ESSENTIAL DIMENSION OF PROJECTIVE ORTHOGONAL AND SYMPLECTIC GROUPS OF SMALL DEGREE

SANGHOON BAEK

Abstract. In this paper, we study the essential dimension of classes of central simple algebras with involutions of index less or equal to 4. Using structural theorems for simple algebras with involutions, we obtain the essential dimension of projective and symplectic groups of small degree.

1. Introduction

Let F be a field, A a central simple F -algebra, and (σ, f) a quadratic pair on A (see [5, 5.B]). A morphism of algebras with quadratic pair ϕ : (A, σ, f) → (A′, σ′, f′) is an F -algebra morphism ϕ : A → A′ such that σ′ ∘ ϕ = ϕ ∘ σ and f ∘ ϕ = f′. For any field extension K/F, we write (A, σ, f)_{K} for (A ⊗_{F} K, σ ⊗ Id_{K}, f_{K}), where f_{K} : Sym(A_{K}, σ_{K}) → K.

For n ≥ 2, let D_{n} denote the category of central simple F -algebras of degree 2n with quadratic pair, where the morphisms are the F -algebra homomorphisms which preserve the quadratic pairs and let A_{1}^{2} denote the category of quaternion algebras over an étale quadratic extension of F, where the morphisms are the F -algebra isomorphism. Then, there is an equivalence of groupoids:

(1)  D_{2} ≡ A_{1}^{2};

see [5, Theorem 15.7].

Moreover, if we consider the full subgroupoid ^{1}A_{1}^{2} of A_{1}^{2} whose objects are F -algebras of the form Q×Q′, where Q and Q′ are quaternion algebras over F, and the full subgroupoid ^{1}D_{n} of D_{n} whose objects are central simple algebras over F with quadratic pair of trivial discriminant, then the equivalence in (1) specialize to the following equivalence of subgroupoids:

(2)  ^{1}D_{2} ≡^{1}A_{1}^{2};

see [5, Corollary 15.12].

For n ≥ 1, we denote by C_{n} be the category of central simple F -algebras of degree 2n with symplectic involution, where the morphisms are the F -algebra isomorphism which preserve the involutions.

By Galois cohomology, there are canonical bijections (see [5, §29.D,F])

(3)  D_{n} ⟷ H^{1}(F, PGO_{2n})

The work has been partially supported from the Fields Institute and from Zainoulline’s NSERC Discovery grant 385795-2010.

and

(4)  C_{n} ⟷ H^{1}(F, PGSp_{2n}).

Let T : Fields/F → Sets be a functor from the category Fields/F of field extensions over F to the category Sets of sets and let p be a prime. We denote by ed(T ) and ed_{p}(T ) the essential dimension and essential p-dimension of T , respectively. We refer to [4, Def. 1.2] and [8, Sec.1] for their definitions. Let G be an algebraic group over F. The essential dimension ed(G) (respectively, essential p-dimension ed_{p}(G)) of G is defined to be ed(H^{1}(−, G)) (respectively, ed_{p}(H^{1}(−, G))), where H^{1}(E, G) is the nonabelian cohomology set with respect to the finitely generated faithfully flat topology (equivalently, the set of isomorphism classes of G-torsors) over a field extension E of F.

A morphism S → T from Fields/F to Sets is called p-surjective if for any E ∈ Fields/F and any α ∈ T (E), there is a finite field extension L/E of degree prime to p such that α_{L} ∈ Im(S(L) → T (L)). A morphism of functors S → T from Fields/F to Sets is called surjective if for any E ∈ Fields/F, S(E) → T (E) is surjective. Obviously, any surjective morphism is p-surjective for any prime p. Such a surjective morphism gives an upper bound for the essential (p)-dimension of T and a lower bound for the essential (p)-dimension of S,

(5)  ed(T ) ≤ ed(S) and ed_{p}(T ) ≤ ed_{p}(S);

see [4, Lemma 1.9] and [8, Proposition 1.3].

Example 1.1. Let (M_{2}(F), γ) ∈ C_{1}, where γ is the canonical involution on M_{2}(F). As (M_{2}(K), γ_{K}) ≃ (M_{2}(F), γ) ⊗_{F} K for any field extension K/F, we have ed((M_{2}(F), γ)) = 0.

Assume that char(F) ≠ 2. The exact sequence

1 → μ_{2} → Sp_{2} → PGSp_{2} → 1

induces the connecting morphism ∂ : H^{1}(−, PGSp_{2}) → Br_{2}(−) which sends a pair (Q, γ) of a quaternion algebra with canonical involution to the Brauer class [Q]. As this morphism is nontrivial, by [4, Corollary 3.6] we have ed(PGSp_{2}) ≥ 2 (or by Lemma 2.6). Consider the morphism 𝔾_{m}^{2} → C_{1} defined by (x, y) ↦ ((x, y), γ), where γ is the canonical involution. As this morphism is surjective, by (5) we have ed(C_{1}) ≤ 2, thus ed(PGSp_{2}) = 2. This can be recovered from the exceptional isomorphism PGSp_{2} ≃ O_{3}^{+}.

Acknowledgements: I am grateful to A. Merkurjev for useful discussions. I am also grateful to J. P. Tignol and S. Garibaldi for helpful comments.

2. Essential dimension of projective orthogonal and symplectic groups associated to central simple algebras of index ≤ 4

First, we compute upper bounds for the essential dimension of certain classes of simple algebras with involutions of index less or equal to 2.

ESSENTIAL DIMENSION OF PROJECTIVE ORTHOGONAL, SYMPLECTIC GROUPS 3

Let (Q, γ) be a pair of a quaternion over a field F and the canonical involution. For any field extension K/F and any integer n ≥ 3, we write QH_{n}^{+}(K) (respectively, QH_{n}^{−}(K)) for the set of isomorphism classes of (M_{n}(Q), σ_{h}), where σ_{h} is the adjoint involution on M_{n}(Q) with respect to a hermitian form (respectively, skew-hermitian form) h (with respect to γ). If n is odd, we write 1QH_{n}^{−}(K) for the set of isomorphism classes of (M_{n}(Q), σ_{h}), where σ_{h} is the adjoint involution on M_{n}(Q) with respect to a skew-hermitian form h with disc(σ_{h}) = 1.

Lemma 2.1. Let F be a field and n ≥ 3 any integer. Then (1) ed(QH_{n}^{+}) ≤ n + 1.

(2) ed(QH_{n}^{−}) ≤ 3n − 3 if char(F) ≠ 2.

(3) ed(^{1}QH_{n}^{−}) ≤ 3n − 4 if char(F) ≠ 2.

Proof. (1) If h is a hermitian form, then h = ⟨t_{1}, t_{2}, ⋯, t_{n}⟩ for some t_{i} ∈ F. We consider the affine variety

X = \begin{cases}𝔾_{m}^{2} ×𝔸_{F}^{n−1} & if char(F) ≠ 2, \\ 𝔾_{m} ×𝔸_{F}^{n} & if char(F) = 2, \end{cases}

and define a morphism X(K) → QH_{n}^{+}(K) by

(a, b, t_{1}, ⋯, t_{n−1}) ↦ \begin{cases}((a, b) ⊗ M_{n}(K), σ_{⟨1,t_{1},··· ,t_{n−1}⟩}) & if char(F) ≠ 2, \\ ([a, b) ⊗ M_{n}(K), σ_{⟨1,t_{1},··· ,t_{n−1}⟩}), & if char(F) = 2. \end{cases}

As a scalar multiplication does not change the adjoint involution, this morphism is surjective. Therefore, by (5), we have ed(QH_{n}^{+}) ≤ n + 1.

(2) From now we assume that char(F) ≠ 2. If h is a skew-hermitian form, then h = ⟨q_{1}, q_{2}, ⋯, q_{n}⟩ for some pure quaternions q_{i} ∈ Q. We consider the affine variety Y = 𝔾_{m}^{2} ×𝔸^{1} × 𝔸^{3(n−2)} with coordinates (a, b, c, t_{1}, ⋯, t_{3n−6}) and the conditions ac^{2} + b ≠ 0, at_{1+3(k−1)}^{2} + bt_{2+3(k−1)}^{2} − abt_{3k}^{2} ≠ 0 for all 1 ≤ k ≤ n − 2. Define a morphism ϕ_{K} : Y (K) → QH_{n}^{−}(K) by

(a, b, c, t_{1}, ⋯, t_{3n−6}) ↦ ((a, b) ⊗ M_{n}(K), σ_{h}),

where p = i, q = ci + j, r_{k} = t_{1+3(k−1)}i + t_{2+3(k−1)}j + t_{3k}ij for 1 ≤ k ≤ n − 2, and h = ⟨p, q, r_{1}, ⋯, r_{3n−2}⟩.

We show that Y is a classifying variety for QH_{n}^{−}. Suppose that we are given a quaternion K-algebra Q = (a, b) and a skew hermitian form h = ⟨p, q, r_{1}, ⋯, r_{n−2}⟩ for some pure quaternions p, q, r_{k}. We can find a scalar c ∈ K such that p and q − cp anticommute, thus we have Q ≃ (p^{2}, (q − cp)^{2}). For 1 ≤ k ≤ n − 2, let

(6)  r_{k} = t_{1+3(k−1)}p + t_{2+3(k−1)}(q − cp) + t_{3k}p(q − cp)

with t_{1}, t_{2}, ⋯, t_{3n−6} ∈ K. Then (M_{n}(Q), σ_{h}) ≃ (M_{n}((p^{2}, (q − cp)^{2})), σ_{h}) is the image of ϕ_{K}. Therefore, by (5), we have ed(QH_{n}^{−}) ≤ 3n − 3.

(3) Assume that n = 2m + 1 for m ≥ 1. We consider the variety Y in (2) with an additional condition

−a(ac^{2}+b) ∏_{k=1}^{m−1} at_{1+3(k−1)}^{2}+bt_{2+3(k−1)}^{2}−abt_{3k}^{2} = ∏_{k=m}^{n−2} at_{1+3(k−1)}^{2}+bt_{2+3(k−1)}^{2}−abt_{3k}^{2}.

We show that this variety with the same morphism ϕ_{K} in (2) is a classifying variety for^{1}QH_{n}^{−}. Suppose that we are given a quaternion K-algebra Q = (a, b) and a skew hermitian form h = ⟨p, q, r_{1}, ⋯, r_{n−2}⟩ for some pure quaternions p, q, r_{k}. We do the same procedure as in (1), so that we have (M_{n}(Q), σ_{h}) ≃ (M_{n}((p^{2}, (q − cp)^{2})), σ_{h}) and (6). As disc(σ_{h}) = 1, there is a scalar d ∈ K^{×} such that

−p^{2}q^{2}r_{1}^{2} ⋯ r_{m−1}^{2} = (\frac{d}{r_{m}^{2} ⋯ r_{n−2}^{2}} )^{2}r_{m}^{2} ⋯ r_{n−2}^{2}.

We set f = d/r_{m}^{2} ⋯ r_{n−2}^{2}. As a scalar multiplication does not change the adjoint involution, we can modify h by the scalar f. As (p^{2}, (q − cp)^{2}) ≃ (f^{2}p^{2}, f^{2}(q − cp)^{2}), (M_{n}(Q), σ_{h}) is the image of ϕ_{K}. Therefore, by (5), we have ed(^{1}QH_{n}^{−}) ≤ 3n − 4.

□

Remark 2.2. The main idea of the proof of the case where h is a skewhermitian form is from Merkurjev’s work on algebras of degree 4 in his private note.

Assume that n is odd. Then we have

QH_{n}^{+} = C_{n}, QH_{n}^{−} = D_{n}, and ^{1}QH_{n}^{−} =^{1} D_{n}.

Hence, by [5, Theorem 4.2] and the exceptional isomorphism PGO_{6} ≃ PGU_{4}, we have

Corollary 2.3. Assume that n ≥ 3 is odd. Then (1) ed(PGSp_{2n}) ≤ n + 1.

(2) ed(PGO_{2n}) ≤ 3n − 3 if char(F) ≠ 2. In particular, ed(PGU_{4}) ≤ 6.

(3) ed(PGO_{2n}^{+}) ≤ 3n − 4 if char(F) ≠ 2.

Remark 2.4. In fact, ed_{2}(PGSp_{2n}) = ed(PGSp_{2n}) = n+1 for n ≥ 3 odd and char(F) ≠ 2. The lower bound was obtained by Chernousov and Serre in [6, Theorem 1] and the exact value was obtained by Macdonald in [7, Proposition 5.1].

Lemma 2.5. [2, Section 2.6] Let F be a field of characteristic different from 2. Then

ed_{2}(PGL_{2}^{×n}) = ed(PGL_{2}^{×n}) = 2n.

Proof. By [4, Lemma 1.11], we have

ed_{2}(PGL_{2}^{×n}) ≤ ed(PGL_{2}^{×n}) ≤ n · ed(PGL_{2}) = 2n.

On the other hand, the natural morphism

(7)  H^{1}(−, PGL_{2}^{×n}) → Dec_{2^{n}}(−)

is surjective, where Dec_{2^{n}}(K) is the set of all decomposable algebras of degree 2^{n} over a field extension K/F, hence, by (5), we have

2n = ed_{2}(Dec_{2^{n}}) ≤ ed_{2}(PGL_{2}^{×n}).
□

Lemma 2.6. Let F be a field of characteristic different from 2. Then

ed_{2}(PGO_{2^{r}}), ed_{2}(PGSp_{2^{r}}) ≥ \begin{cases}2 & if r = 1, \\ 4 & if r = 2, \\ (r − 1)2^{r−1} & if r ≥ 3. \end{cases}

Proof. Consider the forgetful functors

(8)  H^{1}(−, PGO_{2^{r}}) → Alg_{2^{r},2}

and

(9)  H^{1}(−, PGSp_{2^{r}}) → Alg_{2^{r},2},

where Alg_{2^{r},2}(K) is the set of isomorphism classes of simple algebras of degree 2^{n} and exponent dividing 2 over a field extension K/F. These functors are surjective by a theorem of Albert.

It is well known that ed_{2}(Alg_{2,2}) = 2, ed_{2}(Alg_{4,2}) = 4. For r ≥ 3, we have ed_{2}(Alg_{2^{r},2}) ≥ (r − 1)2^{r−1} by [3, Theorem]. Therefore, by (5), we have the above lower bound for ed_{2}(PGO_{2^{r}}) and ed_{2}(PGSp_{2^{r}}). □

The following Lemma 2.7(1) was proved by Rowen in [10, Theorem B] (see also [5, Proposition 16.16]) and Lemma 2.7(2) was proved by Serhir and Tignol in [11, Proposition]. We shall need the explicit forms of involutions on the decomposed quaternions as in (1):

Lemma 2.7. Let F be a field of characteristic different from 2. Let (A, σ) be a central simple F -algebra of degree 4 with a symplectic involution σ.

(1) If A is a division algebra, then we have

(A, σ) ≃ (Q, σ|_{Q}) ⊗ (Q′, γ),

where σ|_{Q} is an orthogonal involution defined by σ|_{Q}(x_{0} + x_{1}i + x_{2}j + x_{3}k) = x_{0} + x_{1}i + x_{2}j − x_{3}k with a quaternion basis (1, i, j, k) for Q and γ is the canonical involution on a quaternion algebra Q′.

(2) If A is not a division algebra, then we have

(A, σ) ≃ (M_{2}(F), ad_{q}) ⊗ (Q′, γ),

where q is a 2-dimensional quadratic form, ad_{q} is the adjoint involution on M_{2}(F), and γ is the canonical involution on a quaternion algebra Q′.

Proof. (1) By [10, Proposition 5.3], we can choose a i ∈ A\F such that σ(i) = i and [F (i) : F] = 2. Let ϕ be the nontrivial automorphism of F (i) over F. By [10, Proposition 5.4], there is a j ∈ A\F such that σ(j) = j and ji = ϕ(i)j. Then i and j generate a quaternion algebra Q, σ|_{Q}(i) = i, σ|_{Q}(j) = j, and σ|_{Q}(k) = −k with k = ij. Hence, σ|_{Q} is an orthogonal involution on Q. By the double centralizer theorem, we have A ≃ Q ⊗ C_{A}(Q), where C_{A}(Q) is the centralizer of Q ⊂ A and is isomorphic to quaternion algebra Q′ over F. By [5, Proposition 2.23], the restriction of σ on Q′ is the canonical involution γ.

(2) See [11, Proposition]. □

Proposition 2.8. Let F be a field of characteristic different from 2. Then (1) ed_{2}(PGO_{4}^{+}) = ed(PGO_{4}^{+}) = 4.

(2) ed_{2}(PGO_{4}) = ed(PGO_{4}) = 4.

(3) ed_{2}(PGSp_{4}) = ed(PGSp_{4}) = 4.

Proof. (1) By the exceptional isomorphism (2), we have

PGO_{4}^{+} = PGL_{2} × PGL_{2} .

The proof follows from Lemma 2.5 with n = 2.

(2) By Lemma 2.6, we have ed_{2}(PGO_{4}) ≥ 4. For the opposite inequality, we consider the affine variety X defined in 𝔸_{F}^{4} with the coordinates (a, b, c, e) by e(a^{2} − b^{2}e)(c^{2} − e) ≠ 0. Define a morphism X →A_{1}^{2} by

(a, b, c, e) ↦ \begin{cases}(a + b\sqrt{e}, c + \sqrt{e}) & if F (\sqrt{e}) is a quadratic field extension, \\ (a, b) × (c, \sqrt{e}) & otherwise, \end{cases}

We show that X is a classifying variety for A_{1}^{2}. Let Q = (a + b\sqrt{e}, c + d\sqrt{e}) be a quaternion algebra over a quadratic extension L = F (\sqrt{e}). If b = d = 0, we can modify c by a norm of L(\sqrt{a})/L, hence we may assume that d ≠ 0. Similarly, we can assume that d = 1, replacing e by ed^{2}. Thus, the morphism X → A_{1}^{2} is surjective. By (5), ed(A_{1}^{2}) ≤ 4. Hence, the opposite inequality ed(PGO_{4}) ≤ 4 comes from the exceptional isomorphism (1) and the canonical bijection (3).

(3) By Lemma 2.6, we have ed_{2}(PGSp_{4}) ≥ 4. For the opposite inequality, we define a morphism 𝔾_{m}^{4} → C_{2} by

(x, y, z, w) ↦ \begin{cases}((x, y), σ) ⊗ ((z, w), γ) & if x ≠ 1, \\ (M_{2}(F), ad_{q}) ⊗ ((z, w), γ) & if x = 1, \end{cases}

where σ is an involution defined by σ(x_{0}+x_{1}i+x_{2}j+x_{3}k) = x_{0}+x_{1}i+x_{2}j−x_{3}k with a quaternion basis (1, i, j, k) of the quaternion algebra (x, y), q = ⟨1, y⟩ is a quadratic form, and γ is the canonical involution on the quaternion algebra (z, w). Note that multiplying any two dimensional quadratic form by a scalar does not change the adjoint involution. By Lemma 2.7, this morphism is surjective. Therefore, by (5), we have ed(C_{2}) ≤ 4, hence the result follows from the canonical bijection (4).

□

Remark 2.9.

(1) Assume that F is a field of characteristic 2. By [1, Corollary 2.2], we have ed_{2}(Alg_{4,2}) = ed_{2}(Dec_{4}) ≥ 3. As the morphisms (7) and (8) are surjective, we get ed_{2}(PGO_{4}^{+}) ≥ 3 and ed_{2}(PGO_{4}) ≥ 3, respectively. On the other hand, the upper bounds in Proposition 2.8 (1) and (2) still hold, hence 3 ≤ ed(PGO_{4}^{+}), ed(PGO_{4}) ≤ 4.

(2) As ed(O_{5}^{+}) = 4 by [9, Theorem 10.3], Proposition 2.8 (3) can be recovered from the exceptional isomorphism PGSp_{4} ≃ O_{5}^{+}.

References

[1] S. Baek, Essential dimension of simple algebras in positive characteristic, C. R. Math. Acad. Sci. Paris 349 (2011) 375–378.

[2] S. Baek and A. Merkurjev, Invariants of simple algebras, Manuscripta Math. 129 (2009), no. 4, 409–421.

[3] S. Baek and A. Merkurjev, Essential dimension of central simple algebras, to appear in Acta Math.

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

[5] M.-A. Knus, A. Merkurjev, M. Rost, and J.-P. Tignol, The book of involutions, American Mathematical Society, Providence, RI, 1998, With a preface in French by J. Tits.

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

[7] M. L. MacDonald, Cohomological invariants of odd degree Jordan algebras, Math. Proc. Cambridge Philos. Soc. 145 (2008), no. 2, 295–303.

[8] A. S. Merkurjev, Essential dimension, Quadratic forms—algebra, arithmetic, and geometry, Contemp. Math., vol. 493, Amer. Math. Soc., Providence, RI, (2009), pp. 299–325.

[9] Z. Reichstein, On the notion of essential dimension for algebraic groups, Transform. Groups 5 (2000), no. 3, 265-304.

[10] L. Rowen, Central simple algebras, Israel J. Math. 29 (1978), no. 2-3, 285–301.

[11] A. Serhir and J.-P. Tignol, The discriminant of a decomposable symplectic involution, J. Algebra 273 (2004), no. 2, 601–607.

Department of Mathematics and Statistics, University of Ottawa, 585 King Edward, Ottawa, ON K1N6N5, Canada

E-mail address: sbaek@uottawa.ca