Elsevier

Indagationes Mathematicae

Volume 32, Issue 5, September 2021, Pages 1139-1151
Indagationes Mathematicae

Special issue to the memory of T.A. Springer
Homogeneous varieties under split solvable algebraic groups

https://doi.org/10.1016/j.indag.2021.04.004Get rights and content

Abstract

We present a modern proof of a theorem of Rosenlicht, asserting that every variety as in the title is isomorphic to a product of affine lines and punctured affine lines.

Introduction

Throughout this note, we consider algebraic groups and varieties over a field k. An algebraic group G is split solvable if it admits a chain of closed subgroups {e}=G0G1Gn=Gsuch that each Gi is normal in Gi+1 and Gi+1/Gi is isomorphic to the additive group Ga or the multiplicative group Gm. 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 X be a homogeneous variety under a split solvable algebraic group G. Then there is an isomorphism of varieties XAm×(A×)n for unique nonnegative integers m, n.

Here Am(A1)m denotes the affine m-space, and A×=A1{0} 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 G 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 X be a variety equipped with an action of a split solvable algebraic group G. Then there exist a dense open G-stable subvariety X0X and an isomorphism of varieties X0Am×(A×)n×Y (where m, n are uniquely determined nonnegative integers and Y is a variety, unique up to birational isomorphism) such that the resulting projection f:X0Y is the rational quotient by G.

By this, we mean that f yields an isomorphism k(Y)k(X)G, where the left-hand side denotes the function field of Y and the right-hand side stands for the field of G-invariant rational functions on X; in addition, the fibers of f are exactly the G-orbits.

As a direct but noteworthy application of Theorem 2, we obtain:

Corollary 3

Let X be a variety equipped with an action of a split solvable algebraic group G. Then k(X) is a purely transcendental extension of k(X)G.

When k 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 X be a variety equipped with a nontrivial action of Ga. Then there exist a variety Y, an open immersion φ:A1×YX and a monic additive polynomial PO(Y)[t] such that gφ(x,y)=φ(x+P(y,g),y)for all gGa, xA1 and yY.

Here P is said to be additive if it satisfies P(y,t+u)=P(y,t)+P(y,u) identically; then Ga acts on A1×Y via g(x,y)=(x+P(y,g),y), and φ is equivariant for this action. If char(k)=0, then we have P=t and hence Ga acts on A1×Y by translation on A1. So Corollary 4 just means that every nontrivial Ga-action becomes a trivial Ga-torsor on some dense open invariant subset. On the other hand, if char(k)=p>0, then P is a p-polynomial, that is, P=a0t+a1tp++antpnfor some integer n1 and a0,,anO(Y). Thus, the map (P,id):Ga×YGa×Y,(g,y)(P(y,g),y)is an endomorphism of the Y-group scheme Ga,Y=prY:Ga×YY; conversely, every such endomorphism arises from an additive polynomial P, see [5, II.3.4.4]. Thus, Corollary 4 asserts that for any nontrivial Ga-action, there is a dense open invariant subset on which Ga 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 G/H by the left action of a normal subgroup scheme NG; here G is a connected algebraic group, and HG a subgroup scheme. In particular, we show that such a quotient is a torsor under a finite quotient of N, if either NGm or NGa and char(k)=0 (Lemma 3.4). The more involved case where NGa and char(k)>0 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 k of characteristic p0 unless otherwise mentioned. Morphisms and products of schemes are understood to be over k as well. A variety is an integral separated scheme of finite type.

An algebraic group G is a group scheme of finite type. By a subgroup HG, we mean a (closed) subgroup scheme. A G-variety is a variety X equipped with a G-action α:G×XX,(g,x)gx.We say that X is G-homogeneous if G is smooth, X is geometrically reduced, and the morphism (id,α):G×XX×X,(g,x)(x,gx)is surjective. If in addition X is equipped with a k-rational point x, then the pair (X,x) is a G-homogeneous space. Then (X,x)(G/StabG(x),x0), where StabG(x)G denotes the stabilizer, and x0 the image of the neutral element eG(k) under the quotient morphism GG/StabG(x0).

Given a field extension K/k and a k-scheme X, the K-scheme X×Spec(k)Spec(K) is denoted by XK.

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 k-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 Gm is isomorphic to G

Quotients of homogeneous spaces by normal subgroups

Let G be an algebraic group, HG a subgroup, and NG a smooth normal subgroup. Then H acts on N by conjugation. The semi-direct product NH defined by this action (as in [11, Sec. 2.f]) is equipped with a homomorphism to G, with schematic image the subgroup NHG. Recall that HNHG and NH/HN/NH. Denote by q:GG/H,r:GG/NHthe quotient morphisms. Then q is an H-torsor, and hence a categorical quotient by H. Since r is invariant under the H-action on G 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 X be a locally noetherian scheme. Let ZA1×X be a closed subscheme such that the projection prX:ZX is finite and flat. Then Z is the zero subscheme of a unique monic polynomial PO(X)[t].

Proof

First consider the case where X=Spec(A), where A is a local algebra with maximal ideal m and residue field K. Denoting by x the closed point of X, the fiber Zx is a finite subscheme of AK1.

Proof of Theorem 1

We first consider the case where X is equipped with a k-rational point x0. Then X=G/H for some subgroup HG. If G is a torus, then G/H has the structure of a split torus, and hence is isomorphic to (A×)n for some integer n0. Otherwise, G admits a normal subgroup NGa by Lemma 2.1. If NH then X(G/N)/(H/N) and we conclude by induction on dim(G). So we may assume that NH. Then we have a morphism f:X=G/HG/NH(G/N)/(NH/N).Moreover, f is a Ga-torsor by Lemma 3.2 (if p=0) and Lemma 4.3 (if p>0).

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