The Stacks project

Example 94.17.1. Let $S$ be a scheme. Let $G$ be a commutative group. Let $X \to S$ be a scheme over $S$. Let $a : G \times X \to X$ be an action of $G$ on $X$. For $g \in G$ we denote $g : X \to X$ the corresponding automorphism. In this case the inertia stack of $[X/G]$ (see Remark 94.15.5) is given by

\[ I_{[X/G]} = \coprod \nolimits _{g\in G} [X^ g/G], \]

where, given an element $g$ of $G$, the symbol $X^ g$ denotes the scheme $X^ g = \{ x \in X \mid g(x) = x\} $. In a formula $X^ g$ is really the fibre product

\[ X^ g = X \times _{(1, 1), X \times _ S X, (g, 1)} X. \]

Indeed, for any $S$-scheme $T$, a $T$-point on the inertia stack of $[X/G]$ consists of a triple $(P/T, \phi , \alpha )$ consisting of an fppf $G$-torsor $P\to T$ together with a $G$-equivariant morphism $\phi : P \to X$, together with an automorphism $\alpha $ of $P\to T$ over $T$ such that $\phi \circ \alpha = \phi $. Since $G$ is a sheaf of commutative groups, $\alpha $ is, locally in the fppf topology over $T$, given by multiplication by some element $g$ of $G$. The condition that $\phi \circ \alpha = \phi $ means that $\phi $ factors through the inclusion of $X^ g$ in $X$, i.e., $\phi $ is obtained by composing that inclusion with a morphism $P \to X^\gamma $. The above discussion allows us to define a morphism of fibred categories $I_{[X/G]} \to \coprod _{g\in G} [X^ g/G]$ given on $T$-points by the discussion above. We omit showing that this is an equivalence.


Comments (2)

Comment #4554 by Leo on

Typo on 9th line : "together with a -equivariant isomorphism " should just be a morphism, not an isomorphism.

There are also:

  • 2 comment(s) on Section 94.17: Examples of inertia stacks

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Your email address will not be published.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 0374. Beware of the difference between the letter 'O' and the digit '0'.