LOWER BOUNDS FOR ESSENTIAL DIMENSIONS IN

CHARACTERISTIC 2 VIA ORTHOGONAL REPRESENTATIONS

A. BABIC AND V. CHERNOUSOV

Abstract. We give a lower bound for the essential dimension of a split simple algebraic group of “adjoint” type over a field of characteristic 2. We also compute the essential dimension of orthogonal and special orthogonal groups in characteristic 2. Keywords: Linear algebraic group, essential dimension, torsor, orthogonal representation, Killing form, quadratic form. MSC 2010 11E04, 11E57, 11E72, secondary 11E81, 14L35, 20G15.

1. Introduction

Informally speaking, the essential dimension of an algebraic object can be thought of as the minimal number of independent parameters needed to define it. Essential dimension assigns a numerical invariant (a non-negative integer) to algebraic objects and allows us to compare their relative complexity. Naturally, the fewer parameters needed for definition, the simpler the object is.

The notion of essential dimension first appeared in a 1997 paper by J. Buhler and Z. Reichstein [BuRe] within the context of finite groups. Later on A. Merkurjev extended this notion to arbitrary functors; see [BF]. For the definition, properties and results on essential dimension of algebraic groups and various functors we refer to recent surveys [Me] and [Re].

In the past 15 years this numerical invariant has been extensively studied by many people. To the best of our knowledge in all publications on this topic the only approach for computing the essential dimension ed(G) of an algebraic group G consisted of finding its upper and lower bounds. If, by lucky circumstance, both bounds for G are equal then of course their common value is ed(G). We remark that this strategy has worked in all cases where ed(G) is known.

Date: November 21, 2014.

V. Chernousov was partially supported by the Canada Research Chairs Program and an NSERC research grant.

The aim of the current paper is two-fold. We recall that a general method for computing lower bounds of the essential dimensions of simple algebraic groups defined over fields of characteristic ≠ 2 via orthogonal representations was developed in [ChSe]. Our first goal is to extend this approach to characteristic 2 case. In Section 12 we prove the incompressibility of the so-called canonical monomial quadratic forms and this result leads us to Theorem 2.1 below which says that for any simple split “adjoint group” G defined over a field of characteristic 2 one has ed(G) ≥ r + 1 where r = rank(G). Second, we show that for an adjoint split group G of type B_{r} one has ed(G) = r + 1. Thus, this result indicates that the lower bound r + 1 of the essential dimension in Theorem 2.1 is optimal for groups of adjoint type in the general case and it seems inevitable that any future progress, if possible, will be based on case by case consideration.

We now pass to the precise description of the main results of the paper.

2. The main Theorems

In what follows, we assume that k is an algebraically closed field of characteristic 2 and all fields and rings under consideration will contain k.

Let G^{∘} be a simple algebraic group over k of adjoint type, and let T be a maximal torus of G^{∘}. Let c ∈ Aut(G^{∘}) be such that c^{2} = 1 and c(t) = t^{−1} for every t ∈ T (it is known that such an automorphism exists, see e.g. [DG], Exp. XXIV, Prop. 3.16.2, p. 355). This automorphism is inner (i.e. belongs to G^{∘}) if and only if −1 belongs to the Weyl group of (G, T). When this is the case, we put G = G^{∘}. If not, we define G to be the subgroup of Aut(G^{∘}) generated by G^{∘} and c. We have

• G = G^{∘} for types A_{1}, B_{r}, C_{r}, D_{r} (r even), G_{2}, F_{4}, E_{7}, E_{8} ;

• (G : G^{∘}) = 2 and G = Aut(G^{∘}) for types A_{r} (r ≥ 2), D_{r} (r

odd), E_{6}.

Let r = dim (T) be the rank of G.

2.1. Theorem. If G is as above, we have ed (G) ≥ r + 1.

Our second main theorem deals with orthogonal and special orthogonal groups.

2.2. Theorem. Let q be a non-degenerate n-dimensional quadratic form over k. We have:

(a) if n = 2r then ed(O(q)) = r + 1;

(b) if n = 2r and r is even then ed(SO(q)) = r + 1; (c) if n = 2r and r is odd then r ≤ ed(SO(q)) ≤ r + 1;

(d) If n = 2r + 1 then ed(O(q)) = ed(SO(q)) = r + 1.

3. Strategy of the proof of main Theorems

For groups of type G_{2} and F_{4} in Theorem 2.1 there is an easy reduction to orthogonal groups (see Section 14 below). For all other adjoint types, orthogonal and special orthogonal groups we follow the same approach as in [ChSe]. Namely,

a) we construct a G-torsor θ_{G} over a suitable extension K/k with tr. deg_{k}(K) = r + 1, see below;

b) we show that there exists a suitable representation ρ : G → O_{N} such that the image of θ_{G} in H^{1}(K, O_{N}) is incompressible; this implies that θ_{G} itself is incompressible, and Theorems 2.1 and 2.2 follow.

Let us start with part a) for an adjoint group G. Let R be the root system of G with respect to T, and let R_{sh} be the (sub) root system formed by the short roots of R. Let Δ = {α_{1}, …, α_{r}} be a basis of R_{sh}. The root lattices of R and R_{sh} are the same; hence Δ is a basis of the character group X(T). This allows us to identify T with G_{m}×⋯×G_{m} using the basis Δ.

Call A_{0} the kernel of “multiplication by 2” on T. Let A = A_{0} ×{1, c} be the subgroup of G generated by A_{0} and by the element c defined above. The group A is isomorphic to μ_{2} × ⋯ × μ_{2} × ℤ/2.

Take K = k(t_{1}, …, t_{r}, x) where t_{1}, …, t_{r} and x are independent indeterminates. We have

H^{1}(K, A) = H^{1}(K, μ_{2}) × … × H^{1}(K, μ_{2}) × H^{1}(K, ℤ/2).

Identify H^{1}(K, μ_{2}) with K^{×}/(K^{×})^{2} and H^{1}(K, ℤ/2) with K/℘(K) as usual. Here ℘ : K → K is the Artin-Schreier map. given by ℘(a) = a^{2} + a. Then x and the t_{i}’s define elements (x) and (t_{i}) of H^{1}(K, ℤ/2) and H^{1}(K, μ_{2}) respectively. Let θ_{A} be the element of H^{1}(K, A) with components ((t_{1}), …, (t_{r}), (x)). Let θ_{G} be the image of θ_{A} in H^{1}(K, G). We will prove in Section 14 :

3.1. Theorem. (K, θ_{G}) is incompressible.

Note that Theorem 3.1 implies Theorem 2.1 since tr. deg. K = r + 1. Its proof relies on studying properties of the so-called monomial quadratic forms (see Section 10 below) which are also crucial for the proof of Theorem 2.2.

4. Review: Quadratic spaces in characteristic 2

The purpose of this section is to review some properties of quadratic forms in characteristic 2 needed for construction of a representation of our group G with the required property explained above. To this end we will introduce the notion of a “normalization” of a quadratic form which may not be standard.

Let K be an arbitrary field of characteristic 2. Recall that a quadratic space over K is a pair (V, q) where V is a vector space over K and q is a quadratic form on V. As usual, for any a, b ∈ K we will denote by [a, b] a 2-dimensional quadratic form given by [a, b] = ax^{2} + xy + by^{2}. The form [0, 0] is called the hyperbolic plane and is denoted by ℍ. Similarly, for a ∈ K we denote by ⟨ a ⟩ the quadratic form ax^{2}.

There is a special class of quadratic forms called n-fold Pfister forms (see [EKM]). Recall that, by definition, a quadratic form [1, a] where a ∈ K is called a 1-fold Pfister form and denoted ⟨⟨a]]. A quadratic form isometric to

⟨⟨a_{1}, …, a_{n}]] : = ⟨⟨a_{1}, …, a_{n−1}⟩⟩_{b} ⊗ ⟨⟨a_{n}]]

for some a_{1}, …, a_{n} ∈ K is called a quadratic n-fold Pfister form. Here ⟨⟨a_{1}, …, a_{n−1}⟩⟩_{b} is a symmetric bilinear form given by

⟨⟨a_{1}, …, a_{n−1}⟩⟩_{b} = ⟨1, a_{1}⟩_{b} ⊗ ⋯ ⊗ ⟨1, a_{n−1}⟩_{b}.

Let K/k be a finitely generated field extension of our base field k and q a quadratic form over K. Then, if there exists another quadratic form g defined over a field L/k satisfying

• k ⊂ L ⊂ K;

• tr. deg_{k} L < tr. deg_{k} K; and
• g ⊗_{L} K ≃ q

we say that q is compressible. Otherwise, it is incompressible.

Given a quadratic form q one associates the bilinear form (called the polar form of q) b_{q} : V × V → K given by

b_{q}(u, v) = q(v + u) − q(u) − q(v).

Its radical is

rad(b_{q}) = {v ∈ V | b_{q}(v, w) = 0 ∀ w ∈ V }

and the quadratic radical of q is defined as

rad(q) = {v ∈ rad(b_{q}) | q(v) = 0}.

Obviously, both rad(b_{q}) and rad(q) are vector subspaces in V.

One says that q is regular if rad(q) = 0 and q is non-degenerate if it is regular over any field extension L/K. Note that non-degeneracy is equivalent to the property dim(rad(b_{q})) ≤ 1.

It is well-known (see [EKM]) that any non-degenerate quadratic form q of even dimension n = 2m is isometric to q ≃ ⊕_{i=1}^{m}[a_{i}, b_{i}] where a_{i}, b_{i} ∈ K. In this case the element c = ∑ a_{i}b_{i} modulo ℘(K) is called the Arf invariant of q. If q is non-degenerate and has odd dimension n = 2m + 1 then q ≃ ⊕_{i=1}^{m}[a_{i}, b_{i}] + ⟨ c ⟩ where c ∈ K^{×} is unique up to squares. This element c (modulo (K^{×})^{2}) is called the determinant (= discriminant) of q.

Let q: V → K be a quadratic form. We denote \overline{V} : = V/rad(q) and let π: V → \overline{V} be the canonical map. It is straightforward to check that the mapping \overline{q}: \overline{V} → K given by \overline{q}(\overline{v}) = q(v) is well defined. Thus a quadratic space (V, q) gives rise to a quadratic space (\overline{V} , \overline{q}). We will see in the example below that \overline{q} is non-degenerate, but first we state the following definition.

4.1. Definition. We will say that \overline{q} is the (non-degenerate) normalization of q.

Example. Let q be a quadratic form over k. Since k is algebraically closed it is isometric to a quadratic form

⟨0⟩ ⊕ ⋯ ⊕ ⟨0⟩ ⊕ ℍ ⊕ ⋯ ⊕ ℍ or
⟨0⟩ ⊕ ⋯ ⊕ ⟨0⟩ ⊕ ⟨1⟩ ⊕ ℍ ⊕ ⋯ ⊕ ℍ.

It easily follows from the definition that its normalization is the following quadratic form:

ℍ ⊕ ⋯ ⊕ ℍ or ⟨1⟩ ⊕ ℍ ⊕ ⋯ ⊕ ℍ;

in particular \overline{q} is non-degenerate.

Lastly, we want to relate the orthogonal group of a quadratic form q to that of its normalization. Recall that given a quadratic space (V, q) the orthogonal group of (V, q) is

O(V, q) = {x ∈ GL(V) | q(x(v)) = q(v) ∀ v ∈ V }.

We define a map

λ: O(V, q) ⟶ O(\overline{V} , \overline{q}).

by x → \overline{x} where \overline{x}(\overline{v}) = \overline{x(v)} for all \overline{v} ∈ \overline{V}.

Let us first show that \overline{x} is well defined, i.e. x(rad(q)) ⊂ rad(q) or equivalently x(v) ∈ rad(b_{q}) for v ∈ rad(q) (because x preserves length of vectors). Let w_{0} ∈ V. Since x is invertible we have x(w) = w_{0} for some w ∈ V. Then

b_{q}(x(v), w_{0}) = q(x(v) + w_{0}) + q(x(v)) + q(w_{0})
= q(x(v) + x(w)) + q(x(v)) + q(x(w))
= q(x(v + w)) + q(x(v)) + q(x(w))
= q(v + w) + q(v) + q(w) = b_{q}(v, w) = 0,

because v ∈ rad(q) ⊂ rad(b_{q}). Thus, x(v) ∈ rad(q) as required. It remains to see that \overline{x} ∈ O(\overline{V} , \overline{q}). However,

\overline{q}(\overline{x}(\overline{v})) = \overline{q} (\overline{x(v)}) = q(x(v)) = q(v) = \overline{q}(\overline{v}).

Thus we have the following result:

4.2. Lemma. The canonical map V → \overline{V} induces a natural morphism

λ: O(V, q) ⟶ O(\overline{V} , \overline{q}).

5. Killing forms of simple Lie algebras over ℤ

Let G be as in Theorem 2.1 and let G̃ be a universal simply connected covering of its connected component G^{∘}. To construct the required orthogonal representation ρ of G (see part (b) of our strategy described in Section 3) we need to know how the “normalized” Killing symmetric bilinear (resp. quadratic) form K_{b} (resp. K_{q}) of the Lie algebra Lie(G̃) looks like.

Since our main field has characteristic 2, we begin by computing K_{q} in a Chevalley basis of the Lie algebra L of a split simple simply connected algebraic group defined over ℤ. We then pass to k by first normalizing K_{b}, i.e. by dividing all its coefficients by their g.c.d., and then applying the base change ℤ → 𝔽_{2} = ℤ/2ℤ ↪ k.

Recall that a Chevalley basis is a canonical basis of L which arises from a decomposition of

L = L_{0} ⊕ ( ∐_{α≠0} L_{α})

into a direct sum of the weight subspaces L_{α} with respect to a split maximal toral subalgebra H = L_{0} ⊂ L. Note that the set of all nontrivial weights in the above decomposition forms a simple root system and that for every root α we have dim(L_{α}) = 1.

In what follows Φ will denote the set of all roots of L with respect to H, Δ ⊂ Φ its basis and Φ^{+} (resp. Φ^{−}) positive (resp. negative) roots.

It is known (see [St]) that there exist elements {H_{α_{i}}| α_{i} ∈ Δ} in H and X_{α} ∈ L_{α}, α ∈ Φ such that the set

(5.0.1)  {H_{α_{i}}| α_{i} ∈ Δ } ∪ {X_{α}| α ∈ Φ^{+}} ∪ {X_{−α}| α ∈ Φ^{+}}

forms a basis for L, known as a Chevalley basis, and these generators are subject to the following relations:

• [H_{α_{i}}, H_{α_{j}}] = 0;

• [H_{α_{i}}, X_{α}] = ⟨α, α_{i}⟩X_{α};

• H_{α} : = [X_{α}, X_{−α}] = ∑_{α_{i}∈Δ} n_{i}H_{α_{i}} where n_{i} ∈ ℤ;

• [X_{α}, X_{β}] = \{ \begin{matrix}0 & if α + β ∉ Φ \\ ±(p + 1)X_{α+β} & otherwise \end{matrix} ,

where p is the greatest positive integer such that α − pβ ∈ Φ. Here for two roots α, β ∈ Φ the scalar ⟨α, β⟩ is given by

⟨α, β⟩ = \frac{2(α, β)}{(β, β)} ,

where (−, −) denotes the standard inner product on the root lattice. It is in this Chevalley basis (5.0.1) that we will compute the Killing form K_{q} of L.

Note that many people addressed computation of Killing forms (see, for example, [GN], [Ma] [Sel], [SpSt]), but we could not find in the literature explicit formulas valid in characteristic 2. Below we produce such formulas for the normalized Killing forms for each type with the use of the following known facts.

Recall that for any X, Y ∈ L one has

K_{b}(X, Y) = Tr(ad(X) ∘ ad(Y)), K_{q}(X) = K_{b}(X, X)

where ad: L → End(L) is the adjoint representation of L. It is straightforward to check that

K_{b}(H_{α_{i}}, X_{α}) = 0, K_{b}(X_{α}, X_{β}) = 0

for all i and all roots α, β ∈ Φ such that α + β ≠ 0; in particular, K_{q}(X_{α}) = K_{b}(X_{α}, X_{α}) = 0. Thus as a vector space L is decomposed into an orthogonal sum of its subspaces H and ⟨ X_{α}, X_{−α} ⟩, α ∈ Φ^{+}.

Another fact which we need is due to Steinberg and Springer [SpSt]: for any long root α ∈ Φ one has

(5.0.2)  K_{b}(H_{α}, H_{α}) = Tr(ad(H_{α}) ∘ ad(H_{α})) = 4ȟ ,

where ȟ is the dual Coxeter number of the given Lie algebra. Also, for any root α ∈ Φ we have

(5.0.3)  K_{b}(X_{α}, X_{−α}) = \frac{1}{2} Tr(ad(H_{α}) ∘ ad(H_{α})).

Lastly, we need one more result from [Ma]:

(5.0.4)  K_{b}(H_{α_{i}}, H_{α_{j}}) = 2ȟ(α̌_{i}, α̌_{j}) ,

where α̌_{i} = \frac{2α_{i}}{(α_{i},α_{i})} and (α̌, β̌) is the Weyl-invariant inner product such that (α̌, α̌) = 2 for a long root α. Note that the above formula requires (α̌, α̌) = 2 for a long root α, so that for groups of type C_{n} and G_{2} we will have to multiply the standard inner product by an appropriate scalar to match this condition.

Combining the above mentioned results we see that for computation of K_{b} we need to know only how K_{b} looks on the Cartan subalgebra H. Indeed, formula (5.0.3) allows us to compute the restriction of K_{b} to each 2-dimensional subspace ⟨ X_{α}, X_{−α} ⟩. Furthermore, for each long root α we know by equation (5.0.2) that K_{b}(H_{α}, H_{α}) = 4ȟ. Similarly, by using (5.0.4) and the fact that the Killing form is W -invariant, where W is the corresponding Weyl group, we see that K_{b}(H_{β}, H_{β}) is a constant value for all short roots β, but this value will depend on the type of Φ. Finally we remark that if α_{i}, α_{j} ∈ Δ ⊂ Φ are non adjacent roots, then

K_{b}(H_{α_{i}}, H_{α_{j}}) = Tr(ad(H_{α_{i}}) ∘ ad(H_{α_{j}})) = 0.

Indeed this is equivalent to saying that (α_{i}, α_{j}) = 0 which is true for non adjacent roots.

Below we skip straightforward computations of K_{b}(H_{α_{i}}, H_{α_{i}}) and K_{b}(H_{α_{i}}, H_{α_{i+1}}) for each type and present the final result only.

5.1. Type A_{n}. We have:

Tr(ad(H_{α_{i}}) ∘ ad(H_{α_{i}})) = 4ȟ and Tr(ad(H_{α_{i}}) ∘ ad(H_{α_{i+1}})) = −2ȟ.

Thus the Killing quadratic form K_{q} restricted to the Cartan subalgebra H of the Lie algebra L of type A_{n} is of the form

K_{q}|_{H} = 4ȟ (∑_{i=1}^{n} x_{i}^{2}) − 4ȟ (∑_{i=1}^{n−1} x_{i}x_{i+1}) .

and the Killing form on all of L is

K_{q} = K_{q}|_{H} + 4ȟ (∑_{|Φ^{+}|} y_{i}y_{i+1}) .

To pass to the main field k we first modify (normalize) K_{q} by dividing all coefficients of K_{q} by 4ȟ. After doing so our modified Killing form (still denoted by K_{q}) becomes

K_{q} = ∑_{i=1}^{n} x_{i}^{2} − ∑_{i=1}^{n−1} x_{i}x_{i+1} + ∑_{|Φ^{+}|} y_{i}y_{i+1}.

Passing to ℤ → ℤ/2ℤ, which is a field of characteristic 2, we finally would like to “diagonalize” our form. Simple computations show that a diagonalization of K_{q} looks as follows:

K_{q} ≃ ⨁_{i=1}^{(n−1)/2} [0, 0] ⊕ ⟨c⟩ ⊕ ⨁_{|Φ^{+}|} [0, 0], if n is odd; c ∈ {0, 1}

and

K_{q} ≃ ⨁_{i=1}^{(n−1)/2}[0, 0] ⊕ ⨁_{|Φ^{+}|} [0, 0], if n is even

5.2. Remark. In the above formula c can be 0 and 1. Its value depends on the parity of m where n = 2m + 1.

Similar arguments work for each type. Below we present the final result only.

5.3. Type B_{n}.

K_{q} ≃ ⨁_{i=1}^{(n−2)/2}[0, 0] ⊕ ⨁_{|Φ_{long}^{+}|}[0, 0] ⊕ ⟨c⟩ ⊕ m⟨0⟩, if n is even; c ∈ {0, 1}

and

K_{q} ≃ ⨁_{i=1}^{(n−1)/2}[0, 0] ⊕ ⨁_{|Φ_{long}^{+}|}[0, 0] ⊕ m⟨0⟩, if n is odd,

where m = 2|Φ_{short}^{+}| + 1.

5.4. Type C_{n}.

K_{q} ≃ ⟨1⟩ ⊕ ⨁_{|Φ_{long}^{+}|}[0, 0] ⊕ m⟨0⟩

where m = (n − 1) + 2|Φ_{short}^{+}|.

10 A. BABIC AND V. CHERNOUSOV

5.5. Type D_{n}.

K_{q} ≃ ⨁_{i=1}^{(n−1)/2}[0, 0] ⊕ ⟨0⟩ ⊕ ⨁_{|Φ^{+}|} [0, 0], if n is odd;

and

K_{q} ≃ ⨁_{i=1}^{(n−2)/2}[0, 0] ⊕ ⟨c_{1}⟩ ⊕ ⟨c_{2}⟩ ⊕ ⨁_{|Φ^{+}|} [0, 0], if n is even; c_{1}, c_{2} ∈ {0, 1}.

where one of c_{1} or c_{2} equals 0. 5.6. Type E_{6}.

K_{q} ≃ [0, 0] ⊕ [0, 0] ⊕ [0, 0] ⊕ ⨁_{|Φ^{+}|} [0, 0].

5.7. Type E_{7}.

K_{q} ≃ [0, 0] ⊕ [0, 0] ⊕ [0, 0] ⊕ ⟨1⟩ ⊕ ⨁_{|Φ^{+}|} [0, 0].

5.8. Type E_{8}.

K_{q} ≃ [0, 0] ⊕ [0, 0] ⊕ [0, 0] ⊕ [0, 0] ⊕ ⨁_{|Φ^{+}|} [0, 0].

5.9. Type F_{4}.

K_{q} ≃ [0, 0] ⊕ ⨁_{|Φ_{long}^{+}|} [0, 0] ⊕ m⟨0⟩

where m = 2 + |Φ_{short}^{+}|. 5.10. Type G_{2}.

K_{q} ≃ [0, 0] ⊕ ⨁_{|Φ^{+}|} [0, 0].

6. An orthogonal representation

6.1. Proposition. Let G^{∘} be a split simple adjoint algebraic group over k of one of the following types: A_{r}, B_{r}, C_{r}, D_{r}, E_{6}, E_{7}, E_{8}. There exists a quadratic space (V, q) over k, and an orthogonal linear representation

ρ : G^{∘} ⟶ O (V, q)

with the following property :

(∗) q is non-degenerate and the nonzero weights of T on V are the short roots and they have multiplicity 1.

Proof. Types A_{n}, D_{n}, E_{6}, E_{7}, E_{8}. Let W = Lie(G̃). The adjoint representation G̃ → O(W, K_{q}) factors through G̃ → G^{∘}. So it induces the representation μ : G^{∘} → O(W, K_{q}). Let ρ be the composition of μ and the map λ : O(W, K_{q}) → O(\overline{W} , \overline{K_{q}}) given in Lemma 4.2. Denote V = \overline{W}. The inspection of the normalized Killing form K_{q} presented in Section 5 shows that ρ has the required property.

Type B_{r}. We take V to be the standard representation of SO_{2r+1} of dimension 2r + 1.

Type C_{r}. The formula for K_{q} presented in 5.4 shows that the adjoint representation doesn’t work. So instead of the adjoint representation of G = PSp_{2r} we consider its representation on the exterior square.

More precisely, let V_{1} be the standard representation of G̃ = Sp_{2r} over ℤ equipped with a standard skew-symmetric bilinear form ω. Choose a standard basis {e_{1}, …, e_{r}, e_{−r}, …, e_{−1}} of V_{1}. There exists a natural embedding ⋀^{2}(V_{1}) → V_{1} ⊗ V_{1} given by v ∧ w → v ⊗ w − w ⊗ v. We extend ω to a symmetric bilinear form on V_{1} ⊗ V_{1} by

ω(v_{1} ⊗ v_{2}, w_{1} ⊗ w_{2}) = ω(v_{1}, w_{1})ω(v_{2}, w_{2})

and take its restriction (still denoted by ω) to V_{2} = ⋀^{2}(V_{1}).

Consider a natural action of G on V_{2}. This action preserves ω and thus we have a natural representation G → O(V_{2}, ω). Let q_{2}(x) = ω(x, x) be the quadratic form on V_{2} corresponding to ω. Denote v_{i} = e_{i} ∧ e_{−i}. Also if i < j let v_{ij} = e_{i} ∧ e_{j}, w_{ij} = e_{−i} ∧ e_{−j} and u_{ij} = e_{i} ∧ e_{−j} for all i ≠ j. It is straightforward to check that the subspaces ⟨v_{i}⟩, ⟨v_{ij}, w_{ij}⟩, ⟨u_{ij}, u_{ji}⟩ of V_{2} are orthogonal to each other and that q_{2} written in the bases v_{i}, v_{ij}, u_{ij}, w_{ij} of V_{2} is of the form

q_{2} = 2 (∑ x_{i}^{2}) ⊕ 4 (∑ y_{ij}z_{ij}).

Note that dividing all coefficients of q_{2} by 2 and passing to ℤ → ℤ/2 we don’t achieve our goal since the resulting quadratic form is “highly degenerate”. So instead of considering the representation of G on V_{2} we do the following. One can easily check that any (hyperplane) reflection τ : V_{1} → V_{1} acts trivially on a 1-dimensional subspace of V_{2} spanned by v = v_{1} + ⋯ + v_{r}. It follows that Sp_{2r} acts trivially on ⟨v⟩ and hence so does G. This implies that G acts on the orthogonal complement V = ⟨v⟩^{⊥} (with respect to ω). This subspace is spanned by v_{1} − v_{2}, v_{2} − v_{3}, …, v_{r−1} − v_{r}, v_{ij}, u_{ij}, w_{ij}. In this basis of V the restriction q of q_{2} to V is of the form

q = 4 (∑ x_{i}^{2} − ∑ x_{i}x_{i+1}) ⊕ 4 (∑ y_{ij}z_{ij}).

Dividing all coefficients of q by 4 and taking the base change ℤ → ℤ/2 ⊂ k we obtain an orthogonal representation of G over k with the required property. □

7. The Witt group in characteristic 2

In this section we summarize Arason’s results [Ar1] on the structure of the Witt group of quadratic forms over complete fields of characteristic 2 used in our present work.

Let K be a field of characteristic 2, π an indeterminate over K and let K((π)) be the field of formal Laurent series over K. If f is a nondegenerate quadratic form over K((π)), we will denote its image in the Witt group W_{q}(K((π))) by f_{W}.

7.1. Theorem. W_{q}(K((π))) is the additive group generated by the elements [α, βπ^{−k}]_{W} and [απ^{−1}, βπ^{−k+1}]_{W} where k ∈ ℤ, k ≥ 0 and α, β ∈ K, with the condition that [α, βπ^{−k}]_{W} and [απ^{−1}, βπ^{−k+1}]_{W} are biadditive as functions of α, β and satisfy the following sets of relations:

(7.1.1a)  [α, βρ^{2}π^{−k}]_{W} + [β, αρ^{2}π^{−k}]_{W} = 0 if k is even
(7.1.1b) [απ^{−1}, βρ^{2}π^{−k+1}]_{W} + [βπ^{−1}, αρ^{2}π^{−k+1}]_{W} = 0 if k is even
(7.1.1c)  [α, βρ^{2}π^{−k}]_{W} + [βπ^{−1}, αρ^{2}π^{−k+1}]_{W} = 0 if k is odd
(7.1.2a)  [α, αρ^{2}π^{−2k}]_{W} + [α, ρπ^{−k}]_{W} = 0
(7.1.2b)  [απ^{−1}, αρ^{2}π^{−2k+1}]_{W} + [απ^{−1}, ρπ^{−k+1}]_{W} = 0

Here k runs through the non-negative integers and α, β and ρ run through K.

7.2. Theorem. Let m ≥ 0 and let W_{q}(K((π)))_{m} be the subgroup of W_{q}(K((π))) generated by the [α, βπ^{−k}]_{W} and [απ^{−1}, βπ^{−k+1}]_{W} where k ∈ ℤ, 0 ≤ k ≤ m and α, β ∈ K. Then:

(1) W_{q}(K((π)))_{0} is isomorphic to W_{q}(K)⊕W_{q}(K). A generator [α, β]_{W} of W_{q}(K((π)))_{0} is sent to [α, β]_{W} in the first summand W_{q}(K), but a generator [απ^{−1}, βs]_{W} corresponds to [α, β]_{W} in the second summand.

(2) If n > 0 then W_{q}(K((π)))_{2n}/W_{q}(K((π)))_{2n−1} is isomorphic to K ∧_{K^{2}} K ⊕ K ∧_{K^{2}} K. The class of a generator [α, βπ^{−2n}]_{W} corresponds to α ∧ β in the first summand, but the class of a generator [απ^{−1}, βπ^{−2n+1}]_{W} corresponds to α ∧ β in the second summand.

(3) If n ≥ 0 then W_{q}(K((π)))_{2n+1}/W_{q}(K((π)))_{2n} is isomorphic to K ⊗_{K^{2}} K. The class of a generator [α, βπ^{−2n+1}]_{W} corresponds to α⊗β, but the class of a generator [απ^{−1}, βπ^{−2n}]_{W} corresponds to β ⊗ α.

By the above theorem one has

W_{q}(K((π)))_{0} ≃ W_{q}(K) ⊕ W_{q}(K),

so that we have two natural projections

∂_{1} : W_{q}(K((π)))_{0} → W_{q}(K) and ∂_{2} : W_{q}(K((π))) → W_{q}(K)

which we will call the first and second residues (of the zero term of Arason’s filtration).

Using the fact that [f, g] ≃ ℍ for all f, g ∈ K((π)) such that fg ∈ πK[[π]], it is straightforward to show that the the zero term of the Witt group of Arason’s filtration and the first residue don’t depend on presentation L = K((π))). In other words, they don’t depend on a choice of a coefficient field K̃ ⊂ L (for the notion of coefficient fields we refer to Section 9 below) nor of a choice of a uniformizer of L and that the second residue is defined up to similarity only. We leave the details of the verification to the reader.

8. Presentation of quadratic forms inside the Witt group

In this section we will work with the Witt group of quadratic forms over a field of Laurent series K((π)) where the coefficient field K is of characteristic 2 and is finitely generated over k. By Theorems 7.1 and 7.2, given a non-degenerate quadratic form f defined over K((π)), we may decompose its image f_{W} in the Witt group as

(8.0.1)  f_{W} = f′_{m,W} + f′_{m−1,W} + ... + f′_{0,W}

where f′_{i,W} ∈ W_{q}(K((π)))_{i} is homogeneous of degree i, i.e. a sum of elements of the form [α, βπ^{−i}] and [απ^{−1}, βπ^{−i+1}] with α, β ∈ K. Such decomposition is not unique. The following lemma allows us to choose the homogeneous components of f_{W} in a canonical way.

8.1. Lemma. Let {α_{i}}_{i=1}^{N} be a basis for K as a K^{2}-vector space and let f be a non-degenerate quadratic form over K((π)). Then f_{W} admits a decomposition f_{W} = f_{m,W} + f_{m−1,W} + ... + f_{0,W} such that it satisfies the following:

if n is even then

f_{n,W} = ∑_{i<j}[α_{i}, u_{j}^{2}α_{j}π^{−n}]_{W} + ∑_{i<j}[α_{i}π^{−1}, v_{j}^{2}α_{j}π^{−n+1}]_{W} ,

where u_{i}, v_{j} ∈ K; if n is odd then

f_{n,W} = ∑_{i,j=1}^{N}[α_{i}, u_{j}^{2}α_{j}π^{−n}]_{W} ,

where u_{j} ∈ K.

Proof. Take decomposition (8.0.1). Suppose first that n = 2s is even. Write f′_{2s,W} in the form

f′_{2s,W} = ∑[p_{i}, q_{i}π^{−2s}]_{W} + ∑[p′_{i}π^{−1}, q′_{i}π^{−2s+1}]_{W}

where p_{i}, q_{i}, p′_{i}, q′_{i} ∈ K. Since {α_{i}}_{i=1}^{N} is a basis for K/K^{2} one has p_{i} = ∑_{i,j=1}^{N} e_{ij}^{2}α_{j} where e_{i,j} ∈ K and similarly for the q_{i}, p′_{i}, q′_{i}. Replacing the p_{i}, q_{i}, p′_{i}, q′_{i} with these expressions and using the biadditivity of [ , ]_{W} and the fact [uv^{2}, w] = [u, v^{2}w] for all u, v, w ∈ K((π)) we get that f′_{2s,W} can be written in the form

f′_{2s,W} = ∑_{i,j=1}^{N}[u_{i}^{2}α_{i}, v_{j}^{2}α_{j}π^{−2s}]_{W} + ∑_{i,j=1}^{N}[u_{i}^{′2}α_{i}π^{−1}, v_{j}^{′2}α_{j}π^{−2s+1}]_{W}

= ∑_{i,j=1}^{N}[α_{i}, w_{ij}^{2}α_{j}π^{−2s}]_{W} + ∑_{i,j=1}^{N}[α_{i}π^{−1}, w_{ij}^{′2}α_{j}π^{−2s+1}]_{W}

where u_{i}, v_{j}, u′_{i}, v′_{j} ∈ K and w_{ij} = u_{i}v_{j} , w′_{ij} = u′_{i}v′_{j}. If i = j we have

[α_{i}, w_{ii}^{2}α_{i}π^{−2s}]_{W} =^{7.1.2a} [α_{i}, w_{ii}π^{−s}]_{W}

and

[α_{i}π^{−1}, w_{ii}^{′2}α_{i}π^{−2s+1}]_{W} =^{7.1.2b} [α_{i}π^{−1}, w′_{ii}π^{−s+1}]_{W} .

If i > j we get

[α_{i}, w_{ij}^{2}α_{j}π^{−2s}]_{W} =^{7.1.1a} [α_{j}, w_{ij}^{2}α_{i}π^{−2s}]_{W}

and

[α_{i}π^{−1}, w_{ij}^{′2}α_{j}π^{−2s+1}]_{W} =^{7.1.1b} [α_{j}π^{−1}, w_{ij}^{′2}α_{i}π^{−2s+1}]_{W} .

If n = 2s − 1 is odd similar arguments shows that f′_{2s−1,W} can be written as a sum of symbols of the form [α_{i}, u^{2}α_{j}π^{−2s+1}]_{W} where u ∈ K. Collecting all summands in the above decompositions of all f′_{2s,W} and f′_{2s−1,W} of the same degree together we obtain the required decomposition of f_{W}. □

The following proposition provides us with the uniqueness of the above decomposition.

8.2. Proposition. Given a quadratic form f, its image in the Witt group can be decomposed uniquely as f_{W} = f_{m,W} + f_{m−1,W} + ... + f_{0,W}, where f_{m,W} , ..., f_{0,W} are as in Lemma 8.1.

Proof. We already know that a decomposition exists, so we only need to prove uniqueness. Suppose

f_{W} = f_{m,W} + f_{m−1,W} + ... + f_{0,W} = g_{n,W} + g_{n−1,W} + ... + g_{0,W}

are 2 different decompositions of f_{W}. We first claim that n = m. Suppose not. Then without loss of generality we may assume m > n. Let us compare the images of these decompositions in the quotient group W_{q}(K((s)))_{m}/W_{q}(K((s)))_{m−1}. Since n < m the image of g_{n,W} + g_{n−1,W} + ... + g_{0,W} equals 0 whereas the other decomposition has image the class of f_{m,W}. We consider separately the cases m is even and odd.

m is even: write

f_{m,W} =^{8.1} ∑_{i<j}[α_{i}, u_{j}^{2}α_{j}s^{−m}]_{W} + ∑_{i<j}[α_{i}s^{−1}, v_{j}^{2}α_{j}s^{−m+1}]_{W}

and let

ϕ: W_{q}(K((s)))_{m}/W_{q}(K((s)))_{m−1} ≃^{7.2} K ∧_{K^{2}} K ⊕ K ∧_{K^{2}} K

be the canonical isomorphism. Then

ϕ(f_{m,W}) = (∑_{i<j} u_{j}^{2}(α_{i} ∧ α_{j}), ∑_{i<j} v_{j}^{2}(α_{i} ∧ α_{j})) .

Since {α_{i} ∧ α_{j}}_{i<j} is a basis for K ∧_{K^{2}} K,

ϕ(f_{m,W}) = 0 ⇔ u_{j}^{2} = v_{j}^{2} = 0 ∀ j.

This would imply that f_{m,W} = 0, a contradiction.

m is odd: let

f_{m,W} =^{8.1} ∑_{i,j=1}^{N}[α_{i}, u_{j}^{2}α_{j}s^{−m}]_{W}

and

ϕ: W_{q}(K((s)))_{n}/W_{q}(K((s)))_{n−1} ≃^{7.2} K ⊗_{K^{2}} K .

Then we have

ϕ(f_{m,W}) = ∑_{i,j=1}^{N} u_{j}^{2}(α_{i} ⊗ α_{j}) .

Since {α_{i} ⊗ α_{j}}_{i,j=1}^{N} is a basis for K ⊗_{K^{2}} K,

ϕ(f_{m,W}) = 0 ⇔ u_{j}^{2} = 0 ∀ j,

a contradiction.

Thus m = n. If m is even, from ϕ(f_{m,W}) = ϕ(g_{m,W}) we conclude that

∑ u_{j}^{2}(α_{i} ∧ α_{j}) = ∑ u_{j}^{′2}(α_{i} ∧ α_{j}),

where u_{j}^{′2} are the corresponding coefficients of g_{m,W}, and similarly

∑ v_{j}^{2}(α_{i} ∧ α_{j}) = ∑ v_{j}^{′2}(α_{i} ∧ α_{j}).

This implies u_{j}^{2} = u_{j}^{′2} and v_{j}^{2} = v_{j}^{′2}, hence f_{m,W} = g_{m,W}. Similarly we can see that f_{m,W} = g_{m,W} if m is odd. Then from the equality

(f_{0,W} + ... + f_{m−1,W}) + f_{m,W} = (g_{0,W} + ... + g_{m−1,W}) + g_{m,W}

it follows that

f_{0,W} + ... + f_{m−1,W} = f′_{0,W} + ... + f′_{m−1,W} .

By induction, the proof is completed. □

9. Differential bases, 2-bases, Cohen Structure Theorem and coefficient fields

Let K/k be a finitely generated field extension. Recall that Ω_{K/k} denotes the K-vector space of Kähler differentials. A differential basis for K/k is a set of elements { α_{i} }_{i∈I} of K such that { dα_{i} } ⊂ Ω_{K/k} is a vector space basis. Recall also that a set of elements { x_{λ} }_{λ∈Λ} of K is a 2-basis for K over k if the set W of monomials in the x_{λ} having degree < 2 in each x_{λ} separately forms a vector space basis for K over its subfield k · K^{2} = K^{2} ⊂ K. The following facts are well-known.

9.1. Theorem. Let B = {x_{1}, …, x_{n}} ⊂ K be a subset. The following are equivalent:

(1) B is a separating transcendence basis for K over k;

(2) B is a 2-basis for K over k:

(3) B is a differential basis for K/k.

Proof. See [Ei, 16.14]. □

Assume now that K is equipped with a discrete valuation trivial on k. We denote its valuation ring by R and residue field by \overline{K}. Since our valuation is trivial on k residue field \overline{K} contains a copy of k. Let π be a uniformizer. Choose a_{1}, …, a_{n} ∈ R such that their images \overline{a}_{1}, …, \overline{a}_{n} under the canonical map R → \overline{K} form a differential basis for \overline{K}/k. Note that, by Theorem 9.1, we have tr. deg_{k}(\overline{K}) = n, hence tr. deg_{k}(K) = n + 1. Then we claim that

(9.1.1)  B = {a_{1}, …, a_{n}, π}

is a differential basis for K/k. Indeed, it is straightforward to check that all monomials a_{1}^{ϵ_{1}} … a_{n}^{ϵ_{n}}π^{ϵ_{n+1}} with ϵ_{1}, …, ϵ_{n+1} = 0, 1 are linearly independent over kK^{2} = K^{2}. This implies that B is a 2-basis for K over k. Hence the claim follows from Theorem 9.1.

Conversely, if (9.1.1) is a differential basis for Ω_{K/k} such that all a_{i} are units in R then da_{1}, …, da_{n} viewed as elements of Ω_{R/k} are linearly independent over R. Then from the conormal sequence

(π)/(π)^{2} ⟶ \overline{K} ⊗_{R} Ω_{R/k} ⟶ Ω_{\overline{K}/k} ⟶ 0,

(see [Ei, page 387]) we conclude that their images d \overline{a}_{1}, …, d \overline{a}_{n} are linearly independent over \overline{K}. By dimension count \{\overline{a}_{1}, …, \overline{a}_{n}\} is a differential basis for \overline{K}/k.

We will say that a differential basis {a_{1}, a_{2}, …, a_{n+1}} for K/k comes from \overline{K} if a_{i} is a uniformizer in K and a_{1}, …, a_{i−1}, a_{i+1}, …, a_{n+1} are units in R for some index i.

Now let R be a complete discrete valuation ring containing a field k. Denote its quotient field by L and residue field by \overline{L}. We will assume throughout that the field extension \overline{L}/k is finitely generated. It follows from the Cohen Structure Theorem [Ei, Theorem 7.7] that R ≃ \overline{L}[[π]] and L ≃ \overline{L}((π)) where π is a uniformizer. Such decompositions are not unique. They depend on a choice of π and a choice of a coefficient field in L, i.e. a subfield of L contained in R that maps isomorphically onto \overline{L} under the canonical map R → \overline{L}. Such coefficient fields do exist because the field extension \overline{L}/k is separable. The following theorem describe all coefficient fields.

9.2. Theorem. Let R be as above. If B is a differential basis for \overline{L} over k then there is one-to-one correspondence between coefficient fields Ẽ ⊂ R containing k and the set B̃ ⊂ R of representatives for B obtained by associating to each Ẽ the set B̃ of representatives for B that it contains.

Proof. See [Ei, Theorem 7.8]. □

10. Monomial quadratic forms

Let K = k(t_{1}, t_{2}, ..., t_{n}, x) be a pure transcendental extension of k of transcendence degree n + 1. We say that a non-degenerate quadratic form f over K is monomial if it is of the form

f = ⊕_{μ∈𝔽_{2}^{n}} m_{f}(μ) t^{μ} [1, x] ⊕ ℍ ⊕ ... ⊕ ℍ

where μ = (μ_{1}, ..., μ_{n}) ∈ 𝔽_{2}^{n}, t^{μ} = t_{1}^{μ_{1}}t_{2}^{μ_{2}} … t_{n}^{μ_{n}} are monomials in t_{1}, …, t_{n} and m_{f}(μ) the number of times a given summand appears. Note that the multiplicity m_{f}(μ) may be 0. Since t^{μ}[1, x] ⊕ t^{μ}[1, x] ≃ ℍ ⊕ ℍ we may assume without loss of generality that m_{f}(μ) = 0 or 1.

Let V be the vector subspace of 𝔽_{2}^{n} generated by all μ such that m_{f}(μ) = 1. Choose a basis of V, say μ_{1}, μ_{2}, ...μ_{s}. Then define u_{i} = t^{μ_{i}} for i = 1, ..., s. It is easy to see that u_{1}, …, u_{s} are algebraically independent over k. Furthermore, any μ ∈ V can be written as μ = ∑_{i=1}^{s} α_{i}u_{i} where α_{i} = 0 or 1 so that t^{μ} = u_{1}^{α_{1}}...u_{s}^{α_{s}}.

Thus f has descent to the subfield K′ = k(u_{1}, ..., u_{s}, x) ⊂ K and viewed over K′ it is of the form

f = u_{1}[1, x] ⊕ u_{2}[1, x] ⊕ ⋯ ⊕ u_{s}[1, x] ⊕ (⊕_{μ∈V} u^{μ}[1, x]) ⊕ ℍ ⊕ ⋯ ⊕ ℍ

where u^{μ} are monomials in u_{1}, …, u_{s} of length at least 2. When a monomial quadratic form f is written in such a way and is viewed over K′ we say that it is a canonical monomial form. We also say that f has rank s.

For later use we need the following easy observation.

10.1. Proposition. Let f be a canonical monomial form without summands isometric to the hyperbolic plane ℍ. Then f is anisotropic.

Proof. The argument is similar to that in [ChSe, Proposition 5] and we leave the details to the reader. □

The main result related to canonical monomial quadratic forms is the following theorem.

10.2. Theorem. Let f be a canonical monomial form over K. Then f is incompressible.

11. Incompressibility of monomial forms in codimension 2

In this section we establish an auxiliary result, Theorem 11.3 below, needed later on to prove Theorem 10.2. Let K = k(x, t_{1}, …, t_{n}) be a pure transcendental extension of k of degree n + 1 and v the valuation on K associated to t_{1}. It is characterized by:

v(t_{1}) = 1 and v(h) = 0 ∀ h ∈ k(x, t_{2}, ..., t_{n})^{×} .

Let R ⊂ K be the corresponding valuation ring. Note that K^{2} ⊂ K is a finite field extension of degree 2^{n+1}. As usual, K^{2}(a_{i_{1}}, …, a_{i_{l}}) ⊂ K denotes the subfield generated by K^{2} and elements a_{i_{1}}, …, a_{i_{l}} ∈ K.

11.1. Proposition. Let F ⊂ K be a subfield containing k such that tr. deg_{k}(F) < n+1. Then there exists a differential basis {a_{1}, …, a_{n+1}} for K/k coming from \overline{K} such that F ⊂ K^{2}(a_{1}, , …, a_{l}) with l ≤ tr. deg_{k}(F) < n + 1.

Proof. Choose any 2-basis {b_{1}, …, b_{s}} for F/k. Since s = tr. deg_{k}(F) we get s < n + 1. Let L = K^{2}(b_{1}, …, b_{s}). Clearly, L contains F and for any α_{1}, …, α_{s} ∈ K one has

L = K^{2}(b_{1}, …, b_{s}) = K^{2}(α_{1}^{2}b_{1}, …, α_{s}^{2}b_{s}).

The restriction w = v|_{F} is either nontrivial or trivial.

Case 1: w is nontrivial and the ramification index e(v/w) is even. Multiplying b_{i} by an appropriate scalar α_{i}^{2} we may assume that c_{1} = α_{1}^{2}b_{1}, …, c_{s} = α_{s}^{2}b_{s} are units in R and F ⊂ L = K^{2}(c_{1}, …, c_{s}). Without loss of generality we may also assume that {c_{1}, …, c_{l}} where l ≤ s is a minimal set of generators of L over K^{2} so that L = K^{2}(c_{1}, …, c_{l}). The set of all monomials c_{1}^{ϵ_{1}} ⋯ c_{l}^{ϵ_{l}} with ϵ_{i} = 0, 1 is linearly independent over K^{2}. Put a_{1} = c_{1}, …, a_{l} = c_{l} and choose units a_{l+1}, …, a_{n} in R such that B = {a_{1}, …, a_{n}, a_{n+1}} where a_{n+1} = t_{1} is a 2-basis for K over k. By Theorem 9.1, B is a differential basis for K over k coming from \overline{K} and by construction it has the required property. Case 2: w is trivial. Take α_{i} = 1 and c_{i} = b_{i} for all i and apply the same argument as above . Case 3: w is nontrivial and the ramification index e(v/w) is odd. Without loss of generality we may assume that b_{1}, …, b_{s−1} are units and b_{s} is a uniformizer for w. Choose scalars α_{1}, …, α_{s} ∈ K such that c_{1} = α_{1}^{2}b_{1}, …, c_{s−1} = α_{s−1}^{2}b_{s−1} are units in R, c_{s} = α_{s}^{2}b_{s} is a uniformizer for v and F ⊂ L = K^{2}(c_{1}, …, c_{s}). Then the same argument as above completes the proof. □

Let f be a canonical monomial quadratic form over K given by

(11.1.1)  f = ⊕_{μ∈𝔽_{2}^{n}} m_{f}(μ) t^{μ} [1, x] ⊕ ℍ ⊕ ... ⊕ ℍ ,

where all multiplicities m_{f}(μ) are 1 or 0. Since f is canonical it contains summands t_{i}[1, x], i = 1, …, n.

Below we will be considering two Witt groups: W_{q}(K) and W_{q}(K̂). Here K̂ ≃ k(x, t_{2}, …, t_{n})((t_{1})). There exists the natural map W_{q}(K) → W_{q}(K̂) and if there is no risk of confusion we will denote the image of f in both groups by f_{W}.

11.2. Lemma. (fK̂)_{W} lives in W_{q}(K̂)_{0}. Its first residue is a canonical monomial form of rank n − 1 and its second residue up to similarity is a nontrivial monomial form of rank ≤ n − 1.

Proof. This follows from the definitions of monomial forms and the first and second residues. □

11.3. Theorem. There exist no differential basis B = {a_{1}, …, a_{n+1}} for K/k coming from \overline{K} and a non-degenerate quadratic form g defined over L = K^{2}(a_{1}, …, a_{n−1}) such that g_{K,W} = f_{W}.

Proof. Assume the contrary. Let B and g be the corresponding differential basis and quadratic form. The differential basis B gives rise to the coefficient field E ⊂ K̂ containing all units from B and presentation K̂ ≃ E((t_{1})).

We argue by induction on n. If n = 1 then L = K^{2}. Hence g can be written as a direct sum of 2-dimensional quadratic forms [u_{i}, v_{i}] = [1, w_{i}^{2}] where u_{i}, v_{i} ∈ K^{2} and u_{i}v_{i} = w_{i}^{2}. We now pass to K̂ = E((t_{1})) and view g over K̂. Writing w_{i} in the form w_{i} = t_{1}^{−s_{i}}(∑_{j≥0} e_{ij}t_{1}^{j}) with e_{ij} ∈ E and using the property [u(t_{1}), v(t_{1})]_{W} = 0 if v(u(t_{1})v(t_{1}) > 0 we conclude that [1, w_{i}^{2}]_{K̂,W} can be written as a sum of symbols of the form [1, e_{kj}^{2}t_{1}^{−2k}]_{W} with e_{kj} ∈ E. Thus, g_{K̂,W} can be written as g_{K̂,W} = g_{n,W} + ⋯ + g_{0,W} where g_{i,W} is of the form

g_{i,W} = ∑_{j} [1, e_{ij}^{2}t_{1}^{−2i}]_{W} = [1, (∑_{j} e_{ij})^{2}t_{1}^{−2i}]_{W}

with e_{ij} ∈ E. Since g_{K̂,W} = f_{W}, it lives in the zero term of Arason’s filtration. Then, by Proposition 8.2, we conclude g_{n} = … = g_{1} = 0. Therefore f_{W} = g_{K̂,W} = [1, α^{2}]_{W} for some α ∈ E. But this implies that the second residue of f_{W} is 0 which contradicts the second assertion in Lemma 11.2.

Now let n be arbitrary and suppose that the statement is true for all canonical monomial forms of rank < n. Assume first that a_{1}, …, a_{n−1} are units in R. Let g = ⊕_{i} [u_{i}, v_{i}] with u_{i}, v_{i} ∈ L. Writing

u_{i} = ∑_{ϵ} u_{ϵ}^{2}a_{1}^{ϵ_{1}} … a_{n−1}^{ϵ_{n−1}}, v_{i} = ∑_{ϵ} v_{ϵ}^{2}a_{1}^{ϵ_{1}} … a_{n−1}^{ϵ_{n−1}}

where we use multi-index notation ϵ = (ϵ_{1}, …, ϵ_{n−1}) and u_{ϵ}, v_{ϵ} ∈ K and arguing as above we conclude that g_{K̂,W} can be written as a sum g_{K̂,W} = g_{n,W} + ⋯ + g_{0,W} where the homogeneous component g_{i,W} is of the form

g_{i,W} = ∑_{j,ϵ,ϵ′}[a_{1}^{ϵ_{1}} … a_{n−1}^{ϵ_{n−1}}, α_{ij}^{2}a_{1}^{ϵ′_{1}} … a_{n−1}^{ϵ′_{n−1}}t_{1}^{−i}]_{W}

with α_{ij} ∈ E. Since g_{K̂,W} lives in the zero term of Arason’s filtration, application of Proposition 8.2 yields g_{n} = … = g_{1} = 0. Then as above we conclude that the second residue of g_{K̂,W} is zero, a contradiction.

Finally, assume that up to numbering a_{n−1} is a uniformizer of v. The same argument as above shows that g_{K̂,W} = g_{0,W} is homogeneous of degree 0 where the component g_{0,W} is a sum of symbols of the form

[a_{1}^{ϵ_{1}} … a_{n−2}^{ϵ_{n−2}}t_{1}, α_{i}^{2}a_{1}^{ϵ′_{1}} … a_{n−2}^{ϵ′_{n−2}}t_{1}^{−1}]_{W} and [a_{1}^{ϵ_{1}} … a_{n−2}^{ϵ_{n−2}}, β_{i}^{2}a_{1}^{ϵ′_{1}} … a_{n−2}^{ϵ′_{n−2}}]_{W}

with α_{i}, β_{i} ∈ E. Then the first residue of g_{K̂,W} (and hence of f_{W}) is a sum of symbols

[a_{1}^{ϵ_{1}} … a_{n−2}^{ϵ_{n−2}}, β_{i}^{2}a_{1}^{ϵ′_{1}} … a_{n−2}^{ϵ′_{n−2}}]_{W}

where β_{i} ∈ E ≃ \overline{K}. But \overline{B} = \{a_{1}, …, a_{n−2}, a_{n}, a_{n+1}\} is a differential basis for E ≃ \overline{K} over k and the first residue of f_{K̂,W} is a canonical monomial form of rank n − 1. This contradicts the induction assumption. □

11.4. Corollary. There exists no subfield k ⊂ F ⊂ K of transcendence degree ≤ n − 1 and a quadratic form g over F such that g_{K} ≃ f.

Proof. This follows from Proposition 11.1 and Theorem 11.3. □

12. Incompressibility of canonical monomial quadratic forms

Proof of Theorem 10.2. We keep the above notation. In particular, K = k(x, t_{1}, t_{2}, ..., t_{n}) is a pure transcendental extension of k of transcendence degree n + 1 equipped with a discrete valuation v associated to t_{1} and R the corresponding discrete valuation ring. As a matter of notation we denote π = t_{1} and K_{1} = k(t_{2}, …, t_{n}, x). Thus K̂ ≃ K_{1}((π)) and \overline{K} ≃ K_{1}.

Consider a canonical monomial quadratic form f over K given by (11.1.1). The proof of incompressibility of f will be carried out by induction on rank n. More precisely, we will prove by induction on n that the image f_{W} of f in W_{q}(K) is incompressible. Of course, this would imply incompressibility of f itself.

The base of induction n = 0 is obvious.

12.1. Lemma. Let K = k(x) and let f = [1, x] ⊕ ℍ ⊕ ⋯ ⊕ ℍ. Then f_{W} is incompressible.

Proof. Any subfield of K of transcendence degree 0 over k coincides with k. Hence, if f_{W} were compressible then it would be represented by a non-degenerate quadratic form defined over k, which is automatically hyperbolic. On the other hand, by Proposition 10.1, f_{W} is represented by an anisotropic form [1, x], a contradiction. □

Now let n > 0 and suppose that for all canonical monomial quadratic forms of rank < n their Witt-equivalence classes are incompressible. Suppose that f_{W} is compressible. Then there exists a subfield F ⊂ K containing k which may be assumed to have transcendence degree n over k, and a non-degenerate quadratic form g over F such that (g_{K})_{W} = f_{W}.

For the restriction w = v|_{F} of v to F there are three possibilities. Case 1: w is trivial. Write g as a direct sum of 2-dimensional forms [b_{i}, c_{i}] with b_{i}, c_{i} ∈ F ⊂ R. Consider Arason’s filtration of W_{q}(K̂) with respect to the presentation K̂ = K_{1}((π)). Since b_{i}, c_{i} are units, g_{W} lives in the zero term of Arason’s filtration and moreover its second residue is trivial. On the other hand, since g_{K̂,W} = f_{K̂,W} it has nontrivial second residue by Lemma 11.2, a contradiction. Case 2: w is nontrivial and the ramification index e(v/w) is even. Arguing as in Proposition 11.1 we can choose a differential basis B = \{a_{1}, …, a_{n}, π\} for K/k coming from \overline{K} such that

(12.1.1)  F ⊂ K^{2}(a_{1}, …, a_{n}).

By Theorem 9.2, B gives rise to the coefficient field E ⊂ K̂ containing a_{1}, …, a_{n} and presentation K̂ ≃ E((π)). Clearly, {a_{1}, …, a_{n}} is a 2basis for E/k. We fix the presentation K̂ ≃ E((π)) and we will apply Arason’s results [Ar1] for this presentation only.

By construction f_{K̂,W} = g_{K̂,W}, hence the first residue of g_{K̂,W} is a canonical monomial form of rank n − 1 and the second residue of gK̂ is nontrivial. We now pass to computing the residues of gK̂ using our presentation K̂ = E((π)) and inclusion (12.1.1).

Since g is non-degenerate it can be written as a direct sum of 2dimensional forms [b_{i}, c_{i}] with b_{i}, c_{i} ∈ F. In turn, in view of (12.1.1) b_{i}, c_{i} can be written as sums of elements of the form α_{i_{1}...i_{s}}^{2}a_{i_{1}}a_{i_{2}} ⋯ a_{i_{s}} with α_{i_{1}...i_{s}} ∈ K. Then arguing as in Theorem 11.3 we conclude that the image of gK̂ in W_{q}(K̂) can be written as a sum of symbols

[a_{i_{1}}a_{i_{2}} ⋯ a_{i_{s}}, \frac{α_{j_{1}...j_{p}}^{2}}{π^{2l}} a_{j_{1}}a_{j_{2}} ⋯ a_{j_{p}}]_{W}

where α_{j_{1}...j_{p}} ∈ E. Thus, we can write g_{K̂,W} as the sum

g_{W} = g_{2n} + g_{2(n−1)} + … + g_{0}

where all homogeneous components g_{2i} have even degree and are sums of symbols

[a_{i_{1}}a_{i_{2}} ⋯ a_{i_{s}}, \frac{α_{j_{1}...j_{p}}^{2}}{π^{2i}} a_{j_{1}}a_{j_{2}} ⋯ a_{j_{p}}]_{W}

with α_{j_{1}...j_{p}} ∈ E. Obviously g_{0} has trivial second residue (because a_{1}, …, a_{n} are units). So to get a contradiction it remains to show that g_{2n} + ⋯ + g_{2} = 0 in W_{q}(K̂).

Let us start from the highest component g_{2n}. Recall that according to Arason’s Theorem we have

W_{q}(E((π)))_{2n}/W_{q}(E((π)))_{2n−1} ≃ E ∧_{E^{2}} E ⊕ E ∧_{E^{2}} E.

The class of a generator [α, βπ^{−2n}]_{W} corresponds to α ∧ β in the first summand, but the class of a generator [απ^{−1}, βπ^{−2n+1}]_{W} corresponds to α ∧ β in the second summand. To simplify writing we introduce multi-indices a_{I_{j}} where I_{j} is a set of some indices i_{1}, …, i_{s} and a_{I_{j}} is the product of the corresponding a_{i_{p}}.

Let a_{I_{1}}, …, a_{I_{2^{n}}} be a E^{2}-basis of E. By construction, g_{2n} is a sum of elements of the form

[a_{I_{j}}, \frac{α_{I_{s}}^{2}}{π^{2n}} a_{I_{s}}]_{W}

with α_{I_{s}} ∈ E. Recall that if I_{j} = I_{s} then

[a_{I_{j}}, \frac{α_{I_{j}}^{2}}{π^{2n}} a_{I_{j}}]_{W} = [a_{I_{j}}, \frac{α_{I_{j}}}{π^{n}}]_{W}

by (7.1.2a)). It follows that g_{2n} can be written as g_{2n} = g′_{2n} + gn^{′′} where

g′_{2n} = ∑_{s<j} [a_{I_{j}}, a_{I_{s}} \frac{α_{I_{s}}^{2}}{π^{2n}}]_{W}

with α_{s} ∈ E and gn^{′′} lives in W_{q}(E((π)))_{n}. But g lives in W_{q}(E((π)))_{0}, hence the image of g′_{2n} in W_{q}(K̂)_{2n}/W_{q}(K̂)_{2n−1} is trivial and this of course implies g′_{2n} = 0, by Theorem 7.2.

Note that arguing in such a way we have eliminated the highest homogeneous component g_{2n} of g living in W_{q}(E((π)))_{2n}, but we possibly acquire the component gn^{′′} with n > 0 in even or odd degree W_{q}(E((π)))_{n} (if n is odd).

We can continue to do the same with the next highest homogeneous component of g. If it has even degree the same argument as above reduces it to a smaller component. If it has odd degree 2m + 1 then it can be written in the form

∑_{j,s} [a_{I_{j}}, \frac{a_{I_{s}}}{π^{2m+1}} α_{I_{s}}^{2}]_{W}

with α_{I_{s}} ∈ E. Then applying Arason’s Theorem 7.2, part (3), and arguing similarly we conclude that this component is automatically 0. This completes the proof of the fact that g_{2n} + ⋯ + g_{2} = 0 and hence the proof of incompressibility of f_{W} in the case e(v/w) is even. Case 3: e = e(v/w) is odd. Let π′ ∈ F be a uniformizer for w. Write π′ = uπ^{e} where u ∈ R^{×}. Our argument below doesn’t depend on a choice of a uniformizer π for v. So replacing π with uπ if necessary we may assume without loss of generality that u = v^{2} for some v ∈ R^{×}.

Choose a differential basis B′ = {a_{1}, …, a_{n−1}, π′} for F/k coming from residue field \overline{F}. Clearly, F ⊂ K^{2}(a_{1}, …, a_{n−1}, π′). We claim that all monomials a_{1}^{ϵ_{1}} … a_{n−1}^{ϵ_{n−1}} with ϵ_{i} = 0, 1 are linearly independent modulo K^{2}. Indeed, assume the contrary. Then up to numbering we may assume that a_{1}, …, a_{l} with l < n − 1 is a minimal system of generators of K^{2}(a_{1}, …, a_{n−1}) over K^{2}. Then there exists units a′_{l+1}, …, a′_{n} in R such that {a_{1}, …, a_{l}, a′_{l+1}, …, a′_{n}, π′} is a 2-basis of K over k. Since

F ⊂ K^{2}(a_{1}, …, a_{n−1}, π′) = K^{2}(a_{1}, …, a_{l}, π′)

and l < n − 1 this contradicts Theorem 11.3.

Now choose a_{n} ∈ R^{×} such that B = {a_{1}, …, a_{n−1}, a_{n}, π} is a 2basis for K over k and hence a differential basis for K/k coming from \overline{K}. It follows that two completions F̂ ⊂ K̂ with respect to w and v respectively admit compatible coefficient fields, i.e. if E′ ⊂ F̂ (resp. E ⊂ K̂) is a coefficient field corresponding to B′ (resp. B) then E′ = E ∩ F̂ and so we may choose compatible presentations

F̂ = E′((π′)) ⊂ K̂ = E((π))

where E′ ⊂ E.

The set of all monomials a_{1}^{ϵ_{1}} … a_{n−1}^{ϵ_{n−1}} where ϵ_{i} = 0, 1 form a basis of E′ over (E′)^{2}. Like before, let us number them in any order using multi-indices a_{I_{1}}, …, a_{I_{2^{n−1}}}. Lemma 8.1 and Proposition 8.2 show that g_{W} viewed over F̂ can be written uniquely in the form g_{W} = g_{n}+⋯+g_{0} where g_{l} is of the form:

if l is even then

g_{l,W} = ∑_{i<j} [a_{I_{i}}, a_{I_{j}} \frac{u_{j}^{2}}{(π′)^{l}}]_{W} + ∑_{i<j} [a_{I_{i}} \frac{1}{π′} , a_{I_{j}} \frac{v_{j}^{2}}{(π′)^{l−1}} ]_{W} ,

where u_{i}, v_{j} ∈ E′; if l is odd then

g_{l,W} = ∑_{i,j=1}^{2^{n−1}} [a_{I_{i}}, a_{I_{j}} \frac{u_{j}^{2}}{(π′)^{l}}]_{W}

where u_{j} ∈ E′.

We claim that n = 0. Indeed, if n ≠ 0 then substituting π′ = v^{2}π^{e} in the above expressions for g_{i} and writing v^{−2n} in the form

v^{−2n} = w_{0}^{2} + w_{1}^{2}π^{2} + w_{2}^{2}π^{4} + ⋯

where w_{i} ∈ E, we easily obtain that the highest term in the decomposition of g_{K̂,W} as a sum of its homogeneous components (with respect to presentation K̂ = E((π))) has degree ne which is impossible because g_{K̂,W} = f_{K̂,W}.

Thus n = 0 and hence g_{W} is of the form

g_{W} = ∑_{i<j} [a_{I_{i}}, u_{j}^{2}a_{I_{j}}]_{W} + ∑_{i<j} [a_{I_{i}}(π′)^{−1}, v_{j}^{2}a_{I_{j}}π′]_{W} ,

where u_{i}, v_{j} ∈ E′. It follows that the first residue of g_{W} (and hence of f_{W}) lives in a subfield E′ ⊂ \overline{K} = E of transcendence degree n−1 over k which contradicts the induction assumption. This completes the proof of incompressibility of f.

13. Orthogonal and special orthogonal groups

Let g be a non-degenerate n-dimensional quadratic form on a vector space V over k and F be any extension of k. Orthogonal groups. It is well known (see [KMRT, §29.E]) that if n = 2r is even then there exists a natural bijection between H^{1}(F, O(V, g)) and the set of isometry classes of n-dimensional non-degenerate quadratic spaces (V′, g′). Similarly, if n = 2r+1 is odd then H^{1}(F, O(V, g)) is in one-to-one correspondence with the set of isometry classes of (2r + 1)-dimensional non-degenerate quadratic spaces (V′, q′) over F such that disc(q′) = 1. Note that any such q′ is isometric to a quadratic form of the shape ([a_{1}, b_{1}] ⊕ ⋯ ⊕ [a_{r}, b_{r}]) ⊕ ⟨1⟩. Then in both cases the incompressibility of canonical monomial quadratic forms provides us with the required lower bound ed(O(V, g)) ≥ r + 1. What is left to finish the proof of Theorem 2.2 for orthogonal groups is to find a “good” upper bound.

13.1. Proposition. In the above notation one has ed(O(V, g)) ≤ r +1.

Proof. It suffices to show that any 2r-dimensional non-degenerate quadratic form depends on at most 2r parameters. Let h be such form over F. Write h = a_{1}[1, b_{1}]⊕⋯⊕a_{r}[1, b_{r}]. Each summand [1, b_{i}] corresponds to a unique element ξ_{i} ∈ H^{1}(F, ℤ/2). Let H = ℤ/2 ⊕ ⋯ ⊕ ℤ/2 be the direct sum of r copies of the constant group scheme ℤ/2 and let ξ = (ξ_{1}, …, ξ_{r}). Choose any embedding H ↪ G_{a,k}, which exists because k is infinite. The exact sequence

0 ⟶ H ⟶ G_{a,k} ⟶^{ϕ} G_{a,k} ⟶ 0

gives rise to

F ⟶^{ϕ} F ⟶^{ψ} H^{1}(F, H) ⟶ 1.

Let a ∈ F be such that ψ(a) = ξ. It follows that ξ has descent to the subfield k(a) of F. This amounts to the fact that there exist b′_{1}, …, b′_{r} ∈ k(a) such that the quadratic form [1, b′_{i}] viewed over F is isometric to [1, b_{i}]. Therefore h is isometric to the quadratic form h′ = a_{1}[1, b′_{1}] ⊕ ⋯ ⊕ a_{r}[1, b′_{r}] defined over the subfield k(a, a_{1}, …, a_{r}) of F of transcendence degree (over k) at most r + 1. □

13.2. Remark. Note that if h has trivial Arf invariant then taking a suitable quadratic subextension of k(a, a_{1}, …, a_{r}) in F, if necessary, we may also assume that h′ has trivial Arf invariant. Thus, our argument for finding a lower bound of the essential dimension of can be applied to special orthogonal groups as well.

Special orthogonal groups. By [KMRT, §29.E]), if n = 2r is even then there exists a natural bijection between H^{1}(F, SO(V, g)) and the set of isometry classes of (2r)-dimensional non-degenerate quadratic spaces (V′, g′) over F such that the Arf invariant of g′ is trivial. Taking into consideration Remark 13.2 it follows that the proof of Theorem 2.2 for special orthogonal groups in even dimensional case is similar to that of for orthogonal groups.

Let n = 2r +1 be odd. Then there exists a natural bijection between H^{1}(F, SO(V, g)) and the set of isometry classes of (2r +1)-dimensional non-degenerate quadratic spaces (V′, g′) over F such that disc(g′) = 1. As we mentioned above any such g′ is isometric to a quadratic form of the shape ([a_{1}, b_{1}] ⊕ ⋯ ⊕ [a_{r}, b_{r}]) ⊕ ⟨1⟩ for some a_{i}, b_{i} ∈ F. It follows that ed(SO(V, g)) ≤ r + 1.

To find a “good” lower bound we recall that SO_{2r+1}(g) = O_{2r+1}(g)_{red}, the reduced subscheme of O_{2r+1}(g). Thus we have a natural closed embedding SO_{2r+1}(g) ↪ O_{2r+1}(g). Fix a decomposition g ≃ h ⊕ ⟨1⟩ where h = ℍ ⊕ ⋯ ⊕ ℍ. It induces a natural closed embedding ϕ_{1} : O_{2r}(h) ↪ SO_{2r+1}(g) (because O_{2r}(h) is smooth). Furthermore, we can view ⟨1⟩ as a subform of [1, 0] ≃ ℍ. This allows us to view g as a subform of a (2r+2)-dimensional split quadratic form q = ℍ⊕⋯⊕ℍ and this induces a natural map

ϕ_{2} : SO_{2r+1}(g) ↪ O_{2r+1}(g) ↪ O_{2r+2}(q).

The maps ϕ_{1} and ϕ_{2}, in turn, induce the natural maps

ψ_{1} : H^{1}(F, O_{2r}(h)) → H^{1}(F, SO_{2r+1}(g))

and

ψ_{2} : H^{1}(F, SO_{2r+1}(g)) → H^{1}(F, O_{2r+2}(q)).

It easily follows from the above discussions that ψ_{1} is surjective. Also, identifying elements in H^{1}(F, O_{2r}(h)) and H^{1}(F, O_{2r+2}(q)) with the isometry classes of the corresponding quadratic spaces we obtain that the isometry class of a quadratic form ⊕_{i=1}^{r}[a_{i}, b_{i}] goes to the isometry class of ⊕_{i=1}^{r}[a_{i}, b_{i}] ⊕ ℍ under the composition ψ_{2} ∘ ψ_{1}.

13.3. Theorem. Let g be a non-degenerate quadratic form of dimension 2r + 1 over k. Then ed (SO_{2r+1})(g) ≥ r + 1.

Proof. Take a pure transcendental extension K = k(x, t_{1}, …, t_{r}) of k of degree r + 1 and a canonical monomial form f = t_{1}[1, x] ⊕ ⋯ ⊕ t_{r}[1, x] of dimension 2r. We claim that its image ξ under ψ_{1} is incompressible. Indeed, if ξ is compressible so is ψ_{2}(ξ). However, ψ_{2}(ξ) is represented by a canonical monomial form t_{1}[1, x]⊕⋯⊕t_{r}[1, x]⊕ℍ which is incompressible by Theorem 10.2, a contradiction. Thus ξ is incompressible itself implying ed (SO_{2r+1}(g)) ≥ r + 1. □

14. Proof of Theorem 3.1

Types A_{r}, B_{r}, C_{r}, D_{r}, E_{6}, E_{7}, E_{8}. Let ρ : G^{∘} → O (V, q) be as in Proposition 6.1. As in [ChSe], we can extend it to ρ_{G} : G → O (V, q) . Let θ_{O} = ρ_{G}(θ_{G}) be the image of θ_{G} in H^{1}(K, O (V, q)). Consider the quadratic form q_{O} on V corresponding to θ_{O}. If dim(q) is even then arguing as in [ChSe] we conclude that q_{O} is a canonical monomial form of rank r. By Theorem 10.2, q_{O} is incompressible and hence so is θ_{G}.

If dim(q) is odd then we can write it as q = ⟨1⟩ ⊕ q′ where q′ is a non-degenerate quadratic form of even dimension. The twist q_{O} of q by θ_{O} is of the form q_{O} = ⟨1⟩ ⊕ g where g is a canonical monomial form of rank r. Then the proof of Theorem 13.3 shows that q_{O} is incompressible as well. Type G_{2}. Let F be a field of an arbitrary characteristic. By [Se, Théorème 11], there is a canonical one-to-one correspondence between H^{1}(F, G_{2}) and the set of isometry classes of 3-fold Pfister forms defined over F where G_{2} denotes a split group of type G_{2} over F. Clearly, any 3fold Pfister form depends on at most 3 parameters implying ed(G_{2}) ≤ 3. Conversely, a generic 3-fold Pfister form is a canonical monomial form of rank 2, hence incompressible. It follows ed(G_{2}) ≥ 3. Type F_{4}. Let F be a field of an arbitrary characteristic. It is known that there is a canonical one-two-one correspondence between H^{1}(F, F_{4}) and the set of isomorphism classes of 27-dimensional exceptional Jordan algebras over F where F_{4} denotes a split group of type F_{4} over F. To each such reduced Jordan algebra J one associates a unique (up to isometry) 5-fold Pfister form f_{5}(J) [Pe, 4.1]. Moreover, it is known that any 5-fold Pfister form over F corresponds to some Jordan algebra J over F. Since a generic 5-Pfister form is incompressible we conclude that ed(F_{4}) ≥ 5.

References

[Ar1] J. Arason, Generators and relations for W_{q}(K((S))) in characteristic 2, Preprint, RH-19-2006 Science Institute University of Iceland Dunhaga 3, IS 107 Reykjavik.

[Ar2] J. Arason, Generators and relations for W_{q}(K) in characteristic 2, Preprint, RH-18-2006 Science Institute University of Iceland Dunhaga 3, IS 107 Reykjavik.

[BF] G. Berhuy, G. Favi, Essential dimension: a functorial point of view (after A. Merkurjev), Documenta Mathematica 8 (2003), 279-330.

[BuRe] J. Buhler, Z. Reichstein, On the essential dimension of a finite group, Compositio Math. 106 (1997), no. 2 159-179.

[ChSe] V. Chernousov, J.-P. Serre, Lower bounds for essential dimensions via orthogonal representations, Journal of Algebra, 305 (2006), no. 2, 1055-1070.

[DG] M. Demazure and A. Grothendieck, Structure des Schémas en Groupes Réductifs, SGA 3 III, LN 153, Springer-Verlag, 1970.

[Ei] D. Eisenbud, Commutative algebra with a view toward algebraic geometry, Springer, 1994.

[EKM] R. Elman, N. Karpenko, A. Merkurjev, The algebraic and geometric theory of quadratic forms, American Mathematical Society, 2008.

[GN] B. H. Gross and G. Nebe, Globally maximal arithmetic groups, J. Algebra 272 (2004), 625-642.

[KMRT] M.-A. Knus, A. Merkurjev, M. Rost, J.-P. Tignol, The Book of Involutions, American Mathematical Society, USA, 1998.

[Ma] A.L. Malagon, Killing forms of Lie algebras, PhD Dissertation at Emory University, 2009.

[Me] A. Merkurjev, Essential dimension: a survey, Transformation groups, 18 (2013), no. 2, 415-481.

[Pe] H. P. Petersson, Structure theorems for Jordan algebras of degree 3 over fields of arbitrary characteristic, Comm. in Algebra 32 (2004), no. 3, 1019-1049.

[Re] Z. Reichstein, Essential dimension, Proceedings of the International Congress of Mathematicians, Hyderabad, India, 2010.

[Sel] G. B. Seligman, Some remarks on classical Lie algebras, J.Math. Mech. 6 (1957), 549-558.

[Se] J.-P. Serre, Cohomologie galoisienne: progrès et problèmes, Séminaire Bourbaki 1993/94, no. 783, Astérisque 227 (1995), 229-257.

[SpSt] T.A. Springer, R. Steinberg, Conjugacy Classes, Lecture Notes in Mathematics 131, 167-266, Springer-Verlag, New York, 1970.

[St] R. Steinberg, Lectures on Chevalley groups, Yale University lecture notes, 1967.

Department of Mathematics, University of Alberta, Edmonton, Alberta T6G 2G1, Canada

E-mail address: ababic@ualberta.ca

Department of Mathematics, University of Alberta, Edmonton, Alberta T6G 2G1, Canada

E-mail address: chernous@math.ualberta.ca