G-Representations Are A-Comodules

Let $k$ be a field, $G$ an affine group scheme over $\operatorname{Spec} k$ with coordinate Hopf algebra $A = \mathcal{O}(G)$, and $V$ a $k$-vector space. The following data are naturally equivalent:

an $A$-comodule structure on $V$ (a coaction $\rho : V \to V \otimes_k A$ satisfying coassociativity and counit);
a natural action of the functor of points $G(-)$ on the functor $R \mapsto V \otimes_k R$ — that is, for each $k$-algebra $R$ a group homomorphism $G(R) \to \operatorname{GL}_R(V \otimes_k R)$, natural in $R$;
when $V$ is finite-dimensional, a morphism of group schemes $\rho^\sharp : G \to \operatorname{GL}(V)$ over $\operatorname{Spec} k$.

Proof sketch.
From a coaction $\rho : V \to V \otimes_k A$ and a $k$-algebra map $g : A \to R$, define the endomorphism $\rho_g : V \otimes_k R \to V \otimes_k R$ by

\[ \rho_g(v \otimes r) = \sum_{(v)} v_{(0)} \otimes g(v_{(1)})\, r \]

(using sumless Sweedler notation for $\rho(v) = \sum v_{(0)} \otimes v_{(1)}$). Counit makes $\rho_g$ become the identity when $g$ is the unit of $G(R)$; coassociativity makes $\rho_{gh}$ equal $\rho_g \circ \rho_h$. Hence $\rho$ induces a group homomorphism $G(R) \to \operatorname{GL}_R(V \otimes_k R)$, natural in $R$.

Conversely, from a natural family of group homomorphisms one recovers the coaction by evaluating at the universal $R$-point $\operatorname{id}_A : A \to A$ in $G(A)$.

When $V$ is finite-dimensional, the functor of points $R \mapsto \operatorname{GL}_R(V \otimes_k R)$ is represented by an affine group scheme $\operatorname{GL}(V)$, and a natural transformation $G(-) \to \operatorname{GL}(V)(-)$ of group-valued functors corresponds by Yoneda to a morphism of group schemes $G \to \operatorname{GL}(V)$.
$\square$

G-Representations Are A-Comodules

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