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. $\square$

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