Affine Group Schemes, Hopf Algebras, and Rational Representations — Examples

The coordinate ring of $T$ is the $n$-fold tensor product ${⨂}_{i=1}^{n}k[t_i,t_i^{-1}]=k[t_1^{\pm 1},\dots ,t_n^{\pm 1}]$ by [F2]. Its componentwise operations satisfy the group identities, and [F5] supplies finite type; hence $T$ is a group scheme and $A$ is a commutative Hopf algebra with structure maps transposed from the group operations (The coordinate ring of an affine group scheme is a commutative Hopf algebra); the product of the ${\mathbf{G}}_m$ structures has comorphisms $\mathrm{\Delta }(t_i)=t_i\otimes t_i$, $\epsilon (t_i)=1$ and $S(t_i)=t_i^{-1}$ on each Laurent generator, extended multiplicatively. These images are units, so they define $k$-algebra homomorphisms out of $A$, and the Hopf axioms hold because they hold componentwise for $k[t_i,t_i^{-1}]$ by [F1]. Points satisfy $T(R)={\mathbf{G}}_m(R{)}^n=(R^{\times }{)}^n$ with pointwise multiplication.

Put $I=(t_1^{r_1}-1,\dots ,t_n^{r_n}-1)\subseteq A$. For each $i$ one has $\mathrm{\Delta }(t_i^{r_i}-1)=t_i^{r_i}\otimes t_i^{r_i}-1=(t_i^{r_i}-1)\otimes t_i^{r_i}+1\otimes (t_i^{r_i}-1)\in I{\otimes }_{k}A+A{\otimes }_{k}I$, also $\epsilon (t_i^{r_i}-1)=1-1=0$ and $S(t_i^{r_i}-1)=t_i^{-r_i}-1=-t_i^{-r_i}(t_i^{r_i}-1)\in I$; hence $I$ is a Hopf ideal by [F3], and $A\mathrm{/}I$ carries the quotient Hopf algebra structure with $A\to A\mathrm{/}I$ a Hopf morphism.

By [F3] the quotient map gives a closed immersion ${\mu }_{r_1}\times \cdots \times {\mu }_{r_n}:=\mathrm{Spec}(A\mathrm{/}I)\hookrightarrow \mathrm{Spec}A=T$, and the quotient Hopf structure gives its group operations. By [F5] it is of finite type, and the inclusion preserves the group operations, hence is a closed subgroup scheme. Its $R$-points are the algebra maps $A\mathrm{/}I\to R$, equivalently the tuples $(a_1,\dots ,a_n)$ with $a_i\in R^{\times }$ and $a_i^{r_i}=1$, and the group law is the restriction of the pointwise product of $T(R)$; this is exactly ${\mu }_{r_1}(R)\times \cdots \times {\mu }_{r_n}(R)$ with ${\mu }_r(R)=\{a\in R^{\times }:a^r=1\}$.

Suppose $\mathrm{char}k=p>0$ and take $n=1$, $r_1=p$. By [F4], $t^p-1=(t-1{)}^p$ in $k[t]$, so $k[t,t^{-1}]\mathrm{/}(t^p-1)\cong k[t]\mathrm{/}(t^p-1)\cong k[s]\mathrm{/}(s^p)$ with $s=t-1$; the class $s$ is a nonzero nilpotent, so ${\mu }_p$ is nonreduced. By [F4] again, ${\mu }_p(k)=\{1\}$, so the $k$-points are trivial while the coordinate ring $k[s]\mathrm{/}(s^p)$ is not the coordinate ring $k$ of the trivial group scheme; in particular the scheme, and with it the group scheme, is not determined by its $k$-points. Every computation used the explicit Laurent generators and finitely many relations.

The map $\rho$ is a comodule structure: by [F1], $(\rho \otimes \mathrm{id})\rho (v)={\sum }_m{v}_m\otimes t^m\otimes t^m=(\mathrm{id}\otimes \mathrm{\Delta })\rho (v)$ and $(\mathrm{id}\otimes \epsilon )\rho (v)={\sum }_m{v}_m\otimes 1=v$.

Conversely, let $\rho$ be any comodule structure on the finite-dimensional space $V$. Writing $\rho (v)={\sum }_m{v}_m\otimes t^m$ with finitely many nonzero terms, coassociativity and [F1] give ${\sum }_m\rho (v_m)\otimes t^m={\sum }_m{v}_m\otimes t^m\otimes t^m$, so ${\sum }_m(\rho (v_m)-v_m\otimes t^m)\otimes t^m=0$; by [F3] each $\rho (v_m)=v_m\otimes t^m$, so $v_m\in V_m$ with $V_m=\{v:\rho (v)=v\otimes t^m\}$. The counit identity gives $v={\sum }_m{v}_m$, so the $V_m$ span $V$, and a relation ${\sum }_m{u}_m=0$ with $u_m\in V_m$ gives ${\sum }_m{u}_m\otimes t^m=0$ and hence $u_m=0$ by [F3]; thus $V={⨁}_m{V}_m$. Applying the construction to a finite basis of $V$, the union of the finitely many supports of its coaction expressions spans $V$ by weight vectors; hence all other $V_m$ vanish. Each $V_m$ is a linear subspace because its defining condition is linear.

By [F2] the associated representation is $r_R(g)(v\otimes 1)=(\mathrm{id}\otimes g)\rho (v)={\sum }_m{v}_m\otimes g^m$ for $g\in R^{\times }$; since $(v\otimes r)$ is mapped to $r\cdot {\sum }_m{v}_m\otimes g^m$, the action on each $V_m{\otimes }_{k}R$ is multiplication by the scalar $g^m$, and $r_R$ is a group homomorphism because $g\mapsto g^m$ is multiplicative; hence $\rho$ defines the stated rational representation.

Let $e_1,\dots ,e_n$ be a basis of weight vectors with $e_i\in V_{m_i}$, so that $a_{ij}={\delta }_{ij}{t}^{m_i}$; by [F2] the matrix coefficients satisfy $\mathrm{\Delta }(a_{ij})={\sum }_l{a}_{il}\otimes a_{lj}$ and $\epsilon (a_{ij})={\delta }_{ij}$, and the associated comorphism sends $x_{ij}$ to ${\delta }_{ij}{t}^{m_i}$. For $V=0$, use ${\mathrm{GL}}_0=\mathrm{Spec}k$ with comorphism $k\to k[t,t^{-1}]$ and no matrix entries. Together with steps 1.2 and 2.1 this proves both directions of the stated dictionary, using only the explicit monomial basis of $k[t,t^{-1}]$ and finitely many weight spaces.