On the Notion of Essential Dimension for Algebraic Groups

Zinovy Reichstein^{1}

Department of Mathematics, Oregon State University, Corvallis, OR 97331-4605

Abstract: We introduce and study the notion of essential dimension for linear algebraic groups defined over an algebraically closed fields of characteristic zero. The essential dimension is a numerical invariant of the group; it is often equal to the minimal number of independent parameters required to describe all algebraic objects of a certain type. For example, if our group G is S_{n}, these objects are field extensions, if G = O_{n}, they are quadratic forms, if G = PGL_{n}, they are division algebras (all of degree n), if G = G_{2}, they are octonion algebras, if G = F_{4}, they are exceptional Jordan algebras. We develop a general theory, then compute or estimate the essential dimension for a number of specific groups, including all of the above-mentioned examples. In the last section we give an exposition of results, communicated to us by J.-P. Serre, relating essential dimension to Galois cohomology.

Contents

1 Introduction 2

2 Preliminaries 5 2.1 Notation and terminology . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2 Primitive G-varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.3 Rational quotients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.4 Generically free varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.5 (G, H)-sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.6 Compressions and fiber products . . . . . . . . . . . . . . . . . . . . . . 9 2.7 The “no-name” lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3 Definition of essential dimension 11 3.1 The essential dimension of a G-variety . . . . . . . . . . . . . . . . . . . 11 3.2 The essential dimension of a group . . . . . . . . . . . . . . . . . . . . . 13 3.3 First properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

4 (G − H)-sections and essential dimension 15 4.1 The essential dimension of a section . . . . . . . . . . . . . . . . . . . . 15 4.2 An application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

5 Groups of low essential dimension 18 5.1 Rational sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 5.2 Special groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 5.3 n-special groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 6 Structured spaces I 20

1Partially supported by NSA grant MDA904-9610022 and NSF grant DMS-9801675

6.1 Definition and examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 6.2 First properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

7 Spaces of rational maps 22 7.1 Equivariant maps into vector spaces . . . . . . . . . . . . . . . . . . . . 22 7.2 Structured spaces of rational maps . . . . . . . . . . . . . . . . . . . . . 24

8 Structured spaces II 25 8.1 Structured spaces of type (V, α) . . . . . . . . . . . . . . . . . . . . . . . 25 8.2 Structured spaces and essential dimension . . . . . . . . . . . . . . . . . 28

9 Projective linear groups 29 9.1 Central simple algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 9.2 Cyclic algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 9.3 Algebras of odd degree . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 9.4 Algebras of composite degree . . . . . . . . . . . . . . . . . . . . . . . . 31

10 Orthogonal groups 32 10.1 Pfister forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 10.2 The essential dimension of O_{n} . . . . . . . . . . . . . . . . . . . . . . . . 33 10.3 The essential dimension of SO_{n} . . . . . . . . . . . . . . . . . . . . . . . 34

11 Some exceptional groups 35 11.1 Octonion algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 11.2 The essential dimension of G_{2} . . . . . . . . . . . . . . . . . . . . . . . . 35 11.3 The Albert algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 11.4 The essential dimension of F_{4} . . . . . . . . . . . . . . . . . . . . . . . . 37 11.5 The essential dimension of E_{6} . . . . . . . . . . . . . . . . . . . . . . . . 39

12 Cohomological invariants 39 12.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 12.2 Cohomological invariants and essential dimension . . . . . . . . . . . . . 41 12.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 12.4 The Rost invariant and simply connected semisimple groups . . . . . . . 44 12.5 Mod p invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

1 Introduction

The purpose of this paper is to introduce and study the notion of essential dimension for linear algebraic groups. All algebraic groups we shall consider, along with all varieties, fields, etc., and all morphism between them will be defined over a fixed algebraically closed base field k of characteristic zero. The essential dimension ed(G) of an algebraic group G is the smallest integer d such that every principal G-bundle X ⟶ B can be obtained (up to birational isomorphism) by pull-back from a diagram

X ≃ B ×_{C} Y ⟶ Y
↓  ↓
B  ⟶ C

where Y ⟶ C is a principal G-bundle, B ⟶ C is a dominant map, and dim(C) ≤ d; cf. Definition 3.5 and Remark 2.17. The essential dimension naturally comes up in many interesting situations; several examples are given below.

(a) The essential dimension of the symmetric group S_{n} is the minimal number of parameters required to write down the general polynomial of degree n in one variable. (Here we allow Tschirnhaus transformations without auxiliary radicals.) Equivalently, ed(S_{n}) is the smallest integer d with the following property: every field extension K ⊂ L of degree n can be defined by (a degree n) polynomial with at most d algebraically independent coefficients (over k). The question of computing ed(S_{n}) is related to the algebraic form of Hilbert’s 13th problem. For more on this we refer the reader to [BR_{1}] and [BR_{2}]; a discussion of essential dimensions of other finite groups can be found in [BR_{1}].

(b) Groups of essential dimension zero are precisely the “special groups” introduced by Serre [Se_{1}] and classified by Grothendieck [G] in the 1950s; see Section 5.

(c) Let D be a division algebra with center K. We say that D is defined over a subfield F of K if D ≃ E⊗_{F} K for some division algebra E with center F. The essential dimension of the projective linear group PGL_{n} is the smallest integer d such that every division algebra D of degree n is defined over some field F with trdeg_{k}(F) ≤ d. This number is of interest in the theory of division algebras; see [Pr_{1}, Thm. 2.1] and [Row]. We will discuss essential dimensions of projective linear groups in Section 9.

(d) The essential dimension of the orthogonal group O_{n} (resp. the special orthogonal group SO_{n}) is the smallest integer d such that every quadratic form (resp. every quadratic form of determinant 1) over every field F is equivalent to one with ≤ d algebraically independent coefficients (over k). We will determine these integers in Section 10.

(e) The essential dimensions of the exceptional group G_{2} (resp. F_{4}) is the smallest integer d such that every octonion algebra (resp. every 27-dimensional exceptional simple Jordan algebra) can be defined over a field of transcendence degree ≤ d. These numbers are discussed in Section 11.

The methods we use for computing, or at least estimating, the essential dimension of a given algebraic group G can be roughly divided into three categories: geometric, algebraic and cohomological. The geometric approach, based on (birational) invariant theory, is discussed in Sections 2-4. The algebraic approach, based on descent of ”structured spaces” is developed in Sections 6-8; some applications are given in Sections 9-11. The cohomological approach is discussed in Sections 5 and 12; Section 5 is mostly concerned with the vanishing of H^{1} and Section 12 deals with the relationship between the essential dimension of G and its cohomological invariants.

In order to illustrate the interplay among the geometric, algebraic, and cohomological methods, we will often prove the same result in several different ways, sometimes in different parts of the paper. For the convenience of the reader, we summarize most of what we now know about essential dimensions of specific groups in the table below and indicate where the proof of each result can be found.

\underline{Group} EssentialDimension

\underline{Proof}

(ℤ/n)^{r}  r  [BR_{1}, 6.1]

S_{n}

\{ \begin{matrix}≥ [n/2] \\ ≤ n − 3 \end{matrix}

[BR_{1}, 6.5(b)], [BR_{2}, 1.1] , Ex. 12.8
[BR_{1}, 6.5(c)]
(k^{∗})^{n}  = 0  Thm. 5.4 , Ex. 3.9(a)
GL_{n}  = 0  Thm. 5.4 , Ex. 3.9(b)
SL_{n}  = 0  Thm. 5.4 , Ex. 3.9(c), 8.9(a)
Sp_{2n}  = 0  Thm. 5.4 , Ex. 8.9(b)

PGL_{n} \{ \begin{matrix}≤ n^{2} − 2n \\ ≤ n + \frac{(n−1)(n−2)}{2}, n odd \\ ≥ 2r if n = n_{0}^{r} \\ = 2 if n = 2, 3, 6 \\ other & results \end{matrix}

Thm. 4.5
Thm. 9.6
Thm. 9.3
Lemma 9.4(c)
Prop. 9.8, Rem. 9.9

O_{n} \{ \begin{matrix}= n \\ ≤ n \\ ≥ n \end{matrix}

Thm. 10.3
Ex. 3.10(a) , 4.2
Ex. 12.6

SO_{n} , n ≥ 3 \{ \begin{matrix}= n − 1 \\ ≤ n − 1 \\ ≥ n − 1 \end{matrix}

Thm. 10.4
Ex. 3.10(b) , 4.2
Ex. 12.7

G_{2} \{ \begin{matrix}= 3 \\ ≤ 3 \\ ≥ 3 \end{matrix}

Thm. 11.2
Rem. 11.4
Rem. 11.3 , Ex. 12.9, Thm. 12.12
F_{4}  ≥ 5  Thm. 11.5 , Ex. 12.10

E_{6} (simply conn.)

\{ \begin{matrix}≥ 3 \\ ≤ ed(F_{4}) + 1 \end{matrix}

Prop. 11.6, Thm. 12.12
Prop. 11.7
E_{7} (simply conn.) ≥ 3  Prop. 12.11(a), Thm. 12.12
E_{8}  ≥ 3  Prop. 12.11(b), Thm. 12.12

For further results see the table in the appendix at the end of this paper.

Acknowledgements

We would like to thank J.-P. Serre for his help and encouragement and for allowing us to include an exposition of some of his unpublished results in this paper. In particular, Section 12 is based entirely on results he communicated to us. He also suggested the statements of Theorems 10.3, 10.4, 11.2, and 11.5; his proofs of these results are outlined in Section 12. (We give somewhat different proofs in Sections 10 and 11.)

The material of Section 5 was brought to our attention by V. L. Popov, who also made a number of other helpful comments after reading an earlier version of this paper. The results of Section 9.3 were suggested to us by L. H. Rowen and D. J. Saltman.

We would also like to express our gratitude to V. E. Kordonsky for sending us his preprints [Ko_{1}] and [Ko_{2}] along with a list of comments on our earlier preprint (see Remark 3.6), to J. Buhler and B. Kahn for asking questions that got us interested in the subject matter of this paper, to R. S. Garibaldi, A. Vistoli, and N. Vonessen for stimulating discussions, and finally to H. Bass, P. Woit, and the Columbia University Mathematics Department for their hospitality during the summer of 1996, when some of the work on this paper was done.

2 Preliminaries

The purpose of this section is to introduce notation, terminology and a number of known results from birational invariant theory. Our emphasis will be on stating these results in a form suitable for subsequent use, not on developing the theory in a coherent and self-contained manner. For an excellent introduction to birational invariant theory we refer the reader to [PV, Sect 2]; see also [Po].

Our basic object of study will be primitive rather than irreducible G-varieties (see Definition 2.1); this (slightly) greater degree of generality will be needed the sequel.

2.1 Notation and terminology

The following notational conventions will be used throughout the paper.

k  algebraically closed base field of characteristic 0
F  usually a field extension of k
G  algebraic group defined over k
X  usually a G-variety
Gx  orbit of x under the action of G
Stab(x)  stabilizer of x
ed(X)  essential dimension X; see Definition 3.1
ed(G)  essential dimension of G; see Definition 3.5
(W, β)  structured space over F; see Definition 6.1
(V, α)  structured space over k
Aut_{k}(V, α)  automorphism group of (V, α), usually denoted by G
τ(W, β)  see Definition 8.7
Z(A)  center of the central simple algebra A
UD(m, n)  universal division algebra of m n × n-matrices
<< a_{1}, …, a_{n} >> n-fold Pfister form; see Section 10.1

Throughout this paper we shall work over a fixed base field k, which we assume to be algebraically closed and of characteristic 0. All algebraic objects we will consider (e.g., rings, fields, algebraic groups, algebraic varieties) and all maps between them will be defined over k.

If X is an algebraic variety, we shall denote the algebra of regular (resp. rational) functions on X by k[X] (resp. k(X)). If X is irreducible then k(X) is a field; in general, k(X) = k(X_{1})⊕…⊕k(X_{n}) is a direct sum of fields; here X_{1}, …, X_{n} are the irreducible components of X. We shall say that a certain property holds for x in general position in X if it holds for all x ∈ U, where U is a dense open subset of X (i.e., an open subset which non-trivially intersects every irreducible component of X).

As usual, a rational map X −→ Y is an equivalence class of regular maps U ⟶ Y, where U is a dense open subset of X; f_{1} : U_{1} ⟶ Y and f_{2} : U_{2} ⟶ Y are considered equivalent if they agree on U_{1} ∩ U_{2}. The domain of a rational map f : X −→ Y is the union of all dense open subsets U where f is defined, and the range f(X) is the union of f(U). We will say that f is dominant if f(X) is dense in Y.

We shall call an algebraic variety X a G-variety if X is equipped with a regular action of G (i.e., an action given by a regular morphism G × X ⟶ X). If X and Y are G-varieties then by a regular map X ⟶ Y of G-varieties we mean a regular G-equivariant map. The same applies to rational maps of G-varieties, biregular and birational isomorphisms of G-varieties, etc.

2.2 Primitive G-varieties

Definition 2.1 We shall call a G-variety X primitive if G transitively permutes the components of X.

Note that if G is connected then X is a primitive G-variety if and only if X is irreducible.

Lemma 2.2 Let X be a G-variety. Then (a) X is birationally isomorphic to a disjoint union of primitive G-varieties,

(b) X is primitive if and only if k(X)^{G} is a field.

Proof. (a) Write X = X_{1} ∪ … ∪ X_{n}, where each X_{i} is a union of components of X transitively permuted by G. Then each X_{i} is primitive and the natural projection from the disjoint union ⨿_{i=1}^{m}X_{i} to X is a G-equivariant birational isomorphism.

(b) Suppose k(X)^{G} is not a field. Then k(X)^{G} has a zero divisor f ≠ 0. Let X_{0} be the union of those irreducible components of X where f = 0 and X_{1} be the union of the remaining irreducible components. Then X_{0} and X_{1} are G-invariant and non-empty. Since G cannot map a component in X_{0} to a component in X_{1}, we conclude that X is not primitive. Conversely, suppose X is not primitive. By part (a) we may assume that X is a disjoint union of m ≥ 2 primitive G-varieties X_{1}, …, X_{m}. Let f ∈ k(X)^{G} be given by f_{|X_{1}} = 1 and f_{|X_{i}} = 0 for all i = 2, …, m. Then f is a zero divisor in k(X)^{G} and thus k(X)^{G} is not a field. □

2.3 Rational quotients

Theorem 2.3 (Rosenlicht [Ros_{1}], [Ros_{2}]) Let X be a G-variety. Suppose k(W) = k(X)^{G} and π : X −→ W is rational map induced by the inclusion k(X)^{G} ↪ k(X). Then there exists an open dense subset U of X such that π_{|U} : U ⟶ W is a regular map and for any x, y ∈ U, π(x) = π(y) iff Gx = Gy. □

We shall denote W by X/G; this variety is uniquely defined up to birational equivalence. By Lemma 2.2(b), X is primitive if and only if X/G is irreducible. We will refer to the map π : X −→ X/G as the rational quotient map or simply the quotient map, since no other kind of quotient map will be considered in this paper.

Remark 2.4 The identity k(X/G) = k(X)^{G}, which we used to define X/G, can be restated as a universal property of X/G as follows. Any rational map f : X −→ Y of G-varieties gives rise to a rational map \overline{f} : X/G −→ Y/G such that the diagram

\begin{aligned}& X & −→^{f} & Y \\ & | & & | \\ π_{X} & | & & | & π_{Y} \\ & ↓ & & ↓ \\ & X/G & −→^{\overline{f}} & Y/G \end{aligned}  (1)

commutes. Here π_{X} and π_{Y} are the rational quotient maps.

Remark 2.5 The rational quotient map π : X −→ X/G is unique in the following sense.

Let X be a primitive G-variety. Suppose f : X −→ X_{0} is a dominant rational map such that f^{−1}(f(x)) = Gx for x_{0} ∈ X in general position. Then there exists a birational isomorphism \overline{f} : X/G −→^{≃} X_{0} such that f = \overline{f} ∘ π.

The map \overline{f} is given by diagram (1) with Y = X_{0}, where X_{0} is viewed as a Gvariety with trivial G-action. Here π_{X} = π, Y/G = X_{0} and π_{Y} : Y −→ Y/G is the identity map. Since X is primitive, both X_{0} and X/G are irreducible. Moreover, by our assumption on f, the map \overline{f} : X/G −→ X_{0} is dominant and \overline{f}^{−1}(x_{0}) is a single point for x_{0} in general position in X_{0}. Thus \overline{f} is a birational isomorphism; see, e.g., [H, 4.6]. □

Remark 2.6 Suppose X is a G-variety and N is a closed normal subgroup of G. Then G/N has a naturally defined rational action on X/N; see [PV, Proposition 2.6]. In general, this action will not be regular; however, by a theorem of Rosenlicht (see [Ros_{1}, Theorem 1], [PV, Corollary 1.1]) there exists a model for X/N such that the induced G/N-action on it is regular, i.e., this model is a G/N-variety. Moreover, this G/Nvariety is uniquely defined up to birational isomorphism.

2.4 Generically free varieties

Definition 2.7 A G-variety X is said to be generically free if G acts freely (i.e., with trivial stabilizers) on a dense open subset of X.

Lemma 2.8 Suppose a rational map f : X −→ Y of generically free G-varieties induces a birational isomorphism \overline{f} : X/G −→ Y/G. Then f is a birational isomorphism.

Proof. We may assume without loss of generality that X/G and Y/G are irreducible, i.e., X and Y are primitive G-varieties. In particular, every irreducible component of X has dimension dim(X) and every irreducible component of Y has dimension dim(Y). Moreover,

dim(X) = dim(X/G) + dim(G) = dim(Y/G) + dim(G) = dim(Y) .

Replacing X/G and Y/G by sufficiently small open subsets we may assume that all four maps in diagram (1) are regular, G acts freely on X and Y, and π_{X} separates the G-orbits in X. Then f(X) is G-invariant subset of Y which intersects every G-orbit in Y; hence, f(X) = Y. On the other hand, f(x_{1}) = f(x_{2}) means that π_{X}(x_{1}) = π_{X}(x_{2}), i.e., x_{2} = g(x_{1}) for some g ∈ G. Consequently, gf(x_{1}) = f(x_{2}) = f(x_{1}). Since G acts freely on Y, we conclude that x_{1} = x_{2}.

We have therefore proved that f is bijective. Let X = X_{1}∪…∪X_{n} be an irreducible decomposition of X. Let Y_{i} be the closure of f(X_{i}) in Y. Since dim(Y_{i}) = dim(X_{i}) = dim(X) = dim(Y), Y_{1}, …, Y_{n} are the irreducible components of Y. Moreover the injectivity of f implies that Y_{i} ≠ Y_{j} if i ≠ j. For every i = 1, …, n, the map f_{|X_{i}} : X_{i} ⟶ Y_{i} is dominant and injective. Hence, each f_{|X_{i}} is a birational isomorphism (see, e.g., [H, 4.6]) and therefore, so is f. □

2.5 (G, H)-sections

Definition 2.9 (cf. [Ka_{1}, Sect. 1], [Po, 1.7.6], [Do, Sect. 3].) Let G be an algebraic group, X be a G-variety, and H be a closed subgroup of G. An irreducible subvariety S ⊂ X is called a (G, H)-section if

(a) GS is dense in X and

(b) there is a dense open subvariety S_{0} ⊂ S such that for any s ∈ S_{0} we have gs ∈ S if and only if g ∈ H.

Example 2.10 Let X be a primitive G-variety and let X_{1} be an irreducible component of X. Then X_{1} is a (G, G_{1})-section, where G_{1} = {g ∈ G | g(X_{1}) = X_{1}}.

Lemma 2.11 Let X be a G-variety and S be a (G, H)-section of X. Then k(S)^{H} = k(X)^{G}.

Proof. Let π : X −→ X/G be the rational quotient map. Then the restriction π_{|S} : S −→ X/G is well-defined and separates H-orbits in S_{0}. By Remark 2.5 π_{|S} is the quotient map for the H-action on S. Equivalently, k(S)^{H} = k(X/G), as claimed.

□

Definition 2.12 Let G be an algebraic group, H be a closed subgroup of G, and W be an H-variety. Then G × W is a G × H-variety via

(g, h) : (g′, w) ↦ (gg′h^{−1}, hw) ,  (2)

for any g, g′ ∈ G, h ∈ H, and w ∈ W. By Remark 2.6 a model Y for (G×W)/H can be chosen so that the G-action on Y is regular and thus the quotient map π: G×W −→ Y is a rational map of G-varieties. (Here we are identifying H with a normal subgroup N = {1_{G}} × H of G × H and G with (G × H)/N).) We shall denote Y by G ∗_{H} W; this G-variety is well-defined up to birational isomorphism.

Remark 2.13 (a) If W is a generically free H-variety then it is easy to see that G∗_{H} W is a generically free G-variety.

(b) W ↪ G ∗_{H} W given by w ↦ (1, w) is an H-equivariant embedding of Hvarieties, whose image is a (G, H)-section of G ∗_{H} W. The following lemma shows that every (G, H)-section is of this form.

Lemma 2.14 (cf. [Po, 1.7.5]) Let X be a G-variety and S be a (G, H)-section of X. Then G ∗_{H} S and X are birationally isomorphic G-varieties.

Proof. Consider the map ϕ : G×S ⟶ X given by (g, s) ↦ gs. Then ϕ is a morphism of G × H varieties, where G × H acts via (2) on G × S and H acts trivially on X. Moreover, this map is dominant and ϕ^{−1}(x) is a single H-orbit for x in general position in X. Thus by Remark 2.5, ϕ is the rational quotient map for the H-action on G × S. In other words, we have the following commutative diagram:

G × S
π | ↘ ϕ
↓
G ∗_{H} S −→^{≃} X

To complete the proof, note that since the G-action on G × S commutes with the Haction, every map in this diagram is G-equivariant. □

2.6 Compressions and fiber products

Definition 2.15 Let X be a generically free G-variety. A G-compression of X is a dominant G-equivariant rational map X −→ Y, where Y is another generically free G-variety.

Let Y be a primitive G-variety, π : Y −→ Y/G be the rational quotient map, and ϕ : X_{0} −→ Y/G be a dominant rational map. Recall that the fiber product Y ×_{Y/G} X_{0} is defined as the closure of

{(y, x_{0}) | π(y) = ϕ(x_{0})}

in Y × X_{0}. This variety inherits a G-action from Y. Clearly Y ×_{Y/G} X_{0} is a generically free G-variety if an only if so is Y.

Lemma 2.16 (a) Suppose Y and ϕ : X_{0} −→ Y/G are as above. Denote the fiber product Y ×_{Y/G} X_{0} by X. Then the natural projection pr : X = Y ×_{Y/G} X_{0} ⟶ X_{0} is the rational quotient map for the G-action on X.

(b) Conversely, suppose X is a generically free G-variety, X_{0} = X/G, π : X −→ X_{0} is the rational quotient map and \overline{f} : X_{0} −→ Y/G is the dominant rational map induced by a G-compression f : X −→ Y. Then

ψ =^{def} f × (\overline{f} ∘ π) : X −→^{≃} Y ×_{Y/G} X_{0}

is a birational isomorphism of G-varieties.

Proof. (a) It is easy to see that pr^{−1}(x_{0}) = Gx_{0} for x_{0} in general position in X_{0}. Part (a) now follows from Remark 2.5.

(b) By part (a), the map \overline{ψ} : X/G −→^{≃} X_{0} = (Y ×_{Y/G} X_{0})/G is a birational isomorphism. The desired conclusion now follows from from Lemma 2.8. □

Remark 2.17 Suppose X is a generically free G-variety. One can show that the rational quotient map π: X −→ X/G is a principal G-bundle over a dense open subset U ⊂ X/G. This means that there exists an etale cover U′ ⟶ U such that U′×_{U} π^{−1}(U) is isomorphic to U′ × G, as a G-variety. If X is primitive, this is equivalent to the following: there exists a finite rational cover X_{0} −→ X/G of irreducible varieties such that X_{0} ×_{X/G} X is birationally isomorphic to X_{0} × G (as a G-variety). The latter assertion follows from the existence of a rational quasisection for X; see [Po, 1.1.2]. Since we shall not use this result, except to motivate Definition 3.1, we omit the details of the argument.

2.7 The “no-name” lemma

Let Y be a G-variety and let ϕ : E ⟶ Y be a G-vector bundle of rank d. This means that ϕ is an algebraic vector bundle of rank d, G acts on both E and Y so that ϕ is a surjective morphism of G-varieties, and g : ϕ^{−1}(y) ↦ ϕ^{−1}(g(y)) is a linear map for each y ∈ Y and g ∈ G; see, e.g., [BK, Section 1].

Lemma 2.18 Let Y be a generically free G-variety and let E be a G-vector bundle on Y. Then there exists a birational isomorphism α so that the diagram

\begin{aligned}& E/G & −→^{α} & Y/G × k^{d} \\ & | \\ \overline{ϕ} & | & ↙ \\ & ↓ \\ & Y/G \end{aligned}  (3)

commutes. Here k^{d} is the d-dimensional affine space and Y/G × k^{d} ⟶ Y/G is projection to the first factor.

Proof. We may assume without loss of generality that Y is a primitive G-variety. Moreover, in view of Example 2.10 and Lemma 2.11, we may assume Y is irreducible (otherwise we can replace Y by an irreducible component Y_{1}, G by the stabilizer G_{1} of Y_{1}, and E by ϕ^{−1}(Y_{1})).

If Y is irreducible then Lemma 2.18 is the usual form of the so-called “no-name lemma”; see [BK, Lemma 1.2], [Ka_{2}, p. 104], or [Do, p. 6]. □

In the sequel we will need the following variant of the no-name lemma.

Lemma 2.19 Let Y be a generically free G-variety and let ϕ : E ⟶ Y be a G-vector bundle of dimension d. Then E is birationally isomorphic to Y × k^{d} (as a G-variety), where k^{d} is the affine d-space with trivial G-action.

Proof. By Lemma 2.16, E is birationally isomorphic to Y ×_{Y/G} (E/G). Let α be as in (3). Then we have the following birational isomorphisms of G-varieties:

E ≃ Y ×_{Y/G} E/G ≃^{1×α} Y ×_{Y/G} (Y/G × k^{d}) ≃ Y × k^{d}. □

Corollary 2.20 Let Y be a primitive generically free G-variety and let V be a ddimensional linear representation of G. Then there exists a G-equivariant dominant rational map f : Y × k^{d} −→ V. Here G acts on Y × k^{d} via g(y, a) = (gy, a), as in the previous lemma.

Proof. Let E = Y × V and let ϕ : Y × k^{d} −→^{≃} Y × V be the birational isomorphism of Lemma 2.19. Now set f = pr ∘ϕ, where pr : Y ×V ⟶ V is projection to the second component. □

3 Definition of essential dimension

3.1 The essential dimension of a G-variety

Recall that, up to birational isomorphism, a generically free G-variety X can be viewed as a principal G-bundle over the base X/G; see Remark 2.17. Moreover, Lemma 2.16(b) says that if X −→ Y is a compression then X is obtained from Y by a (dominant) base extension X/G −→ Y/G. Informally speaking, the G-bundle structure of X −→ X/G is completely determined by the G-bundle structure of Y −→ Y/G; the base extension X/G −→ Y/G simply “spreads” this structure over a larger base. Thus given a generically free G-variety X one can ask for a ”minimal possible” base space B = Y/G over which it is defined. Of course, such a base space cannot be expected to be unique, so we will limit ourselves to studying its dimension.

Definition 3.1 The essential dimension of a primitive generically free G-variety X is the smallest possible value of dim(Y/G), where X −→ Y is a compression; see Definitions 2.1 and 2.15. We shall denote this number by ed(X, G) or simply ed(X) if the reference to G is clear from the context.

Remark 3.2 If X is primitive and X −→ Y is a G-compression then Y is primitive as well. Indeed, k(Y)^{G} is contained in k(X)^{G}, which is a field; see Lemma 2.2(b).

Note also that dim(Y/G) = dim(Y) − dim(G), where dim(Y) is the dimension of any irreducible component of Y. Thus we could have defined ed(X, G) as the minimal value of dim(Y) rather than of dim(Y/G) = dim(Y) − dim(G). Subtracting dim(G) is simply a matter of convention; we will later find it useful.

We now record several simple observations for future reference.

Lemma 3.3 Let X and Y be primitive generically free G-varieties. Then (a) ed(X, G) ≤ dim(X) − dim(G).

(b) Suppose there exists a G-compression X −→ Y. Then ed(X, G) ≤ ed(Y, G).

(c) Let Y_{0} be the union of ϕ(X), as ϕ ranges over all G-equivariant rational maps ϕ : X −→ Y. (As usual, ϕ(X) is defined as ϕ(U), where U is the domain of ϕ.) Suppose Y_{0} is dense in Y. Then ed(X, G) ≤ ed(Y, G).

(d) Let S be a G-variety with trivial G-action. Then ed(X × S, G) = ed(X, G).

(e) If G = {1} then ed(X, G) = 0 for any X.

Proof. Parts (a) and (b) are immediate consequences of the definition.

(c) Let f : Y −→ Z be a G-compression with dim(Z/G) = ed(Y, G). Suppose G acts freely on a dense G-invariant open subset V of Z. Let U = ϕ^{−1}(V) ⊂ Y. By our assumption Y_{0} ∩ U ≠ ∅. In other words, there exists a rational G-equivariant map ϕ : X −→ Y such that ϕ(X)∩U ≠ ∅. Then f ∘ϕ : X −→ Z is a well-defined G-equivariant rational map. Denote the closure of the image of this map in Z by Z_{0}. By our choice of U, Z_{0} is a generically free G-variety. Thus f ∘ ϕ can be viewed as a compression X −→ Z_{0}. Consequently, ed(X, G) ≤ dim(Z_{0}/G) ≤ dim(Z/G) = ed(Y, G), as claimed.

(d) Since the natural projection X × S ⟶ X is a G-compression, part (b) says that ed(X × S, G) ≤ ed(X, G). On the other hand, given s ∈ S, let ϕ_{s} : X ⟶ X × S be the (regular) G-equivariant map given by ϕ_{s}(x) = (x, s). Since the images of ϕ_{s} cover X × S, as s ranges over S, part (c) implies ed(X, G) ≤ ed(X × S, G).

(e) If G = {1} then we can (G-equivariantly) compress X to a point. □

3.2 The essential dimension of a group

Theorem 3.4 Let G be an algebraic group, X be a generically free primitive G-variety, E ⟶ X be a G-vector bundle over X and V, V′ be two generically free linear representations of G. Then

(a) ed(E, G) = ed(X, G),

(b) ed(V, G) = ed(V′, G) and

(c) ed(X, G) ≤ ed(V, G).

Proof. (a) By Lemma 2.19, E is birationally equivalent to X × k^{d}, as a G-variety. Thus ed(E, G) = ed(X × k^{d}, G) = ed(X, G); see Lemma 3.3(d).

(b) Let E = V × V′. Applying part (a) to the vector bundle E ⟶ V given by projection to the first factor, we obtain the equality ed(E, G) = ed(V, G). Similarly, ed(E, G) = ed(V′, G), and thus ed(V, G) = ed(V′, G).

(c) By Corollary 2.20 there exists a G-compression X × k^{d} −→ V, where d = dim(V). Now combining parts (b) and (d) of Lemma 3.3, we obtain

ed(X, G) = ed(X × k^{d}, G) ≤ ed(V, G) ,

as claimed. For an alternative proof, see Remark 7.2. □

Note that every linear algebraic group can be embedded in some GL_{n} and thus has a generically free linear representation. Thus we can now give the following definition.

Definition 3.5 The essential dimension of an algebraic group G is defined to be the essential dimension of a generically free linear G-representation. By Theorem 3.4(b) this number is is independent of the choice of the representation; we shall denote it by ed(G). Note that ed(G) = max{ed(X, G)}, as X ranges over all primitive generically free G-varieties; see Theorem 3.4(c).

Remark 3.6 In an earlier version of this paper the inequality ed(G, X) ≤ ed(G) was only established for reductive groups G. Theorem 3.4(c) in its current form was first pointed out to us by V. E. Kordonsky, who proved it by reducing the general case to one where G is reductive and then appealing to our earlier result. His argument also yields the following theorem [Ko_{1}, Theorem 1]:

If G is an algebraic group and L is the Levi subgroup of G then ed(G) = ed(L). □

3.3 First properties

Lemma 3.7 Suppose an algebraic subgroup G is an (algebraic) semidirect product of its subgroups N and H, with N ◁ G. Then ed(G) ≥ ed(H).

Proof. Let V be a generically free linear representation of G and let ϕ : V −→ Y be a G-compression with dim(Y/G) = ed(G). Then we can choose a regular model for Y/N which is a generically free H-variety; see Remark 2.6. We may thus view V as a generically free linear representation of H and the composite map V −→ Y/N below as an H-compression:

V −→ Y
|
↓
Y/N
|
↓
Y/G

Thus ed(G) = dim(Y) − dim(G) = dim(Y/N) − dim(H) ≥ ed(H). □

Lemma 3.8 ed(G_{1} × G_{2}) ≤ ed(G_{1}) + ed(G_{2}) for any two algebraic groups G_{1} and G_{2}.

Proof. For i = 1, 2 let V_{i} be a generically free linear representation of G_{i} and let ϕ_{i} : V_{i} −→ Y_{i} be a G_{i}-compression such that ed(G_{i}) = dim(Y_{i}) − dim(G_{i}). Then V_{1} × V_{2} is a generically free linear representation of G_{1} × G_{2} and

ϕ_{1} × ϕ_{2} : V_{1} × V_{2} −→ Y_{1} × Y_{2}

is a G_{1} ×G_{2}-compression. Thus ed(G) ≤ dim(Y_{1} ×Y_{2})−dim(G) = dim(Y_{1})+dim(Y_{2})− dim(G_{1}) − dim(G_{2}) = ed(G_{1}) + ed(G_{2}), as claimed. □

3.4 Examples

Example 3.9 (a) Let T_{d} = (k^{∗})^{d} be a d-dimensional torus. Then ed(T_{d}) = 0 for any d ≥ 1.

(b) ed(GL_{n}) = 0,

(c) ed(SL_{n}) = 0.

Proof. (a) The natural representation of T_{d} on V = k^{d} is generically free. Hence by Lemma 3.3(a), ed(T_{d}) = ed(V, T_{d}) ≤ dim(V) − dim(T_{d}) = 0.

(b) Consider the linear representation of GL_{n} on V = M_{n}(k) given by left multiplication. By Lemma 3.3(a), ed(GL_{n}) ≤ dim(V) − dim(GL_{n}) = 0.

(c) Let SL_{n} act on V = M_{n}(k) by left multiplication, as above. Let Y ⊂ V be the set of matrices of determinant 1 and let f : V −→ Y be given by

f(A) = ( 1 ; \overline{det(A)} v_{1}, v_{2}, …, v_{n}) ,

where A = (v_{1}, …, v_{n}) is the matrix whose columns are v_{1}, …, v_{n}. It is easy to see that f is a SL_{n}-compression; thus ed(SL_{n}) ≤ dim(Y) − dim(SL_{n}) = 0. □

Example 3.10 (a) ed(O_{n}) ≤ n, (b) ed(SO_{n}) ≤ n − 1.

Proof. Let W = k^{n}, q : W × W ⟶ k be a non-degenerate symmetric bilinear form, and O_{n} be the group of q-preserving linear transformations W ⟶ W. If v ∈ W and 0 ≠ v_{1}, …, v_{r} ∈ W, we define P (v; v_{1}, …, v_{r}) by

P (v; v_{1}, …, v_{r}) = v − ∑_{i=1}^{r} \underline{q(v_{i}, v)} ; q(v_{i}, v_{i}) v_{i}  (4)

Note that if v_{1}, …, v_{r} are mutually orthogonal then P (v, v_{1}, …, v_{r}) is just the component of v orthogonal to v_{1}, …, v_{r}.

For d = 1, …, n let Y_{d} ⊂ W^{d} be the set of elements (w′_{1}, …, w′_{d}) such that q(w′_{i}, w′_{j}) = 0 and q(w′_{i}, w′_{i}) ≠ 0 for any 1 ≤ i < j ≤ d. Note that Y_{d} is a locally closed O_{n}-invariant subvariety of W^{d}. An easy induction argument, using the projection Y_{d+1} ⟶ Y_{d} to the first d components, shows that Y_{d} is an irreducible variety of dimension n + (n − 1) + … + (n − d + 1) = d(2n − d + 1)/2.

(a) Consider the linear representation of O_{n} on V = W^{n}. Both W^{n} and Y_{n} are generically free O_{n}-varieties. We now observe that the usual diagonalization process gives rise to a compression f : W^{n} −→ Y_{n}. That is, we define f(w_{1}, …, w_{n}) = (w′_{1}, …, w′_{n}), where w′_{1} = w_{1} and

w′_{i} = P (w_{i}; w′_{1}, …, w′_{i−1})  (5)

for i = 2, …, n. Thus ed(O_{n}) ≤ dim(Y_{n}/O_{n}) = dim(Y_{n}) − dim(O_{n}) = n(n + 1)/2 − n(n − 2)/2 = n, as claimed.

(b) Note that W^{n−1} is a generically free linear representation of SO_{n} (but not of O_{n}) and Y_{n−1} is a generically free SO_{n}-variety of W^{n−1}. Moreover, the map W^{n−1} −→ Y_{n−1} given by (w_{1}, …, w_{n−1}) ↦ (w′_{1}, …, w′_{n−1}), with w′_{1}, …, w′_{n−1} as in (5), is an SO_{n}- compression. Thus

ed(SO_{n}) ≤ dim(Y_{n−1}) − dim(SO_{n}) = (n − 1)(n + 2)/2 − n(n − 1)/2 = n − 1. □

4 (G − H)-sections and essential dimension

4.1 The essential dimension of a section

Lemma 4.1 Let X be a generically free G-variety and let S ⊂ X be a (G, H)-section; see Definition 2.9. Then ed(X, G) ≤ ed(S, H).

Proof. Recall that X ≃ G ∗_{H} S; see Definition 2.12 and Lemma 2.14.

Choose a generically free H-variety Y and an H-compression α : S −→ Y such that dim(Y/H) = ed(S, H). Then G∗_{H} Y is a generically free G-variety; see Remark 2.13(a). It is easy to see that α lifts to a G-compression β : X ≃ G ∗_{H} S −→ G ∗_{H} Y given by β : (g, s) ↦ (g, α(s)). Thus ed(X, G) = ed(G ∗_{H} S, G) ≤ dim(G ∗_{H} Y) − dim(G) = dim(Y) − dim(H) = ed(S, H). □

Example 4.2 To illustrate Lemma 4.1 we will give an alternative proof of the inequalities ed(O_{n}) ≤ n and ed(SO_{n}) ≤ n − 1 of Example 3.10.

Let W = k^{n}, q be a non-degenerate symmetric bilinear form of W, e_{1}, …, e_{n} be an orthonormal basis with respect to q, and V = W^{n} be a generically free linear O_{n}- variety, as in Example 3.10. Set V_{i} = Span(e_{1}, …, e_{i}). Then S = V_{1} × … × V_{n} is a (O_{n}, (ℤ/2)^{n})-section of V = W^{n}. Thus by Lemma 4.1

ed(O_{n}) = ed(V, O_{n}) ≤ ed(S, (ℤ/2)^{n}) = ed((ℤ/2)^{n}) ≤ n .

Note that ed(S, (ℤ/2)^{n}) = ed((ℤ/2)^{n}) because S is a linear space with a generically free (ℤ/2)^{n}-action. The last inequality follows from the fact that the group (ℤ/2)^{n} has a generically free n-dimensional representation. (We remark that, in fact, ed((ℤ/2)^{n}) = n by [BR_{1}, Theorem 6.1]; however, all we need here is the inequality ed((ℤ/2)^{n}) ≤ n, which is immediate from the definition of essential dimension.)

Note that S can also be viewed as a (SO_{n}, (ℤ/2)^{n−1})-section for the SO_{n}-action on V. Applying Lemma 4.1 to this section yields the inequality ed(SO_{n}) ≤ n − 1 of Example 3.10(b). □

Another application of Lemma 4.1 can be found in the proof of Proposition 11.7.

Proposition 4.3 Let G be a connected semisimple group whose center is trivial. Let N be the normalizer of a maximal torus in G. Then ed(G) ≤ ed(N).

Proof. Let T be a maximal torus of G and let Lie(G) and Lie(T) be the Lie algebras of G and T respectively. Then the linear representation of G on

V = Lie(G) × Lie(G)

given by the adjoint action in each component is generically free and

S = Lie(T) × Lie(G)

is a (G, N)-section for this action; see [Po, 1.7.17]. Thus ed(N) = ed(S, N); see Definition 3.5. By Lemma 4.1 we have ed(G) = ed(V, G) ≤ ed(S, N) = ed(N), as claimed.

□

Remark 4.4 Note that ed(G) can be strictly less than ed(N). Indeed, let G = SL_{n}. Then ed(G) = 0; see Example 3.9. On the other hand, ed(N) > 0, since N is not connected; see Theorem 5.4.

We also remark that the group N cannot be replaced by the Weyl group W of G in the statement of Proposition 4.3. In other words, the inequality ed(G) ≤ ed(W) is false in general. For example, if G = PGL_{4} then W = S_{4} and thus ed(W) = 2; see [BR_{1}, Thm. 6.5]. On the other hand, ed(PGL_{4}) ≥ 4; see Theorem 9.3. □

4.2 An application

Theorem 4.5 ed(PGL_{n}) ≤ n^{2} − 2n for every n ≥ 4.

Proof. Let T be a maximal torus in PGL_{n} and let N be the normalizer of T. By Proposition 4.3 it is sufficient to show that ed(N) ≤ n^{2} −2n. We therefore take a closer look at N. The maximal torus T can be thought of as (k^{∗})^{n}/Δ, where Δ ≃ k^{∗} is the diagonal subgroup of (k^{∗})^{n}. Then N can be written as a semidirect product T ×_{ϕ} S_{n}, where ϕ : S_{n} ⟶ Aut(T) is given by permuting the n factors of k^{∗}.

We will now construct a linear representation V of N. Let V be the k-vector space freely spanned by the n(n − 1) basis vectors e_{ij}, where i, j = 1, …, n and i ≠ j. We shall denote ∑ x_{ij}e_{ij} by (x_{ij}). The action of N on V is defined as follows. Write an element of N as (t, σ), where t = (t_{1}, …, t_{n}) (modulo Δ) and σ ∈ S_{n}. Then

(t, σ) : e_{ij} ↦ t_{σ(i)}t_{σ(j)}^{−1}e_{σ(i)σ(j)} .

One can check that this action is generically free and thus

ed(N) = ed(V, N) ≤ dim(V/N) = dim(V) − dim(N) = n^{2} − 2n + 1 .

We shall, instead, derive the slightly better bound of Theorem 4.5 by considering the N-action on the projective space P (V).

Lemma 4.6 The action of N on P (V) is generically free.

Proof. Let a, a′, b, b are four distinct integers between 1 and n. Define a rational function z_{(a,a′)(b,b′)} on P (V) by

z_{(a,a′)(b,b′)} = \underline{x_{ab}x_{a′b′}} ; x_{ab′}x_{a′b} .

Note that z_{(a,a′)(b,b′)} = z_{(c,c′)(d,d′)} implies {a, a′} = {c, c′} and {b, b′} = {d, d′}. Let U be the dense N-invariant open subset of V consisting of those p ∈ P (V) satisfying (i) x_{ij}(p) ≠ 0 for every distinct i, j = 1, …, n, and (ii) z_{(a,a′)(b,b′)}(p) ≠ z_{(c,c′)(d,d′)}(p) whenever {a, a′} ≠ {c, c′} or {b, b′} ≠ {d, d′}.

We claim that N acts freely on U. Indeed, let p ∈ U and suppose (t, σ)p = p for some (t, σ) ∈ N. Note that the functions z_{(a,a′)(b,b′)} are T -invariant and

(t, σ)z_{(a,a′)(b,b′)} = z_{(σ(a),σ(a′))(σ(b)σ(b′))} .

Thus if p ∈ U, then {σ(a), σ(a′)} = {a, a′} and {σ(b), σ(b′)} = {b, b′} for every 4-tuple a, a′, b, b′ of distinct integers between 1 and n. If n ≥ 4 then the only σ ∈ S_{n} with this property is the identity element.

We have thus proved that if p ∈ U and (t, σ)p = p then σ = 1 in S_{n}. We now want to show that t = 1 in T. Indeed, since every coordinate x_{ij} of p is assumed to be non-zero, (t, 1)p = p implies t_{a}t_{b}^{−1} = t_{c}t_{d}^{−1} for every 1 ≤ a, b, c, d ≤ n with a ≠ b and c ≠ d. Letting b = d = 1, we see that t_{2} = t_{3} = … = t_{n}. Similarly for b = d = 2 we obtain t_{1} = t_{3} = … = t_{n}. Thus t_{1} = … = t_{n}, i.e., t = 1 in T. This completes the proof of the lemma. □

To finish the proof of the theorem, consider the natural (N-equivariant) projection map V −→ P (V). By Lemma 4.6 this map is an N-compression. Thus by Lemma 3.3(a-b) ed(N) ≤ dim(P (V)) − dim(N) = (n^{2} − n − 1) − (n − 1) = n^{2} − 2n, as claimed. □

Remark 4.7 In Section 9 we will strengthen the upper bound of Theorem 4.5 for every n which is not a power of 2; see Theorem 9.6 and Remark 9.9.

5 Groups of low essential dimension

The results of this section have been suggested to us by V. L. Popov.

5.1 Rational sections

Let X be a primitive G-variety (see Definition 2.1) and let π : X −→ X/G be the rational quotient map for the G-action on X. A subvariety S ⊂ X is said to be a rational section if π_{|S} : S −→ X/G is a birational isomorphism.

Remark 5.1 S is a rational section of X if and only if S is a (G, H)-section with H = {1_{G}}. Indeed, if S is a (G, {1})-section then π_{|S} : S −→ X/G is a birational isomorphism by Lemma 2.11. The opposite implication follows from Theorem 2.3.

□

Lemma 5.2 Let X be a primitive generically free G-variety. Then ed(X, G) = 0 if and only if X has a rational section.

Proof. Suppose ed(X, G) = 0. Then there exists a G-compression f : X −→ Y, where Y/G is a point. In other words, there is a point y ∈ Y such that Gy is dense in Y and Stab(y) = {1_{G}}. It is now easy to see that S = f^{−1}(y) is a (G, {1})-section, and, hence, a rational section; see Remark 5.1.

Conversely, suppose S ⊂ X is a rational section. Then S is a (G, {1})-section; hence, by Lemmas 4.1 and 3.3(e), ed(X, G) ≤ ed(S, {1}) = 0. □

5.2 Special groups

Let G be an algebraic group defined over k. Recall that G is called special if H^{1}(K, G) = {1} for every field K containing k. Special groups were introduced by Serre [Se_{1}] and classified by Grothendieck [G].

Proposition 5.3 The following conditions are equivalent:

(a) G is a special group,

(b) ed(X, G) = 0 for every primitive generically free G-variety X,

(c) ed(G) = 0.

Proof. By Lemma 5.2 condition (b) is equivalent to

(b′) every primitive generically free G-variety has a rational section and (c) is equivalent to

(c′) every generically free linear representation of G has a rational section. It is well known that conditions (a), (b’) and (c’) are equivalent; see e.g., [Po, 1.4], [PV, Section 2.6], or [Do, Sect. 7]. Hence, so are (a), (b), and (c). □

Combining Proposition 5.3 with Grothendieck’s classification, of special groups (see [G, Section 5] or [PV, Theorem 2.8]), we obtain the following result.

Theorem 5.4 Let G be an algebraic group defined over k. Then ed(G) = 0 if and only if G is connected and its maximal connected semisimple subgroup is a direct product G_{1} × … × G_{r}, where each G_{i} is of type SL or Sp. □

5.3 n-special groups

Definition 5.5 We will say that an algebraic group G defined over k is n-special if H^{1}(K, G) = {1} for any extension K/k of transcendence degree ≤ n.

Note that a special group is n-special for every n.

Proposition 5.6 Let G be an n-special algebraic group and let X be a generically free primitive G-variety. Then

(a) either ed(X, G) = 0 or ed(X, G) ≥ n + 1 and

(b) either ed(G) = 0 or ed(G) ≥ n + 1.

Proof. (a) Assume the contrary: ed(X, G) = i, where 1 ≤ i ≤ n. Then there exists a compression α : X −→ Y such that trdeg_{k} k(Y)^{G} = dim(Y/G) = i. Recall that the generically free action of G on Y determines an element of H^{1}(k(Y/G), G); following Popov [Po, 1.3], we shall denote this class by cl(G : Y). By our assumption H^{1}(k(Y/G), G) = {1} and thus cl(G, Y) = 1. Consequently, Y has a rational section; see [Po, 1.4.1]. By Lemma 5.2, this implies that ed(Y, G) = 0. Thus ed(X, G) ≤ ed(Y, G) = 0 (see Lemma 3.3(b)), contradicting our assumption.

Part (b) is an immediate consequence of (a) and Definition 3.5. □

Corollary 5.7 Let G be a connected algebraic group. Then (a) ed(X, G) ≠ 1 for any irreducible G-variety X.

(b) ed(G) ≠ 1.

Proof. By a theorem of Steinberg [St, Thm. 1.9] every connected group is 1-special.

□

Remark 5.8 Note that Corollary 5.7 fails if G is not assumed to be connected. For example, ed(G) = 1 if G is a (finite) cyclic or odd dihedral group; see [BR_{1}, 6.2].

Remark 5.9 A special case of Conjecture II of Serre says that every simply connected semisimple group is 2-special (see [Se_{3}, III.3.1]; note that in our setting G is a split group). This assertion has been proved for all simple groups other than E_{8}; see [Se_{3}, Sect 5], [Se_{4}, III.3], [BP], [Ko_{2}], and [Gi].

Remark 5.10 It is shown in [RY_{2}] that every 3-special group (and, hence, every nspecial group for n ≥ 3) is special.

6 Structured spaces I

6.1 Definition and examples

Definition 6.1 Let F be a field.

(a) A structured space over F is a pair (W, β), where W is a finite-dimensional F - vector space and β is a tensor on W, i.e., β ∈ T (W)⊗T (W^{∗}). If F′/F is a field extension then (W, β) ⊗_{F} F′ is defined to be the structured space (W′, β′), where W′ = W ⊗_{F} F′ and β′ = β ⊗ 1 ∈ T (W) ⊗ T (W^{∗}) ⊗ F′ = T (W′) ⊗ T (W^{′∗}).

(b) Two structured spaces (W, β) and (W′, β′) over F are isomorphic if there exists an isomorphism ϕ : W ⟶ W′ of F -vector spaces such that ϕ^{∗}(β) = β′, where ϕ^{∗} is the isomorphism T (W) ⊗ T (W^{∗}) ⟶ T (W′) ⊗ T (W^{′∗}) induced by ϕ.

(c) In the special case where A is a finite-dimensional F -algebra, W is the underlying F -vector space of A and the multiplicative structure is given by a tensor β ∈ W^{∗} ⊗ W^{∗} ⊗ W, we will write A = (W, β). Note that here the algebra A is not assumed to be commutative or associative, or to have an identity element.

Remark 6.2 The above definition is essentially the same as the one in [Se_{2}, X.2] or [Se_{4}, III.1.1]. The only difference is that we do not require α to be homogeneous, i.e., to lie in T^{p}(W) ⊗ T^{q}(W^{∗}) for some p, q ≥ 0; see Example 6.3(e) below.

We will always assume (as we do elsewhere in this paper) that F is a field extension of k; if F = k we will usually denote our structured spaces by (V, α). In this case we will be interested in the algebraic group Aut_{k}(V, α) =^{def} {g ∈ GL(V) | g^{∗}(α) = α}. The following examples will be of special interest to us in the sequel.

Example 6.3 (a) V = k^{n} and α = 0. Then Aut_{k}(V, α) = GL_{n}.

(b) V = k^{n} and 0 ≠ α ∈ Λ^{n}(V^{∗}) is a volume form on V. Then Aut_{k}(V, α) = SL_{n}.

(c) V = k^{2n} and α ∈ Λ^{2}(V^{∗}) is a symplectic (i.e., non-degenerate skew-symmetric bilinear) form. Then Aut_{k}(V, α) = Sp_{2n}.

(d) V = k^{n} and α ∈ S^{2}(V^{∗}) is a non-degenerate symmetric bilinear form. Then Aut_{k}(V, α) = O_{n}.

(e) V = k^{n} and α = α_{1}+α_{2}, where α_{1} ∈ Λ^{n}(W^{∗}) is a volume form and α_{2} ∈ S^{2}(W^{∗}) is a non-degenerate symmetric bilinear form. Then any element g ∈ GL(V) preserving α has to preserve both α_{1} and α_{2}. Thus Aut_{k}(V, α) = SL_{n} ∩ O_{n} = SO_{n}.

(f) (V, α) is the algebra k ⊕…⊕k (n times); see Definition 6.1(c). Then Aut_{k}(V, α) is the symmetric group S_{n}.

(g) (V, α) = M_{n}(k) is the algebra of n × n-matrices over k. Then Aut_{k}(V, α) = PGL_{n}.

(h) (V, α) = O, where O is the (split) octonion algebra over k; see Section 11.1. Then Aut_{k}(O) is the exceptional group G_{2}.

(i) (V, α) = A, where A is the (split) Albert algebra over k; see Section 11.3. Then Aut_{k}(V, α) is the exceptional group F_{4}.

(j) V = k^{27} is the underlying vector space of the Albert algebra A and α ∈ V^{∗} ⊗ V^{∗} ⊗ V^{∗} be the trilinear form associated to the cubic norm on A; see Sections 11.3 and 11.5. Then Aut(V, α) is the (simply connected) exceptional group E_{6}.

6.2 First properties

Let (W, β) be a structured space defined over a field F containing k. Suppose B = {b_{1}, …, b_{n}} is an F -basis of W. Denote the dual basis of W^{∗} by B^{∗} = {b_{1}^{∗}, …, b_{n}^{∗}}. Then the tensors b_{I} = b_{i_{1}} ⊗ … ⊗ b_{i_{r}} form an F -basis of T (W), as r ranges over the positive integers, I = (i_{1}, …, i_{r}), and each i_{j} ranges over {1, …, n}. Similarly, the tensors b_{J}^{∗} = b_{j_{1}}^{∗} ⊗ … ⊗ b_{j_{s}}^{∗} form an F -basis of T^{∗}(W). Thus we can write

β = ∑_{I,J} β_{I}^{J}(B) b_{I} ⊗ b_{J}^{∗} ,  (6)

where each coefficient β_{I}^{J}(B) lies in F. Note that only finitely many of these coefficients are non-zero.

Let (V = k^{n}, α) be a structured space and G = Aut_{k}(V, α). Note that the G-action on V may not be generically free; however, the (diagonal) G-action on V^{n} is always free on the open subset (V^{n})′ consisting of n-tuples of linearly independent elements. (We prefer the notation (V^{n})′ to GL(V), since we view this set as a Zariski open subset of V^{n} and ignore the group structure.)

Let B ∈ (V^{n})′ be a basis of V. Write α = ∑_{I,J} α_{I}^{J}(B) b_{I} ⊗ b_{J}^{∗}, as in (6). Then each α_{I}^{J} can be viewed as a function (V^{n})′ ⟶ k, which takes B ∈ (V^{n})′ to α_{I}^{J}(B). It is easy to see that the functions α_{I}^{J} are regular on (V^{n})′ and thus rational on V^{n}.

Lemma 6.4 Let V = k^{n}, α ∈ T (V) ⊗ T (V^{∗}), G = Aut_{k}(α) and (V^{n})′ ⊂ V^{n} be as above. Suppose B, B′ ∈ (V^{n})′. Then

(a) B′ = gB for some g ∈ G if and only if α_{I}^{J}(B) = α_{I}^{J}(B′) for every I and J.

(b) If Z is a G-invariant subvariety of V^{n} such that Z ∩ (V^{n})′ ≠ ∅ then k(Z)^{G} = k(α_{I}^{J} _{|Z}).

Proof. (a) Write B′ = gB for some g ∈ GL(V) and observe that g ∈ G = Aut_{k}(V, α) if and only if α_{I}^{J}(B) = α_{I}^{J}(B′) for every I and J.

(b) Each α_{I}^{J} defines a regular map (V^{n})′ ⟶ k; together they define a regular map f : (V^{n})′ ⟶ k^{N}, where N is the number of non-zero α_{I}^{J}’s. Denote f(Z ∩ (V^{n})′) by Z_{0}. By part (a) f separates the G-orbits in (V^{n})′ and, hence, in Z ∩ (V^{n})′. Thus by Remark 2.5, Z_{0} is birationally isomorphic to Z/G; hence, k(Z)^{G} = k(Z/G) = k(Z_{0}) = k(α_{I}^{J} _{|Z}), as claimed. □

7 Spaces of rational maps

7.1 Equivariant maps into vector spaces

Proposition 7.1 Let G be an algebraic group and let V be a linear representation of G. Then for every generically free G-variety X and every v ∈ V, there exists a G-equivariant rational map f : X −→ V whose image contains v.

Proof. Suppose the action of G on V is given by the group homomorphism ϕ : G ⟶ GL_{n}, where n = dim(V). We may assume without loss of generality that ϕ is injective. Indeed, otherwise we can replace V by V′ = V ⊕ W, where W is a generically free linear representation of G. If we can construct a G-equivariant rational map X −→ V′ whose image contains (v, 0), then the composition of this map with the projection V′ = V ⊕ W ⟶ V will have the desired properties.

Note that we may also assume without loss of generality that X is primitive. Indeed, by Lemma 2.2 X is isomorphic to a disjoint union of primitive varieties. It is enough to define f on one of them; after that the others can be sent to 0.

Thus we can think of G as a subgroup of GL_{n} = GL(V); the G-action on V is then given by left multiplication. Let X′ = GL_{n} ∗_{G} X; see Definition 2.12. Recall that X′ is a generically free GL_{n}-variety; see Remark 2.13. Since GL_{n} is a special group, Proposition 5.3 tells us that ed(X′, GL_{n}) = 0. This means that there exists a GL_{n}-compression α : X′ −→ O, where O is a single GL_{n}-orbit with trivial stabilizer. In other words, up to birational isomorphism of GL_{n}-varieties, O ≃ V^{n} ≃ M_{n}, where GL_{n} acts on M_{n} by left multiplication.

Note that the image of α contains every non-singular matrix in M_{n} (these matrices form a single dense GL_{n}-orbit). In particular, there exists an x′ = (g, x) ∈ X′ such that α(x′) = I_{n} = n × n-identity matrix. Then α(1_{G}, x) = g^{−1}. Let v_{1}, …, v_{n} be the columns of g^{−1}; since they are linearly independent, we can write

v = c_{1}v_{1} + … + c_{n}v_{n}

for some c_{1}, …, c_{n} ∈ k. Let β : M_{n} ⟶ V be the linear map given by

β(A) = A ( c_{1} ; ⋮ ; c_{n} ) .

Note that β is GL_{n}-equivariant and β(g^{−1}) = v. Now the desired map f : X −→ V is obtained as a composition

X ↪ X′ = GL_{n} ∗_{G} X −→^{α} M_{n} ⟶^{β} V ,

where the inclusion X ↪ X′ is given by x ↦ (1_{G}, x); see Remark 2.13. This inclusion is clearly G-equivariant; the maps α and β are GL_{n}-equivariant (and, hence, G-equivariant) by our construction. Hence, f is G-equivariant and f(x) = v, as desired. □

Remark 7.2 Let G be an algebraic group, X be a generically free primitive G-variety and V be a generically free linear representation of G. Then combining Proposition 7.1 with Lemma 3.3(c), we obtain the inequality ed(X, G) ≤ ed(V, G) =^{def} ed(G). This gives another proof of Theorem 3.4(c). □

Definition 7.3 Let G be an algebraic group, G ⟶ GL(V) be a linear representation of G and X be a G-variety. We shall denote the k(X)^{G}-vector space of G-equivariant rational maps X −→ V by RMaps_{G}(X, V).

Equivalently, RMaps_{G}(X, V) = (V ⊗_{k} k(X))^{G}.

Lemma 7.4 Let V = k^{n}, let G ⟶ GL(V) be a linear representation of G, and let X be a primitive G-variety. Then

(a) b_{1}, …, b_{m} ∈ RMaps_{G}(X, V) are linearly independent over k(X)^{G} if and only if b_{1}(x), …, b_{m}(x) are linearly independent over k for x in general position in X.

(b) If X is generically free then dim_{k(X)^{G}} (RMaps_{G}(X, V)) = n.

Proof. (a) Suppose b_{1}(x), …, b_{m}(x) are linearly independent for x ∈ X in general position and ∑_{i=1}^{m} f_{i}b_{i} = 0 for some f_{1}, …, f_{m} ∈ k(X)^{G}. Then evaluating both sides at x ∈ X in general position, we conclude that f_{1} = … = f_{m} = 0 in k(X)^{G} and thus b_{1}, …, b_{m} are k(X)^{G}-linearly independent.

Conversely, suppose b_{1}(x), …, b_{m}(x) are k-linearly dependent for x ∈ X in general position. Then we may assume without loss of generality that b_{1}(x), …, b_{m−1}(x) are linearly independent for x in general position; otherwise, we can simply replace b_{1}, …, b_{m} by b_{1}, …, b_{m−1}. Thus for each x in general position in X, there exist uniquely defined f_{1}(x), …, f_{m−1}(x) ∈ k such that

b_{m} = f_{1}b_{1} + … + f_{m−1}b_{m−1} .  (7)

It remains to prove that f_{1}, …, f_{m−1} ∈ k(X)^{G}. By Cramer’s rule f_{1}, …, f_{m−1} ∈ k(X). To prove these functions are G-invariant, choose x ∈ X and g ∈ G, then substitute gx into (7) and apply g^{−1} to both sides. Since each b_{i} is a G-equivariant map X −→ V, we have g^{−1}b_{i}(gx) = b_{i}(x) and thus

b_{m}(x) = f_{1}(gx)b_{1}(x) + … + f_{m−1}(gx)b_{m−1}(x) .

By uniqueness of f_{i}(x), we conclude that f_{i}(gx) = f_{i}(x), i.e., f_{i} ∈ k(X)^{G}, as claimed.

(b) The inequality dim_{k(X)^{G}} RMaps(X, V) ≤ n is a direct consequence of part (a). To prove the opposite inequality, let v_{1}, …, v_{n} be a k-basis of V, v = (v_{1}, …, v_{n}) ∈ V^{n}. By Proposition 7.1 there exists a rational G-equivariant map b : X −→ V^{n} whose image contains v. Write b = (b_{1}, …, b_{n}), where each b_{i} is a rational G-equivariant map X −→ V, i.e., and element of RMaps_{G}(X, V). In order to complete the proof of part (b), it is enough to show that b_{1}, …, b_{n} are k(X)^{G}-linearly independent. Let S be the set of x ∈ X such that b_{1}(x), …, b_{n}(x) are k-linearly independent. Clearly S is an open G-invariant subset of X; moreover, since the image of b contains v, S ≠ ∅. Thus S is open and dense in X; the desired conclusion now follows from part (a). □

7.2 Structured spaces of rational maps

Let (V = k^{n}, α) be a structured space, let G = Aut_{k}(V, α), and let H be an algebraic subgroup of G, as above. In this setting RMaps_{H}(X, V) carries a naturally defined tensor, which we shall denote by α_{H}^{X}. To define α_{H}^{X}, choose a k(X)^{H}-basis B = {b_{1}, …, b_{n}} of RMaps_{H}(X, V). Then

α_{H}^{X} =^{def} ∑_{I,J}(α_{H}^{X})_{I}^{J} b_{I} ⊗ b_{J}^{∗} ,

where we define (α_{H}^{X})_{I}^{J} ∈ k(X)^{H} (as a rational function on X) by

(α_{H}^{X})_{I}^{J}(x) = α_{I}^{J}(B(x)) .  (8)

Here B(x) = {b_{1}(x), …, b_{n}(x)} is a basis of V (for x in general position in X). Note that since H is a subgroup of G = Aut_{k}(V, α), each function x ↦ α_{I}^{J}(B(x)) is, indeed, an element of k(X)^{H}. Moreover, it is easy to see that our definition of α_{H}^{X} does not depend on the choice of B.

Example 7.5 Suppose α ∈ V^{∗} ⊗ V^{∗} ⊗ V defines the structure of an algebra A on V, i.e., (V, α) = A; see Definition 6.1(c). Then the structured space (RMaps_{H}(X, V), α_{H}^{X}) is the algebra of H-equivariant rational maps X −→ V, with pointwise multiplication.

Lemma 7.6 Let (V = k^{n}, α) be a structured space, H be an algebraic subgroup of G = Aut_{k}(V, α), and X be a generically free primitive G-variety.

(a) Suppose f : X −→ Y is an H-compression. Then

(RMaps_{H}(X, V), α_{H}^{X}) ≃ (RMaps_{H}(Y, V), α_{H}^{Y}) ⊗_{k(Y)^{H}} k(X)^{H} ,

where the field extension k(X)^{H}/k(Y)^{H} is induced by f.

(b) Suppose S is an (H, H′)-section of X. Then

(RMaps_{H}(X, V), α_{H}^{X}) ≃ (RMaps_{H′}(S, V), α_{H′}^{S}) .

(c) Suppose X = H × X_{0}, where X_{0} is an irreducible variety and H acts on X by h(h′, x_{0}) = (hh′, x_{0}). Then (RMaps_{H}(X, V), α_{H}^{X}) ≃ (V, α) ⊗_{k} k(X_{0}).

Proof. (a) Let ϕ : RMaps_{H}(Y, V)⊗_{k(Y)^{H}} k(X)^{H} ⟶ RMaps_{H}(X, V) be k(X)^{H}-linear map given by a ⊗ 1 ⟶ a ∘ f. If C = {c_{1}, …, c_{n}} is a k(Y)^{H}-basis of RMaps_{H}(Y, V) then Lemma 7.4 shows that ϕ(C) is a k(X)^{H}-basis of RMaps_{H}(Y, V). This proves that ϕ is an isomorphism of k(X)^{H}-vector spaces. Comparing the coefficients of α_{H}^{X} in the basis C to the coefficients of α_{H}^{Y} in the basis ϕ(C), we conclude that ϕ is an isomorphism of structured spaces.

(b) Let B = {b_{1}, …, b_{n}} be a k(X)^{H}-basis of RMaps_{H}(X, V). Recall that by Lemma 2.11, the restriction a ↦ a_{|S} induces an isomorphism between the fields k(X)^{H} and k(S)^{H′}. We will therefore identify these fields. Moreover, by Lemma 7.4, B_{|S} = {(b_{1})_{|S}, …, (b_{n})_{|S}} is a basis of RMaps_{H′}(S, V). Hence, ϕ : a ↦ a_{|S} gives rise to an isomomorphism ϕ : RMaps_{H}(X, V) ⟶ RMaps_{H′}(S, V) of k(X)^{H}-vector spaces. To show that ϕ induces the desired isomorphism of structured spaces, observe that if we restrict the coefficients of α_{H}^{X} in the basis B to S, we obtain the coefficients of α_{H′}^{S}; see (8).

(c) Note that {1} × X_{0} is an (H, {1})-section of X. Thus by part (b), we may assume H = {1} and X = X_{0}. Now apply part (a) to the {1}-compression X ⟶ {pt}, where {pt} is a single point. Since (RMaps_{{1}}({pt}, V), α_{{1}}^{{pt}}) = (V, α), this completes the proof. □

8 Structured spaces II

8.1 Structured spaces of type (V, α)

Definition 8.1 Let F be a field containing k and let (V = k^{n}, α) be a structured space. We shall say that (W = F^{n}, β) is a structured space of type (V, α) if

(W, β) ⊗_{F} E ≃ (V, α) ⊗_{k} E  (9)

for some field extension E/F.

Remark 8.2 A structured space (W = F^{n}, β) of type (V, α) is usually called “an F/kform of (V, α)”; see e.g., [Se_{4}, III.1]. We will use the term “structured space” in order to avoid linguistic confusion in those cases where α and β are themselves multilinear forms, as in, e.g., Example 6.3(b), (c), (d), (j).

Remark 8.3 If (W, β) is a structured space of type (V, α) then we can always choose the field extension E/F in Definition 8.1 to be finitely generated. Indeed, if v_{1}, …, v_{n} is a k-basis of V, w_{1}, …, w_{n} is an F -basis of W and the isomorphism (9) is given by w_{i} ↦ ∑_{j} a_{ij}v_{ij} with a_{ij} ∈ E then we can replace E by F (a_{ij}).

Example 8.4 (a) Suppose V = k^{n} and α is a volume (resp. non-degenerate symmetric bilinear) form on V. Then (W, β) is of type (V, α) if and only if W = F^{n} and β is a a volume (resp. non-degenerate symmetric bilinear) form on W.

(b) Let V = k^{n}, let e_{1}, …, e_{n} be a k-basis of V, α_{1} = e_{1}^{∗} ∧ … ∧ e_{n}^{∗} ∈ Λ^{n}(V), α_{2} = (e_{1}^{∗})^{2} + … + (e_{n}^{∗})^{2} ∈ S^{2}(V), and α = α_{1} + α_{2}. Recall that Aut_{k}(V, α) = SO_{n}; see Example 6.3(e). Now let F be a field extension of k and W = F^{n}. Then an easy linear algebra argument shows that β_{2} ↦ (W, α_{1} ⊗_{k} F + β_{2}) defines a bijection between equivalence classes of bilinear forms of determinant 1 on W (modulo SL(W)) and isomorphism classes of structured spaces (W, β) of type (V, α); cf. [KMRT, (29.29), p. 407].

(c) Suppose (V, α) is the algebra k ⊕ … ⊕ k (n times), as in Example 6.3(f). Then (W, β) is of type (V, α) if and only if β defines the structure of an n-dimensional etale F -algebra on W, i.e., W = F^{n} and (W, β) is isomorphic, as an F -algebra, to a direct sum of field extensions of F; cf. [KMRT, (29.9), p. 395].

(d) Suppose (V = k^{27}, α) = A, where A is the (split) Albert algebra; see Section 11.3. Then (W, β) is a structured space of type (V, α) if and only if W = F^{27} and β defines the structure of a central simple exceptional Jordan algebra on W; see [J, Sect V.7].

For other examples see Lemma 9.1 and Remark 11.4.

Lemma 8.5 Suppose (V = k^{n}, α) is a structured space with automorphism group G = Aut_{k}(V, α). Let H be an algebraic subgroup of G, and let X be a primitive generically free H-variety. Then (RMaps_{H}(X, V), α_{H}^{X}) is a structured space of type (V, α).

Proof. We may assume without loss of generality that X is irreducible. Indeed, otherwise an irreducible component X_{1} of X is an (H, H_{1})-section (see Example 2.10); we can thus replace X by X_{1}, H by H_{1} and then appeal to Lemma 7.6(b).

For irreducible X, consider the H-compression f : H × X ⟶ X, where H acts on H × X by h′(h, x) = (h′h, x) and f(h, x) = hx. Now apply Lemma 7.6(a) to this compression; the desired conclusion then follows from Lemma 7.6(c). □

Proposition 8.6 Let (V = k^{n}, α) be a structured space with automorphism group G = Aut_{k}(V, α), and let F be a finitely generated field extension of k. Then

(a) a structured space (W = F^{n}, β) is of type (V, α) if and only if there exists a generically free primitive G-variety X such that (W, β) ≃ (RMaps_{G}(X, V), α_{G}^{X}).

(b) Suppose X and X′ are generically free primitive G-varieties such that k(X)^{G} = k(X′)^{G} and (RMaps_{G}(X, V), α_{G}^{X}) ≃ (RMaps_{G}(X′, V), α_{G}^{X′}) (as structured spaces). Then X ≃ X′ (as G-varieties).

Proof. (a) The “if” assertion follows from Lemma 8.5. To prove the converse, assume (W = F^{n}, β) is a structured space of type (V = k^{n}, α). Choose E/F as in Definition 8.1. By Remark 8.3 we may assume that E is finitely generated over F and, hence, over k. Thus there exist irreducible varieties X_{0} and Y_{0} such that E = k(Y_{0}), F = k(X_{0}), and the field extension E/F is induced by a dominant rational morphism ϕ : Y_{0} −→ X_{0}.

Let B = {b_{1}, …, b_{n}} be an F -basis of W. Then B is also an E-basis of V ⊗_{k} E; see (9). Recall that (V, α) ⊗_{k} E ≃ (RMaps_{G}(Y, V), α_{G}^{Y}), where Y = G × Y_{0}; see Lemma 7.6(c). Thus we can view b_{1}, …, b_{n} as G-equivariant rational maps Y −→ V; together they define a G-equivariant rational map b : Y −→ V^{n}. Let Z be the closure of b(Y) in V^{n}. Note that Z ∩ (V^{n})′ ≠ ∅ (see Lemma 7.4) and thus the rational map \overline{b} : Y_{0} −→ Z_{0} = Z/G induced by b : Y −→ Z is given by (α_{I}^{J})_{|Z} ↦ β_{I}^{J}(B); see Remark 2.4 and Lemma 6.4(b). Since every β_{I}^{J}(B) lies in F, \overline{b} factors through X_{0} (via ϕ). Consequently b factors through X = X_{0} ×_{Z_{0}} Z. These maps are shown in the following diagram.

b : Y = G × Y_{0} −→ X = X_{0} ×_{Z_{0}} Z −→^{c} Z ⊂ V^{n}
↓  ↓  ↓
\overline{b} : Y_{0}  −→^{ϕ}  X_{0}  −→^{\overline{c}} Z_{0} ⊂ (V^{n})/G

Here the vertical arrows represent quotient maps and c (resp. \overline{c}) is the map X −→ Z ⊂ V^{n} (resp. X_{0} −→ Z_{0} ⊂ V^{n}/G) induced by b (resp. \overline{b}). Let c = (c_{1}, …, c_{n}), where each c_{i} is a G-equivariant rational map X −→ V and let C = {c_{1}, …, c_{n}}. Since Z ∩(V^{n})′ ≠ ∅, C is an F -basis of RMaps_{G}(X, V); see Lemma 7.4. By our construction (α_{G}^{X})_{I}^{J}(C) = β_{I}^{J}(B) for every I and J. This means that the structured spaces (W, β) and RMaps_{G}(X, V) are isomorphic via the linear map W ⟶ RMaps_{G}(X, V) which takes b_{i} to c_{i}.

(b) Denote the structured space (RMaps_{G}(X, V), α_{G}^{X}) by (W, β). We want to show that X can be uniquely recovered from (W, β). Let C = {c_{1}, …, c_{n}} be a basis of W = RMaps_{G}(X, V), c = (c_{1}, …, c_{n}) be the rational map X −→ V^{n} defined by C, and Z be the closure of c(X) in V^{n}. Then by Lemma 7.4, Z ∩(V^{n})′ ≠ ∅; thus c induces a rational map \overline{c} : X_{0} −→ Z_{0}, where Z_{0} = π(Z) ≃ Z/G and π : V^{n} ⟶ V^{n}/G is the quotient map. The maps c and \overline{c} are shown in the diagram below.

X −→^{c} Z ⊂ V^{n}
↓  ↓ π
X_{0} −→^{\overline{c}} Z_{0} ⊂ (V^{n})/G

Here the vertical maps are quotient maps and \overline{c} is given by α_{I}^{J} ↦ β_{I}^{J}(C). (Recall that k(V^{n}/G) = k(α_{I}^{J}); see Lemma 6.4(b).) Now note that X ≃ X_{0} ×_{Z_{0}} Z (see Lemma 2.16(b)) and Z = π^{−1}(Z_{0}).

This shows that X can be uniquely recovered from (W, β). More precisely, suppose

(RMaps_{G}(X, V), α_{G}^{X}) =^{def} (W, β) ≃ (RMaps_{G}(X′, V), α_{G}^{X′})

via a map which sends the basis C of (W, β) to a basis D of (RMaps_{G}(X′, V), α_{G}^{X′}). Denote the G-equivariant rational morphism (d_{1}, …, d_{n}) : X′ −→ V^{n} by d. Then the induced map \overline{d} : X′/G = X_{0} −→ V^{n}/G of quotient spaces is the same as \overline{c}; thus X′ ≃ X_{0} ×_{Z_{0}} Z ≃ X, as claimed. □

8.2 Structured spaces and essential dimension

Definition 8.7 Let F be a field, (W = F^{n}, β) be a structured space and F_{0} be a subfield of F.

(a) We say that (W, β) is defined over F_{0} if there exists an F -basis B of W such that every coefficient β_{I}^{J}(B) of β with respect to this basis lies in F_{0}. Equivalently, (W, β) is defined over F_{0} if and only if it is isomorphic to (W_{0}, β_{0}) ⊗_{F_{0}} F for some structured space (W_{0} = F_{0}^{n}, β_{0}).

(b) τ(W, β) =^{def} min{trdeg_{k}(F_{0})}, where (W, β) is defined over F_{0}. Equivalently, τ(W, β) = min{trdeg_{k}k(β_{I}^{J}(B) | I, J)}, as B ranges over all F -bases of W.

Theorem 8.8 Let (V, α) be a structured space and G = Aut_{k}(V, α).

(a) If X is a generically free primitive G-variety then ed(X, G) = τ(RMaps_{G}(X, V), α_{G}^{X}).

(b) ed(G) = max{τ(W, β)}, where the maximum is taken over all structured spaces (W, β) of type (V, α).

Proof. (a) Denote the structured space (RMaps_{G}(X, V), α_{G}^{X}) by (W, β).

If X −→ Y is a G-compression then by Lemma 7.6(a) the structured space (W, β) is defined over k(Y)^{G}. If we choose Y to be of minimal possible dimension, then trdeg_{k}k(Y)^{G} = dim(Y/G) = ed(X, G). This proves that τ(W, β) ≤ ed(X, G).

To prove the opposite inequality, write (W, β) ≃ (W̃ , β̃) ⊗F̃ F, where (W̃ , β̃) is a structured space over F̃ and trdeg_{k}(F̃) = τ(W, β). Note that since (W, β) is a structured space of type (V, α), so is (W̃ , β̃). Thus by Proposition 8.6(a),

(W̃ , β̃) ≃ (RMaps_{G}(X̃), αX̃ ; G)

for some primitive generically free G-variety X̃ with k(X̃)^{G} = F̃. On the other hand, the extension F/F̃ is given by a rational morphism X/G −→ X̃/G of algebraic varieties. Using this morphism we can construct the fiber product Y = X/G ×_{X̃/G} X̃. By Lemma 7.6(a),

\begin{matrix}(RMaps_{G}(Y, V), α_{G}^{Y}) ≃ (RMaps_{G}(X̃, V), αX̃ ; G) ⊗ k(X)^{G} ≃ \\ (W̃ , β̃) ⊗F̃ F = (W, β) = (RMaps_{G}(X, V), α_{G}^{X}) . \end{matrix}

Hence, by Proposition 8.6(b), X is birationally isomorphic to Y = X/G×_{X̃/G} X̃. Since the natural projection Y ⟶ X̃ is a G-compression, we conclude that

ed(X, G) = ed(Y, G) ≤ dim(X̃/G) = trdeg_{k} k(X̃/G) = trdeg_{k} F̃ = τ(W, β) ,

as claimed.

(b) Recall that ed(G) = max{ed(X, G)}, as X ranges over all generically free primitive G-varieties; see Definition 3.5. The desired conclusion now follows from part (a) and Proposition 8.6. □

We conclude this section with two simple applications of Theorem 8.8. Other applications will be given in Sections 9-11.

Example 8.9 (a) Suppose V = k^{n} and α be a volume form on V, so that Aut_{k}(V, α) = SL_{n}. Then every volume form β on W = F^{n} can be written as b_{1} ∧ … ∧ b_{n} for some basis b_{1}, …, b_{n} of W. Thus every structured space (W, β) of type (V, α) is isomorphic to (V, α) ⊗_{k} F, and, consequently, τ(W, α) = 0. Applying Theorem 8.8, we conclude that ed(SL_{n}) = 0; cf. Example 3.9(c) and Theorem 5.4.

(b) (cf. [Se_{4}, III.1.2(a)]) Suppose V = k^{2n} and α ∈ Λ^{2}(V^{∗}) is symplectic form on V. Then every symplectic form β form on W = F^{2n} can be written as

β = b_{1}^{∗} ∧ b_{2}^{∗} + … + b_{2n−1}^{∗} ∧ b_{2n}^{∗}

in some basis b_{1}, …, b_{2n} of W. This means that every structured space (W, β) of type (V, α) is isomorphic to (V, α)⊗_{k} F and thus τ(W, α) = 0. By Theorem 8.8, we conclude that ed(Sp_{2n}) = 0; cf. Theorem 5.4.

9 Projective linear groups

9.1 Central simple algebras

Recall that PGL_{n} = Aut_{k}(M_{n}), where we view the matrix algebra M_{n} = M_{n}(k) as a structured space (k^{n^{2}}, α); see Definition 6.1(c).

Lemma 9.1 (Wedderburn; see e.g., [Se_{2}, X.5 Proposition 7] or [KMRT, 1.1]) Let F be a field containing k and let W = F^{n^{2}}. Then (W, β) is a structured space of type M_{n} if and only if β ∈ W^{∗} ⊗ W^{∗} ⊗ W defines the structure of a central simple F -algebra on W. □

If A is a central simple algebra, we will denote the center of A by Z(A). Division algebras will usually be denoted by D. Recall that UD(m, n) is the division algebra generated by m generic n × n matrices X_{1} = (x_{ij}^{(1)}), …, X_{m} = (x_{ij}^{(m)}). For a more detailed discussion of these algebras we refer the reader to [Pr_{1}, Part II] or [Pi, 20.8].

Lemma 9.2 (a) ed(PGL_{n}) = max{τ(A)}, as A ranges over all central simple algebras of degree n whose center contains k.

(b) ed(PGL_{n}) = τ(UD(m, n)) for any m ≥ 2.

(c) ed(PGL_{n}) = max{τ(D)}, as D ranges over all division algebras of degree n whose center contains k.

Proof. Part (a) follows from Lemma 9.1 and Theorem 8.8(b). To prove part (b), consider the representation of PGL_{n} on V_{m} = (M_{n})^{m} given by

g(a_{1}, …, a_{m}) = (ga_{1}g^{−1}, …, ga_{m}g^{−1}) .

The structured space (RMaps_{PGL_{n}}(V_{m}, V_{1}), αPGL_{n}^{V_{m}}) is the algebra of rational PGL_{n}- equivariant maps V_{m} −→ V_{1} = M_{n}; see Example 7.5. By a theorem of Procesi this algebra is isomorphic to UD(m, n); see [Sa, Theorem 14.16], cf. also [Pr_{2}, Thm. 2.1]. (The isomorphism identifies X_{i} ∈ UD(m, n) with the projection map π_{i} : V_{m} = (M_{n})^{m} ⟶ M_{n} given by π_{i}(a_{1}, …, a_{m}) = a_{i}). Now part (b) follows from Theorem 8.8(a). Part (c) is an immediate consequence of (a) and (b). □

Theorem 9.3 ed(PGL_{n^{r}}) ≥ 2r for any n ≥ 2 and r ≥ 1.

Proof. [Re, Theorem 16.1(b)] shows that τ(UD(2, n^{r})) ≥ 2r. The desired conclusion now follows from Lemma 9.2(b). □

9.2 Cyclic algebras

Lemma 9.4 (a) If D is a division algebra of degree n ≥ 2 then τ(D) ≥ 2. (b) If D is a cyclic division algebra then τ(D) = 2.

(c) ed(PGL_{n}) = 2 if n = 2, 3, and 6.

Proof. (a) Assume the contrary: D = D_{0} ⊗_{K_{0}} K, where K is the center of D and K_{0} is the center of D_{0} and trdeg_{k}(K_{0}) ≤ 1. Then D_{0} cannot be a division algebra by Tsen’s theorem (see e.g., [Pi, 19.4]), a contradiction. Another proof of part (a) can be deduced from Corollary 5.7(a) (with G = PGL_{n}).

(b) Since D is cyclic, there are elements x, y ∈ D such that x^{n} = u ∈ K, y^{n} = v ∈ K, xy = ζyx where ζ is a primitive n-th root of unity. (Recall that we are assuming k is algebraically closed and of characteristic 0; in particular, ζ ∈ k.) Examining the structure constants of D in the basis B = {x^{i}y^{j} | i, j = 0, …, n − 1}, we see that D is defined over the field k(u, v). Thus τ(D) ≤ trdeg_{k}(u, v) ≤ 2. Part (b) follows from this inequality and part (a).

(c) Follows from part (b) and Lemma 9.2(c), since every division algebra of degree 2, 3, or 6 is cyclic; see e.g., [Pi, 15.6]. □

9.3 Algebras of odd degree

In this section we use a theorem of Rowen [Row] to give an upper bound on ed(PGL_{n}) for odd n.

Recall that by a theorem of Wedderburn every central simple algebra A with center F is of the form A = M_{d}(D), where D is a division algebra with center F. The algebra D is unique up to isomorphism (of F -algebras); we shall refer to it as the underlying division algebra of A. The degree of D is called the index of A and is denoted by Ind(A). Two central simple algebras with center F are called Brauer equivalent if they have F -isomorphic underlying division algebras. Brauer equivalence classes of central simple algebras form an abelian group under tensor products; this group is called the Brauer group of F and is denoted by Br(F). For a more detailed discussion of the structure theory of central simple algebras and the Brauer group we refer the reader to [Pi].

Lemma 9.5 Let D_{1} and D_{2} be division algebras with center F. Suppose D_{1} and D_{2} generate the same cyclic subgroup of Br(F). Then τ(D_{1}) = τ(D_{2}).

Proof. By symmetry it is enough to show

τ(D_{1}) ≥ τ(D_{2})  (10)

Note that our assumption forces D_{1} and D_{2} to have the same degree, which we shall denote by n; see [Pi, Proposition 13.4(viii)]. Write D_{1} = E_{1} ⊗_{K} F, where E_{1} is a subalgebra of D_{1} of degree n, K is the center of E_{1} and trdeg_{k}(K) = τ(D_{1}). Suppose D_{1}^{⊗ i} = M_{n^{i−1}}(D_{2}), where i relatively prime to n. We claim that the central simple algebra E_{1}^{⊗ i} has index n. Indeed, on the one hand, Ind(E_{1}^{⊗ i}) ≤ Ind(E_{1}) = n and, on the other hand, Ind(E_{1}^{⊗ i}) ≥ Ind(D_{1}^{⊗ i}) = n. Thus we can write E_{1}^{⊗ i} = M_{n^{i−1}}(E_{2}) for some division algebra E_{2} of degree n, with center K. We now claim that

D_{2} = E_{2} ⊗_{K} F .  (11)

Note that (11) immediately implies (10), since it says that D_{2} is defined over K. To prove (11), observe that M_{n^{i−1}}(E_{2}) ⊗_{K} F = E_{1}^{⊗ i} ⊗_{K} F = D_{1}^{⊗ i} = M_{n^{i−1}}(D_{2}). Now (11) follows from the uniqueness part of Wedderburn’s theorem; see [Pi, Section 3.5]. □

Theorem 9.6 Let n ≥ 3 be an odd integer. Then ed(PGL_{n}) ≤ n + \frac{(n−1)(n−2)}{2}.

Proof. Let D_{1} = UD(2, n) and D_{2} be the underlying division algebra of D_{1}^{⊗ 2}. By a theorem of Rowen [Row], τ(D_{2}) ≤ n + \frac{(n−1)(n−2)}{2}. Combining Lemma 9.5 with Lemma 9.2(b), we obtain ed(PGL_{n}) = τ(D_{1}) = τ(D_{2}) ≤ n + \frac{(n−1)(n−2)}{2}, as claimed.

□

9.4 Algebras of composite degree

Lemma 9.7 Suppose n_{1} and n_{2} are relatively prime positive integers. For i = 1, 2 let D_{i} be a division algebra of degree n_{i} with center F and let D = D_{1} ⊗_{F} D_{2}. Then

(a) τ(D_{i}) ≤ τ(D) for i = 1, 2.

(b) τ(D) ≤ τ(D_{1}) + τ(D_{2}).

Proof. (a) Let E be a division subalgebra of D of degree n_{1}n_{2} such that trdeg_{k} Z(E) is as small as possible, i.e., is equal to τ(D). Then we can write E as E_{1}⊗_{Z(E)}E_{2}, where deg(E_{i}) = n_{i} for i = 1, 2. Thus D = E ⊗_{Z(E)} F = (E_{1} ⊗_{Z(E)} F) ⊗_{Z(D)} (E_{2} ⊗_{Z(E)} F). Since the decomposition D = D_{1} ⊗_{Z(D)} D_{2} is unique, we have D_{1} ≃ E_{1} ⊗_{Z(E)} F and D_{2} ≃ E_{2} ⊗_{Z(E)} F. This means that D_{1} and D_{2} are both defined over Z(E); consequently, τ(D_{i}) ≤ τ(D) for i = 1, 2.

(b) Suppose the algebra D_{i} is defined over F_{i} ⊂ F with trdeg_{k}(F_{i}) = τ(D_{i}); here i = 1, 2. In other words, D_{i} has a subalgebra E_{i} of degree n_{i} whose center is F_{i}. Let E = E_{1}E_{2} be the division subalgebra generated by E_{1} and E_{2} inside D. Then D = E ⊗_{Z(E)} F and therefore D is defined over Z(E). Note that Z(E) is the subfield of F generated (over k) by F_{1} and F_{2}. Hence, trdeg_{k} Z(E) ≤ trdeg_{k} F_{1} + trdeg_{k} F_{2} ≤ τ(D_{1}) + τ(D_{2}) , as claimed. □

Proposition 9.8 Let n_{1}, n_{2} ≥ 2 be relatively prime integers. Then (a) ed(PGL_{n_{i}}) ≤ ed(PGL_{n_{1}n_{2}}) for i = 1, 2.

(b) ed(PGL_{n_{1}n_{2}}) ≤ ed(PGL_{n_{1}}) + ed(PGL_{n_{2}}).

Proof. (a) It is enough to show that ed(PGL_{n_{1}}) ≤ ed(PGL_{n_{1}n_{2}}). Let D_{1} = UD(2, n_{1}), D_{2} = M_{n_{2}}(Z(D_{1})), and D = D_{1} ⊗_{Z(D_{1})} D_{2} = M_{n_{2}}(D_{1}). The desired inequality now follows from Lemmas 9.2 and 9.7(a).

(b) The algebra D = UD(2, n_{1}n_{2}) can be written as D_{1} ⊗_{Z(D)} D_{2}, where D_{1} and D_{2} are subalgebras of D of degrees n_{1} and n_{2} respectively; see, e.g., [Pi, 14.4]. Applying Lemmas 9.2 and 9.7(b), we obtain

ed(PGL_{n_{1}n_{2}}) = τ(D) ≤ τ(D_{1}) + τ(D_{2}) ≤ ed(PGL_{n_{1}}) + ed(PGL_{n_{2}}) ,

as claimed. □

Remark 9.9 Proposition 9.8(b) can often be used to strengthen the upper bounds on ed(PGL_{n}) given by Theorem 4.5, Lemma 9.4(c) and Theorem 9.6. For example, since ed(PGL_{5}) ≤ 11 (by Theorem 9.6) and ed(PGL_{6}) = 2 (by Lemma 9.4(c)), we have

ed(PGL_{30}) ≤ ed(PGL_{6}) + ed(PGL_{5}) ≤ 13 .

On the other hand, a direct application of Theorem 4.5 with n = 30 gives a much weaker bound ed(PGL_{30}) ≤ 840.

More generally, Proposition 9.8, in combination with Theorem 9.6, allows us to strengthen the bound given by Theorem 4.5 for every n which is not a power of 2.

10 Orthogonal groups

The main results of the next two sections (namely, Theorems 10.3, 10.4, 11.2, and 11.5) were communicated to us by J.-P. Serre. The proofs we present here are based on Tsen-Lang theory; they are somewhat different from Serre’s original proofs, which will be given in Section 12.

10.1 Pfister forms

Let F be a field. As usual, we shall denote the quadratic form ∑_{i=1}^{n} a_{i}x_{i}^{2} on F^{n} by < a_{1}, …, a_{n} >; the n-fold Pfister form < 1, a_{1} > ⊗ … ⊗ < 1, a_{n} > will be denoted by << a_{1}, …, a_{n} >>.

Lemma 10.1 (see, e.g., [Pf, Ch. 8, Ex. 1.2.6]) Suppose a_{1}, …, a_{n} are independent indeterminates over k and F = k(a_{1}, …, a_{n}). Then the Pfister form << a_{1}, …, a_{n} >> is anisotropic over F. □

Remark 10.2 Pfister forms naturally arise in the following context. Suppose β is a symmetric bilinear form on W. Recall that β gives rise to the symmetric bilinear Λ^{i}(β) on Λ^{i}(W) given by

Λ^{i}(β)(w_{1} ⊗ … ⊗ w_{i}, w′_{1} ⊗ … ⊗ w′_{i}) = ∑_{σ∈S_{n}}(−1)^{σ}β(w_{1}, w′_{σ(1)}) … β(w_{i}, w′_{σ(i)}) . (12)

Then Λ(β) =^{def} Λ^{1}(β) ⊕ … ⊕ Λ^{n}(β) is a symmetric bilinear form on Λ(W) = Λ^{1}(W) ⊕ … ⊕ Λ^{n}(W). An easy direct computation shows that if β(x, x) =< a_{1}, …, a_{n} > then < 1 > ⊕Λ(β)(x, x) =<< a_{1}, …, a_{n} >>.

10.2 The essential dimension of O_{n}

Theorem 10.3 ed(O_{n}) = n for every n ≥ 1.

Proof. We proved the inequality ed(O_{n}) ≤ n in Example 3.10(a) (and then again in Example 4.2); thus we only need to show that ed(O_{n}) ≥ n. Suppose α is a nondegenerate symmetric bilinear form on V = k^{n}, so that Aut_{k}(V, α) = O_{n}. A structured space (W = F^{m}, β) is of type (V, α) if and only if β is a non-degenerate symmetric bilinear form on W. Thus in view of Theorem 8.8 it suffices to construct a field extension F/k and a symmetric bilinear form β on W = F^{n} such that τ(W, β) ≥ n.

We now proceed to construct F and β. Let a_{1}, …, a_{n} be algebraically independent indeterminates over k, F = k(a_{1}, …, a_{n}), W = F^{n}, e_{1}, …, e_{n} be a basis of W and β = a_{1}(e_{1}^{∗})^{2} + … + a_{n}(e_{n}^{∗})^{2} be a symmetric bilinear form on W. We claim that τ(W, β) ≥ n (or, equivalently, = n, since trdeg_{k}(F) = n). Indeed, extend β to a symmetric bilinear form Λ(β) to Λ(V). Recall that by Remark 10.2

< 1 > ⊕Λ(β)(x, x) =<< a_{1}, …, a_{n} >> ;

hence, < 1 > ⊕Λ(β)(x, x) is an anisotropic quadratic form over F; see Lemma 10.1. On the other hand, suppose β = β_{0} ⊗_{F_{0}} F where β_{0} is a symmetric bilinear form on W_{0} = (F_{0})^{n} and trdeg_{k}(F_{0}) = n − 1. By the Tsen-Lang theorem the quadratic form < 1 > ⊕Λ(β_{0})(x, x) defined on the 2^{n}-dimensional space F_{0} ⊕ Λ(W_{0}) is isotropic; see e.g., [Pf, Sect. 5.1]. Since, Λ(β) = Λ(β_{0}) ⊗_{F_{0}} F, we conclude that the quadratic form < 1 > ⊕Λ(β)(x, x) is isotropic as well, a contradiction. □

10.3 The essential dimension of SO_{n}

Theorem 10.4 ed(SO_{n}) = \{ \begin{matrix}0 if n = 2 \\ n − 1, if n ≠ 2 \end{matrix}

Note that Theorem 10.4 says, in particular, that ed(SO_{n}) ≠ 1 for any n. This is consistent with Corollary 5.7(b).

Proof. Recall that we proved the inequality

ed(SO_{n}) ≤ n − 1  (13)

for every n ≥ 1 in Example 3.10(b) (and then again in Example 4.2). If n is odd then O_{n} ≃ SO_{n} × ℤ/2 and thus ed(O_{n}) ≤ ed(SO_{n}) + ed(ℤ/2). Since ed(O_{n}) = n (see Theorem 10.3) and ed(ℤ/2) = 1, this means n − 1 ≤ ed(SO_{n}). Combining this inequality with (13), we conclude that ed(SO_{n}) = n − 1 for every odd integer n ≥ 1.

We now assume that n is even. If n = 2 then SO_{2} is isomorphic to the 1-dimensional torus k^{∗} and thus ed(SO_{2}) = 0; see Example 3.9(a).

It remains to show that ed(SO_{n}) ≥ n − 1 for n = 2m, where m ≥ 2. Let V = k^{n}, let e_{1}, …, e_{n} be a k-basis of V, α_{1} = e_{1}^{∗} ∧ … ∧ e_{n}^{∗}, α_{2} = (e_{1}^{∗})^{2} + … + (e_{n}^{∗})^{2}, and α = α_{1} + α_{2}, as in Example 8.4(b). Recall that Aut_{k}(V, α) = SO_{n}; see Example 6.3(e). By Theorem 8.8 it is enough to construct a structured space (W = F^{n}, β) of type (V, α) such that τ(W, β) ≥ n − 1.

We now proceed to construct F, W, and β. Let F = k(a_{1}, …, a_{n−1}), where a_{1}, …, a_{n−1} are independent indeterminates over k, W = V ⊗_{k} F and β = β_{1} + β_{2}, where β_{1} = e_{1}^{∗} ∧ … ∧ e_{n}^{∗} = α_{1} ⊗_{k} F ∈ Λ^{n}(W^{∗}) and

β_{2} = a_{1}(e_{1}^{∗})^{2} + … + a_{n−1}(e_{n−1}^{∗})^{2} + (a_{1} … a_{n−1})^{−1}(e_{n}^{∗})^{2} ∈ S^{2}(W^{∗}) .

Since det(β_{2}) = 1, (W, β) is a structured space of type (V, α); see Example 8.4(b).

It is enough to show that bilinear form β_{2} cannot be defined over a field F_{0} with trdeg_{k}(F_{0}) < n − 1. Indeed, this will immediately imply that β cannot be defined over F_{0}, thus proving τ(W, β) ≥ n − 1. Note that β_{2} is equivalent to the bilinear form

γ = a_{1}(e_{1}^{∗})^{2} + … + a_{n−1}(e_{n−1}^{∗})^{2} + (a_{1} … a_{n−1})(e_{n}^{∗})^{2} ∈ S^{2}(V^{∗}) ;

thus it is enough to show that γ cannot be defined over any field F_{0} with trdeg_{k}(F_{0}) < n − 1.

We now argue as in the proof of Theorem 10.3. Let Λ^{i}(γ) be the symmetric bilinear form on Λ^{i}(W) defined in Remark 10.2. An easy computation shows that

\begin{matrix}Λ^{i}(γ)(x, x) = ⊕_{1≤j_{1}<...<j_{i}≤n−1} < a_{j_{1}}, …, a_{j_{i}} > \\ ⊕_{1≤h_{1}<...<h_{n−1−i}≤n−1} < a_{h_{1}}, …, a_{h_{n−1−i}} > . \end{matrix} (14)

Consider the form λ(γ) = Λ^{1}(γ) ⊕ … Λ^{m−1}(γ) on Λ^{1}(W) ⊕ … Λ^{m−1}(W). (Recall that we are assuming n = 2m and m ≥ 2.) Adding up the forms (14), we see that < 1 > ⊕λ(γ)(x, x) is equivalent to << a_{1}, …, a_{n−1} >> and therefore, is anisotropic over F; see Lemma 10.1.

On the other hand, suppose γ = δ ⊗_{F_{0}} F, where δ is a symmetric bilinear form on (F_{0})^{n} and trdeg_{k}(F_{0}) < n − 1. Then λ(γ) = λ(δ) ⊗_{F_{0}} F, where

λ(δ) = Λ^{1}(δ) ⊕ … Λ^{m−1}(δ) .

The quadratic form < 1 > ⊕λ(δ)(x, x) is then defined on F_{0} ⊕ Λ^{1}(W_{0}) ⊕ … ⊕ Λ^{m−1}(W_{0}) ≃ F_{0}^{2^{n−1}}. Thus the Tsen-Lang theorem says that this form is isotropic over F_{0}; see [Pf, Sect. 5.1]. Hence, so is < 1 > ⊕λ(γ)(x, x), a contradiction. This completes the proof of the theorem. □

11 Some exceptional groups

11.1 Octonion algebras

Let F be a field. Recall that for any 0 ≠ a, b, c ∈ F, the octonion (or Cayley-Dickson) algebra O_{F} (a, b, c) is defined as follows. Let

Q = (a, b)_{2} = F {i, j}/(i^{2} = a, j^{2} = b, ji = −ij)

be a 4-dimensional (associative) quaternion algebra over F. This algebra is equipped with an involution x → \overline{x} given by

\overline{x_{0} + x_{1}i + x_{2}j + x_{3}ij} = x_{0} − x_{1}i − x_{2}j − x_{3}ij .  (15)

(Here x_{0}, …, x_{3} ∈ F.) Now O_{F} (a, b, c) =^{def} Q + Ql is an 8-dimensional F -algebra with (non-associative) multiplication given by (x+yl)(z+wl) = (xz+c\overline{w}y)+(wx+y\overline{z})l. The involution (15) extends from Q to O_{F} (a, b, c) via \overline{x + yl} = \overline{x}−yl. The algebra O_{F} (a, b, c) is also equipped with F -valued trace and norm functions given by t(x) = x + \overline{x} and n(x) = x\overline{x} = \overline{x}x; moreover, x^{2} − tr(x)x + n(x) = 0 for any x ∈ O_{F} (a, b, c). Using this identity, it is easy to show that every automorphism of O_{F} (a, b, c) preserves the trace and the norm. For a more detailed description of octonion algebras we refer the reader to [J, I.5], [Sc, III.4] or [SSSZ, 2.2].

If F = k, we shall write O for O_{F} (1, 1, 1).

11.2 The essential dimension of G_{2}

Recall that Aut_{k}(O) is the 14-dimensional exceptional group G_{2}.

Lemma 11.1 Suppose F is a field containing k. Then for any 0 ≠ a, b, c ∈ F, we have

O_{F} (a, b, c) ⊗_{F} E = O ⊗_{k} E ,

where E = F (\sqrt{a}, \sqrt{b}, \sqrt{c}). In particular, O_{F} (a, b, c) is an algebra of type O.

Proof. The E-linear map O_{F} (a, b, c) ⊗_{k} E ⟶ O ⊗_{F} E given by

\begin{matrix}i ↦ \sqrt{a} i \\ j ↦ \sqrt{b} j \\ l ↦ \sqrt{c} l \end{matrix}

is an isomorphism of E-algebras. □

Theorem 11.2 ed(G_{2}) = 3.

Proof. We first prove that ed(G_{2}) ≥ 3. Let F = k(a_{1}, a_{2}, a_{3}), where a_{1}, a_{2} and a_{3} are algebraically independent indeterminates over k. In view of Lemma 11.1 and Theorem 8.8(b) it is sufficient to show that τ(O_{F} (a_{1}, a_{2}, a_{3})) ≥ 3.

Expressing the trace form tr(x^{2}) in the basis {(i^{ϵ_{1}}j^{ϵ_{2}})l^{ϵ_{3}} | ϵ_{1}, ϵ_{2}, ϵ_{3} = 0, 1}, we see that this form is equivalent to the 3-fold Pfister form << a_{1}, a_{2}, a_{3} >> and thus is anisotropic over F; see Lemma 10.1. Thus by the Tsen-Lang theorem the form tr(x^{2}) (and, hence, the F -algebra O_{F} (a_{1}, a_{2}, a_{3})) cannot be defined over a field F_{0} with trdeg_{k}(F_{0}) ≤ 2. This proves that τ(O_{F} (a_{1}, a_{2}, a_{3})) and therefore ed(G_{2}) ≥ 3.

Our proof of the opposite inequality will rely on an argument analogous to the one we used in Example 3.10. Let V = k^{8} be the underlying vector space of the split octonion algebra O. Since O is generated by three elements, the natural linear representation of G_{2} on V^{3} = k^{24} is generically free. Let

Y = {(x_{1}, x_{2}, x_{3}) | tr(x_{r}) = tr(x_{r}x_{s}) = tr((x_{1}x_{2})x_{3}) = 0, tr(x_{r}^{2}) ≠ 0, 1 ≤ r < s ≤ 3} .

Then Y is a (locally closed) G_{2}-invariant subvariety of V^{3}. Moreover, Y irreducible and dim(Y) = 17; this can be proved by considering the projection Y ⟶ V^{2} to the first two components. Since Y contains a triple of generators of O (e.g., (i, j, l)), the action of G_{2} on Y is generically free.

Now let f : V −→ Y be the G_{2}-equivariant rational map given by f(x_{1}, x_{2}, x_{3}) = (y_{1}, y_{2}, y_{3}), where

y_{1} = P (x_{1}; 1)
y_{2} = P (x_{2}; 1, y_{1})
y_{3} = P (x_{3}; 1, y_{1}, y_{2}, y_{1}y_{2}) ;

here P (v; v_{1}, …, v_{m}) is defined by equation (4) of Example 3.10 with q(x, y) = tr(xy). Since f_{|Y} = id_{|Y}, f is a compression. Thus ed(G_{2}) ≤ dim(Y) − dim(G_{2}) = 17 − 14 = 3, as claimed. □

Remark 11.3 (a) By [Se_{3}, Sect 8.2] G_{2} is 2-special; see Definition 5.5. In view of Proposition 5.6(b) this gives an alternative proof of the inequality ed(G_{2}) ≥ 3. Moreover, by Proposition 5.6(a), ed(X, G_{2}) = 0 or 3 for every primitive (or, equivalently, irreducible) generically free G_{2}-variety X.

Remark 11.4 The converse of Lemma 11.1 is also true, namely

(W = F^{8}, β) is structured space of type O if and only if (W, β) = O_{F} (a, b, c) for some a, b, c ∈ F. This gives an alternative proof of the inequality ed(G_{2}) ≤ 3.

The first assertion is a consequence of a theorem of Zorn [Sc, III.3.17] (see also [SSSZ, 7.3] for a more general result of Kleinfeld). To prove the second assertion, note that O_{F} (a, b, c) is defined over the field k(a, b, c), which has transcendence degree ≤ 3 over k. Thus τ(O_{F} (a, b, c)) ≤ 3 for every F and every a, b, c ∈ F^{∗}. By Theorem 8.8(b) this implies ed(G_{2}) ≤ 3. □

11.3 The Albert algebra

Recall that the (split) Albert algebra A = H_{3}(O) is the Jordan algebra of 3 × 3 hermitian matrices over the octonion algebra O. Elements of A are of the form

X = ( \begin{matrix}a & x & y \\ \overline{x} & b & z \\ \overline{y} & \overline{z} & c \end{matrix} ) ,  (16)

where a, b, c ∈ k and x, y, z ∈ O, and multiplication is given by X ·Y = 1/2(XY +Y X), where XY is the usual matrix product of X and Y. Every element X of A satisfies a “Cayley-Hamilton identity” of the form

X^{3} − tr(X)X^{2} + \underline{tr(X)^{2} − tr(X^{2})} ; 2 X − n(X) = 0 ,

where tr(X) = a + b + c ∈ k is the trace of X and n(X) is the cubic norm of X; see [J, p. 233]. Both the trace and the norm are preserved under automorphisms of A.

Let e_{αβ} be the (α, β) matrix unit in M_{3}(k). We shall denote the element xe_{αβ} +\overline{x}e_{αβ} by [x]_{αβ}. Then the 27 elements e_{11}, e_{22}, e_{33} and [(i^{ϵ_{1}}j^{ϵ_{2}})l^{ϵ_{3}}]_{αβ} form a k-basis of A; here ϵ_{1}, ϵ_{2}, ϵ_{3} = 0, 1 and 1 ≤ α < β ≤ 3.

11.4 The essential dimension of F_{4}

Recall that Aut_{k}(A) is the exceptional group F_{4}.

Theorem 11.5 ed(F_{4}) ≥ 5.

Proof. By Theorem 8.8 it is sufficient to construct a field F containing k and an F -algebra B of type A with τ(B) ≥ 5. We now proceed to construct such an algebra. Let s_{1}, s_{2}, s_{3}, t_{1}, t_{2} be five independent variables over k,

t_{3} =^{def} t_{1}^{−1}t_{2}^{−1} ,  (17)

S_{α} = s_{α}^{2} and T_{α} = t_{α}^{2} for α = 1, 2, 3. Let E and F be the fields given by F = k(S_{1}, S_{2}, S_{3}, T_{1}, T_{2}) and E = k(s_{1}, s_{2}, s_{3}, t_{1}, t_{2}), and let B be the F -subspace of A ⊗_{k} E spanned by the 27 elements

e_{11}, e_{22}, e_{33}, and b_{α}^{ϵ_{1},ϵ_{2},ϵ_{3}} = t_{α}s_{1}^{ϵ_{1}}s_{2}^{ϵ_{2}}s_{3}^{ϵ_{3}}[(i^{ϵ_{1}}j^{ϵ_{2}})l^{ϵ_{3}}]_{βγ} ,  (18)

where ϵ_{1}, ϵ_{2}, ϵ_{3} = 0, 1, {α, β, γ} = {1, 2, 3} and β < γ. We claim that B is an F - subalgebra of of A ⊗_{k} E. Since the elements (18) form an E-basis of A ⊗_{k} E, it is sufficient to check that all structure constants of A ⊗_{k} E in this basis lie in F. Indeed,

e_{αα} · e_{αα} = e_{αα} e_{αα} · e_{ββ} = 0 e_{αα} · b_{α}^{ϵ_{1},ϵ_{2},ϵ_{3}} = 0 e_{αα} · b_{α}^{ϵ_{1},ϵ_{2},ϵ_{3}} = \frac{1}{2}b_{β}^{ϵ_{1},ϵ_{2},ϵ_{3}}

b_{α}^{ϵ_{1},ϵ_{2},ϵ_{3}} · b_{α}^{δ_{1},δ_{2},δ_{3}} = \{ \begin{matrix}±T_{α}S_{1}^{ϵ_{1}}S_{2}^{ϵ_{2}}S_{3}^{ϵ_{3}}(e_{ββ} + e_{γγ}) if (ϵ_{1}, ϵ_{2}, ϵ_{3}) = (δ_{1}, δ_{2}, δ_{3}) \\ 0 otherwise . \end{matrix}

b_{α}^{ϵ_{1},ϵ_{2},ϵ_{3}} · b_{β}^{δ_{1},δ_{2},δ_{3}} = ±T_{γ}^{−1}S_{1}^{ϵ_{1}δ_{1}}S_{2}^{ϵ_{2}δ_{2}}S_{3}^{ϵ_{3}δ_{3}}b_{γ}^{ϵ_{1}+δ_{1},ϵ_{2}+δ_{2},ϵ_{3}+δ_{3}} .

(cf. [Sc, p. 105]). Here we always assume that {α, β, γ} = {1, 2, 3}; in the last formula the subscripts ϵ_{n} and δ_{n} are added and multiplied modulo 2. All structure constants appearing in this table are, indeed, elements of F. We have therefore verified that B is a 27-dimensional F -algebra. Since the elements (18) form an E-basis of A ⊗_{k} E, we have B ⊗_{F} E ≃ A ⊗_{k} E. Thus B is an algebra of type A.

It remains to prove that τ(B) ≥ 5 (or, equivalently, = 5, since trdeg_{k}(F) = 5). We shall do so by examining the trace form of B and showing that it cannot be defined over a field F_{0} with trdeg_{k}(F_{0}) ≤ 4 (and, consequently, B cannot be defined over such a field).

Our multiplicaton table shows that the 27 basis elements (18) are mutually orthogonal with respect to the trace form. Moreover, taking the trace of the square of each element and remembering that (i) \sqrt{2}, \sqrt{−1} ∈ k ⊂ F by our assumption on k and (ii) T_{3} = T_{1}^{−1}T_{2}^{−1} by (17), we obtain

tr(x^{2}) ≃ < 1, 1 > ⊕ ϕ ,  (19)

where

ϕ =< 1 > ⊕ T_{1} << S_{1}, S_{2}, S_{3} >> ⊕ T_{2} << S_{1}, S_{2}, S_{3} >> ⊕ T_{1}T_{2} << S_{1}, S_{2}, S_{3} >> .

Note that

ϕ ⊕ < S_{1}, S_{2}, S_{3}, S_{1}S_{2}, S_{1}S_{3}, S_{2}S_{3}, S_{1}S_{2}S_{3} > ≃ << S_{1}, S_{2}, S_{3}, T_{1}, T_{2} >> . (20)

Since the Pfister form << S_{1}, S_{2}, S_{3}, T_{1}, T_{2} >> is anisotropic over F (see Lemma 10.1), so is ϕ. Thus (19) gives the Witt decomposition of the quadratic form tr(x^{2}) over F = k(S_{1}, S_{2}, S_{3}, T_{1}, T_{2}); the anisotropic part of this form equals ϕ and has dimension 25. On the other hand, if B were defined over a field F_{0} with trdeg_{k}(F_{0}) ≤ 4 then tr(x^{2}) would also be defined over F_{0} and, consequently, by the Tsen-Lang theorem, its anisotropic part would have dimension ≤ 2^{4} = 16. This contradiction proves that τ(B) ≥ 5 and thus ed(F_{4}) ≥ 5, as claimed. □

11.5 The essential dimension of E_{6}

Recall that the simply connected exceptional group E_{6} is isomorphic to Aut_{k}(V, α), where V = k^{27} is the underlying vector space of the Albert algebra A and α ∈ S^{3}(V^{∗}) is the symmetric trilinear form associated to the cubic norm in A.

Proposition 11.6 ed(E_{6}) ≥ 3.

Proof. Let F = k(a, b, c). By a theorem of Albert there exists an exceptional Jordan division F -algebra B; see [J, IX.12 Thm. 21]. Let W = F^{27} be the underlying vector space of B and let β ∈ S^{3}(V^{∗}) be the symmetric trilinear form associated to the cubic norm in B. Since B is an algebra of type A, (W, β) is a structured space of type (V, α). (Here V = k^{27} and α ∈ S^{3}(V^{∗}) are as above.)

By Theorem 8.8 it suffices to show that τ(W, β) ≥ 3. Assume the contrary: (W, β) is defined over a field F_{0} with trdeg_{k}(F_{0}) ≤ 2. Then the cubic norm n_{B}(x) = β(x, x, x) on B can also be defined over F_{0}. Note that n_{B} is a cubic form in 27 variables; thus by the Tsen-Lang theorem n_{B} is isotropic, i.e. n_{B}(x) = 0 for some 0 ≠ x ∈ B. This contradicts our assumption that B is a division algebra; see [J, VI.3 Cor. 3]. We therefore conclude that τ(W, β) ≥ 3, as claimed. □

Proposition 11.7 ed(E_{6}) ≤ ed(F_{4}) + 1.

Proof. E_{6} is by definition a subgroup of GL_{27} = GL(A); thus its linear representation on V = A^{27} is generically free. (Using [J, VI.7 Thm. 7], one can prove that the action of E_{6} on A^{n} is generically free for every n ≥ 4; however, we will not need this fact here.)

Let S = {(x_{1}, …, x_{27}) | x_{1} ∈ k1_{A}} ⊂ V. Then the proof of [I, Thm. 1] shows that S is an (E_{6}, H)-section of V (see Definition 2.9), where H ≃ F_{4}×ℤ/3 is the subgroup of E_{6} generated by F_{4} and multiplication by cube roots of 1. Note that S is a linear generically free H-variety. Thus by Lemma 4.1, ed(E_{6}) = ed(V, E_{6}) ≤ ed(S, H) = ed(H) and by Lemma 3.8, ed(H) = ed(F_{4} × ℤ/3) ≤ ed(F_{4}) + ed(ℤ/3) = ed(F_{4}) + 1, as claimed. □

12 Cohomological invariants

In this section we discuss the relationship between the essential dimension of an algebraic group G and cohomological invariants associated to G. This material is based entirely on results communicated to us by J.-P. Serre.

12.1 Preliminaries

We begin by recalling several definitions and results related to Galois cohomology and cohomological invariants. For details we refer the reader to [Se_{4}], [KMRT] and [B].

Let H^{i}(F) = H^{i}(Γ_{F} , ℤ/2), where Γ_{F} = Gal(\overline{F} /F), \overline{F} is the algebraic closure of F, and ℤ/2 is viewed as a trivial Γ_{F} -module.

By Kummer theory the first cohomology group H^{1}(F) is canonically isomorphic to F^{∗}/(F^{∗})^{2}; as usual, we will denote the class of a ∈ F^{∗} modulo (F^{∗})^{2} by (a).

Lemma 12.1 (see e.g., [BR_{2}, Lemma 8.2]) Let a_{1}, …, a_{n} be independent variables over k and let F = k(a_{1}, …, a_{n}). Then the cup product (a_{1}) … (a_{n}) is non-zero in H^{n}(F). □

Suppose F is a field, 0 ≠ a_{1}, …, a_{n} ∈ F and ϕ =< a_{1}, …, a_{n} > is a quadratic form. Then r-th Stiefel-Whitney class w_{r}(ϕ) ∈ H^{r}(F) is given by

w_{r}(ϕ) = ∑_{1≤i_{1}<...<i_{r}≤n} (a_{i_{1}})(a_{i_{2}}) … (a_{i_{r}}) .  (21)

By a theorem of Delzant w_{r}(ϕ) depends only on the equivalence class of ϕ and not on the specific presentation of ϕ as a sum of squares; see [De].

We shall now consider the following categories:

F ields - category of finitely generated field extensions of k,

Sets′ - category of pointed sets, i.e., sets with a marked element,

AGrps - category of abelian groups,

and the following covariant functors:

H^{i}( · ), from F ields to AGrps.

H^{1}( · , G), from F ields to Sets′. Here G is a (fixed) algebraic group.

G − V ar( · ), from F ields to Sets′. Here again, G is a fixed algebraic group. If F ∈ F ields then G − V ar(F) is the set of birational equivalence classes of generically free primitive G-varieties X such that k(X/G) ≃ F (as field extensions of k). Note that for each F ∈ F ields the marked element in G − V ar(F) is the G-variety X_{0} × G; here the variety X_{0} is chosen so that k(X_{0}) = F. An inclusion F ↪ F′ of fields induces the map G − V ar(F) ⟶ G − V ar(F′) of pointed sets given by X ↦ X′_{0} ×_{X/G} X, where k(X′_{0}) = F′.

Spaces_{(V,α)}( · ), from F ields to Sets′. Here (V = k^{n}, α) is a (fixed) structured space. If F ∈ F ields then Spaces_{(V,α)}(F) is defined as the set of all structured spaces (W = F^{n}, β) of type (V, α); see Definition 8.1. The marked element in this set is (V, α) ⊗_{k} F. An inclusion F ↪ F′ of fields induces a morphism Spaces_{(V,α)}(F) ⟶ Spaces_{(V,α)}(F′) given by (W, β) ↦ (W, β) ⊗_{F} F′.

Definition 12.2 ([Se_{3}, 6.1], [B, 4.1], [KMRT, 31.B]) Let G be an algebraic group. A cohomological invariant of dimension n associated to G is a morphism of functors f : H^{1}( · , G) ⟶ H^{n}( · ).

12.2 Cohomological invariants and essential dimension

Lemma 12.3 (a) The functors H^{1}( · , G) and G − V ar( · ) are isomorphic.

(b) If (V, α) is a structured space and G = Aut_{k}(V, α) then the functors H^{1}( · , G), G − V ar( · ) and Spaces_{(V,α)}( · ) are isomorphic.

Proof. Part (a) is proved in [Se_{4}, I.5.2]; see also [Po, 1.3] and Remark 2.17. To prove part (b), define

f_{F} : G − V ar( · )(F) ⟶ Spaces_{(V,α)}(F)

by f_{F} : X ↦ (RMaps_{G}(X, V), α_{G}^{X}) for every F ∈ F ields. Now Lemma 7.6(c) shows that f_{F} is a map of pointed sets for each F, Lemma 7.6(a) shows that the maps f_{F} collectively define a morphism of functors f from G − V ar( · ) to Spaces_{(V,α)}( · ), and Proposition 8.6 shows that f is an isomorphism. □

In view of the above lemma we can think of a cohomological invariant of degree n associated to G as a morphism of functors G − V ar( · , G) ⟶ H^{n}( · ) or, if G is of the form Aut(V, α), as a morphism of functors Spaces_{(V,α)}( · ) ⟶ H^{n}( · ).

Proposition 12.4 Let f : G − V ar( · ) ⟶ H^{n}( · ) be a cohomological invariant associated to G and let X be a primitive generically free G-variety such that f_{F} (X) ≠ 0. (Here F = k(X)^{G}.) Then

(a) ed(X, G) ≥ n.

(b) Let V be a generically free linear representation of G and let K = k(V)^{G}. Then f_{K}(V) ≠ 0.

Proof. (a) Assume the contrary. Then there exists a G-compressions X −→ Y such that dim(Y/G) ≤ n − 1, i.e.,

trdeg_{k}(L) ≤ n − 1 , where L = k(Y/G)^{G} = k(Y)^{G} .  (22)

Since X is birationally isomorphic to X/G ×_{Y/G} Y (see Lemma 2.16(b)) and f is a morphism of functors, we conclude that f_{L}(Y) ⟶^{res} f_{F} (X) where res is the restriction map H^{n}(L) ⟶ H^{n}(F). However, in view of (22), the cohomological dimension of L is ≤ n − 1 (see [Se_{4}, II.4.2]) and thus f_{L}(Y) ∈ H^{n}(L) = (0). Consequently, f_{F} (X) = (0) in H^{n}(F), a contradiction.

(b) By Corollary 2.20, there exists a compression X ×k^{d} −→ V, where d = dim(V). Then f_{F′}(X × k^{d}) is a homomorphic image of f_{K}(V); here F′ = k(X × k^{d})^{G}; hence it is enough to show that f_{F′}(X × k^{d}) ≠ 0. Note that f_{F′}(X × k^{d}) is the image of f_{F} (X) under the restriction map H^{n}(F) ⟶ H^{n}(F′). Since F′ is a purely transcendental extension of F, this restriction map is injective; see [Se_{4}, Remark 1, p. 85]. Thus f_{F′}(X × k^{d}) ≠ 0, as claimed. □

Note that we can associate a trivial cohomological invariant to any group G by setting f_{F} to be the zero map for every F ∈ F ields. We shall denote this invariant by f = 0 and refer to it as the zero invariant.

Corollary 12.5 If there exists a non-zero cohomological invariant of dimension n associated to G then ed(G) ≥ n.

Proof. Immediate from Proposition 12.4; see Definition 3.5. □

12.3 Examples

We now apply Corollary 12.5 to give alternative proofs of several lower bounds on ed(G), which we previously established by other methods. Every group G we shall consider in Examples 12.6-12.10 is of the form G = Aut_{k}(V, α); thus we will interpret a cohomological invariant of degree n associated to G as a morphism of functors f : Spaces_{(V,α)}( · ) ⟶ H^{n}( · ); see Lemma 12.3.

Example 12.6 G = O_{n}. Let V = k^{n} and α be a non-degenerate symmetric bilinear form on V so that Aut_{k}(V, α) = O_{n}. Recall that for any F ∈ F ields, (W = F^{n}, β) is of type (V, α) if and only if β is a non-degenerate symmetric bilinear form on W. For each i ≥ 1 we can now define a cohomological invariant f_{i} : Spaces_{(V,α)}( · ) ⟶ H^{i}( · ) of degree i associated to O_{n} by (f_{i})_{F} (W, β) = w_{i}(\overline{β}), where \overline{β} is the quadratic form such that \overline{β}(x) = β(x, x) and w_{i}(\overline{β}) is the Stiefel-Whitney class of \overline{β}; see (21).

It is easy to see that f_{i} = 0 for all i ≥ n + 1. We claim that f_{i} ≠ 0 for any i = 1, …, n. Indeed, suppose F = k(a_{1}, …, a_{i}) and \overline{β} =< a_{1}, …, a_{i}, 1, …, 1 >, where a_{1}, …, a_{i} are independent variables over k. Then w_{i}(\overline{β}) = (a_{1}) · . . . · (a_{i}) ≠ 0 in H^{i}(F); see Lemma 12.1. Applying Proposition 12.4 to f_{n}, we obtain an alternative proof the inequality ed(O_{n}) ≥ n (cf. Theorem 10.3).

Example 12.7 G = SO_{n}. Let V = k^{n} and let α = α_{1} + α_{2}, where α_{1} is a volume form and α_{2} is a non-degenerate symmetric bilinear form on V, as in the beginning of Section 10.3. Recall that Aut_{k}(V, α) = SO_{n} (see Example 6.3(e)) and that for any F ∈ F ields, Spaces_{(V,α)}(F) can be identified with the set of isomorphism classes of symmetric bilinear forms on F^{n} of determinant 1 (see Example 8.4(b)). For every i ≥ 1 we can define a cohomological invariant f_{i} : Spaces_{(V,α)}( · ) ⟶ H^{i}( · ) of degree i associated to SO_{n} by (f_{i})_{F} (β_{2}) = w_{i}(\overline{β_{2}}), where w_{i} is the ith Stiefel-Whitney class, β_{2} is a symmetric bilinear form of determinant 1 on F^{n}, and \overline{β_{2}}(x) = β_{2}(x, x). Note that

\overline{β_{2}} ≃< a_{1}, …, a_{n−1}, (a_{1} … a_{n−1})^{−1} >≃< a_{1}, …, a_{n−1}, a_{1} … a_{n−1} >

for some a_{1}, …, a_{n} ∈ F. It is easy to see that f_{i} = 0 for all i ≥ n. If n is odd then w_{n−1}(\overline{β}) = (a_{1}) … (a_{n−1}), which is non-zero if a_{1}, …, a_{n} are independent variables over k and F = k(a_{1}, …, a_{n}); see Lemma 12.1. This gives an alternative proof of the inequality ed(SO_{n}) ≥ n − 1 for n odd; cf. Theorem 10.4.

If n is even then f_{n−1} = 0. Taking a_{1}, …, a_{n−2} to be independent variables over k and a_{n−1} = 1 and applying Lemma 12.1, we see that f_{n−2} ≠ 0. This proves the inequality ed(SO_{n}) ≥ n − 2 for n even. To recover the stronger inequality

ed(SO_{n}) ≥ n − 1  (23)

of Theorem 10.4 for even integers n ≥ 4, we consider the cohomological invariant

g_{n−1} : Spaces_{(V,α)}( ·) ⟶ H^{n−1}( ·)

given by (a_{1})(a_{2}) … (a_{n−1}); see [Se_{4}, Remark 2, p. 188]. Here a_{1}, …, a_{n−1} are the coefficients of \overline{β_{2}}, as above. Taking a_{1}, …, a_{n−1} to be independent variables over k and applying Lemma 12.1 once again, we conclude that g_{n−1} ≠ 0. This gives an alternative proof of the inequality (23).

Note that this argument is not as different from the proof of Theorem 10.4 as it may seem. The key step here is proving that g_{n−1} is well-defined. One way of doing it involves passing to a Pfister form on ⊕_{i=1}^{n/2−1} Λ^{i}(k^{n}), in a manner similar to the argument we used in Section 10.3.

Example 12.8 G = S_{n}. Let A be the (split) etale k-algebra k ⊕ … ⊕ k, so that Aut_{k}(A) = S_{n}; see Example 6.3(f). Recall that (W = F^{n}, β) is a structured space of type A if and only if β defines the structure of an n-dimensional etale F -algebra on W; see Example 8.4(c). Thus a cohomological invariant associated to S_{n} is a cohomological invariant of etale algebras. These invariants were completely described by Serre; see [B, Section 4]. In particular, for each i = 1, …, n, one can define a cohomological invariant f_{i} : Spaces_{A}( · ) ⟶ H^{i}( · ) by (f_{i})_{F} (B) = w_{i}(tr_{B}(x^{2})), where B is an n-dimensional etale F -algebra, tr_{B}(x^{2}) is the trace form on B, and w_{i} is the i-th Stiefel-Whitney class. Then f_{i} = 0 for any i > [n/2] (see [B, 4.6]) but f_{i} ≠ 0 for i = [n/2] (see [BR_{2}, Theorem 8.4]). Consequently, by Proposition 12.4, ed(S_{n}) ≥ [n/2]; cf. [BR_{1}, Thm. 6.5].

Example 12.9 G = G_{2}. A non-zero cohomological invariant f_{3} of dimension 3 associated to G_{2} is constructed in [Se_{3}, Sect. 8]. By Proposition 12.4 this construction gives an alternative proof of the inequality ed(G_{2}) ≥ 3; cf. Theorem 11.2 and Remark 11.3.

For the reader’s convenience, we briefly recall the definition of f_{3}. Recall that G_{2} = Aut_{k}(O), where O is the split octonion algebra over k. Moreover, for F ∈ F ields the structured space (W = F^{n}, β) is of type O if and only if (W, β) = O_{F} (a, b, c) for some a, b, c ∈ F; see Remark 11.4. The trace form of O_{F} (a, b, c) is isomorphic to a 3fold Pfister form << a, b, c >>. We now define f_{3} as the Arason invariant of this form, i.e., f_{F} : SpacesO(F) ⟶ H^{3}(F) is given by O_{F} (a, b, c) ⟶ (a)(b)(c). This invariant is non-zero because (a)(b)(c) ≠ 0 in H^{3}(F) if a, b, c are algebraically independent over k and F = k(a, b, c); see Lemma 12.1.

Example 12.10 G = F_{4}. Non-zero cohomological invariants f_{3} and f_{5}, of dimension, respectively 3 and 5, associated to F_{4} are constructed in [Se_{3}]; we briefly recall their definition below. The existence of f_{5} gives an alternative proof of the inequality ed(F_{4}) ≥ 5; cf. Theorem 11.5.

Recall that F_{4} = Aut_{k}(A), where A is the split Albert algebra over k and that structured spaces of type A are precisely 27-dimensional exceptional simple Jordan algebras; see Example 8.4(d). If B is a F -algebra of type A than the trace form of B satisfies the identity

tr_{B}(x^{2})⊕ << a_{1}, a_{2}, a_{3} >>≃< 1, 1, 1 > ⊕ << a_{1}, a_{2}, a_{3}, b_{1}, b_{2} >>

for some a_{1}, a_{2}, a_{3}, b_{1}, b_{2} ∈ F, where the Pfister forms << a_{1}, a_{2}, a_{3} >> and << a_{1}, a_{2}, a_{3}, b_{1}, b_{2} >> are uniquely defined by B; see [Se_{3}, Theorem 10]. (The proof of this assertion for reduced Jordan Algebras is, in fact, similar to our computation in Section 11.4.) Then (f_{3})_{F} (B) =^{def} (a_{1})(a_{2})(a_{3}) ∈ H^{3}(F) and (f_{5})_{F} (B) =^{def} (a_{1})(a_{2})(a_{3})(b_{1})(b_{2}) ∈ H^{5}(F). To see that these invariants are non-zero, let F = k(S_{1}, S_{2}, S_{3}, T_{1}, T_{2}) and B be as in the proof of Theorem 11.5. Then in view of (19) and (20), we have (f_{3})_{F} = (S_{1})(S_{2})(S_{3}) ≠ 0 in H^{3}(F) and (f_{5})_{F} = (S_{1})(S_{2})(S_{3})(T_{1})(T_{2}) ≠ 0 in H^{5}(F); see Lemma 12.1. Thus f_{3}, f_{5} ≠ 0, as claimed.

12.4 The Rost invariant and simply connected semisimple groups

The Rost invariant H^{1}( · , G) ⟶ H^{3}( ·) is defined in [KMRT, 31B]. This invariant is non-zero for every simply connected exceptional group G; see [KMRT, (31.40) and (31.47)]. For G = G_{2} and F_{4}, this invariant is the invariant f_{3} of (respectively) Examples 12.9 and 12.10. In the case G = E_{6}, the non-vanishing of the Rost invariant gives an alternative proof of the inequality ed(E_{6}) ≥ 3; cf. Proposition 11.6. For G = E_{7} and E_{8}, we obtain the following result.

Proposition 12.11 (a) ed(E_{7}) ≥ 3, (b) ed(E_{8}) ≥ 3. □

Recall that a special case of of Serre’s Conjecture II [Se_{4}, III.3, p. 139] asserts that every simply connected semisimple group G is 2-special; cf. Definition 5.5 and Remark 5.9. This is equivalent to the assertion that ed(X, G) = 0 or ≥ 3 for every irreducible generically free G-variety X; cf. Proposition 5.6. Theorem 12.12 below proves this in the case where X = V is a vector space with a generically free linear action of G.

Theorem 12.12 Let G be a simply connected semisimple group. Then either G is special (and thus ed(G) = 0) or ed(G) ≥ 3.

Proof. We may assume without loss of generality that G is simple and simply connected. Indeed, if ed(G_{i}) = 0 for every simple factor G_{i} of G then ed(G) = 0 (see Lemma 3.8) and if ed(G_{i}) ≥ 3 for one of the factors G_{i} then ed(G) ≥ 3 (see Lemma 3.7).

Every connected and simply connected simple group, other than E_{8}, is known to be 2-special (see Definition 5.5 and Remark 5.9); for these groups the desired conclusion follows from Proposition 5.6 (cf. Theorems 11.2, 11.5, and Proposition 11.6). If G = E_{8} then ed(G) ≥ 3 by Proposition 12.11. □

12.5 Mod p invariants

Definition 12.2 and Proposition 12.4 remain valid if H^{i}( ·) is interpreted as H^{i}( · , C), where C is a cyclic group of prime order (not necessarily ℤ/2). Moreover, Proposition 12.4(b) can be strengthened as follows.

Proposition 12.13 Suppose there exists a non-zero cohomological invariant f : G − V ar( · ) ⟶ H^{n}( · , C). Let V be a generically free linear representation of G, X be a primitive generically free G-variety, and X −→ V be a G-compression which is generically m : 1. (In other words, [k(X)^{G} : k(V)^{G}] = m.) If m is relatively prime to the order of C then ed(X, G) ≥ n.

Proof. Since m = [k(X)^{G} : k(V)^{G}] is relatively prime to |C|, the restriction map H^{n}(k(V)^{G}, C) ⟶ H^{n}(k(X)^{G}, C) is injective; see [Se_{4}, Sect. 2.4]. The desired conclusion now follows from Proposition 12.4(b). □

Example 12.14 Applying Proposition 12.13 to the cohomological invariants we discussed above (all defined with C = ℤ/2), we arrive at the following result.

Suppose V is a generically free linear representation of G and, X −→ V is an m : 1-compression, where m is an odd integer. Then

(a) ed(X, G) = n, if G = O_{n},

(b) ed(X, G) = n − 1, if G = SO_{n} with n ≥ 3,

(c) ed(X, G) ≥ [n/2] if G = S_{n} (cf. [BR_{2}, Thm. 7.1]),

(d) ed(X, G) = 3 if G = G_{2},

(e) ed(X, G) ≥ 5 if G = F_{4}, and

(f) ed(X, G) ≥ 3 if G = E_{6}, E_{7} or E_{8}.

Example 12.15 Let V be a generically free linear representation of the exceptional group F_{4}, X be a generically free F_{4}-variety, and X −→ V be an F_{4}-compression, which is generically m : 1, where m is not divisible by 3. Then ed(X, F_{4}) ≥ 3.

This follows from the non-vanishing of the Serre-Rost invariant

H^{1}( · , F_{4}) ≃ Spaces_{A}( ·) ⟶ H^{3}( · , ℤ/3) ;

see [Se_{3}, Sect. 9.3] and [Rost_{1}]. (Here A is the Albert algebra and Spaces_{A}(F) is the set of 27-dimensional exceptional Jordan algebras over F, as in Examples 8.4(d) and 12.10.) □

Remark 12.16 We do not know whether the inequality ed(P GL_{n^{r}}) ≥ 2r of Theorem 9.3 can be proved by the methods of this section. However, if n = p is a prime then the above bound is “stable under prime to p-extensions” in the following sense. Let V be a generically free linear representation of PGL_{p^{r}} and X be a generically free irreducible PGL_{p^{r}}-variety. If X admits an m : 1-compression X −→ V, with m relatively prime to p, then ed(X, PGL_{p^{r}}) ≥ 2r. Equivalently, if D is a prime-to-p extension of a universal division algebra UD(m, p^{r}) then τ(D) ≥ 2r; see [Re, Theorem 16.1(b)].

Appendix

Since the time that this paper was first submitted for publication, a number of new results on essential dimension have been obtained by this and other authors. We summarize some of them in the table below.

\underline{Group} EssentialDimension

\underline{Proof}

P O_{n}  ≥ n − 1  [RY_{1}]

Spin_{n}  \{ \begin{matrix}= 4, if n = 7^{(∗)}, 10 \\ = 5, if n = 8, 9, 11 \\ = 6, if n = 12, 13 \\ = 7, if n = 14 \\ ≥ [n/2] + 1 if n ≡ 0, ±1 mod 8 \end{matrix}  [Rost_{2}] [Rost_{2}] [Rost_{2}] [Rost_{2}] [RY_{1}]

F_{4}  = 5  [Ko_{2}]
E_{6} simply conn. \{ \begin{matrix}≥ 4 \\ ≤ 6 \end{matrix}  \begin{matrix}[RY_{1}] & (∗∗) \\ [Ko_{2}] \end{matrix}

E_{7} simply conn. \{ \begin{matrix}≥ 7 \\ ≤ 9 \end{matrix}  {[RY_{1}] \atop [Ko_{2}]}
E_{7} adjoint  ≥ 8  [RY_{1}]
E_{8}  ≥ 9  [RY_{1}]

(∗) For n = 2, …, 6, the group Spin_{n} is special and, hence, has essential dimension 0; see Proposition 5.3.

(∗∗) Independent proofs of the inequality ed(E_{6}) ≥ 4 were communicated to us by R. S. Garibaldi and M. Rost.

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