Affine finite-type group schemes have faithful finite-dimensional representations

Fix $v\in M$ and write $\rho (v)={\sum }_{i=1}^r{v}_i\otimes a_i$ with the $a_i$ linearly independent. Let $C\subset A$ be the finite-dimensional space containing the $a_i$ and every second-factor coefficient of the finitely many $\mathrm{\Delta }(a_i)$. Choose linear functionals ${\lambda }_j:C\to k$ with ${\lambda }_j(a_i)={\delta }_{ij}$ by extending this finite independent list to a basis of $C$. Apply $1\otimes 1\otimes {\lambda }_j$ to coassociativity. It gives $\rho (v_j)={\sum }_i{v}_i\otimes (1\otimes {\lambda }_j)\mathrm{\Delta }(a_i)$. Thus $W={\mathrm{span}}_k\{v_1,\dots ,v_r\}$ is a finite-dimensional subcomodule. The counit gives $v={\sum }_iϵ(a_i)v_i\in W$. Finite sums of subcomodules are subcomodules, proving the filtered-union assertion. Both sides lie in $M\otimes A\otimes C$, so all these contractions are defined on finite coefficient spaces and require only finite choices.

Apply step 1.1 to the comodule $(A,\mathrm{\Delta })$ and to a finite algebra-generating list for $A$. Taking the sum gives a finite-dimensional subcomodule $V$ containing that list. In a basis $e_1,\dots ,e_n$ of $V$, write $\mathrm{\Delta }(e_j)={\sum }_i{e}_i\otimes a_{ij}$. Coassociativity and counit give $\mathrm{\Delta }(a_{ij})={\sum }_l{a}_{il}\otimes a_{lj}$ and $ϵ(a_{ij})={\delta }_{ij}$. The antipode gives an inverse for the matrix $(a_{ij})$. Hence these entries define a homomorphism $G\to \mathrm{GL}(V)$: for any $k$-algebra $R$ and point $g:A\to R$, its matrix is $(g(a_{ij}))$. This is the right regular action $(gf)(x)=f(xg)$, formulated on all algebras.

The image of $k[\mathrm{GL}(V)]\to A$ contains every $a_{ij}$. Applying $ϵ\otimes 1$ to $\mathrm{\Delta }(e_j)$ gives $e_j={\sum }_iϵ(e_i)a_{ij}$, so that image contains $V$, hence the algebra generators of $A$. The ring map is onto and therefore the group morphism is a closed immersion. In particular it is injective on $R$-points for every $R$, including nonreduced algebras.