Unipotent and Solvable Groups and Borel Fixed Points — Examples

The coordinate algebra of $U_n$ is coconnected by the explicit polynomial filtration in [F2], so the algebraic implication(c) implies(a) gives unipotence without a geometric closed-subgroup conversion. Its explicit central series has additive quotients by [F2]. As a scheme $U_n\cong {\mathbf{A}}_k^{n(n-1)\mathrm{/}2}$ by its upper entries, so it is smooth. Its polynomial coordinate ring is a domain, hence its spectrum is irreducible and connected, including $n=1$, where it is the trivial group.

Formula [F3] is a natural group-functor isomorphism ${\mathbf{G}}_a\cong U_2$. Thus ${\mathbf{G}}_a$ is smooth connected unipotent, with coordinate Hopf algebra $k[x]$. This uses the explicit coordinate construction, not a choice of faithful representation.

In characteristic $p$, the Frobenius homomorphism $x\mapsto x^p$ has kernel ${\alpha }_p=\mathrm{Spec}k[x]\mathrm{/}(x^p)$. Its coordinate Hopf algebra is the explicit surjective quotient in [F4], hence coconnected and unipotent by [F2]. The class of $x$ is nonzero nilpotent, so this scheme is not reduced and therefore not smooth over $k$.

The homomorphism $x\mapsto x^p-x$ has kernel $\mathrm{Spec}k[x]\mathrm{/}(x^p-x)$, the constant additive group ${\mathbf{F}}_p$ by [F4]. It is again an explicit coconnected Hopf quotient, hence unipotent by [F2], and has $p>1$ disjoint rational points, so is not connected. This is not the image of the $p$-torsion of ${\mathbf{G}}_a$, which is all of ${\mathbf{G}}_a$ in this characteristic.

Thus the complete Example is verified: the smooth connected $U_n$ have the stated filtration and central quotients, ${\mathbf{G}}_a=U_2$ is its first positive-dimensional case, and ${\alpha }_p$ and constant ${\mathbf{F}}_p$ exhibit nonsmooth and disconnected unipotent groups. Every unipotence assertion here follows from an explicitly presented Hopf algebra, so no AC-qualified geometric embedding or quotient conversion is used.

A Borel subgroup $B$ of $G$ is solvable, hence trigonalizable by [F1]: there is a basis $v_1,\dots ,v_n$ of $V$ in which $B$ acts through upper triangular matrices, so $B\subseteq T_n$ relative to that basis; maximality of $B$ gives $B=T_n$ in that basis. Conversely $T_n$ is a Borel subgroup by [F1]: it is connected solvable, and a connected solvable subgroup strictly containing $T_n$ would be trigonalizable in a basis of its own, contradicting maximality of the dimension of $T_n$. Hence the Borel subgroups of $G$ are exactly the conjugates of $T_n$, and by [F2] any two of them are conjugate by an element of $G(k)$.

A conjugate $g{T}_n{g}^{-1}$ is exactly the stabilizer of the flag $g{F}_{\mathrm{s}\mathrm{t}\mathrm{d}}$, where $F_{\mathrm{s}\mathrm{t}\mathrm{d}}$ is the standard flag with $i$-th step $⟨e_1,\dots ,e_i⟩$: an element $h$ stabilizes the flag $F_{\mathrm{s}\mathrm{t}\mathrm{d}}$ if and only if its matrix is upper triangular, so $g{T}_n{g}^{-1}$ is the scheme-theoretic stabilizer of $g{F}_{\mathrm{s}\mathrm{t}\mathrm{d}}$, and conversely every maximal flag is $g{F}_{\mathrm{s}\mathrm{t}\mathrm{d}}$ for some $g$ because $G$ acts transitively on bases and maximal flags correspond to bases. Therefore the Borel subgroups of $G$ are exactly the stabilizers of maximal flags in $V$, and each is a semidirect product $U_n⋊D_n$ by [F1].

By [F4] the standard representation makes $G$, and hence its subgroup $T_n$, act rationally on the space of lines $\mathbf{P}(V)$; an upper triangular matrix satisfies $g{e}_1=g_{11}{e}_1$ with $g_{11}\in k^{\times }$, so $g$ preserves the line $⟨e_1⟩$. Thus $⟨e_1⟩$ is a line fixed by all of $T_n(k)$: the Borel fixed point theorem is realized concretely, the fixed line being the first step of the standard flag.

By [F3] the orbit of the standard flag is all of $\mathrm{F}\mathrm{l}(V)$, its scheme-theoretic stabilizer is $T_n$, and the orbit map $\mathrm{G}\mathrm{L}(V)\to \mathrm{F}\mathrm{l}(V)$ is faithfully flat and locally of finite presentation. The coset-quotient proposition [F5] therefore applies and shows that $\mathrm{F}\mathrm{l}(V)$ represents the fppf quotient $\mathrm{G}\mathrm{L}(V)\mathrm{/}{T}_n$. Since $\mathrm{F}\mathrm{l}(V)$ is complete by [F3], this illustrates the completeness of $G\mathrm{/}B$ from [F2] in the special case of $\mathrm{G}\mathrm{L}(V)$.

Collecting: the Borel subgroups of $\mathrm{G}\mathrm{L}(V)$ are the flag stabilizers, equivalently the conjugates of $T_n=U_n⋊D_n$; the standard maximal flag provides a $T_n$-fixed line in $\mathbf{P}(V)$, and $\mathrm{G}\mathrm{L}(V)\mathrm{/}{T}_n\cong \mathrm{F}\mathrm{l}(V)$ is the complete flag variety.

The morphism $\alpha :{\mathbf{G}}_a{\times }_{k}X\to X=\mathrm{Spec}k[x]$, dual to $k[x]\to k[t]{\otimes }_{k}k[x]=k[t,x]$, $x\mapsto x+t$, defines an action: on $R$-points it is $(a,z)\mapsto z+a$, and the identities $0\cdot z=z$ and $a\cdot (b\cdot z)=z+b+a=(a+b)\cdot z$ hold in every $k$-algebra $R$. Hence ${\mathbf{G}}_a$ acts on $X$ by translation, algebraically.

The group ${\mathbf{G}}_a$ is smooth connected unipotent by [F1], and is commutative since addition commutes on every algebra-valued point; its commutator is the identity, so its derived series terminates after one step and it is solvable. The scheme $X$ is nonempty of finite type over $k$, and it is not complete by [F3].

The action has no fixed point: for $z\in X(k)=k$ and $a\in k$, the equation $a\cdot z=z$ reads $z+a=z$, i.e. $a=0$. Hence for every $z\in X(k)$ and every $a\mathrm{\ne }0$ the translate differs from $z$, and $X(k)$ contains no point fixed by all of ${\mathbf{G}}_a(k)$.

Therefore the statement refuted is false: the action of the smooth connected solvable group ${\mathbf{G}}_a$ on the nonempty finite-type scheme ${\mathbf{A}}_k^1$ has no fixed point, so the completeness hypothesis in the Borel fixed point theorem is indispensable even in this minimal example, in contrast with [F4].