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. $\square$

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).