Processing math: 100%

The Stacks project

Definition 101.5.3. Let f : \mathcal{X} \to \mathcal{Y} be a morphism of algebraic stacks. Let Z be an algebraic space.

  1. Let x : Z \to \mathcal{X} be a morphism. We set

    \mathit{Isom}_{\mathcal{X}/\mathcal{Y}}(x, x) = Z \times _{x, \mathcal{X}} \mathcal{I}_{\mathcal{X}/\mathcal{Y}}

    We endow it with the structure of a group algebraic space over Z by pulling back the composition law discussed in Remark 101.5.2. We will sometimes refer to \mathit{Isom}_{\mathcal{X}/\mathcal{Y}}(x, x) as the relative sheaf of automorphisms of x.

  2. Let x_1, x_2 : Z \to \mathcal{X} be morphisms. Set y_ i = f \circ x_ i. Let \alpha : y_1 \to y_2 be a 2-morphism. Then \alpha determines a morphism \Delta ^\alpha : Z \to Z \times _{y_1, \mathcal{Y}, y_2} Z and we set

    \mathit{Isom}_{\mathcal{X}/\mathcal{Y}}^\alpha (x_1, x_2) = (Z \times _{x_1, \mathcal{X}, x_2} Z) \times _{Z \times _{y_1, \mathcal{Y}, y_2} Z, \Delta ^\alpha } Z.

    We will sometimes refer to \mathit{Isom}_{\mathcal{X}/\mathcal{Y}}^\alpha (x_1, x_2) as the relative sheaf of isomorphisms from x_1 to x_2.

If \mathcal{Y} = \mathop{\mathrm{Spec}}(\mathbf{Z}) or more generally when \mathcal{Y} is an algebraic space, then we use the notation \mathit{Isom}_\mathcal {X}(x, x) and \mathit{Isom}_\mathcal {X}(x_1, x_2) and we use the terminology sheaf of automorphisms of x and sheaf of isomorphisms from x_1 to x_2.


Comments (0)

There are also:

  • 2 comment(s) on Section 101.5: Inertia 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Your email address will not be published.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.