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Conjugacy of Borel subgroups and of maximal tori over an algebraically closed field

Statement

Assume the Axiom of Choice. Let $k$ be an algebraically closed field and let $G$ be a connected affine group variety over $k$ , i.e. a smooth connected affine algebraic group of finite type over $k$ (Smooth morphism of schemes, Group schemes of finite type over a field). Then: (a) for every Borel subgroup $B \subseteq G$ the quotient $G / B$ is complete; (b) any two Borel subgroups of $G$ are conjugate by an element of $G (k)$ ; (c) any two maximal tori of $G$ are conjugate by an element of $G (k)$ ; (d) any two Borel pairs are conjugate. The assertions here are made under the stated smooth connected and algebraically closed hypotheses; no necessity claim for each hypothesis is made.

Facts & Assumptions

Given: The Axiom of Choice, an algebraically closed field $k$ , and a smooth connected affine algebraic $k$ -group $G$ .

[F1]

Borel subgroups are smooth connected solvable subgroup varieties, geometrically maximal among such subgroups. Over algebraically closed $k$ , choosing a smooth connected solvable subgroup of largest dimension gives a Borel: a strict inclusion of smooth connected subgroup varieties increases dimension, since they are irreducible. Conjugation preserves this class. A closed subscheme of a smooth finite-type scheme containing all its $k$ -points is the whole scheme, by schematic density. (Borel subgroups, maximal tori and Borel pairs, Smooth morphism of schemes, Rational points of smooth finite-type schemes over a separably closed field are schematically dense)

[F2]

Assume AC. For a Borel subgroup $B_{0}$ of largest possible dimension, the quotient $G / B_{0}$ is a nonempty complete finite-type $k$ -scheme, and the fppf quotient $G / B$ is representable by a separated finite-type $k$ -scheme for every closed subgroup scheme $B$ , with quotient morphism faithfully flat and locally of finite presentation; the left translation action of $G$ on $G / B$ is rational and restricts to an action of any closed subgroup. Every $k$ -point of $G / B$ lifts to $G (k)$ : its fibre is nonempty and finite type by faithful flatness and finite presentation; a maximal ideal in a nonempty affine chart exists under AC and has residue field $k$ by the weak Nullstellensatz. (The quotient of a connected group by a Borel subgroup of maximal dimension is complete, Homogeneous spaces of smooth affine groups are separated schemes, A faithfully flat orbit map represents the coset quotient sheaf, Smooth orbits are locally closed and their orbit maps are faithfully flat over every field, In a nonzero commutative ring, every proper ideal is contained in a maximal ideal, Over an algebraically closed field, every maximal ideal is an evaluation ideal)

[F3]

Assume AC. Let $H$ be a smooth connected solvable affine algebraic group over $k$ and let $X$ be a nonempty complete finite-type $k$ -scheme with a rational action of $H$ . Then there is a point $x \in X (k)$ fixed by $H (k)$ . (Borel fixed point theorem for complete schemes)

[F4]

Assume AC. Let $H$ be a smooth connected solvable affine algebraic group over $k$ and let $T \subseteq H$ be a maximal torus. Then $H = H_{u} ⋊ T$ , and any two maximal tori of $H$ are conjugate by an element of $H (k)$ ; equivalently every closed subgroup of multiplicative type of $H$ is conjugate into $T$ . (Maximal tori of a smooth connected solvable group are conjugate)

[F5]

A maximal torus $T$ of $G$ is a smooth connected commutative subgroup variety. Among smooth connected solvable subgroup varieties containing $T$ , choose one of largest dimension. No strictly larger smooth connected solvable subgroup can contain it, so it is a Borel. Thus each maximal torus of $G$ is contained in a Borel; no assertion is made that a maximal torus of an arbitrary solvable subgroup is maximal in $G$ . (Borel subgroups, maximal tori and Borel pairs, Groups of multiplicative type and tori)

Proof

Given: The Axiom of Choice, an algebraically closed field $k$ , and a smooth connected affine $k$ -group $G$ .

1.1F1F2

By [F1] choose a Borel subgroup $B_{0}$ of largest possible dimension; [F2] makes $G / B_{0}$ a nonempty complete finite-type $k$ -scheme with a rational $G$ -action. This proves (a) in the case $B = B_{0}$ .

1.2F1F2F3

Let $B$ and $B^{'}$ be Borel subgroups with $B$ of largest possible dimension. By [F1] and [F3] the smooth connected solvable group $B^{'}$ acts on the complete variety $G / B$ , so there is a fixed point in $(G / B) (k)$ . Its fibre under the faithfully flat finite-type map $G \to G / B$ is nonempty and has a $k$ -point by the weak Nullstellensatz (choose a maximal ideal in a nonempty affine chart). Thus the fixed point is represented by $g B$ for some $g \in G (k)$ , with $B^{'} g B = g B$ ; the closed subgroup $B^{'} \cap g B g^{- 1}$ contains every point of $B^{'} (k)$ , so smoothness and schematic density [F1] give $B^{'} \subseteq g B g^{- 1}$ scheme-theoretically, and $g B g^{- 1}$ is a connected solvable closed subgroup scheme by [F1]. Since $B^{'}$ is a Borel subgroup, maximality gives $B^{'} = g B g^{- 1}$ .

2.1F1F2step 1.1step 1.2

In particular every Borel subgroup is conjugate to the largest-dimensional one, so all Borel subgroups have the same dimension $\dim B_{0}$ and every Borel subgroup is of largest possible dimension. Hence (a) holds for every Borel subgroup: $G / B ≅ G / B_{0}$ as $k$ -schemes (conjugate subgroups give isomorphic quotients), so $G / B$ is complete. This proves (a) and (b).

3.1F4F5step 2.1

For (c), let $T, T^{'}$ be maximal tori of $G$ . By [F5] there are Borel subgroups $B \supseteq T$ and $B^{'} \supseteq T^{'}$ ; by (b) there is $g \in G (k)$ with $g B^{'} g^{- 1} = B$ , so $g T^{'} g^{- 1} \subseteq B$ . Both $T$ and $g T^{'} g^{- 1}$ are maximal tori of the smooth connected solvable group $B$ : they are tori of $B$ , and a torus of $G$ strictly containing one of them would strictly contain a maximal torus of $G$ . By [F4] applied to $B$ there is $b \in B (k)$ with $b g T^{'} g^{- 1} b^{- 1} = T$ , so $T$ and $T^{'}$ are conjugate by $b g \in G (k)$ .

4.1F4step 2.1step 3.1

For (d), let $(B, T)$ and $(B^{'}, T^{'})$ be Borel pairs. By (b) choose $g \in G (k)$ with $g B^{'} g^{- 1} = B$ ; then $g T^{'} g^{- 1}$ and $T$ are maximal tori of the smooth connected solvable group $B$ , by the same maximality argument as in [step 3.1], so by [F4] there is $b \in B (k)$ with $b g T^{'} g^{- 1} b^{- 1} = T$ . Then $(b g) (B^{'}, T^{'}) (b g)^{- 1} = (B, T)$ , so any two Borel pairs are conjugate.

5.1step 2.1step 3.1step 4.1∎

Therefore (a) holds for every Borel subgroup, Borel subgroups are pairwise conjugate, maximal tori are pairwise conjugate, and Borel pairs are pairwise conjugate, as claimed.

Depends on

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Sources

J. S. Milne, Algebraic Groups (corrected 2022 printing, Cambridge University Press) (standard reference, not scraped)
Florian Herzig, Linear Algebraic Groups (University of Toronto lecture notes, 2013) (standard reference, not scraped)
J. S. Milne, Algebraic Groups (v2.00, 20 December 2015 author-hosted preliminary edition) (standard reference, not scraped)