The Stacks project

39.16 Groupoids and group schemes

There are many ways to construct a groupoid out of an action $a$ of a group $G$ on a set $V$. We choose the one where we think of an element $g \in G$ as an arrow with source $v$ and target $a(g, v)$. This leads to the following construction for group actions of schemes.

Lemma 39.16.1. Let $S$ be a scheme. Let $Y$ be a scheme over $S$. Let $(G, m)$ be a group scheme over $Y$ with identity $e_ G$ and inverse $i_ G$. Let $X/Y$ be a scheme over $Y$ and let $a : G \times _ Y X \to X$ be an action of $G$ on $X/Y$. Then we get a groupoid scheme $(U, R, s, t, c, e, i)$ over $S$ in the following manner:

  1. We set $U = X$, and $R = G \times _ Y X$.

  2. We set $s : R \to U$ equal to $(g, x) \mapsto x$.

  3. We set $t : R \to U$ equal to $(g, x) \mapsto a(g, x)$.

  4. We set $c : R \times _{s, U, t} R \to R$ equal to $((g, x), (g', x')) \mapsto (m(g, g'), x')$.

  5. We set $e : U \to R$ equal to $x \mapsto (e_ G(x), x)$.

  6. We set $i : R \to R$ equal to $(g, x) \mapsto (i_ G(g), a(g, x))$.

Proof. Omitted. Hint: It is enough to show that this works on the set level. For this use the description above the lemma describing $g$ as an arrow from $v$ to $a(g, v)$. $\square$

Lemma 39.16.2. Let $S$ be a scheme. Let $Y$ be a scheme over $S$. Let $(G, m)$ be a group scheme over $Y$. Let $X$ be a scheme over $Y$ and let $a : G \times _ Y X \to X$ be an action of $G$ on $X$ over $Y$. Let $(U, R, s, t, c)$ be the groupoid scheme constructed in Lemma 39.16.1. The rule $(\mathcal{F}, \alpha ) \mapsto (\mathcal{F}, \alpha )$ defines an equivalence of categories between $G$-equivariant $\mathcal{O}_ X$-modules and the category of quasi-coherent modules on $(U, R, s, t, c)$.

Proof. The assertion makes sense because $t = a$ and $s = \text{pr}_1$ as morphisms $R = G \times _ Y X \to X$, see Definitions 39.12.1 and 39.14.1. Using the translation in Lemma 39.16.1 the commutativity requirements of the two definitions match up exactly. $\square$


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Your email address will not be published.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 03LK. Beware of the difference between the letter 'O' and the digit '0'.