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

1. set of objects is $X(T)$,

2. set of morphisms is $G(T) \times X(T)$,

3. $s : G(T) \times X(T) \to X(T)$ is the projection map,

4. $t : G(T) \times X(T) \to X(T)$ is $a(T)$, and

5. 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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