Cohomological invariants of odd degree Jordan algebras

Mark L. MacDonald

October 22, 2007

Abstract

In this paper we determine all possible cohomological invariants of Aut(J)-torsors in Galois cohomology with mod 2 coefficients (characteristic of the base field not 2), for J a split central simple Jordan algebra of odd degree n ≥ 3. This has already been done for J of orthogonal and exceptional type, and we extend these results to unitary and symplectic type. We will use our results to compute the essential dimensions of some groups, for example we show that ed(PSp_{2n}) = n + 1 for n odd.

1 Introduction

The Stiefel-Whitney classes of quadratic forms over k define invariants in Galois cohomology H^{∗}(k, ℤ/2ℤ) up to isometry [De62], [Mi70]. It is shown in [GMS03, ch. VI] that the even Stiefel-Whitney classes form a basis of all cohomological invariants of SO_{n}-torsors for n ≥ 3 odd. This is done by identifying the SO_{n}- torsors with isomorphism classes of determinant 1 quadratic forms of dimension n. These torsors may be further identified with isomorphism classes of algebras with orthogonal involution of degree n, by sending q to its adjoint involution ad_{q} (see [KMRT, Thm. 4.2, p.42]). These classes may further be identified with isomorphism classes of central simple Jordan algebras of degree n whose associated composition algebra C is one dimensional (see [Ja68, p.210]). We wish to extend these results to dim(C) = 2 or 4, which is to say, odd degree algebras with unitary and symplectic involutions. In fact, in the octonion case dim(C) = 8, we only have a Jordan algebra when the degree is 3, and then it is called an Albert algebra. The mod 2 cohomological invariants of Albert algebras have been determined in [GMS03, ch. VI]. Nevertheless, we include this case here for completeness.

For any n ≥ 3 we will define J_{n}^{r} := (M_{n}(C), −)_{+} over k, as the split Jordan algebra of hermitian elements, where C is the split composition algebra of dim(C) = 2^{r}. If r = 3 then we insist that n = 3. Here − denotes the conjugate transpose involution. The following table summarizes, for n = 2m + 1, the (split) automorphism groups Aut(J_{n}^{r}), together with their mod 2 cohomological invariants (see Thm. 4.7). We list the degrees of invariants which form an H^{∗}(k_{0})-basis of all invariants.

r  Aut(J_{2m+1}^{r})  Degrees of H^{∗}-basis of Inv(Aut(J_{n}^{r}))
0  SO_{2m+1}  0, 2, 4, ⋯, 2m
1 ℤ/2ℤ ⋉ PGL_{2m+1}  0, 1, 3, ⋯, 2m + 1
2  PSp_{2(2m+1)}  0, 2, 4, ⋯, 2m + 2
3  F_{4}  0, 3, 5

Here m ≥ 1, and F_{4} denotes the split simple group of type F_{4}. To specify the semi-direct product in r = 1, we just need to describe how the non-trivial element of ℤ/2ℤ acts on PGL_{2m+1}. This action is defined by sending any [a] ∈ PGL_{2m+1} to its inverse transpose [(a^{t})^{−1}] (see [KMRT, 29.20]).

We will show (Thm. 4.7) that for r = 1, 2 and 3 and n ≥ 3 odd, the normalized invariants of Aut(J_{n}^{r}) in degree greater than zero are in one-to-one correspondence with the even Stiefel-Whitney classes of n-dimensional quadratic forms (which are the invariants of Aut(J_{n}^{0})). Under this bijection the degree zero Stiefel-Whitney class corresponds to the degree r invariant which classifies the associated composition algebra.

The Aut(J_{3}^{1})-invariants are discussed in detail in [KMRT, §19.B, §30.C] or [HKRT], including a mod 3 invariant of degree 2. These invariants determine an Aut(J_{3}^{1})-torsor up to isomorphism (considered there as a degree 3 algebra with unitary involution).

Recently in [GPT07] Garibaldi, Parimala and Tignol have classified mod 2 invariants of degree ≤ 3 for Aut(J_{n}^{2})-torsors for n even.

In the final section we determine the essential dimension at 2 for our groups Aut(J_{n}^{r}) for n ≥ 3 odd. In each case it is equal to the lower bound given by [CS06, Theorem 1]. In particular, we find that ed(PSp_{2n}) = n + 1 for n ≥ 3 odd, where previously the best upper bound was given by 2n^{2} −3n−1 in [Le04].

2 Preliminaries

Throughout we will let k be a field extension of a fixed base field k_{0} of characteristic not 2, and k_{s} will be a separable closure of k.

For any k-algebra A (by which we will mean finite dimensional, not necessarily associative, with identity), we will use the usual notions from Galois cohomology [KMRT, §29] to identify Aut_{alg}(A)-torsors over k with k-isomorphism classes of k-algebras B such that A_{k_{s}} ≅ B_{k_{s}}. Similarly for algebras with involution.

In the Introduction we defined the split Jordan algebras J_{n}^{r} := (M_{n}(C), −)_{+} for r = 0, 1, 2 when n ≥ 3, and also for r = 3 when n = 3. Here, and throughout this paper we will write dim(C) = 2^{r}. These Jordan algebras are pairwise nonisomorphic, and over k_{s} they represent nearly all simple Jordan algebras by a theorem of Albert (see [Ja68, Ch. V.6, p.204]). We will say that a simple Jordan algebra is of degree n (for some n ≥ 3), if it becomes isomorphic to J_{n}^{r} over k_{s}. These are the only kind of Jordan algebras that we will consider.

Furthermore, for r = 0, 1, 2 we have that (M_{n}(C), −) is a central simple algebra with orthogonal (resp. unitary, symplectic) involution, where C is again the split composition algebra of dimension 2^{r}. They are pairwise non-isomorphic, and over k_{s} they form all k_{s}-isomorphism classes of central simple algebras with involution. We say that a central simple algebra with involution has degree n if it becomes isomorphic to (M_{n}(C), −) over k_{s}.

One must be careful with the potentially confusing terminology here. A central simple algebra with involution over k is defined to be central and simple as an algebra-with-involution, and might not be central or simple as an algebra over k (see [Ja68, p.208] or [KMRT, §2] for precise definitions).

For r = 0, 1, 2 we have a 1-to-1 correspondence between isomorphism classes of these two types of objects given by (A, σ) ↔ (A, σ)_{+}. So we can view Aut(J_{n}^{r})-torsors as either isomorphism classes of algebras with involution or as isomorphism classes of Jordan algebras (see [Ja68, Ch. V.7, p.210]). An advantage of the Jordan algebra point of view is that it includes the exceptional r = 3 case.

2.1 Invariants of quadratic forms

We will follow the notation of [GMS03] and write H^{∗}(k) or even H^{∗} for the Galois cohomology ring H^{∗}(Gal(k_{s}/k), ℤ/2ℤ). For a ∈ k^{∗}/(k^{∗})^{2}, we will denote the corresponding element (a) ∈ H^{1}(k) so that we have (a·b) = (a)+(b). We let Quad_{n,1}(k) be the pointed set of k-isometry classes of n-dimensional quadratic forms of determinant 1. And we let Pf_{r}(k) be the pointed set of k-isometry classes of r-Pfister forms. We will write Inv(G) = Inv_{k_{0}}(H^{1}(−, G)) for the group of cohomological invariants in H^{∗} of G-torsors.

To define the Stiefel-Whitney classes of a quadratic form q over k, take a diagonalization q ≅ ⟨a_{1}, ⋯, a_{n}⟩. Then define w_{i}(q) to be the ith degree part of the product

w(q) = ∏_{j=1}^{n}(1 + (a_{j})) ∈ H^{∗}(k).

This product is called the total Stiefel-Whitney class. It is independent of the diagonalization, which can be shown by a chain equivalence argument (see [Mi70]).

For an r-Pfister form,

q = ⟨⟨a_{1}, ⋯, a_{r}⟩⟩ := ⟨1, −a_{1}⟩ ⊗ ⋯ ⊗ ⟨1, −a_{r}⟩,

define an invariant by e_{r}(q) = (a_{1}) ⋯ (a_{r}) ∈ H^{r}(k). It is shown in [GMS03, VI] that the H^{∗}-module of invariants of r-Pfister forms, Inv(Pf_{r}), has an H^{∗}(k_{0})- basis consisting of {1, e_{r}}.

3 An upper bound for the invariants

In this section we show that Inv(Aut(J_{n}^{r})) injectively embeds into the tensor product of two groups of invariants that we understand well (see Cor. 3.6).

For any associative composition algebra C over k, and hermitian form h on a free n-dimensional C-module V, the trace form, q_{h} is the quadratic form over k on V such that q_{h}(x) = h(x, x). If ϕ is the norm of C, and V ≅ C ⊗ V_{0} as k-vector spaces, then q_{h} ≅ ϕ ⊗ q_{0} for some n-dimensional quadratic form q_{0} on V_{0}.

Lemma 3.1. Let C be an associative composition algebra over k with conjugation involution, and let h, h′ be hermitian forms over C. Then h ≅ h′ iff q_{h} ≅ q_{h′}.

Proof. This is shown in [Sch85, 10.1.1,10.1.7].

Proposition 3.2. Let C be an associative composition algebra over k of dimension 1 (resp. 2, 4). Then isomorphism classes of involutions on M_{n}(C) of orthogonal (resp. unitary, symplectic) type correspond to similarity classes of n-dimensional hermitian forms over C, under ad_{h} ↔ h.

Proof. This is proved in [KMRT, Prop. 4.2, p.43].

We will call a Jordan algebra over k reduced if it is isomorphic to one of the form (M_{n} ⊗ C, ad_{q} ⊗ −)_{+} for some composition k-algebra C, and n-dimensional quadratic form q over k.

Notice this implies Jacobson’s [Ja68] definition of reduced in terms of orthogonal idempotents, and his definition implies this one, by his Coordinatization theorem for n ≥ 3.

Lemma 3.3. Let J be a central simple Jordan algebra of odd degree n and of type r = 0 or 2. Then J is reduced.

Proof. So J ≅ (A, σ)_{+}, where (A, σ) is a central simple algebra with an involution of the first kind. So its index is a power of 2 (see [KMRT, p.18, 2.8]).

Case r = 0: The degree of A as a central simple algebra is n, so the index divides n, and hence A is split. So J is reduced by Prop. 3.2 (or [KMRT, p.1]).

Case r = 2: The degree of A as a central simple algebra is 2n, so the index divides 2, and hence J is reduced by Prop. 3.2 and the remarks at the beginning of this section.

Lemma 3.4. Let J be a central simple Jordan algebra of odd degree. Then J becomes reduced after extending scalars by a field extension of odd degree.

Proof. The only non-reduced algebras are when r = 3 or r = 1 by Lemma 3.3. For the r = 3 case, as stated in the introduction, we must have n = 3. By [SV00, 6.1] any non-reduced Albert algebra becomes reduced after a cubic extension.

For r = 1 there are two cases, as follows.

Case J ≅ (B × B^{op}, σ)_{+}: Here σ is the exchange involution, and B is a central simple algebra over k of degree n odd. Then any maximal subfield L is a splitting field for B of degree dividing n. Then J_{L} is reduced by Prop. 3.2.

Case J ≅ (A, σ)_{+}: Here A is a central simple algebra over K of degree n odd, where K is a quadratic extension of k, and σ is an involution of the second kind (i.e. unitary) over k. Since the Brauer group of a finite field is trivial, we may assume that k is infinite.

Let L/k be an odd degree extension such that ind(A_{L}) = d is minimal, where A_{L} is a simple associative algebra with centre K_{L} = K ⊗_{k} L. d is odd since d|n. Let D be the division K_{L}-algebra that is Brauer equivalent to A_{L}, and hence of degree d as a central simple algebra. Then D has an involution of the second kind τ that fixes L by [KMRT, 3.1, p.31].

We want to show that d = 1, so assume that d > 1. Then we can take a non-scalar element a in the degree d Jordan algebra (D, τ)_{+}. Since k is infinite, we may choose a to be of maximal degree in the sense of [Ja68, p.224]. In other words, a is such that deg(m_{a}) = d, where m_{a}(λ) ∈ L[λ] is the minimal polynomial of a in (D, τ)_{+}, for some indeterminate λ. Here we are using the fact from [Ja68, p.233] that the degree of the generic minimal polynomial of a generic element is equal to the degree of the Jordan algebra as defined in the Preliminaries. Also, the coefficients of m_{a} are in L by [KMRT, 32.1.2, p.452].

If we let α be a root of m_{a} in k_{s}, then the minimal polynomial of α is m_{a}. This is because D is a division algebra and hence m_{a} is irreducible. So E = L(α) is a field extension of degree d over L, and in particular, of odd degree over k. Then by considering the generic norm of a′ := α1 − a_{E} ∈ D_{E}, we see that n(a′) = m_{a}(α) = 0 [Ja68, p.224]. So a′ ≠ 0 and is non-invertible, and hence D_{E} is not a division algebra. So ind(A_{E}) = ind(D_{E}) < d, which contradicts the minimality of d.

Therefore d = 1, and hence A_{L} ≅ M_{n}(K_{L}). So by Prop. 3.2 we see that J_{L} is reduced.

Proposition 3.5. For n ≥ 3 odd, let J be an Aut(J_{n}^{r})-torsor over k. Then there is an odd-degree extension L/k such that J_{L} is in the image of

H : Pf_{r}(L) × Quad_{n,1}(L) → H^{1}(L, Aut(J_{n}^{r}))
(ϕ, q) ↦ (M_{n} ⊗ C_{ϕ}, ad_{q} ⊗ −)_{+}.

Moreover, if r = 2 the map is a surjection, and for r = 0 it is a bijection.

Proof. From Lemma 3.3 and 3.4 we get L/k such that J_{L} is reduced, and since n is odd, we can scale q so that det(q) = 1. Lemma 3.3 gives the r = 2 surjection, and the r = 0 bijection is well-known.

Corollary 3.6. We have an injective map of invariants

Inv(Aut(J_{n}^{r})) ↪ Inv(Pf_{r}) ⊗ Inv(Quad_{n,1}).

Proof. By [Ga07, Lemma 5.3] we can use the surjectivities from Prop. 3.5 to induce an injective map on invariants. Then from [GMS03, Ex. 16.5], we can express the invariants of the direct product Pf_{r} × Quad_{n,1} as the tensor product of the invariants of each factor.

4 Construction of the invariants

Now it is a matter of deciding which of these invariants occur. In other words, we wish to determine the image of the injective map in Cor. 3.6. It turns out to be the constant invariants together with all multiples of e_{r}, the degree r invariant of Pf_{r} (Thm. 4.7).

Theorem 4.1. For n odd, r = 0, 1, 2 or 3, the invariants e_{r} ⊗ w_{2i} ∈ Inv(Pf_{r}) ⊗ Inv(Quad_{n,1}) extend uniquely to Aut(J_{n}^{r})-invariants of degree r + 2i, which we will call v_{i}.

If the invariants extend, then by Cor. 3.6 they are unique. First we will show how to construct the invariants on reduced Jordan algebras, and then use [Ga07, Prop. 7.2] to extend them to all Jordan algebras.

For a reduced Jordan algebra J = (M_{n}(C), ad_{q} ⊗ −)_{+}, we call C the composition algebra associated to J. It is determined up to isomorphism by the isomorphism class of J (see [Ja68] or [Mc04, 16]). We will usually denote its norm form ϕ, which is a Pfister form.

Lemma 4.2. The quadratic form ϕ ⊗ q is determined up to similarity by the isomorphism class of the reduced Jordan algebra J = (M_{n}(C), ad_{q} ⊗ −)_{+}.

Proof. First consider the case when C is associative, which is to say, r ≠ 3. By [Ja68, p.210] the Jordan algebra J determines the isomorphism class of the algebra with involution (M_{n}(C), σ). Then Prop. 3.2 lets us associate up to a scalar, the n-dimensional hermitian form h on C. Finally, Lemma 3.1 allows us to determine its trace form ϕ ⊗ q up to similarity.

In the case r = 3, we use the following argument. For any reduced Jordan algebra of degree n, the quadratic (reduced) trace form, T_{J}(x) = T rd(x^{2}) = trace(x^{2}) is determined up to isometry by the isomorphism class of J, and is of the form

T_{J} ≅ n⟨1⟩ ⊥ ⟨2⟩ϕ ⊗ ∧^{2}q.

But since q is similar to ∧^{2}(q) for 3-dimensional forms, we see that ϕ ⊗ q is determined up to similarity for n = 3, and in particular when r = 3.

Remark 4.3. In the r = 2 case, this observation was noted in [GQMT01, Lemma 4.2].

Lemma 4.4. Let ϕ be an r-fold Pfister form, and q, q′ quadratic forms over k. Then ϕ ⊗ q ≅ ϕ ⊗ q′ implies e_{r}(ϕ)w(q) = e_{r}(ϕ)w(q′) ∈ H^{∗}(k).

Proof. This is an extension of [De62] or [Mi70] where it is shown for r = 0. We need the fact from [WS77] that says if ϕ ⊗ q ≅ ϕ ⊗ q′ then q and q′ are ϕ-chain equivalent.

Two quadratic forms are simply ϕ-equivalent if they can both be diagonalized in such a way that q ≅ ⟨a_{1}, ⋯, a_{n}⟩ and q′′ ≅ ⟨λa_{1}, a_{2}, ⋯, a_{n}⟩, where λ is represented by ϕ. Then two forms are ϕ-chain equivalent if there is a finite chain of simple ϕ-equivalences from one to the other.

This immediately reduces the problem to showing that equality holds at each stage of the chain equivalence. This is the same as showing e_{r}(ϕ)w(⟨a⟩) = e_{r}(ϕ)w(λ⟨a⟩) for λ represented by ϕ. For such a λ, we have ϕ ⊗ ⟨1, −λ⟩ is isotropic, and hence e_{r}(ϕ) · (λ) = 0 ∈ H^{r+1}(k). Expanding w(λ⟨a⟩) = 1 + (λ) + (a), the result clearly follows.

So the following Lemma shows that the quadratic form ϕ⊗q, where det(q) = 1, is in fact determined up to isometry by the isomorphism class of a reduced Jordan algebra. Since q is odd dimensional, there is always a determinant 1 quadratic form similar to it. We will write d(q) = det(q) for the element of k^{∗}/(k^{∗})^{2} corresponding to w_{1}(q) ∈ H^{1}(k).

Lemma 4.5. Let ϕ be an r-Pfister form, λ ∈ k^{∗}, and q, q′ quadratic forms. If ϕ ⊗ q ≅ ϕ ⊗ λq′ then ϕ ⊗ d(q)q ≅ ϕ ⊗ d(q′)q′.

Proof.

e_{r}(ϕ) · w_{1}(q) = e_{r}(ϕ) · w_{1}(λq′) ∈ H^{r+1}(k) ⇔ e_{r}(ϕ) · (λd(q)d(q′)) = 0
⇔ ϕ ⊗ ⟨⟨λd(q)d(q′)⟩⟩ is hyperbolic
⇔ d(q)d(q′) = λ mod D(ϕ)^{∗}
⇒ ϕ ⊗ d(q)q ≅ ϕ ⊗ d(q′)q′.

Proof of Thm. 4.1. First we will show that the invariants e_{r} ⊗ w_{2i} extend to invariants on k-isomorphism classes of reduced Jordan algebras.

Consider the reduced Jordan algebra J = (M_{n}(C), ad_{q} ⊗ −)_{+} with n = 2m + 1. Then we can assume det(q) = 1. Lemma 4.2 together with Lemma 4.5 show that ϕ ⊗ q is determined up to isometry by the isomorphism class of J. Then by Lemma 4.4 we can define v_{i}(J) = e_{r}(ϕ)w_{2i}(q) ∈ H^{r+2i}(k) on k-isomorphism classes of reduced Jordan algebras, for 1 ≤ i ≤ m. This clearly extends e_{r} ⊗ w_{2i}.

Finally, by Lemma 3.4, any odd degree Jordan algebra becomes reduced after an odd degree extension. So by [Ga07, Prop. 7.2] these invariants may be extended to non-reduced Jordan algebras as well, and in other words, to all Aut(J_{n}^{r})-torsors. By Cor. 3.6, v_{i} is the unique invariant extending e_{r} ⊗ w_{2i}.

Remark 4.6. For r = 1, there is an invariant closely related to v_{1} defined on conjugacy classes of algebras with unitary involution of odd degree in [KMRT, p.438, 31.44]. They related it to the Rost invariant.

Now we can state and prove our main theorem.

Theorem 4.7. Inv_{k_{0}}(Aut(J_{n}^{r})) is a free H^{∗}(k_{0})-module with a basis consisting of the invariants {1, v_{0}, v_{1}, v_{2}, ⋯, v_{m}}.

Proof. For r = 0 this is shown in [GMS03, ch. VI], noting that in this case v_{0} = 1, causing a redundancy in the set of basis elements. So take r > 0. From Cor. 3.6 we know that every Aut(J_{n}^{r})-invariant restricts to some

1 ⊗ a + e_{r} ⊗ b ∈ Inv(Pf_{r}) ⊗ Inv(Quad_{n,1}),

for some uniquely defined a, b ∈ Inv(Quad_{n,1}). We know from [GMS03, ch. VI] that any b ∈ Inv(Quad_{n,1}) is in the H^{∗}(k)-span of the even Stiefel-Whitney classes, so by Thm. 4.1, e_{r} ⊗ b is the restriction of some Aut(J_{n}^{r})-invariant in the H^{∗}(k)-span of {v_{0}, v_{1}, ⋯, v_{m}}. So all that remains to show is that if 1 ⊗ a is the restriction of an Aut(J_{n}^{r})-invariant, then a is constant.

Let a′ be an Aut(J_{n}^{r})-invariant that restricts to 1 ⊗ a for some Quad_{n,1}- invariant a. If we let C_{s} be the split composition algebra of dimension 2^{r}, then

J = (M_{n}(C_{s}), ad_{q} ⊗ −)_{+}

is isomorphic to the split algebra J_{n}^{r} (by Prop. 3.2 for r ≠ 3 and by [SV00, Cor. 5.8.2] for r = 3). So for any such J, we must have that a′(J) = a(q) is a constant, independent of q. Since we can take q to be an arbitrary n-dimensional form of determinant 1, this implies a is constant. This completes the proof.

Remark 4.8. One may ask to what extent do these v_{i} determine the Aut(J_{n}^{r})- torsors? There are examples of non-isometric quadratic forms with determinant 1 in each dimension ≥ 4 that have equal total Stiefel-Whitney classes [Sch67, Beispiel 3.4.1]. So one can use these examples to write down two different reduced Aut(J_{n}^{r})-torsors for n ≥ 4 odd, whose invariants agree.

In the case of n = 3, on the other hand, for r = 0 or 2, the torsors are determined completely by v_{0} and v_{1}. This is because they are determined by their quadratic trace form [SV00, §5]. But for r = 1 and r = 3 this is not the case, because (for n = 3) the trace form of any non-reduced algebra is isometric to the trace form of some reduced algebra. Nevertheless, for r = 1 one may define a degree 2, mod 3 invariant, which together with v_{0} and v_{1}, classify Aut(J_{3}^{1})-torsors [KMRT, §19.B, §30.C]. For r = 3, one may define a degree 3, mod 3 invariant, but it is an open problem whether this invariant together with v_{0} and v_{1}, classify Aut(J_{3}^{3})-torsors [Se95, 9.4].

5 Essential dimension

Given an algebraic group G over k_{0}, and a G-torsor E over k, the essential dimension of E is defined to be the minimum transcendence degree over k_{0} of all fields of definition of E. The essential dimension of an algebraic group is defined to be the supremum of the essential dimensions of all of its torsors ([RY00], [BF03], [CS06]). The essential dimension of most simple algebraic groups is unknown. We will determine the value of the essential dimension at 2 for some groups G, which we will denote by ed(G; 2) (see [CS06] for a definition of the essential dimension at a prime). In all cases that we consider, ed(G; 2) is equal to the lower bound given in [CS06, Theorem 1] (or [RY00, Thm. 7.8] for characteristic 0).

Proposition 5.1. For n ≥ 3 odd, we have ed(Aut(J_{n}^{r}); 2) = r + n − 1.

Proof. By the surjectivity in Lemma 3.5 we have that for any Aut(J_{n}^{r})-torsor J over k, there is an odd degree extension L/k such that J_{L} is reduced. So by using [BF03, Lemma 1.11] we have that ed(J; 2) ≤ ed_{L}(J_{L}) ≤ ed(Pf_{r})+ed(Quad_{n,1}) = r + n − 1. This gives us the upper bound ed(Aut(J_{n}^{r}); 2) ≤ r + n − 1.

The lower bound follows from [CS06, Theorem 1]. Alternatively, we could deduce the lower bound by using the non-triviality of the degree r + n − 1 cohomological invariant v_{m}. This follows from a slight modification of [BF03, Cor. 3.6], that if there is a degree d invariant mod 2, then the essential dimension at 2 is at least d.

Let us consider what Prop. 5.1 says for different r and n ≥ 3 odd.

For r = 0 we get the well known fact that ed(SO_{n}) = ed(SO_{n}; 2) = n − 1. For r = 1 we get ed(ℤ/2ℤ ⋉ PGL_{n}) ≥ ed(ℤ/2ℤ ⋉ PGL_{n}; 2) = n. The exact

value of ed(ℤ/2ℤ ⋉ PGL_{n}) is unknown to the author for any n ≥ 3.

For r = 2 we get ed(PSp_{2n}) = ed(PSp_{2n}; 2) = n + 1, since all PSp_{2n}- torsors are reduced. Previously, the best known upper bound for ed(PSp_{2n}) was 2n^{2} − 3n − 1, which holds for n even as well [Le04].

For r = 3 we get ed(F_{4}) ≥ ed(F_{4}; 2) = 5, which is the best known lower bound for ed(F_{4}). The best published upper bound for the essential dimension is ed(F_{4}) ≤ 19 in [Le04]. [Ko00] claimed to show that ed(F_{4}) = 5, but there was a mistake in the proof.

6 Acknowledgements

I would like to thank Jean-Pierre Tignol for pointing out the r = 2 case of Lemma 3.3, and Skip Garibaldi for suggesting the application to essential dimension, and catching several mistakes in a previous version of this paper. I would also like to thank my Ph.D. supervisor Burt Totaro, as well as Carl Mc-Tague and Arthur Prendergast-Smith for many useful discussions. I also thank the referee for their comments and suggestions.

References

[BF03] G. Berhuy and G. Favi. Essential dimension: a functorial point of view (after A. Merkurjev), Doc. Math. 8 (2003), 279-330.

[CS06] V. Chernousov and J.-P. Serre. Lower bounds for essential dimensions via orthogonal representations. J. Algebra 305 (2006), 1055-1070.

[De62] A. Delzant. Définition des classes de Stiefel-Whitney d’un module quadratique sur un corps de caractristique différente de 2. (French) C. R. Acad. Sci. Paris 255 (1962), 1366-1368.

[Ga07] S. Garibaldi. Cohomological invariants: exceptional groups and spin groups. Mem. Amer. Math. Soc., to appear.

[GMS03] S. Garibaldi, A. Merkurjev and J.-P. Serre. Cohomological invariants in Galois cohomology, AMS University Lecture Series 28 (American Mathematical Society, 2003).

[GPT07] S.  Garibaldi,  R.  Parimala and J.-P.  Tig-
nol.  Discriminant  of  symplectic  involutions.  Preprint,

www.mathematik.uni-bielefeld.de/lag/man/264.html.

[GQMT01] S. Garibaldi, A. Queguiner-Mathieu and J.-P. Tignol. Involutions and trace forms on exterior powers of a central simple algebra. Doc. Math. 6 (2001), 99-120.

[HKRT] D. Haile, M.-A. Knus, M. Rost and J.-P. Tignol. Algebras of odd degree with involution, trace forms and dihedral extensions. Israel J. Math. 96 (1996), 299-340.

[Ja68] N. Jacobson. Structure and representations of Jordan algebras. Amer. Math. Soc. Colloq. Publ. vol. XXXIX (American Mathematical Society, 1968).

[Ko00] V. È. Kordonskiĭ. On the essential dimension and Serre’s conjecture II for exceptional groups. (Russian. Russian summary) Mat. Zametki (4) 68 (2000), 539-547; English translation in Math. Notes (3-4) 68 (2000), 464-470.

[KMRT] M.-A. Knus, A. Merkurjev, M. Rost and J.-P. Tignol. The book of involutions. Colloquium Publications (American Mathematical Society, 1998).

[Le04] N. Lemire. Essential dimension of algebraic groups and integral representations of Weyl groups. Transform. Groups 9 (2004), 337-379.

[Mc04] K. McCrimmon. A taste of Jordan algebras. Universitext (Springer, 2004).

[Mi70] J. Milnor. Algebraic K-theory and quadratic forms. Invent. Math. 9 (1970), 318-344.

[RY00] Z. Reichstein and B. Youssin. Essential dimensions of algebraic groups and a resolution theorem for G-varieties (with an appendix by János Kollár and Endre Szabó). Canad. J. Math. 52 (2000), 1018-1056.

[Ro99] M. Rost. A descent property for Pfister forms. J. Ramanujan Math. Soc. 14 (1999), 55-63.

[Sch67] W. Scharlau. Quadratische Formen und Galois-Cohomologie. Invent. Math. 4 (1967), 238-264.

[Sch85] W. Scharlau. Quadratic and hermitian forms. Grundlehren der Mathematischen Wissenschaften, vol. 270 (Springer-Verlag, 1985).

[Se95] J.-P. Serre. Cohomologie galoisienne: progrès et problèmes. Astérisque 227 (1995), 229-257.

[SV00] T.A. Springer and F.D. Veldkamp. Octonions, Jordan algebras and exceptional groups. Springer Monographs in Mathematics (Springer, 2000).

[WS77] A.R. Wadsworth and D.B. Shapiro. On multiples of round and Pfister forms. Math. Z. 157 (1977), 53-62.

DPMMS, University of Cambridge, Wilberforce Road, Cambridge, CB3 0WB, United Kingdom

e-mail: M.MacDonald@dpmms.cam.ac.uk