha '} & & F(g^*x') = F((g')^*x') \\ } \]

implies that $b|_ Z = b'|_ Z$. Since $Z \to U$ is finite locally free of degree $d$ we see $\{ Z \to U\} $ is an fppf covering, hence $b = b'$. $\square$

[1] This means the data gives rise, via the $2$-Yoneda lemma (Categories, Lemma 4.41.2), to a $2$-commutative diagram
\[ \xymatrix{ (\mathit{Sch}/Z)_{fppf} \ar[r]_-x \ar[d] & \mathcal{X} \ar[d]^ F \\ (\mathit{Sch}/U)_{fppf} \ar[r]^-y & \mathcal{Y} } \]
of stacks in groupoids over $(\mathit{Sch}/S)_{fppf}$. Alternatively, we may picture $\alpha $ as a $2$-morphism
\[ \xymatrix{ (\mathit{Sch}/Z)_{fppf} \rrtwocell ^{y \circ (Z \to U)}_{F \circ x}{\alpha } & & \mathcal{Y}. } \]

Comments (2)

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 05WA. Beware of the difference between the letter 'O' and the digit '0'.