Lemma 97.13.1. The functor in groupoids FA_ d defined in (97.13.0.2) is isomorphic (!) to the functor in groupoids which associates to a scheme T the category with
set of objects is X(T),
set of morphisms is G(T) \times X(T),
s : G(T) \times X(T) \to X(T) is the projection map,
t : G(T) \times X(T) \to X(T) is a(T), and
composition G(T) \times X(T) \times _{s, X(T), t} G(T) \times X(T) \to G(T) \times X(T) is given by ((g, m), (g', m')) \mapsto (gg', m').
Proof. We have seen the rule on objects in (97.13.0.1). We have also seen above that g \in G(T) can be viewed as a morphism from m to a(g, m) for any free d-dimensional algebra m. Conversely, any morphism m \to m' is given by an invertible linear map \varphi which corresponds to an element g \in G(T) such that m' = a(g, m). \square
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