Processing math: 100%

The Stacks project

Lemma 100.9.14. Let \mathcal X be an algebraic stack and \mathcal{X}_ i \subset \mathcal X a collection of open substacks indexed by i \in I. Then there exists an open substack, which we denote \bigcup _{i\in I} \mathcal{X}_ i \subset \mathcal X, such that the \mathcal{X}_ i are open substacks covering it.

Proof. We define a fibred subcategory \mathcal{X}' = \bigcup _{i \in I} \mathcal{X}_ i by letting the fiber category over an object T of (\mathit{Sch}/S)_{fppf} be the full subcategory of \mathcal{X}_ T consisting of all x \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{X}_ T) such that the morphism \coprod _{i \in I} (\mathcal{X}_ i \times _{\mathcal X} T) \to T is surjective. Let x_ i \in \mathop{\mathrm{Ob}}\nolimits ((\mathcal{X}_ i)_ T). Then (x_ i, 1) gives a section of \mathcal{X}_ i \times _{\mathcal X} T \to T, so we have an isomorphism. Thus \mathcal{X}_ i \subset \mathcal{X}' is a full subcategory. Now let x \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{X}_ T). Then \mathcal{X}_ i \times _{\mathcal X} T is representable by an open subscheme T_ i \subset T. The 2-fibred product \mathcal{X}' \times _{\mathcal X} T has fiber over T' consisting of (y \in \mathcal{X}_{T'}, f : T' \to T, f^*x \simeq y) such that \coprod (\mathcal{X}_ i \times _{\mathcal X, y} T') \to T' is surjective. The isomorphism f^*x \simeq y induces an isomorphism \mathcal{X}_ i \times _{\mathcal X, y} T' \simeq T_ i \times _ T T'. Then the T_ i \times _ T T' cover T' if and only if f lands in \bigcup T_ i. Therefore we have a diagram

\xymatrix{ T_ i \ar[r] \ar[d] & \bigcup T_ i \ar[r] \ar[d] & T \ar[d] \\ \mathcal{X}_ i \ar[r] & \mathcal{X}' \ar[r] & \mathcal{X} }

with both squares cartesian. By Algebraic Stacks, Lemma 94.15.5 we conclude that \mathcal{X'} \subset \mathcal{X} is algebraic and an open substack. It is also clear from the cartesian squares above that the morphism \coprod _{i \in I} \mathcal{X}_ i \to \mathcal{X}' which finishes the proof of the lemma. \square


Comments (0)

There are also:

  • 2 comment(s) on Section 100.9: Immersions of algebraic stacks

Your email address will not be published.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.