Excellent algebraic groups I

Ina Kersten and Ulf Rehmann

Fakultät für Mathematik Universität Bielefeld Postfach 100131 D 33501 Bielefeld Germany

(Published in J. Algebra 200 (1998), no. 1, 334–346)

Abstract: The notion of an excellent quadratic form has been introduced by M. Knebusch [10, 11]. It has an obvious analogue for a semisimple linear algebraic group G over an arbitrary field k: the group G is excellent if, for any field extension K of k, the anisotropic kernel of the K-group G ×_{k} K can be defined by a k-group. The aim of this paper is to investigate excellence properties of special linear and orthogonal groups.

Introduction Not much is known about anisotropic semisimple linear algebraic groups over arbitrary fields. In order to reduce this gap of knowledge we study an important class of semisimple groups which we call excellent algebraic groups.

If k is any field, a semisimple k-group G is excellent, if, for any field extension K of k, there exists an algebraic group H over k such that H ×_{k} K is the anisotropic kernel of the K-group G ×_{k} K.

A corresponding notion for quadratic forms was introduced by M. Knebusch [11, 7.14, p. 6]. A regular quadratic form q over k is excellent if, for any field extension K of k, there exists a form η over k such that η⊗_{k}K is the anisotropic kernel of the quadratic form q_{K} := q⊗_{k}K.

The notion of excellence also has an obvious analogue for an Azumaya algebra A over k (cf. Def. 1.1 below). It is proved here (Th. 1.10 and Th. 1.6) that the following three statements are equivalent: i) A is excellent. ii) The special linear group G = SL_{1}(A) is excellent. iii) The index of A is squarefree. We like to emphasize that the squarefreeness of the index is a crucial condition for several algebraic properties of Azumaya algebras related to rationality questions discussed in the past and recently, cf. [23, 15, 19, 20, 13, 4].

Using the index reduction formula of Schofield and van den Bergh [17, Th. 1.3], we obtain in Cor. 1.7 that an Azumaya algebra is excellent if and only if its index and its exponent coincide over every field extension of k.

In section 2, the notion of excellence for the special orthogonal group of a regular quadratic form over a field k of characteristic ≠ 2 is discussed. We prove in Th. 2.2, that the projective quadrics of two regular forms over k are isomorphic if and only if their corresponding special orthogonal groups are isomorphic as k-groups. This and the generic splitting theory [8] yield in Th. 2.5 a necessary and sufficient criterion for the special orthogonal group of a regular form q to be excellent. It shows that the class of quadratic forms which give rise to excellent groups is bigger than the class of excellent forms in the sense of Knebusch. There are non excellent quadratic forms which have excellent orthogonal groups. For example, the so called “good forms” as introduced by Fitzgerald [5] have an excellent special orthogonal group, but are not always excellent (cf. Ex. 2.7 below). On the other hand, if q is excellent, then G also is excellent. It would be interesting to get an overview about the class of quadratic forms described in Th. 2.5.

Knebusch [10] introduced a generic splitting tower k = F_{0} ⊂ F_{1} ⊂ … ⊂ F_{h} of a non-split quadratic form q over k and proved in [11, Th. 7.14], that an anisotropic quadratic form q over k is excellent if and only if there is a quadratic form η_{i} over k such that η_{i}⊗_{k}F_{i} is the anisotropic kernel of q⊗_{k}F_{i} for all i = 1, …, h. We prove analogous statements for the special orthogonal group G = SO(q) and for the special linear group G = SL_{1}(A) in 2.4 and 1.9, 1.10 below.

We would like to thank D. Hoffmann, O. Izhboldin and J-P. Serre for useful comments given on the occasion of the conference on “Linear algebraic groups and related structures” at Louvain-la-Neuve, 1996.

1. Azumaya algebras and excellent groups of type ^{1}A_{n} An Azumaya algebra A over k is a finite dimensional central simple k-algebra, and, by Wedderburn’s theorem, there is a unique integer m ≥ 1 and a central division algebra D over k which is unique up to k-isomorphism such that A ≃ M_{m}(D), where M_{m}(D) denotes the full matrix ring of m by m matrices over D. The index of A is defined by ind A = \sqrt{dim_{k}D}, and the exponent exp A of A is the order of the equivalence class [A] in the Brauer group Br(k). It is known that exp A divides ind A and that every prime divisor of ind A divides exp A, cf. [16, 14.4, Prop. b, p. 261].

We write A^{m} for the m-fold tensor product A⊗_{k} ⋯ ⊗_{k}A and set A^{0} = k. Let A^{op} be the opposite algebra of A so that [A^{op}] = [A]^{−1} in Br(k). We write A ∼ A′ if [A] = [A′] and set K^{∗} = K \ {0} for any field K.

1.1 Definition. An Azumaya algebra A over k is said to be excellent if for any field K ⊇ k, there is a k-algebra C such that C⊗_{k}K is the underlying central division K-algebra of A ⊗_{k} K.

If A is excellent and D is the underlying central division algebra of A⊗_{k}K then there is a canonical way to find a k-algebra C such that D ≃ C⊗_{k}K as follows from 1.5 and 1.6 below.

1.2 Lemma. Let A be an Azumaya algebra over k, let K be a field extension of k, and let D be the underlying central division algebra of A⊗_{k}K. Then the following two statements are equivalent.

i) There is a k-algebra C such that C⊗_{k}K ≃ D. ii) There is an Azumaya algebra B over k such that B⊗_{k}K is split and ind (A⊗_{k}K) =

ind (A⊗_{k}B).

Proof. i) implies ii): Define B := A^{op}⊗_{k}C. Then B⊗_{k}K ≃ A^{op}⊗_{k}C⊗_{k}K ≃ (A^{op}⊗_{k}K) ⊗_{K} D ∼ D^{op} ⊗_{K} D is split over K, and ind (A⊗_{k}B) = ind C = ind D = ind (A⊗_{k}K). ii) implies i): Let C denote the underlying central division algebra of A⊗_{k}B over k. Then ind C = ind (A⊗_{k}B) = ind (A⊗_{k}K) = ind D. Since B⊗_{k}K is split, the three K-algebras C⊗_{k}K, A⊗_{k}B⊗_{k}K and A⊗_{k}K are equivalent in the Brauer group of K and represented by the central division algebra D, hence C⊗_{k}K ≃ D.

Let F be the function field of the Severi-Brauer variety of an Azumaya algebra B over k. In [1, 9.3], Amitsur showed that the kernel of the canonical homomorphism Br(k) → Br(F) is cyclic generated by the class [B]. The following lemma is an easy consequence of this result and the index reduction formula of Schofield/van den Bergh [17, Th. 1.3].

1.3 Lemma. Let A be an Azumaya algebra over k of exponent e and index d with prime power decomposition d = ∏ p^{v_{p}(d)}. For any divisor δ of d let F^{(δ)} denote the function field of the Severi-Brauer variety of A^{δ}. Then

exp(A⊗_{k}F^{(δ)}) = gcd (e, δ) and ind (A⊗_{k}F^{(δ)}) = ∏_{p|δ} p^{v_{p}(d)} .

In particular, we have ind (A⊗_{k}F^{(δ)}) = ind A if and only if δ is divisible by any prime divisor of d.

Proof. Since F^{(δ)} ∼ A^{δ}⊗_{k}F^{(δ)} ∼ (A⊗_{k}F^{(δ)})^{δ} it follows that r := exp(A⊗_{k}F^{(δ)}) divides δ, and hence r divides δ′ := gcd (e, δ). Since A^{r} splits over F^{(δ)} there is an integer i such that [A^{r}] = [A^{δ}]^{i}, cf. [1, 9.3, p. 34]. Thus r ≡ δi mod e, and δ′ divides r. Hence exp(A⊗_{k}F^{(δ)}) = gcd (e, δ).

Since every prime divisor of the index of a central simple algebra is also a divisor of its exponent it follows from the above that p divides ind (A⊗_{k}F^{(δ)}) if and only if it divides gcd (e, δ), which holds if and only if p divides δ. Hence we have to show that p^{v_{p}(d)} | ind (A⊗_{k}F^{(δ)}) for every such p. There is a finite separable field extension L of k such that ind (A⊗_{k}L) = p^{v_{p}(d)} (cf. [16, 14.4 Lemma b, p. 260]). Since k is algebraically closed in F^{(δ)} we may replace L by k and assume ind A = p^{v_{p}(d)}. The index reduction formula [17, Th. 1.3, p. 733] implies that for l = ind A^{δ}

ind (A⊗_{k}F^{(δ)}) = min_{i=1,...,l}{ind (A⊗_{k}A^{δi})} = min_{i=1,...,l}{ind (A^{1+δi})}.

For p dividing δ the latter is equal to ind A = p^{v_{p}(d)} for every i, since 1 + δi is coprime to p for every i.

1.4 Remark. Lemma 1.3 yields examples of so called anisotropic splitting of an Azumaya algebra A: Assume that ind A is not squarefree. Let δ denote the product of all prime divisors of ind A, and assume that δ is a proper divisor of exp A. Then ind (A⊗_{k}F^{(δ)}) = ind A and exp(A⊗_{k}F^{(δ)}) = δ. Hence field extension by F^{(δ)} keeps the index invariant, while the exponent goes down.

For an Azumaya algebra A and any divisor δ of ind A, there is a generic partial splitting field F_{δ} of A such that ind (A⊗_{k}F_{δ}) = δ (cf. [2, Th. 3, p. 104]). Hence scalar extension by F_{δ} brings the index down to δ, while extension by F^{(δ)} keeps the index at its highest possible value, as follows from 1.3.

The kernel Br(F_{δ}/k) of the homomorphism Br(k) → Br(F_{δ}) is cyclic and generated by the class of A^{δ} by [2, Th. 7, p. 115].

1.5 Lemma. Let A_{ε}, A_{δ} be central division algebras over k with coprime indices ε, δ, and let A := A_{ε}⊗_{k}A_{δ}. For any field extension K of k with δ = ind (A⊗_{k}K), the underlying central division algebra of A⊗_{k}K is isomorphic to A_{δ}⊗_{k}K. For K = F_{δ}, the following uniqueness statement holds: Any k-algebra C satisfying C⊗_{k}F_{δ} ≃ A_{δ}⊗_{k}F_{δ} is isomorphic to A_{δ}.

Proof. We have δ = ind (A⊗_{k}K) = ind (A_{ε}⊗_{k}K) ind (A_{δ}⊗_{k}K) = ind (A_{δ}⊗_{k}K), since ε, δ are coprime, hence A_{δ}⊗_{k}K is a division algebra. By assumption, A_{δ}⊗_{k}C^{op} is split by F_{δ}. Hence A_{δ}⊗_{k}C^{op} ∼ A^{δj} for some integer j by [2, Th. 7, p. 115]. This yields A_{δ} ∼ C⊗_{k}A^{δj} ∼ C⊗_{k}A_{ε}^{δj}⊗_{k}A_{δ}^{δj} ∼ C⊗_{k}A_{ε}^{δj}. Since δ = ind C and ind A_{ε}^{δj} are coprime, we conclude that A_{ε}^{δj} splits over k, and A_{δ} ≃ C.

1.6 Theorem. An Azumaya algebra A over k is excellent if and only if its index is squarefree.

Proof. We may assume that A is a central division algebra.

Let i := ind A be squarefree, and let K be a field extension of k. Set δ = ind (A⊗_{k}K) and ε = i/δ. Then ε and δ are coprime, and A = A_{ε}⊗_{k}A_{δ} (cf. [16, Primary Decomposition Th., p. 260]). Thus we can apply 1.5.

Let now i be not squarefree. Then, by 1.3, there is a field extension F^{(δ)} of k such that exp(A⊗_{k}F^{(δ)}) is a proper divisor of ind (A⊗_{k}F^{(δ)}). Since it suffices to show that A⊗_{k}F^{(δ)} is not excellent we may replace k by F^{(δ)}, hence e := exp A is a proper divisor of i. We have ind (A⊗_{k}F_{e}) = e. The kernel Br(F_{e}/k) of the homomorphism Br(k) → Br(F_{e}) is cyclic and generated by the class of A^{e} by [2, Th. 7, p. 115], hence Br(F_{e}/k) is trivial, which means that every Azumaya algebra over k which is split by F_{e} is trivial. Since ind (A⊗_{k}F_{e}) = e ≠ i it follows from 1.2 that A is not excellent.

1.7 Corollary. An Azumaya algebra over k is excellent if and only if over every field extension of k its index and its exponent are equal.

Proof. Let A be excellent. Since every prime divisor of the index of a central simple algebra divides its exponent it follows from 1.6 that ind (A⊗_{k}K) = exp(A⊗_{k}K) for every field extension K of k.

Let δ = Π_{p|d} p with d = ind A. Then δ|e|d. Thus 1.3 implies that ind (A⊗_{k}F^{(δ)}) = ind A and exp(A⊗_{k}F^{(δ)}) = δ. If ind (A⊗_{k}F^{(δ)}) = exp(A⊗_{k}F^{(δ)}) then ind A = δ is squarefree.

We say that a field k′ ⊇ k is a k-specialization of a field L ⊇ k if there is a k-place L → k′ ∪ {∞}. If A⊗_{k}L is excellent then so is A⊗_{k}k′ for every k-specialization k′ of L as follows from 1.6 and [8, 4.2, p. 53]. Moreover, there is a divisor δ of ind A such that F_{δ} has the following universal property.

1.8 Corollary. Let A be an Azumaya algebra over k and let ind A = p_{1}^{e_{1}} ⋯ p_{r}^{e_{r}} be the prime number decomposition of its index. If δ = p_{1} ⋯ p_{r} then A⊗_{k}F_{δ} is excellent and for any field extension K of k the following statements are equivalent.

i) A⊗_{k}K is excellent. ii) K is a k-specialization of F_{δ}.

Proof. It follows from [8, 4.2, p. 53 and 4.8, p. 55] that ind (A⊗_{k}F_{δ}) = δ and that every field extension K of k with ind (A⊗_{k}K) dividing δ is a k-specialization of F_{δ}. Conversely, if K is a k-specialization of a field extension L of k then ind (A⊗_{k}K) divides ind (A⊗_{k}L) (cf. [8, 4.2]). Thus 1.8 follows from 1.6.

1.9 Lemma. Let A be an Azumaya algebra over k. Then A is excellent if and only if there is a k-algebra C_{δ} such that C_{δ}⊗_{k}F_{δ} is the underlying central division algebra of A⊗_{k}F_{δ} for every divisor δ of ind A.

Proof. If A is excellent then certainly the condition stated is true by Definition 1.1. Suppose K is a field extension of k such that δ := ind (A⊗_{k}K) is a proper divisor of ind A. We have to show that there is a k-algebra C and an integer r > 0 such that A⊗_{k}K ≃ M_{r}(C⊗_{k}K) and C⊗_{k}K is a division algebra.

It follows from [8, 4.2 and 4.8] that ind (A⊗_{k}F_{δ}) = ind (A⊗_{k}K) and that there is a k-place φ : F_{δ} → K ∪ {∞}. Set F := F_{δ}. Let R be the valuation ring of φ. Then the canonical homomorphism Br(R) → Br(F), B ↦ B ⊗_{R} F, is injective, where Br(R) and Br(F) are the Brauer groups of R and F respectively [18, Lemma 1.2]. By assumption, there is a k-algebra C and an integer r > 0 such that A⊗_{k}F ≃ M_{r}(C⊗_{k}F) and C⊗_{k}F is a central division algebra. Since we have canonical F -algebra isomorphisms

(A⊗_{k}R) ⊗_{R} F ≃ A⊗_{k}F ≃ M_{r}(C⊗_{k}F) ≃ M_{r}(C⊗_{k}R ⊗_{R} F) ≃ M_{r}(C⊗_{k}R) ⊗_{R} F ,

it follows that A⊗_{k}R and M_{r}(C⊗_{k}R) represent the same equivalence class in Br(R) and that the R-ranks of A⊗_{k}R and M_{r}(C⊗_{k}R) coincide. Since R is a local ring, equivalence in Br(R) means that there are integers m, n > 0 such that

(A⊗_{k}R) ⊗_{R} M_{m}(R) ≃ M_{r}(C⊗_{k}R) ⊗_{R} M_{n}(R),

and for rank reasons we must have m = n. By the cancellation property for Azumaya algebras over semilocal rings it follows A⊗_{k}R ≃ M_{r}(C⊗_{k}R) (cf. [7, Chap. III, Prop. 5.2.3, p. 144]). Viewing K as an R-algebra via φ we obtain K-algebra isomorphisms

A⊗_{k}K ≃ A⊗_{k}R ⊗_{R} K ≃ M_{r}(C⊗_{k}R) ⊗_{R} K ≃ M_{r}(C⊗_{k}R ⊗_{R} K) ≃ M_{r}(C⊗_{k}K).

This implies that C⊗_{k}K is a division algebra since ind (A⊗_{k}F) = ind (A⊗_{k}K).

1.10 Theorem. An Azumaya algebra A over k is excellent if and only if the special linear group SL_{1}(A) is excellent.

Proof. Let G = SL_{1}(A). The absolute Dynkin diagram of G is given by

{∘ \atop α_{1}} {∘ \atop α_{2}} ⋯ {∘ \atop α_{n}},

where Δ = {α_{1}, …, α_{n}} is a system of simple roots of G. Here n is such that A⊗_{k} k′ ≃ M_{n+1}(k′) for a splitting field k′ of A.

Let K be an arbitrary field extension of k. Then there is a central division algebra D over K such that A ⊗_{k} K ≃ M_{r+1}(D) where d(r + 1) = n + 1 for d = ind D. The anisotropic kernel (G_{K})_{an} of G_{K} is given by (G_{K})_{an} ≃ SL_{1}(D) ×_{K} … ×_{K} SL_{1}(D) with r + 1 factors and it has an absolute Dynkin diagram

(∗)  {∘ \atop α_{1}} ⋯ {∘ \atop α_{d−1}} {∘ \atop α_{d+1}} ⋯ {∘ \atop α_{2d−1}}  ⋯ {∘ \atop α_{n−d+1}} ⋯ {∘ \atop α_{n}}.

If C is a k-algebra such that D ≃ C ⊗_{k} K then SL_{1}(D) ≃ SL_{1}(C) ×_{k} K, hence (G_{K})_{an} ≃ (SL_{1}(C) ×_{k} … ×_{k} SL_{1}(C))×_{k}K. It follows that G is excellent if A is so.

Conversely, assume that G is excellent. Let d be a divisor of ind A and K = F_{d}. There is a k-group H such that (G_{K})_{an} ≃ H ×_{k} K. Since G is of inner type, k is algebraically closed in K by [8, 3.16]. This forces that H is of inner type, since each field extension k′ of k such that H×_{k}k′ is of inner type contains a finite Galois extension k_{inn} of k by [8, 3.5 ii)]. It follows that (G_{K})_{an} and H have the same absolute Dynkin diagram (∗) whose set of vertices we denote be Δ′. Let K̄ be an algebraic closure of K and let G′ ≃ (SL_{1}(D))K̄ be the first factor of HK̄ corresponding to {α_{1}, …, α_{d−1}}. There is a parabolic subgroup P of HK̄ of type Δ′ \ {α_{1}} and a parabolic subgroup P′ of G′ of type {α_{2}, …, α_{d−1}} such that P ∩ G′ = P′ and HK̄ = G′ · P. We therefore obtain an isomorphism of K̄-varieties HK̄/P ≃ G′/P′. Since H is a k-group of inner type there is a unique k-variety V such that H_{k′}/Q ≃ V ×_{k} k′ for any field extension k′ of k and any parabolic subgroup Q of H_{k′} of type Δ′ \ {α_{1}} (cf. [8, 3.4, 3.7, p. 43f]), in particular, HK̄/P ≃ V ×_{k} K̄. Similarly, since SL_{1}(D) is a K-group of inner type, there is a unique K-variety V′ such that G′/P′ ≃ V′ ×_{K} K̄. This implies that there is an isomorphism of K-varieties V′ ≃ V ×_{k} K. The K-variety V′ is the Severi-Brauer variety of D (cf. [8, 4.7 and 4.10, p. 55f]), hence V is a Severi-Brauer variety over k. It follows that there is an Azumaya algebra C over k such that D ≃ C⊗_{k}K. Thus A is excellent by 1.9.

2. Quadratic forms and excellent special orthogonal groups Let k be an infinite field with char k ≠ 2 and let (M, q) be a regular quadratic k-space of dimension m ≥ 3, hence M is an m-dimensional k-vector space and q is a quadratic form with non-degenerate associated bilinear form ( , ) such that q(x+y) = q(x)+q(y)+(x, y) holds for all x, y ∈ M. The discriminant d(q) of (M, q) is defined to be the square class (−1)^{[m/2]} det((u_{i}, u_{j})_{i,j=1,...,m})k^{∗}^{2} ∈ k^{∗}/k^{∗}^{2}. (Here {u_{1}, …, u_{m}} denotes a k-basis of M.) We have a Witt decomposition of M into mutually orthogonal subspaces

M = H_{1} ⊥ … ⊥ H_{r} ⊥ M_{an}

where H_{i} is a hyperbolic plane for i = 1, …, r and (M_{an}, q_{an}) with q_{an} := q|_{Man} is a maximal anisotropic subspace of (M, q) which is unique up to k-isometry and is called an anisotropic kernel of the quadratic space (M, q). The integer r ≥ 0 is the Witt index of (M, q), and we have m = 2r + dim_{k}M_{an} and d(q) = d(q_{an}). We will write i(q) for the Witt index of q and q_{K} for the form q⊗_{k}K in case K is a field extension of k.

Let G := SO(q) be the special orthogonal group of q. Then the k-rational points G(k) of G are given by G(k) = {g ∈ SL(M) | q(gx) = q(x) ∀ x ∈ M}.

The k-rational points of G_{an} ≃ SO(q_{an}) are given by

G_{an}(k) = {g ∈ G(k) | gM_{an} ⊆ M_{an} and g|_{ℍ^{r}} = id}

where ℍ^{r} = H_{1} ⊥ … ⊥ H_{r} is the hyperbolic part of M.

Using the definitions from the introduction we observe that excellent quadratic forms q have excellent orthogonal groups SO(q). But the converse is not always true. This follows from 2.7 below.

2.1 Lemma. Let M be a k-vector space, let q, q′ : M → k be regular quadratic forms with bilinear forms (., .) and (., .)′ respectively. Assume that, for every quadratic field extension L of k and every x ∈ M_{L}, we have q_{L}(x) = 0 if and only if q′_{L}(x) = 0. Then we have (x, y) = 0 if and only if (x, y)′ = 0 for all x, y ∈ M.

The lemma says, in other words: If the notion of isotropy for q and q′ is the same for all quadratic field extensions of k, then the notion of orthogonality is the same for both forms.

Proof. Let x, y ∈ M. If x, y are linearly dependent, then we may assume that y = λx for some λ ∈ k. We then have (x, y) = 2λq(x) and (x, y)′ = 2λq′(x) and we are done in this case.

Hence we may assume that W := kx⊕ky is a plane. Let L be a quadratic extension of k such that q_{L}|_{W_{L}} is isotropic. Let e, f ∈ W_{L} be a hyperbolic pair for q_{L}. We then have, by our assumption, the formulas q_{L}(e) = 0 = q′_{L}(e), q_{L}(f) = 0 = q′_{L}(f), (e, f) = 1, (e, f)′ = γ for some γ ∈ L^{∗}.

If we write x = ξ_{x}e + η_{x}f and y = ξ_{y}e + η_{y}f with ξ_{x}, ξ_{y}, η_{x}, η_{y} ∈ L we obtain (x, y) = (ξ_{x}η_{y} + ξ_{y}η_{x})(e, f) = γ^{−1}(ξ_{x}η_{y} + ξ_{y}η_{x})(e, f)′ = γ^{−1}(x, y)′ and our claim follows. In fact, the computation shows that γ is necessarily in k^{∗}, but we do not need that here.

2.2 Theorem. Let q and q′ be regular quadratic forms over k of dimensions ≥ 3. Then the following statements are equivalent.

i) SO(q) and SO(q′) are isomorphic as k-groups. ii) The projective quadrics defined by q and q′ are isomorphic as k-varieties. iii) q and q′ are similar over k, that is, there is a γ ∈ k^{∗} such that q ≃ γq′.

Proof. i) implies ii): Any k-isomorphism between the semisimple groups G := SO(q) and G′ := SO(q′) preserves the notion of parabolic subgroups and their types. The quadric of q is the unique (cf. [8, 3.7, p. 44]) k-variety Q_{q} such that, for every field extension L of k with q_{L} isotropic, we have (Q_{q})_{L} ≃ (SO_{q})_{L}/P, where P ⊂ G_{L} is the stabilizer of some isotropic line of q_{L}. The type of the parabolic subgroup P of G_{L} is defined by this description (cf. [8, §5, p. 56 ff]). If P′ ⊂ G′ denotes the image of P under some k-isomorphism G → G′ we obtain for the quadric Q_{q′} of q′ the Lisomorphism (Q_{q′})_{L} ≃ G′_{L}/P′ ≃ G_{L}/P. The uniqueness statement in [8, 3.7, p. 44] asserts that Q_{q} ≃ Q_{q′} over k.

ii) implies iii): The dimension of the quadric of q is dim q−2, hence we may assume that q and q′ are defined on the same k-vector space M. Since the quadrics of q and q′ are k-isomorphic there is an element u ∈ GL_{M}(k) such that (q′ ∘ u)_{L} and q_{L} vanish, for every field extension L of k, on the same points x ∈ M_{L}.

Let e_{1}, …, e_{m} be an orthogonal basis of (M, q). Then, by 2.1, it is an orthogonal basis for (M, q′) as well.

Let α_{i} := q(e_{i}) and α′_{i} := q′(e_{i}) and define γ := α_{1}/α′_{1}. Let i be maximal such that γ ≡ α_{i}/α′_{i} mod k^{∗2}. If i = m we are done. Assume i < m, and let W := ke_{i} ⊥ ke_{i+1}. The forms q_{W} and q′_{W} are either both isotropic or both anisotropic. By the assumption on i we have

(∗)  d := d(q|_{W}) = −α_{i}α_{i+1}k^{∗2} ≠ −α′_{i}α′_{i+1}k^{∗2} = d(q′|_{W}) =: d′

hence q|_{W} , q′|_{W} both have to be anisotropic, and therefore both discriminants are not trivial. But, for L = k\sqrt{d}, both forms (q|_{W})_{L}, (q′|_{W})_{L} are isotropic, hence −α′_{i}α′_{i+1} ∈ L^{∗2} so that k\sqrt{d′} = L and therefore −α_{i}α_{i+1} ≡ −α′_{i}α′_{i+1} mod k^{∗2} contradicting (∗). Hence i = m and iii) follows. Trivially iii) implies i).

Define n := [m/2] where m is the dimension of M. Let Δ = {α_{1}, …, α_{n}} denote the set of vertices of the Dynkin diagram of Gk̄ where k̄ is an algebraic closure of k. Then define for i ∈ {1, …, n − 1} the following subset Θ_{i} of Δ:

Θ_{i} := \{ \begin{matrix}Δ \ \{α_{i}\} & if G is of inner type or i ≤ n − 2; \\ Δ \ \{α_{n−1}, α_{n}\} & if G is of outer type and i = n − 1. \end{matrix}

For i ∈ {1, …, n − 1} let F_{i} := F_{Θ_{i}} be the generic Θ_{i}-splitting field of q as introduced in [8, §§3 and 5]. Then q_{F_{i}} has Witt index i(q_{F_{i}}) ≥ i, and every field extension L of k satisfying i(q_{L}) ≥ i is a k-specialization of F_{i}. Moreover, if there is a field K ⊃ k with index i(q_{K}) = i, then i(q_{F_{i}}) = i by [8, 5.9].

2.3 Theorem. Let G = SO(q) and K be a field extension of k. Let F := F_{i} be a generic Θ_{i}-splitting field of q such that i(q_{F}) = i(q_{K}) = i. If (G_{F})_{an} ≃ H ×_{k} F for some k-group H, then there is an anisotropic quadratic form τ over k, such that H ≃ SO(τ) and (G_{K})_{an} ≃ H ×_{k} K.

Proof. We first show that H ≃ SO(τ) for some anisotropic quadratic form τ over k. If dim q is odd, then this is clearly true since H is of Dynkin type B_{n}. Let now dim q be even. Let k_{s} be the separable closure of k contained in k̄ and let Γ = Gal (k_{s}/k) be its Galois group. Then there is a homomorphism β_{G} : C_{s}^{∗Γ} → Br(k) defined by Tits [22, §3, p. 202ff] where C_{s}^{∗} is the dual of the center C_{s} of G̃_{k_{s}}, and where G̃ denotes a simply connected covering of G. We have β_{G} ×_{k} F = β_{H} ×_{k} F. Since H is of Dynkin type D_{n} we have H ≃ SU(D, h) for some division k-algebra D and some hermitian form h over D [21, p. 56f]. It follows from [22, 6.2, p. 212] that D⊗_{k}F is trivial in Br(F). The kernel of Br(k) → Br(F) was computed by Merkurjev and Tignol [14, Cor. 2.11, p. 43f]. From this we obtain that D = k, hence H ≃ SO(τ) for some anisotropic quadratic form τ over k.

By [8, 5.3, p. 59], there is a k-place φ : F → K ∪ {∞}. Let O denote the valuation ring of φ with maximal ideal 𝔪. Since the canonical homomorphism of Witt rings W(O) → W(F) is injective (cf. [9, p. 283]) it follows that q_{O} := q⊗_{k}O ≃ i × ⟨1, −1⟩ ⊥ (q_{O})_{an}, hence q_{K} = q_{O} ⊗_{O} K ≃ (i × ⟨1, −1⟩ ⊥ (q_{O})_{an}) ⊗_{O} K. The Witt cancellation theorem implies (q_{K})_{an} ≃ (q_{O})_{an} ⊗_{O} K, since i(q_{K}) = i(q_{F}).

We are going to show that SO((q_{O})_{an}) ≃ H ×_{k} O. Then the result will follow since SO((q_{K})_{an}) ≃ SO((q_{O})_{an}) ×_{O} K ≃ (H ×_{k} O) ×_{O} K ≃ H ×_{k} K.

Let V := (M⊗_{k}O)_{an}. Then V_{F} := V ⊗_{O} F ≃ (M⊗_{k}F)_{an}. We define

X := { x ∈ V_{F} | (q_{F})_{an}(x) ∈ O }

and first show X = V.

We may choose a suitable orthogonal basis e_{1}, …, e_{s} of V such that (q_{F})_{an}(e_{i}) is a unit in O for all i = 1, …, s (cf. [12, Prop. 6.4, p. 17]). Thus X ⊇ V.

Assume that there is an x ∈ X \ V. Choose x = ∑_{i=1}^{s} λ_{i}e_{i} with λ_{i} ∈ F such that the number t of λ_{i}’s for which λ_{i} ∉ O is minimal. By our assumption we have t ≥ 1 and we may assume that X ∋ x = ∑_{i=1}^{t} λ_{i}e_{i} with λ_{i} ∈ F \ O for all i = 1, …, t. Since O is a valuation ring we can arrange that λ_{1}O ⊆ λ_{2}O ⊆ … ⊆ λ_{t}O. Thus λ_{1}/λ_{i} ∈ O for all i = 1, …, t. We have λ_{1}^{−1}x = e_{1} + ∑_{i=2}^{t} λ_{i}λ_{1}^{−1}e_{i} with λ_{1}^{−1} ∈ 𝔪 and, since e_{1}, …, e_{s} is an orthogonal basis, (q_{F})_{an}(λ_{1}^{−1}x − e_{1}) = (q_{F})_{an}(λ_{1}^{−1}x) − (q_{F})_{an}(e_{1}) ∈ O. Thus, by minimality, λ_{1}^{−1}x − e_{1} ∈ V and λ_{i}λ_{1}^{−1} ∈ O for all i. We have shown that ε_{i} := λ_{i}λ_{1}^{−1} is a unit in O for all i = 1, …, r and λ_{1}^{−1}x = ∑_{i=1}^{t} ε_{i}e_{i}. Since (q_{O})_{an}(λ_{1}^{−1}x) = λ_{1}^{−2}(q_{F})_{an}(x) ∈ 𝔪 we conclude that λ_{1}^{−1}x becomes isotropic in V ⊗_{O} O/𝔪. Since O/𝔪 ⊆ K we obtain (q_{K})_{an}(y) = 0 for some non-zero vector y ∈ (M⊗_{k}K)_{an} ≃ V ⊗_{O} K and hence a contradiction. Thus X = V.

Let σ : V_{F} → V_{F} be an isometry. Then we have (q_{F})_{an}(σv) = (q_{F})_{an}(v) ∈ O for all v ∈ V. Since X = V this implies that σ(V) ⊆ V, hence SO((q_{O})_{an})(O) = SO((q_{O})_{an})(F). Since H(k) ⊆ H(O) is Zariski-dense in H(F̄), cf. [3, 18.3, p. 220], and since H(O) ⊆ H(F) ≃ SO((q_{O})_{an})(F) = SO((q_{O})_{an})(O), it follows that SO((q_{O})_{an}) ≃ H ×_{k} O.

2.4 Corollary. Let I be the set of all integers i ∈ {1, …, n − 1} which occur as Witt index of q_{K} for some field K ⊃ k. Then G = SO(q) is excellent if and only if there is a quadratic form τ_{i} over k such that (G_{F_{i}})_{an} ≃ SO(τ_{i}) ×_{k} F_{i} for all i ∈ I.

2.5 Theorem. Let G = SO(q), let I be as in 2.4. Then the following statements are equivalent.

i) G is excellent. ii) For every field extension K of k, there is an element a ∈ K^{∗} and a quadratic form

τ over k such that a(q_{K})_{an} ≃ τ_{K}. iii) For each i ∈ I, there is an element a ∈ F_{i}^{∗} and a quadratic form τ over k such

that a(q_{F_{i}})_{an} ≃ τ_{F_{i}}. If dim q is odd, then the elements a can be chosen in k^{∗}, hence in this case G is excellent if and only if q is excellent.

Proof. i) implies ii): Let G be excellent and let K be a field extension of k such that i(q_{K}) > 0. By 2.3, there is an anisotropic form τ over k such that SO((q_{K})_{an}) ≃ SO(τ_{K}). By 2.2, we obtain a(q_{K})_{an} ≃ τ_{K} for some a ∈ K^{∗}. This proves ii). Trivially ii) implies iii), and it follows from 2.4 that iii) implies i).

Let now dim q be odd. Then looking at the discriminants d(τ_{K}) = d(a(q_{K})_{an}) = a d(q_{K}) we observe that, up to a square, a must be in k and the last statement follows.

2.6 Remark. If dim q is odd, we easily obtain the following uniqueness property: If H ×_{k} F_{i} ≃ (G_{F_{i}})_{an} ≃ H′ ×_{k} F_{i} for some k-groups H, H′ and some i ∈ I then H ≃ H′.

Proof. By 2.3, there are quadratic forms τ, τ′ over k such that H ≃ SO(τ) and H′ ≃ SO(τ′). Hence, by 2.2, there are a, a′ ∈ F_{i}^{∗} such that a τ_{F_{i}} ≃ (q_{F_{i}})_{an} ≃ a′ τ′_{F_{i}}. If dim q is odd we find, by computing discriminants, that, up to squares, a, a′ can be chosen in k^{∗}. The collection (F_{j}, j ∈ I) yields a generic splitting tower of q as introduced by Knebusch, cf. [8, 5.8, p. 61]. Proposition 7.2 in [11, p. 1] implies aτ ≃ a′τ′. This forces H ≃ H′.

In [11, Ex. 9.11, p. 24], Knebusch gives an example of an anisotropic non excellent quadratic form q of height 2 with leading form defined over k. For the definition of the height and the leading form of a quadratic form we refer to [10, 5.4, p. 79 and 5.9, p. 82].

2.7 Example. Let q be anisotropic of height 2 with leading form defined over k. Then the special orthogonal group SO(q) is excellent. If, in addition, dim q is a 2-power then q is not excellent.

Proof. If m is odd then q is excellent [11, p. 27] and thus SO(q) is excellent. Assume that m is even and q is not excellent. Let F = k(q) denote the function field of q and let τ denote a quadratic form over k such that τ_{F} is the leading form of q. Then there is an a ∈ F^{∗} such that (q_{F})_{an} ≃ aτ_{F} by the properties of the leading form. Since q has height 2, Th. 2.5 implies that G is excellent.

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