A finite-type affine algebraic group has a faithful rational representation

Let $a\in A$ and let $\mathrm{\Delta }(a)={\sum }_{i=1}^n{u}_i\otimes v_i$ be a finite expression in which $u_1,\dots ,u_n$ are linearly independent; such an expression exists because $\mathrm{\Delta }(a)$ is a finite sum, and deleting redundant terms (replacing $v_i$ by $v_i+c_i{v}_n$ when $u_n={\sum }_{i<n}{c}_i{u}_i$) keeps the sum equal to $\mathrm{\Delta }(a)$. Put $V_a=\mathrm{span}(v_1,\dots ,v_n)$, a finite-dimensional subspace of $A$ with $a\in V_a$, since $a=(\epsilon \otimes \mathrm{id})\mathrm{\Delta }(a)={\sum }_i\epsilon (u_i)v_i$.

In the situation of step 1.1 one has $\mathrm{\Delta }(V_a)\subseteq A\otimes V_a$: writing $q:A\to A\mathrm{/}{V}_a$ for the quotient map, coassociativity gives ${\sum }_i\mathrm{\Delta }(u_i)\otimes v_i={\sum }_i{u}_i\otimes \mathrm{\Delta }(v_i)$, whence $q$ applied to the third factor yields ${\sum }_i{u}_i\otimes (\mathrm{id}\otimes q)\mathrm{\Delta }(v_i)=0$, and the linear independence of the $u_i$ forces $(\mathrm{id}\otimes q)\mathrm{\Delta }(v_i)=0$ for each $i$, that is $\mathrm{\Delta }(v_i)\in A\otimes V_a$.

Choose a finite generating family $g_1,\dots ,g_m$ of $A$ [F1] and, for each $k$, a finite-dimensional subspace $V_{g_k}$ with $g_k\in V_{g_k}$ and $\mathrm{\Delta }(V_{g_k})\subseteq A\otimes V_{g_k}$ as in steps 1.1 and 2.1. Set $V=\mathbb{C}\cdot 1+V_{g_1}+\cdots +V_{g_m}$. Then $V$ is finite-dimensional, contains $1$ and every $g_k$, and satisfies $\mathrm{\Delta }(V)\subseteq A\otimes V$, because $\mathrm{\Delta }(1)=1\otimes 1$ and $\mathrm{\Delta }$ is additive.

Put $N=V\cap \ker \epsilon$, so that $V=\mathbb{C}\cdot 1\oplus N$ because $\epsilon (1)=1$ makes $\epsilon {\mathrm{\mid }}_V:V\to \mathbb{C}$ surjective. Choose a basis $e_1=1,e_2,\dots ,e_n$ of $V$ with $e_i\in N$ for $i\ge 2$.

Write $\mathrm{\Delta }(e_j)={\sum }_i{b}_{ij}\otimes e_i$. These coefficients are unique because $e_i$ is a basis of the second factor. For this left-coaction convention the associated left action evaluates at the inverse: $r_R(g)v=(g\circ S\otimes \mathrm{id})\rho (v)$. Put $a_{ij}=S(b_{ij})$, the matrix coefficients of that action.

Applying the two counit identities gives $\epsilon (b_{ij})={\delta }_{ij}$ and $b_{1j}=e_j$. Coassociativity, with all three tensor factors retained, gives ${\sum }_k\mathrm{\Delta }(b_{kj})\otimes e_k={\sum }_{i,k}{b}_{ij}\otimes b_{ki}\otimes e_k$, hence $\mathrm{\Delta }(b_{kj})={\sum }_i{b}_{ij}\otimes b_{ki}$.

The antipode is the comorphism of inversion on $G(R)={\mathrm{Hom}}_{\mathbb{C}\text{-alg}}(A,R)$: the two antipode identities give the inverse under convolution. Thus $(g^{-1}{)}^{-1}=g$ and $(gh{)}^{-1}=h^{-1}{g}^{-1}$ imply $S^2=\mathrm{id}$, $\epsilon S=\epsilon$ and $\mathrm{\Delta }S=\tau (S\otimes S)\mathrm{\Delta }$, where $\tau$ switches factors. These are identities of coordinate-ring maps, as can be checked on the universal $A$-point and the two universal $A\otimes A$-points. Applying them to step 6.1 gives $\mathrm{\Delta }(a_{kj})={\sum }_i{a}_{ki}\otimes a_{ij}$, $\epsilon (a_{ij})={\delta }_{ij}$ and $a_{1j}=S(e_j)$.

Applying $m(S\otimes \mathrm{id})$ and $m(\mathrm{id}\otimes S)$ to the identity of step 7.1 and using the Hopf algebra identities gives, for all $k,j$, ${\sum }_{i}S(a_{ki})a_{ij}=\epsilon (a_{kj})={\delta }_{kj}$ and ${\sum }_i{a}_{ki}S(a_{ij})=\epsilon (a_{kj})={\delta }_{kj}$; hence the matrix $(a_{ij})$ over the commutative ring $A$ is invertible with two-sided inverse $(S(a_{ij}))$, and $\det (a_{ij})$ is a unit of $A$.

Let $B=\mathcal{O}(GL(V))=\mathbb{C}[t_{ij}:1\le i,j\le n][\det (t_{ij}{)}^{-1}]$ with comultiplication ${\mathrm{\Delta }}_B(t_{kj})={\sum }_i{t}_{ki}\otimes t_{ij}$ and counit ${\epsilon }_B(t_{ij})={\delta }_{ij}$. The assignments $t_{ij}\mapsto a_{ij}$ and $\det (t_{ij}{)}^{-1}\mapsto \det (a_{ij}{)}^{-1}$ define a $\mathbb{C}$-algebra homomorphism $\mathrm{\Phi }:B\to A$ by step 8.1, and $\mathrm{\Phi }$ is a coalgebra homomorphism by step 7.1 and step 6.1; hence $\mathrm{\Phi }$ is a homomorphism of Hopf algebras and $\mathrm{Spec}(\mathrm{\Phi }):G\to GL(V)$ is a homomorphism of affine group schemes.

The image of $\mathrm{\Phi }$ contains $a_{1j}=S(e_j)$ for every $j$ by step 7.1, hence contains $S(V)$ and therefore $S(g_1),\dots ,S(g_m)$. Since $S$ is an involutive algebra automorphism, these also generate $A$; since the image of a ring homomorphism is a subring, $\mathrm{im}\mathrm{\Phi }=A$, so $\mathrm{\Phi }$ is surjective.

By step 10.1 and [F4], $A\cong B\mathrm{/}\ker \mathrm{\Phi }$ and the morphism $\mathrm{Spec}(\mathrm{\Phi })$ is, under this isomorphism, the morphism $\mathrm{Spec}(B\mathrm{/}\ker \mathrm{\Phi })\to \mathrm{Spec}B$ induced by the quotient map; by [F3] that morphism is a closed immersion, and by [F2] the morphism of affine schemes attached to $\mathrm{\Phi }$ is $\mathrm{Spec}(\mathrm{\Phi })$ up to this isomorphism. Hence $G\to GL(V)$ is a closed immersion.

The coaction $\rho =\mathrm{\Delta }{\mathrm{\mid }}_V:V\to A\otimes V$ is a finite-dimensional rational representation in the sense of the statement: its target lies in $A\otimes V$ by step 3.1, coassociativity of $\mathrm{\Delta }$ gives $(\mathrm{\Delta }\otimes \mathrm{id})\rho =(\mathrm{id}\otimes \rho )\rho$, and $(\epsilon \otimes \mathrm{id})\rho ={\mathrm{id}}_V$. Under inverse evaluation for a left coaction, the matrix of $r_R(g)$ is $(g(a_{ij}))$, so the comorphism attached to $\rho$ is $\mathrm{\Phi }$, so the representation is faithful (a closed immersion is in particular a monomorphism of group functors) and $V\mathrm{\ne }0$ because $1\in V$.

Finally, the argument is choice-free: steps 1.1, 4.1 and 5.1 make finitely many finite-dimensional selections, and the only infinite-dimensional linear algebra used is the identification of the second tensor factor with a finite direct sum via the basis $e_1,\dots ,e_n$ of $V$, together with the quotient $A\mathrm{/}{V}_a$ of step 3.1. No basis of $A$ or of any infinite-dimensional space is chosen.