Special issue to the memory of T.A. SpringerHomogeneous varieties under split solvable algebraic groups
Introduction
Throughout this note, we consider algebraic groups and varieties over a field . An algebraic group is split solvable if it admits a chain of closed subgroups such that each is normal in and is isomorphic to the additive group or the multiplicative group . This class features prominently in a series of articles by Rosenlicht on the structure of algebraic groups, see [14], [15], [16]. The final result of this series may be stated as follows (see [16, Thm. 5]):
Theorem 1 Let be a homogeneous variety under a split solvable algebraic group . Then there is an isomorphism of varieties for unique nonnegative integers , .
Here denotes the affine -space, and the punctured affine line.
Rosenlicht’s articles use the terminology and methods of algebraic geometry à la Weil, and therefore have become hard to read. In view of their fundamental interest, many of their results have been rewritten in more modern language, e.g. in the book [5] by Demazure & Gabriel and in the second editions of the books on linear algebraic groups by Borel and Springer, which incorporate developments on “questions of rationality” (see [2], [18]). The above theorem is a notable exception: the case of the group acting on itself by multiplication is handled in [5, Cor. IV.4.3.8] (see also [18, Cor. 14.2.7]), but the general case is substantially more complicated.1
The aim of this note is to fill this gap by providing a proof of Theorem 1 in the language of modern algebraic geometry. As it turns out, this theorem is self-improving: combined with Rosenlicht’s theorem on rational quotients (see [14, Thm. 2], and [1, Sec. 2] for a modern proof) and some “spreading out” arguments, it yields the following stronger version:
Theorem 2 Let be a variety equipped with an action of a split solvable algebraic group . Then there exist a dense open -stable subvariety and an isomorphism of varieties (where , are uniquely determined nonnegative integers and is a variety, unique up to birational isomorphism) such that the resulting projection is the rational quotient by .
By this, we mean that yields an isomorphism , where the left-hand side denotes the function field of and the right-hand side stands for the field of -invariant rational functions on ; in addition, the fibers of are exactly the -orbits.
As a direct but noteworthy application of Theorem 2, we obtain:
Corollary 3 Let be a variety equipped with an action of a split solvable algebraic group . Then is a purely transcendental extension of .
When is algebraically closed, this gives back the main result of [13]; see [3] for applications to the rationality of certain homogeneous spaces.
The proof of Theorem 2 also yields a version of [18, Prop. 14.2.2]:
Corollary 4 Let be a variety equipped with a nontrivial action of . Then there exist a variety , an open immersion and a monic additive polynomial such that for all , and .
Here is said to be additive if it satisfies identically; then acts on via , and is equivariant for this action. If , then we have and hence acts on by translation on . So Corollary 4 just means that every nontrivial -action becomes a trivial -torsor on some dense open invariant subset. On the other hand, if , then is a -polynomial, that is, for some integer and . Thus, the map is an endomorphism of the -group scheme ; conversely, every such endomorphism arises from an additive polynomial , see [5, II.3.4.4]. Thus, Corollary 4 asserts that for any nontrivial -action, there is a dense open invariant subset on which acts by a trivial torsor twisted by such an endomorphism. These twists occur implicitly in the original proof of Theorem 1, see [16, Lem. 3]. 2
This note is organized as follows. In Section 2, we gather background results on split solvable algebraic groups. Section 3 presents further preliminary material, on the quotient of a homogeneous space by the left action of a normal subgroup scheme ; here is a connected algebraic group, and a subgroup scheme. In particular, we show that such a quotient is a torsor under a finite quotient of , if either or and (Lemma 3.4). The more involved case where and is handled in Section 4; we then show that the quotient is a “torsor twisted by an endomorphism” as above (Lemma 4.3). The proofs of our main results are presented in Section 5.
Notation and conventions. We consider schemes over a field of characteristic unless otherwise mentioned. Morphisms and products of schemes are understood to be over as well. A variety is an integral separated scheme of finite type.
An algebraic group is a group scheme of finite type. By a subgroup , we mean a (closed) subgroup scheme. A -variety is a variety equipped with a -action We say that is -homogeneous if is smooth, is geometrically reduced, and the morphism is surjective. If in addition is equipped with a -rational point , then the pair is a -homogeneous space. Then , where denotes the stabilizer, and the image of the neutral element under the quotient morphism .
Given a field extension and a -scheme , the -scheme is denoted by .
We will freely use results from the theory of faithfully flat descent, for which a convenient reference is [7, Chap. 14, App. C].
Section snippets
Split solvable groups
We first recall some basic properties of these groups, taken from [5, IV.4.3] where they are called “groupes -résolubles” (see also [11, §16.g]). Every split solvable group is smooth, connected, affine and solvable. Conversely, every smooth connected affine solvable algebraic group over an algebraically closed field is split solvable (see [5, IV.4.3.4]).
Clearly, every extension of split solvable groups is split solvable. Also, recall that every nontrivial quotient group of is isomorphic to
Quotients of homogeneous spaces by normal subgroups
Let be an algebraic group, a subgroup, and a smooth normal subgroup. Then acts on by conjugation. The semi-direct product defined by this action (as in [11, Sec. 2.f]) is equipped with a homomorphism to , with schematic image the subgroup . Recall that and . Denote by the quotient morphisms. Then is an -torsor, and hence a categorical quotient by . Since is invariant under the -action on by right multiplication, there exists a
Quotients by the additive group
We first record two preliminary results, certainly well-known but for which we could locate no appropriate reference.
Lemma 4.1 Let be a locally noetherian scheme. Let be a closed subscheme such that the projection is finite and flat. Then is the zero subscheme of a unique monic polynomial .
Proof First consider the case where , where is a local algebra with maximal ideal and residue field . Denoting by the closed point of , the fiber is a finite subscheme of .
Proof of Theorem 1
We first consider the case where is equipped with a -rational point . Then for some subgroup . If is a torus, then has the structure of a split torus, and hence is isomorphic to for some integer . Otherwise, admits a normal subgroup by Lemma 2.1. If then and we conclude by induction on . So we may assume that . Then we have a morphism Moreover, is a -torsor by Lemma 3.2 (if ) and Lemma 4.3 (if ).
References (20)
- et al.
On a dynamical version of a theorem of rosenlicht
Ann. Sci. Norm. Super. Pisa Cl. Sci. (5)
(2017) - et al.
Rationality of homogeneous varieties
Trans. Amer. Math. Soc.
(2017) The structure of solvable algebraic groups over general fields
- et al.
Groupes algébriques
(1970) - et al.
- et al.
Algebraic Geometry I
(2010) Éléments de géométrie algébrique (rédigés avec la collaboration de J. Dieudonné)
Publ. Math. Inst. Hautes Études Sci.
(1961–1967)Weylgruppe und Momentabbildung
Invent. math.
(1993)- et al.
Über Bewertungen welche unter einer reduktiven Gruppe invariant sind
Math. Ann.
(1993)
Cited by (2)
A vanishing theorem for varieties with finitely many solvable group orbits
2024, Bulletin of the London Mathematical Society