Prime-to-characteristic torsion bound for affine commutative groups

By [F1] fix a closed immersion $j:N\to {\mathrm{GL}}_m$ of $k$-group schemes, and for $g\in N(k)$ let $M(g)\in {\mathrm{GL}}_m(k)$ be its matrix. Since $j$ is a group homomorphism and $N$ is commutative, $M(g)M(h)=M(gh)=M(h)M(g)$, so the $k$-span $A\subseteq {\mathrm{End}}_k(k^m)$ of the set $\{M(g):g\in N(k)\}$ is a finite-dimensional commutative subalgebra. Let $W\subseteq k^m$ be a nonzero $A$-stable subspace of minimal dimension. If some $A$-element has an eigenspace $C$ inside $W$ with $0\mathrm{\ne }C\mathrm{\ne }W$, then $C$ is a smaller nonzero $A$-stable subspace, a contradiction; if every element of $A$ acts as a scalar on $W$, then every line in $W$ is $A$-stable; hence in either case ${\dim }_{k}W=1$ and $W$ is a common eigenvector. Applying this argument to $k^m$ and then to the successive quotients by the lines produced yields a filtration $0=V_m\subset V_{m-1}\subset \cdots \subset V_0=k^m$ with $M(g)V_i\subseteq V_i$ for all $i$ and all $g\in N(k)$; in a basis adapted to this filtration every $M(g)$ is upper triangular.

Put $P=k[x_{ij}]$, let $J=\ker (P\to k[N])$ be the kernel of the map sending each matrix coordinate to its pullback under $j$, and write $\delta =\det (x_{ij})$. The map $P_{\delta }=k[x_{ij},{\delta }^{-1}]\to k[N]$ is surjective because $j$ is a closed immersion into ${\mathrm{GL}}_m$, and its kernel is $J_{\delta }$. For every $i>j$, $x_{ij}$ vanishes at every common zero of $J$ and $z\delta -1$ in the polynomial ring $P[z]$: such a zero is an invertible matrix satisfying the equations of the closed subscheme $N\subseteq {\mathrm{GL}}_m$, hence is a $k$-point of $N$ and is upper triangular by step 1.1. Applying [F3] in $P[z]$ gives $x_{ij}\in \sqrt{(J,z\delta -1)}$. After quotienting by $z\delta -1$ and identifying $P[z]\mathrm{/}(z\delta -1)\cong P_{\delta }$, this says $x_{ij}\in \sqrt{J_{\delta }}$. Since $P_{\delta }\mathrm{/}{J}_{\delta }\cong k[N]$ is a domain by [F2], $J_{\delta }$ is radical and therefore $x_{ij}\in J_{\delta }$. Thus the coordinate functions below the diagonal vanish scheme-theoretically on $N$, so $j$ factors through the closed subgroup scheme $B\subseteq {\mathrm{GL}}_m$ of upper triangular matrices. In particular the diagonal characters ${\chi }_i:=u_{ii}$ (the images in $k[N]$ of the diagonal coordinate functions) are units and satisfy $\mathrm{\Delta }({\chi }_i)={\chi }_i\otimes {\chi }_i$ in the Hopf algebra $k[N]$.

Let $R\subseteq k[N]$ be the subalgebra generated by ${\chi }_1^{\pm 1},\dots ,{\chi }_m^{\pm 1}$. It is a Hopf subalgebra because the ${\chi }_i$ are group-like units. Its group-like elements are the monomials ${\chi }^{(n)}={\prod }_i{\chi }_i^{n_i}$ for $n\in {\mathbb{Z}}^m$, and distinct group-like elements of a Hopf algebra over a field are linearly independent: a shortest nontrivial relation ${\sum }_{i\in S}{c}_i{h}_i=0$ with distinct $h_i$ and all $c_i\mathrm{\ne }0$ becomes, after applying $\mathrm{\Delta }$ and subtracting the tensor product of the relation with $h_s$, the relation ${\sum }_{i\in S\setminus \{s\}}{c}_i{h}_i\otimes (h_i-h_s)=0$; the $h_i$ for $i\mathrm{\ne }s$ are linearly independent by minimality of $S$, so $h_i=h_s$ for all $i$, a contradiction. Hence $R=k[M]$ where $M={\mathbb{Z}}^m\mathrm{/}L$ is the quotient of ${\mathbb{Z}}^m$ by the relations $L=\{n:{\chi }^{(n)}=1\}$, with $k$-basis the distinct ${\chi }^{(n)}$; by [F4] this is the coordinate Hopf algebra of the diagonalizable group $D_k(M)$, and $N\to D_k(M)$ is the morphism dual to the inclusion $R\subseteq k[N]$.

The ring $k[M]$ is a domain, being a subalgebra of the domain $k[N]$ by [F2]. If $M$ had a nonzero element $n$ of finite order $d>1$, then $({\chi }^{(n)}-1)({\chi }^{(n{)}^{d-1}}+\cdots +{\chi }^{(n)}+1)={\chi }^{(n{)}^d}-1={\chi }^{(dn)}-1=0$ in $k[M]$, and both factors are nonzero because distinct group-like elements are linearly independent by step 3.1 and the second factor is a sum of distinct group-likes with coefficient $1$. This contradicts that $k[N]$ is a domain, so $M$ is torsion-free; as a finitely generated torsion-free abelian group, $M\cong {\mathbb{Z}}^t$ for some $t\ge 0$. Since $R=k[M]$ is a finite-type $k$-domain contained in $k[N]$, its fraction field embeds in $\mathrm{Frac}(k[N])=k(N)$, whence $t=\dim k[M]={\mathrm{trdeg}}_k\mathrm{Frac}(k[M])\le {\mathrm{trdeg}}_{k}k(N)=\dim k[N]=a$ by [F5].

Let $\phi :N(k)\to D_k(M)(k)=\mathrm{Hom}(M,k^{\times })$ be the homomorphism induced by $N\to D_k(M)$. A point $g\in N(k)$ lies in $\ker \phi$ precisely when ${\chi }_i(g)=1$ for all $i$, that is, precisely when its matrix $M(g)$ is upper unitriangular; write $M(g)=I+U$ with $U$ strictly upper triangular, so $U^m=0$. If $\mathrm{char}k=0$ and $g$ has finite order $d$, then $(M(g)-I{)}^m=0$ and $M(g{)}^d=I$, so the minimal polynomial of $M(g)$ divides both $(X-1{)}^m$ and $X^d-1$; in characteristic zero $X^d-1$ is squarefree, so the gcd is $X-1$ and $M(g)=I$. If $\mathrm{char}k=p>0$ and $p^e\ge m$, then $M(g{)}^{p^e}=I+U^{p^e}=I$, so $M(g)$ has $p$-power order. Hence an element of $N(k)$ of order dividing ${\mathrm{\ell }}^{\nu }$ with $\mathrm{\ell }\mathrm{\ne }p$ that lies in $\ker \phi$ equals the identity: it has order dividing both ${\mathrm{\ell }}^{\nu }$ and a $p$-power, hence order $1$. Consequently the restriction $N[{\mathrm{\ell }}^{\nu }](k)\to D_k(M)(k)[{\mathrm{\ell }}^{\nu }]$ is injective.

Since $M\cong {\mathbb{Z}}^t$, evaluation gives $D_k(M)(k)=\mathrm{Hom}({\mathbb{Z}}^t,k^{\times })\cong (k^{\times }{)}^t$, whose ${\mathrm{\ell }}^{\nu }$-torsion is $\mathrm{Hom}({\mathbb{Z}}^t,{\mu }_{{\mathrm{\ell }}^{\nu }}(k))\cong (\mathbb{Z}\mathrm{/}{\mathrm{\ell }}^{\nu }\mathbb{Z}{)}^t$ because $k$ is algebraically closed and $\mathrm{\ell }\mathrm{\ne }\mathrm{char}k$, of cardinal ${\mathrm{\ell }}^{\nu t}$. By step 5.1, $\mathrm{\mid }N[{\mathrm{\ell }}^{\nu }](k)\mathrm{\mid }\le \mathrm{\mid }{D}_k(M)(k)[{\mathrm{\ell }}^{\nu }]\mathrm{\mid }={\mathrm{\ell }}^{\nu t}$, and $t\le a$ by step 4.1, so $\mathrm{\mid }N[{\mathrm{\ell }}^{\nu }](k)\mathrm{\mid }\le {\mathrm{\ell }}^{\nu a}$. AC is used through [F2] and [F3]; DC is inherited from the faithful-representation supplier chain [F1] as declared.