ON A PROPERTY OF SPECIAL GROUPS

Z. REICHSTEIN AND B. YOUSSIN

Abstract. Let G be an algebraic group defined over an algebraically closed field k of characteristic zero. We give a simple proof of the following result: if H^{1}(K_{0}, G) = {1} for some finitely generated field extension K_{0}/k of transcendence degree ≥ 3 then H^{1}(K, G) = {1} for every field extension K/k.

1. Introduction

Let G be an algebraic group. J.-P. Serre stated the following conjectures in [Se_{2}] (see also [Se_{3}, Chapter III]).

Conjecture I: If G is connected then H^{1}(K, G) = {1} for every perfect field K of cohomological dimension ≤ 1.

Conjecture II: If G is semisimple, connected and simply connected then H^{1}(K, G) = {1} for every perfect field K of cohomological dimension ≤ 2.

Conjecture I was proved by Steinberg [St_{1}]. Conjecture II remains open, though significant progress has been made in recent years; see [BP] and [Gi].

Our main result is a partial converse of Conjectures I and II. Recall that an algebraic group G is called special if H^{1}(K, G) = {1} for every field K. Special groups were introduced by Serre [Se_{1}] and classified by Grothendieck [Gr, Section 5]; cf. [PV, Section 2.6].

Theorem 1. Let G be an algebraic group defined over an algebraically closed field k of characteristic zero. Suppose H^{1}(K, G) = {1} for some finitely generated field extension K of k of transcendence degree d.

(a) If d ≥ 1 then G is connected.

(b) If d ≥ 2 then G is simply connected.

(c) If d ≥ 3 then G is special.

Note that the cohomological dimension of K equals d; see [Se_{3}, Section II.4]. Thus, informally speaking, the theorem may be interpreted as saying that Conjectures I and II cannot be extended or strengthened in a meaningful way.

Our proof of Theorem 1 is rather simple: the idea is to use nontoral finite abelian subgroups of G as obstructions to the vanishing of H^{1}. We remark that our argument (and, in particular, the proof of Lemma 3) does not rely on canonical resolution of singularities; cf. [RY, Remark 4.4].

Ph. Gille recently showed us an alternative proof of Theorem 1, based on case by case analysis and properties of the Rost invariant. We would like to thank him, J.-L. Colliot-Thélène and R. Parimala for informative discussions.

1991 Mathematics Subject Classification. 14L30, 20G10.

Z. Reichstein was partially supported by NSF grant DMS-9801675 and (during his stay at MSRI) by NSF grant DMS-9701755.

2. Preliminaries

Throughout this note k will denote an algebraically closed base field of characteristic zero. All fields, varieties, morphisms, algebraic groups, etc., will be assumed to be defined over k.

Let G be an algebraic group. An abelian subgroup A of G is called toral if A is contained in a torus of G and nontoral otherwise.

Lemma 2. Let G be an algebraic group, L be a Levi subgroup of G and A be a finite abelian subgroup of L. If A is nontoral in L then A is nontoral in G.

Proof. Assume the contrary: A ⊂ T for some torus T of G. Since T is reductive, it lies in a Levi subgroup L_{1} of G; see [OV, Theorem 6.5]. Denote the unipotent radical of G by U; then L and L_{1} project isomorphically onto G/U. Since A is toral in L_{1}, it is toral in G/U, and hence, in L, as claimed.

Recall that a G-variety X is an algebraic variety with a G-action; X is generically free if G acts freely on a dense open subset of X and primitive if k(X)^{G} is a field (note that X is allowed to be reducible). Elements of H^{1}(K, G) are in 1—1 correspondence with G-torsors over K, i.e., birational classes of primitive generically free G-varieties X such that k(X)^{G} = K; see e.g., [Po, Section 1.3]. If X is a primitive generically free G-variety, we shall write cl(X) for the class in H^{1}(k(X)^{G}, G) given by X.

Our proof of Theorem 1 is based on the following result.

Lemma 3. ([RY, Lemma 4.3]) Let G be an algebraic group, A be a nontoral finite abelian subgroup of G and X be a generically free primitive G-variety. Suppose A fixes a smooth point of X. Then cl(X) ≠ 1 in H^{1}(k(X)^{G}, G).

3. Construction of a nontrivial torsor

Lemma 4. Let A be an abelian group of rank r and let K be a finitely generated field extension of k of transcendence degree d ≥ r. Then there exists an A-variety Y such that (i) k(Y)^{A} = K and (ii) Y has a smooth A-fixed point.

Proof. Since k is algebraically closed, A has a faithful r-dimensional representation V_{1}. Let V_{2} be the trivial (d − r)-dimensional representation of A, and V = V_{1} ⊕ V_{2}. Then the (geometric) quotient V/A is isomorphic to the affine space 𝔸^{d}. Denote the origin of V by 0, and its image in V/A by \overline{0}.

Let Y_{0} be an affine variety over k such that k(Y_{0}) = K; dim(Y_{0}) = d. Let y_{0} ∈ Y_{0} be a smooth point. Identifying V/A = 𝔸^{d}, we can find a dominant projection f: Y_{0} ⟶ V/A such that f(y_{0}) = \overline{0} and f is étale at y_{0}.

Now set Y = Y_{0} ×_{V/A} V; the A-action on Y is induced from V. The natural projection Y ⟶ Y_{0} is a rational quotient map for this action; see, e.g., [R, Lemma 2.16(a)]. Thus Y satisfies (i). To prove (ii), set y = (y_{0}, 0); y is fixed by A. The morphism Y ⟶ V is obtained from f by a base change, and hence, is étale at y; the smoothness of V implies then that Y is smooth at y. Thus, y ∈ Y is a smooth point fixed by A.

Proposition 5. Let G be an algebraic group, A be a nontoral abelian subgroup of G of rank r, and K/k be a field extension of transcendence degree d. If d ≥ r then H^{1}(K, G) ≠ {1}.

Proof. Choose an A-variety Y and a smooth A-fixed point y ∈ Y, as in Lemma 4. We claim that the image of cl(Y) under the natural map H^{1}(K, A) ⟶ H^{1}(K, G) is nontrivial. Indeed, recall that the image of cl(Y) in H^{1}(K, G) is cl(X), where

X = G ∗_{A} Y = (G × Y)/A

is the (geometric) quotient for the A-action on G×Y given by a(g, y′) = (ga^{−1}, ay′); see [PV, Section 4.8]. By [PV, Proposition 4.22], G ∗_{A} Y is smooth at x = (1_{G}, y) since Y is smooth at y. Moreover, x is an A-fixed point of X; thus Lemma 3 tells us that cl(X) ≠ 1 in H^{1}(K, G), as claimed.

4. Proof of Theorem 1

In view of Proposition 5 it is sufficient to show that G contains a nontoral finite abelian subgroup A, where

(a′) rank(A) = 1, if G is not connected,

(b′) rank(A) ≤ 2, if G is not simply connected and

(c′) rank(A) ≤ 3, if G is not special. Moreover, in view of Lemma 2 we only need to prove (a′), (b′), and (c′) under the assumption that G is reductive (otherwise we may replace G by its Levi subgroup).

Proof of (a′): Write G = F G_{0}, where G_{0} is the identity component of G and F is a finite group; see [V, Proposition 7]. Since G is disconnected, F is not contained in G_{0}. Choose a ∈ F − G_{0} and set A = <a>. Then A is cyclic, finite (because a ∈ F) and nontoral (because every torus of G in contained in G_{0}), as desired.

Proof of (b′): In view of (a′) we may assume without loss of generality that G is connected. Now the desired conclusion follows from [St_{2}, Theorem 2.27].

Proof of (c′): Suppose G is not special. By [Se_{4}, 1.5.1], G has a torsion prime p, and by [St_{2}, Theorem 2.28] G has a nontoral elementary p-abelian subgroup A of rank ≤ 3. see also [Se_{4}, 1.3].

References

[BP] E. Bayer-Fluckiger, R. Parimala, Galois cohomology of classical groups over fields of cohomological dimension ≤ 2, Invent. Math. 122 (1995), 195–229.

[Gi] Ph. Gille, Cohomologie galoisienne des groupes quasi-déployés sur des corps de dimension cohomologique ≤ 2, Compositio Math., to appear.

[Gr] A. Grothendieck, La torsion homologique et les sections rationnelles, Exposé 5, Séminaire C. Chevalley, Anneaux de Chow et applications, 2nd année, IHP, 1958.

[OV] A. L. Onishchik, E. B. Vinberg. Lie Groups and Algebraic Groups. Springer-Verlag, 1990.

[Po] V. L. Popov, Sections in Invariant Theory, Proceedings of the Sophus Lie Memorial Conference, Scandinavian University Press, 1994, 315–362.

[PV] V. L. Popov, E. B. Vinberg, Invariant Theory, in Encyclopaedia of Math. Sciences 55, Algebraic Geometry IV, edited by A. N. Parshin and I. R. Shafarevich, Springer-Verlag, 1994.

[R] Z. Reichstein, On the notion of essential dimension for algebraic groups, Transformation Groups, to appear. Preprint available at http://ucs.orst.edu/ reichstz/pub.html .

[RY] Z. Reichstein, B. Youssin, Splitting fields of G-varieties, preprint.

Available at http://ucs.orst.edu/ reichstz/pub.html .

[Se_{1}] J.-P. Serre, Espaces fibrés algébriques, Exposé 1, Séminaire C. Chevalley, Anneaux de

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[Se_{2}] J.-P. Serre, Cohomologie galoisienne des groups algébriques linéaires, Colloque de Brux-

elles (1962), 53–67.

[Se_{3}] J.-P. Serre, Galois Cohomology, Springer, 1997.

[Se_{4}] J.-P. Serre, Sous-groupes finis des groupes de Lie, Séminaire Bourbaki, 1998–99, n^{∘} 864,

Juin 1999.

[St_{1}] R. Steinberg, Regular elements of semisimple groups, Publ. Math. I.H.E.S. 25 (1965),

281–312. Reprinted in [Se_{3}], pp. 155–186.

[St_{2}] R. Steinberg, Torsion in reductive groups, Advances in Math. 15 (1975), 63–92.

[V] E. B. Vinberg, On invariants of a set of matrices, J. of Lie theory 6 (1996), 249–269.

Department of Mathematics, Oregon State University, Corvallis, OR 97331 Current mailing address: PMB 136, 333 South State St., Lake Oswego, OR 97034-3961,

USA

   E-mail address: zinovy@math.orst.edu

Department of Mathematics and Computer Science, University of the Negev, Be’er Sheva’, Israel

Current mailing address: Hashofar 26/3, Ma’ale Adumim, Israel

   E-mail address: youssin@math.bgu.ac.il