Rational representations of an affine group scheme are comodules of its coordinate Hopf algebra

(From a comodule to a representation.) Let $\rho \text{\hspace{0.17em}⁣}:V\to V{\otimes }_{k}A$ satisfy the comodule axioms. For a commutative unital $k$-algebra $R$ and $g\in G(R)$, define $r_R(g)$ to be the $R$-linear endomorphism of $V{\otimes }_{k}R$ with $r_R(g)(v\otimes x)=({\mathrm{id}}_V\otimes g)\rho (v)\cdot x$. This is natural in $R$. If $\rho (v)={\sum }_i{v}_i\otimes a_i$, then $r_R(g)r_R(h)(v\otimes 1)={\sum }_i({\mathrm{id}}_V\otimes g)\rho (v_i)\cdot h(a_i)=({\mathrm{id}}_V\otimes (gh))\rho (v)=r_R(gh)(v\otimes 1)$ by coassociativity and the convolution product of [F2], and by $R$-linearity this proves multiplicativity; the counit identity gives $r_R(e_R)=\mathrm{id}$, since $e_R$ factors through $\epsilon$. Hence each $r_R(g)$ is invertible with inverse $r_R(g^{-1})$ and $r$ is a rational representation.

(From a representation to a comodule.) Let $r\text{\hspace{0.17em}⁣}:G\to {\mathrm{GL}}_V$ be a rational representation. By [F3] it corresponds to the element $\phi =r_A({\mathrm{id}}_A)\in {\mathrm{GL}}_V(A)={\mathrm{Aut}}_A(V{\otimes }_{k}A)$, and naturality at $g\text{\hspace{0.17em}⁣}:A\to R$ gives $r_R(g)(v\otimes 1)=({\mathrm{id}}_V\otimes g)\phi (v\otimes 1)$. Indeed $(V{\otimes }_{k}A){\otimes }_{A}R\cong V{\otimes }_{k}R$ by $(v\otimes a)\otimes x\mapsto v\otimes g(a)x$, so the base-changed automorphism sends $v\otimes 1$ to $({\mathrm{id}}_V\otimes g)\phi (v\otimes 1)$. Put $\rho (v)=\phi (v\otimes 1)$; then $\phi (v\otimes a)=\rho (v)a$ by $A$-linearity, and $r_R(g)(v\otimes 1)=({\mathrm{id}}_V\otimes g)\rho (v)$. The identity element of $G(k)$ is $\epsilon$, so $r_k(\epsilon )=\mathrm{id}$ gives $({\mathrm{id}}_V\otimes \epsilon )\rho ={\mathrm{id}}_V$. For coassociativity, $p_1\ast p_2=\mathrm{\Delta }$ in $G(A{\otimes }_{k}A)$ by [F2], so $r_{A{\otimes }_{k}A}(\mathrm{\Delta })=r_{A{\otimes }_{k}A}(p_1)r_{A{\otimes }_{k}A}(p_2)$, and evaluating at $v\otimes 1$ gives ${\sum }_i\rho (v_i)\otimes a_i={\sum }_i{v}_i\otimes \mathrm{\Delta }(a_i)$, that is, $(\rho \otimes {\mathrm{id}}_A)\rho =({\mathrm{id}}_V\otimes \mathrm{\Delta })\rho$. Hence $\rho$ is a comodule structure.

(Matrix coefficients.) Let $V$ be finite dimensional with basis $e_1,\dots ,e_n$ and $\rho (e_j)={\sum }_i{e}_i\otimes a_{ij}$. Comparing the components of $e_i$ under the finite coordinate functionals of $V$ in the coassociativity identity gives $\mathrm{\Delta }(a_{ij})={\sum }_l{a}_{il}\otimes a_{lj}$, and comparing them in $({\mathrm{id}}_V\otimes \epsilon )\rho (e_j)=e_j$ gives $\epsilon (a_{ij})={\delta }_{ij}$.

(The two constructions are inverse.) If $\rho$ gives $r$ by step 1.1 and $r$ gives ${\rho }^{\mathrm{\prime }}$ by step 1.2, then ${\rho }^{\mathrm{\prime }}(v)=r_A({\mathrm{id}}_A)(v\otimes 1)=({\mathrm{id}}_V\otimes {\mathrm{id}}_A)\rho (v)=\rho (v)$; conversely if $r$ gives $\rho$ and $\rho$ gives $r^{\mathrm{\prime }}$, then $r_R^{\mathrm{\prime }}(g)$ and $r_R(g)$ are $R$-linear and agree on all $v\otimes 1$ by step 1.2, hence $r^{\mathrm{\prime }}=r$. For a linear map $T\text{\hspace{0.17em}⁣}:V\to W$ between two such structures, $(T\otimes {\mathrm{id}}_A){\rho }_V={\rho }_{W}T$ implies $(T\otimes {\mathrm{id}}_R)r_{V,R}(g)=r_{W,R}(g)(T\otimes {\mathrm{id}}_R)$ by evaluation; conversely the latter identities at $R=A$, $g={\mathrm{id}}_A$ give the former. Pullback along a group morphism $f\text{\hspace{0.17em}⁣}:H\to G$ is $({\mathrm{id}}_V\otimes \mathcal{O}(f))\rho$ on coactions and $r\circ f$ on actions. These formulas specify the asserted naturality.

(Subcomodules and subrepresentations.) If $N\subseteq V$ is a subcomodule, then $r_R(g)(N{\otimes }_{k}R)=({\mathrm{id}}_V\otimes g)\rho (N)\cdot R\subseteq N{\otimes }_{k}R$ for every $R$ and $g$ by step 1.1, so $N$ is a subrepresentation. Conversely, if $N$ is stable under every $r_R(g)$, then taking $R=A$ and $g={\mathrm{id}}_A$ in step 1.2 gives $\rho (N)=r_A({\mathrm{id}}_A)(N\otimes 1)\subseteq N{\otimes }_{k}A$, so $N$ is a subcomodule. The correspondence preserves inclusions.

(The comorphism of the associated morphism.) If $V=0$, both structures are unique and the associated morphism is $G\to {\mathrm{GL}}_0=\mathrm{Spec}k$, with comorphism the structure map $k\to A$; the coefficient identities are empty. Otherwise $n\ge 1$. In the situation of step 1.3, the two antipode identities give ${\sum }_{l}S(a_{il})a_{lj}={\delta }_{ij}={\sum }_l{a}_{il}S(a_{lj})$, so the matrix $(a_{ij})$ over the commutative ring $A$ has two-sided inverse $(S(a_{ij}))$ and therefore unit determinant by [F4]. Hence $\phi \text{\hspace{0.17em}⁣}:k[x_{ij},d^{-1}]\to A$, $x_{ij}\mapsto a_{ij}$, $d^{-1}\mapsto \det (a_{ij}{)}^{-1}$, is a well-defined $k$-algebra homomorphism, and by [F5] it is the comorphism of a morphism $G\to {\mathrm{GL}}_n$. On $R$-points this morphism sends $g$ to the matrix $(\phi (x_{ij})(g))=(g(a_{ij}))$, which by step 1.1 and step 1.3 is exactly the matrix of $r_R(g)$ in the basis $e_1,\dots ,e_n$; by [F4] this identifies the morphism associated with $r$ under the basis with ${\mathrm{GL}}_n$ and its comorphism with $x_{ij}\mapsto a_{ij}$. All constructions used the given coaction or point-action, explicit finite bases and the functorialities of [F3]-[F5]; no choice principle is used.