Rational representations and comodules of an affine group scheme

Let $k$ be a field, let $G$ be an affine group scheme of finite type over $k$ (Group schemes of finite type over a field) with coordinate Hopf algebra $(A,\mathrm{\Delta },\epsilon ,S)$ (The coordinate Hopf algebra of an affine group scheme), and let $V$ be a $k$-vector space (Vector space over a field).

(a) For a commutative unital $k$-algebra $R$ put $V_R=V{\otimes }_{k}R$ and ${\mathrm{GL}}_V(R)={\mathrm{Aut}}_R(V_R)$ (Invertible linear maps, linear isomorphisms, and inverse linear maps); a rational representation of $G$ on $V$ is a morphism of group functors $r\text{\hspace{0.17em}⁣}:G\to {\mathrm{GL}}_V$, that is, a natural family of group homomorphisms $G(R)\to {\mathrm{Aut}}_R(V_R)$ in $R$. When $V$ is finite dimensional, a choice of basis identifies ${\mathrm{GL}}_V$ with the group functor represented by the affine scheme ${\mathrm{GL}}_n$ of The general linear group scheme and its coordinate ring, using ${\mathrm{GL}}_0=\mathrm{Spec}k$ when $V=0$. The identification uses its choice-free point formulas.

(b) An $A$-comodule structure on $V$ is a $k$-linear map $\rho \text{\hspace{0.17em}⁣}:V\to V{\otimes }_{k}A$ (Linear map between vector spaces over the same field, The tensor product $M{\otimes }_{R}N$ from the additive group underlying the free $\mathbb{Z}$-module on $M\times N$, elementary tensors, and finite tensor sums) with $(\rho \otimes {\mathrm{id}}_A)\rho =({\mathrm{id}}_V\otimes \mathrm{\Delta })\rho ,({\mathrm{id}}_V\otimes \epsilon )\rho ={\mathrm{id}}_V;$ a subspace $W\subseteq V$ is a subcomodule if $\rho (W)\subseteq W{\otimes }_{k}A$ (Linear subspace of a vector space).

(c) The two notions correspond: a comodule structure $\rho$ gives the representation by $r_R(g)(v\otimes 1)=({\mathrm{id}}_V\otimes g)\rho (v)$, extended $R$-linearly, and this assignment is a bijection onto the rational representations of $G$ on $V$ under which subcomodules correspond to subrepresentations; the proof is Rational representations of an affine group scheme are comodules of its coordinate Hopf algebra ↗.

(d) The coaction $\mathrm{\Delta }\text{\hspace{0.17em}⁣}:A\to A{\otimes }_{k}A$ makes $A$ itself an $A$-comodule, the regular representation of $G$. A representation $r$ is faithful if every $r_R$ is injective.

The axioms in (b) are exactly the two comodule diagrams; no smoothness, reducedness or finite-dimensionality of $V$ is imposed, and a basis of $V$ is chosen only to name the matrix group ${\mathrm{GL}}_n$.