ESSENTIAL DIMENSION OF INVOLUTIONS AND SUBALGEBRAS

ROLAND LÖTSCHER

Abstract. We use theorems of N. Karpenko about the incompressibility of Severi Brauer varieties and quadratic Weil transfers of Severi Brauer varieties to compute the minimal number of parameters needed to define conjugacy classes of involutions of given type on division algebras of 2-primary index. Similarly we compute the minimal number of parameters needed to define conjugacy classes of étale subalgebras of given type and central simple subalgebras of given degree in division algebras of p-primary index.

Contents

1. Introduction 1 2. A general strategy 3 3. Involutions 5 4. Subalgebras 7 4.1. Étale subalgebras 8 4.2. Central simple subalgebras 9 5. Essential dimension of related group schemes 10 Acknowledgments 12 References 12

1. Introduction

Essential dimension is a certain numerical invariant of algebraic objects measuring their complexity. Roughly, it is defined as the minimal number of independent parameters needed to define algebraic objects of a given type, e.g. degree n central simple algebras, n-dimensional quadratic forms, torsors of an algebraic group etc.

Essential dimension was introduced by J. Buhler and Z. Reichstein around 1995 when studying Tschirnhaus transformations of polynomials in one variable. They defined the essential dimension of a finite group G (relative to a base field F) in invariant theoretic terms and showed that the essential dimension of the symmetric group G = S_{n} can be understood as the minimal number of algebraically independent parameters in a polynomial p(x) = x^{n} + a_{1}x^{n−1} + … + a_{n} obtained by means of Tschirnhaus transformations from the generic degree n polynomial (where a_{1}, …, a_{n} are algebraically independent over F). Later Reichstein extended this notion to algebraic groups G and showed, that this concept can be used to measure the complexity of algebraic objects like quadratic forms (when G = O_{n}), central simple algebras (when G = PGL_{n}), octonion algebras (when G = G_{2}) etc.

Date: October 20, 2010.

2000 Mathematics Subject Classification. 16W10, 16K20.

Key words and phrases. essential dimension, involutions, central simple algebras, canonical dimension, Severi-Brauer variety, étale algebras.

The most general definition of essential dimension, due to A. Merkurjev, assigns to each (covariant) functor F: Fields/F → Sets a non-negative integer, called essential dimension of F. We refer to [BF03] and [Me09] for its definition. Briefly, the essential dimension ed F of the functor F is less or equal to n ∈ ℕ if and only if every element a ∈ F(L) (where L ∈ Fields/F) can be defined over a subfield L_{0} ∈ Fields/F of transcendence degree at most n.

To each functor F: Fields/F → Sets we can associate its detection functor D_{F}: Fields/F → Sets, defined by

D_{F}(L) = \begin{cases}∅ & if F(L) = ∅ \\ \{L\} & otherwise. \end{cases} .

The essential dimension of D_{F} is called canonical dimension of F, denoted cdim F. The concept of canonical dimension was introduced by G. Berhuy and Z. Reichstein [BR05]. See also [Me09]. It is easy to see that cdim F ≤ ed F. The fields L ∈ Fields/F with F(L) ≠ ∅ are called splitting fields of F. The functor F has canonical dimension ≤ n if and only if every splitting field L of F contains a subfield L_{0} ∈ Fields/F of transcendence degree ≤ n that splits F as well.

There is a variant of essential (and canonical) dimension relative to a prime number p, called essential p-dimension (resp. canonical p-dimension). Essential pdimension measures the complexity of algebraic objects up to prime to p extensions. We refer to [Me09] for its definition.

If G is a group scheme (always assumed of finite type) over F the essential dimension (resp. p-dimension) of G is defined as

ed G = ed H^{1}(−, G), resp. ed_{p} G = ed_{p} H^{1}(−, G),

where H^{1}(−, G) is the flat cohomology functor, which takes a field extension L/F to the set of isomorphism classes of G-torsors over L (in the finitely presented faithfully flat topology).

A variety X over F can be viewed as functor of points

X: Fields/F → Sets, L ↦ X(L) = Mor(Spec L, X).

The essential dimension and p-dimension of X are easy to compute: They are equal to the dimension of X, see [BF03, Proposition 1.17], [Me09, Proposition 1.2]. On the other hand, the canonical dimension (or p-dimension) of X is an interesting invariant of X in case that X does not have F -rational points. Interesting examples include quadrics, Severi-Brauer varieties and torsors of an algebraic group G.

In this paper we are going to make use of the following results due to N. Karpenko:

Theorem 1.1 ([Ka00, Ka09]). Let D be a central division K-algebra of degree p^{r} for some r ≥ 0 and a prime p. Then

cdim SB(D) = cdim_{p} SB(D) = dim D = p^{r} − 1.

Moreover if K/F is a quadratic separable field extension, p = 2, and the norm algebra N_{K/F} (D) is split, then

cdim R_{K/F} (SB(D)) = cdim_{2} R_{K/F} (SB(D)) = dim R_{K/F} (SB(D)) = 2 · (2^{r} − 1).

We will adopt and freely use the notations and conventions of [KMRT98] with just a few exceptions, that will be made explicit. Most significantly H^{1}(−, G) (and, when G is abelian, H^{2}(−, G)) will denote flat cohomology rather than Galois cohomology. We refer to [Wa79, §17, 18] for basics on flat cohomology. The difference only shows up for non-smooth group schemes G. For example the flat cohomology set H^{1}(F, GO(M_{n}(F), τ)) (where τ is transposition of matrices) stands in bijection with the set of (all) conjugacy classes of orthogonal involutions σ on M_{n}(F), whereas the Galois cohomology set H_{Gal}^{1}(F, GO(M_{n}(F), τ)) consists only of those classes, for which σ and τ become conjugate over F_{sep}. If F_{sep} is not quadratically closed, this is a non-trivial restriction.

The rest of this paper is organized as follows: In section 2 we describe our strategy of applying Karpenko’s incompressibility results to the computation of the exact values of the essential dimension (and p-dimension) of certain group schemes. In section 3 we apply this strategy to the essential dimension of conjugacy classes of involutions on central simple algebras. In section 4 we consider the essential dimension of conjugacy classes of certain subalgebras of a central simple algebra. We do this in two particular cases. In the first case we consider étale subalgebras of a given type and in the second case we consider central simple subalgebras of given degree. Our main results are given in Theorems 3.1, 4.1 and 4.2.

In section 5 we give close (in certain cases exact) estimates of the essential dimension of group schemes which are related to the previously considered ones.

2. A general strategy

Let E/F be an étale F -algebra and A be an Azumaya E-algebra. We have an exact sequence

1 → R_{E/F} (G_{m}) → R_{E/F} (GL_{1}(A)) →^{Int} Aut_{E}(A) → 1

of group schemes over F, where Int takes an element a of (A ⊗_{F} L)^{×} (for a field extension L/F) to the E ⊗_{F} L-algebra automorphism of A_{L} := A⊗_{F} L sending x to axa^{−1}. Let H be an arbitrary (closed) subgroup of Aut_{E}(A) and set G = Int^{−1}(H). Then we have a commutative diagram

[figure: 1; R_{E/F} (G_{m}); G; H; 1; 1; R_{E/F} (G_{m}); R_{E/F} (GL_{1}(A)); Int; Aut_{E}(A); 1]

of group schemes over F with exact rows, where the vertical arrows are closed embeddings.

Let L/F be a field extension. The set H^{1}(L, Aut_{E}(A)) stands in natural bijection with the set of isomorphism classes of Azumaya E ⊗_{F} L-algebras B which become isomorphic to A_{L} over a seperable closure L_{sep}. Its distinguished element is the class of A_{L}. The group H^{2}(L, R_{E/F} (G_{m})) is naturally isomorphic to the Brauer group of E ⊗_{F} L. The connection map

δ^{1} : H^{1}(L, Aut_{E}(A)) → H^{2}(L, R_{E/F} (G_{m})) = Br(E ⊗_{F} L)

takes the isomorphism class of an Azumaya algebra B to the Brauer class of B⊗_{E⊗_{F} L}A_{L}^{op} = B⊗_{E} A^{op}. Take an H-torsor T over some field extension L of F. Let B be an Azumaya L⊗_{F} E-algebra representing the image of T in H^{1}(L, Aut_{E}(A)). Let

X = R_{E⊗_{F} L/L}(SB(B ⊗_{E} A^{op})).

The following result will be our main tool for computing essential and canonical dimensions of algebraic groups.

Proposition 2.1.

cdim X − dim H ≤ ed G ≤ dim_{F} A − dim G

and

cdim_{p} X − dim H ≤ ed_{p} G ≤ ed G

for every prime p. In particular if X is incompressible (resp. p-incompressible) then ed G = dim_{F} A − dim G (resp. ed_{p} G = ed G = dim_{F} A − dim G).

Proof. We will first show the inequality cdim X−dim H ≤ ed G, by adapting [Me09, Theorem 4.8] (cf. [BRV08, Corollary 4.2]) to our situation.

The splitting fields of X are precisely the field extensions M of L for which T_{M} lifts to a G-torsor. Let M be a splitting field and let E be a G-torsor lifting T_{M}, i.e., such that E ∗_{G} H ≃ T_{M}. Let E_{0} be a G-torsor over an intermediate field L ⊆ M_{0} ⊆ M such that tdeg_{L} M_{0} ≤ ed G_{L} ≤ ed G and E_{0} ×_{M_{0}} M ≃ E. The induced H-torsor T_{0} = E_{0}∗_{G}H and the H-torsor T_{M_{0}} do not need to be isomorphic. However they become isomorphic over an intermediate field M_{0} ⊆ M_{1} ⊆ M of transcendence degree tdeg_{M_{0}} M_{1} ≤ dim H. The field M_{1} is a splitting field of X. Since tdeg_{L} M_{1} ≤ ed G + dim H this proves the inequality cdim X − dim H ≤ ed G. The inequality cdim_{p} X − dim H ≤ ed_{p} G is obtained with standard modifications of the above argument.

The inequality ed G ≤ dim_{F} A − dim G follows from [Me09, Corollary 4.3] and the fact that G embeds in the special group scheme R_{E/F} (GL_{1}(A)) of dimension dim_{F} A. The inequalites ed_{p} G ≤ ed G and cdim_{p} H ≤ cdim H ≤ dim H are clear. For the last claim note that dim X − dim H = dim_{F} A − dim G. □

Interesting subgroups of Aut_{E}(A) are actually easy to find: Just endow A with some additional structure (an involution, a quadratic pair, an étale or central simple subalgebra in our examples) and take the subgroup of Aut_{E}(A) consisting of Ealgebra automorphisms preserving that additional structure.

Our goal now is to find interesting examples where X = R_{E⊗_{F} L/L}(SB(B⊗_{E}A^{op})) is in fact p-incompressible. We will do this in case that E/F is either a separable quadratic field extension (and p = 2) or E = F (and p is arbitrary). We assume that deg A is a power of p. Note that in these cases in view of Theorem 1.1 all we need for p-incompressibility of X is that D = B ⊗_{E} A^{op} is a division algebra and that N_{E⊗_{F} L/L}(D) splits when [E : F] = 2.

We fix a prime p and use the following notation (which slightly differs from the notation used in [KMRT98]):

Definition 2.1. Let L be a field.

Case char L ≠ p: Let ζ be a primitive pth root of unity in L (we assume its existence) and let a, b ∈ L^{×}. We denote by (a, b)_{L} the cyclic L-algebra generated by symbols u and v with relations u^{p} = a, v^{p} = b and vu = ζuv. Case char L = p: Let a, b ∈ L with b invertible. We denote by (a, b)_{L} the cyclic L-algebra generated by symbols u and v with relations u^{p} − u = a, v^{p} = b and vu = uv + u.

Note that the definition of (a, b)_{L} in the first case depends on the choice of ζ. For us this choice will not matter, hence we do not include it in the notation. Different choices of ζ for different algebras (a, b)_{L} will be fine for us, too. The following lemma will help us to produce examples where D = B ⊗_{E} A^{op} is a division algebra:

Lemma 2.1. Let a_{1}, b_{1}, …, a_{r}, b_{r} be algebraically independent variables over F and let L = F′(a_{1}, b_{1}, …, a_{r}, b_{r}), where F′ is F adjoined a primitive pth root of unity if char F ≠ p resp. F′ = F if char F = p. Let

B_{0} = (a_{1}, b_{1})_{L} ⊗_{L} (a_{2}, b_{2})_{L} ⊗_{L} ⋯ ⊗_{L} (a_{r}, b_{r})_{L}.

Let E/F be a (finite) separable field extension which is linearly disjoint from F′ and let A be a central simple E-algebra of p-power degree. Set M = E ⊗_{F} L = (E ⊗_{F} F′)(a_{1}, b_{1}, …, a_{r}, b_{r}). Then B_{0} ⊗_{F} A^{op} is a central simple M-algebra with

ind B_{0} ⊗_{F} A^{op} = p^{r} ind A.

Proof. First of all note that ind A ⊗_{F} F′ = ind A since F′/F has degree prime to p. Hence we may replace F′ by F, A by A ⊗_{F} F′ and E by the field E ⊗_{F} F′ and hence assume F′ = F. Using induction on r ≥ 0 one easily reduces to the case r = 1. The claim then follows from the index formula given in [Me10, (11) §4.1] (cf. [JW90, Prop. 1.15(a)] and [Ti78, Prop. 2.4]). □

We summarize this section in a corollary, which will later be invoked for various choices of group schemes H.

Corollary 2.1. Let either E = F or E/F be a separable quadratic field extension and let A be a division E-algebra of degree p^{r}. If [E : F] = 2 assume that p = 2 and that N_{E/F} (A) splits. Let L/F and B_{0} be as in Lemma 2.1 and set B = B_{0} ⊗_{F} E.

Assume that the class of B lies in the image of H^{1}(L, H) → H^{1}(L, Aut_{E}(A)) for a subgroup H of Aut_{E}(A). Then

ed G = ed_{p} G = dim_{F} A − dim G,

where G is the inverse image of H under Int: R_{E/F} (GL_{1}(A)) → Aut_{E}(A).

Proof. Lemma 2.1 implies that B ⊗_{E} A^{op} = B_{0} ⊗_{F} A^{op} is a division algebra. In case [E : F] = 2 note that N_{E⊗_{F} L/L}(B ⊗_{E} A^{op}) splits since N_{E/F} (A) and N_{E⊗_{F} L/L}(B) split. Theorem 1.1 shows that X = R_{E⊗_{F} L/L}(SB(B ⊗_{E} A^{op})) is p-incompressible. Hence the claim follows from Proposition 2.1. □

3. Involutions

Let A be a central simple E-algebra admitting an involution σ. Let F = E^{σ} ⊆ E be the fixed field of central elements under σ. We call (A, σ) a central simple F - algebra with involution. For unitary involutions we include the case E = F × F in this definition by allowing A to be a direct product A = A_{1} × A_{2} with A_{1} and A_{2} central simple F -algebras of the same degree.

Two involutions σ′ and σ on A are said to be conjugate if there exists a ∈ A such that σ′ = Int(a) ∘ σ ∘ Int(a)^{−1}. Equivalently (A, σ) and (A, σ′) are isomorphic as F -algebras with involution.

Conjugate involutions are necessarily of the same type (unitary, symplectic or orthogonal). When F is algebraically closed there are precisely 3 conjugacy classes of involutions on A = M_{n}(E), one for each type.

For the case of characteristic 2 we will also use the notion of quadratic pair which extends the distinction between quadratic forms and symmetric bilinear forms to non-split central simple algebras. A quadratic pair on a central simple F -algebra A is given by a couple (σ, f) where σ is a symplectic involution (we assume char F = 2) and f: Sym(A, σ) → F is an F -linear map subject to the condition f(x + σ(x)) = Trd_{A}(x) for all x ∈ A.

We want to compute the essential dimension of (conjugacy classes) of involutions and of quadratic pairs. These are the essential dimensions of the functors UInv_{A}, OInv_{A}, SInv_{A}, QPairs_{A}: Fields/F → Sets which takes a field extension L/F to the set of conjugacy classes of unitary L-linear involutions, resp. orthogonal involutions, symplectic involutions and quadratic pairs on A_{L} = A ⊗_{F} L.

Example 3.1. Let Q = (a, b)_{L} be a quaternion algebra and denote by γ the canonical involution on Q. Let L/F be a field extension. The only symplectic involution on Q_{L} is γ_{L}. Hence ed SInv_{Q} = 0.

Every orthogonal involution σ on Q_{L} is of the form σ = Int(s) ∘ γ_{L} for some s ∈ Skew(Q_{L}, γ_{L}) \ L. The conjugacy class of σ is determined uniquely by the discriminant of σ [KMRT98, Example 7.4], given by disc σ = s^{2}·(L^{×})^{2} ∈ L^{×}/(L^{×})^{2}. When Q is split every element of L^{×} is the square of some s ∈ Skew(Q_{L}, γ_{L}) \ L, hence OInv_{Q} is isomorphic to the functor H^{1}(−, μ_{2}): L ↦ L^{×}/(L^{×})^{2}. This implies that ed_{2} OInv_{Q} = ed OInv_{Q} = 1 for a split quaternion algebra Q. We will see that if Q is non-split then ed_{2} OInv_{Q} = ed OInv_{Q} = 2.

Theorem 3.1. Let n be a power of 2 and A be a division algebra of degree n over its center. Then:

(a) Assume that the center K of A is a quadratic separable field extension of F and N_{K/F} (A) is split (i.e., there exist F -linear unitary involutions on A). Then

ed UInv_{A} = ed_{2} UInv_{A} = n^{2} − 1.

(b) Assume that Z(A) = F and A ⊗_{F} A is split (i.e., there exist orthogonal involutions on A). Then

ed OInv_{A} = ed_{2} OInv_{A} = \frac{n(n + 1)}{2} − 1.

(c) Assume that Z(A) = F, A ⊗_{F} A is split and n > 1 (i.e., there exist symplectic involutions on A). Then

ed SInv_{A} = ed_{2} SInv_{A} = \frac{n(n − 1)}{2} − 1.

(d) Assume that char F = 2, Z(A) = F, A ⊗_{F} A is split and n > 1 (i.e., there exist quadratic pairs on A). Then

ed QPairs_{A} = ed_{2} QPairs_{A} = \frac{n(n + 1)}{2} − 1.

Proof. The functors UInv_{A}, OInv_{A} and SInv_{A} when σ is a unitary F -linear, orthogonal, resp. symplectic involution are isomorphic to the functors

H^{1}(−, Sim(A, σ)): Fields/F → Sets,

where Sim(A, σ) is the group scheme of similitudes of (A, σ), see [KMRT98, §29.D]. Similarly the functor QPairs_{A} is isomorphic to the functor H^{1}(−, Sim(A, σ, f)) when (σ, f) is a quadratic pair on A. Notice that

dim Sim(A, σ) = \begin{cases}n^{2} + 1 & if σ is unitary, \\ \frac{n(n−1)}{2} + 1 & if σ is orthogonal, \\ \frac{n(n+1)}{2} + 1 & if σ is symplectic. \end{cases}

and

dim Sim(A, σ, f) = \frac{n(n − 1)}{2} + 1.

So in fact we need to show that

ed Sim(A, σ) = ed_{2} Sim(A, σ) = dim_{F} A − dim Sim(A, σ)

for an involution σ (of arbitrary type) on A, and similarly we must show that

ed Sim(A, σ, f) = ed_{2} Sim(A, σ, f) = dim_{F} A − dim Sim(A, σ, f).

The group Aut_{K}(A, σ) naturally embeds in Aut_{K}(A) and Sim(A, σ) is the inverse image of Aut_{K}(A, σ) under Int: R_{K/F} (GL_{1}(A)) → Aut_{K}(A). The image of the map

H^{1}(L, Aut_{K}(A, σ)) → H^{1}(L, Aut_{K}(A))

consists of those isomorphism classes of Azumaya K ⊗_{F} L-algebras which admit an L-linear involution of the same type as σ. Similarly the image of the map H^{1}(L, Aut_{F} (A, σ, f)) → H^{1}(L, Aut_{F} (A)) consists of those isomorphism classes of central simple L-algebras that admit a quadratic pair. In view of Corollary 2.1 it suffices to show that the algebra B = (a_{1}, b_{1})_{L} ⊗_{L} ⋯ ⊗_{L} (a_{r}, b_{r})_{L} ⊗_{F} K from Corollary 2.1 admits an involution of the same type as σ, resp. a quadratic pair. This is easy to see directly, and can also be proven by looking at B ⊗_{F} B (when K = F), resp. N_{K/F} (B) (when [K : F] = 2). Hence the claim follows. □

4. Subalgebras

Let A be a central simple F -algebra and E be a subalgebra of A. We call (A, E) a couple of F -algebras. If (A′, E′) is another couple of F -algebras we say that (A, E) and (A′, E′) are isomorphic if there exists an isomorphism φ: A →^{∼} A′ of F -algebras which restricts to an isomorphism E →^{∼} E′. We call two subalgebras E and E′ of A conjugate if (A, E) and (A, E′) are isomorphic. Equivalently there exists a ∈ A^{×} such that E′ = aEa^{−1}.

Let G be the inverse image of H = Aut_{F} (A, E) under Int: GL_{1}(A) → Aut_{F} (A). It is the normalizer of GL_{1}(E) in GL_{1}(A). We will compute the essential dimension of G in some examples. For this section we use [KMRT98, §29.C] as a reference for cohomological statements. We start with an additional result, which gives us an interpretation of the functor H^{1}(−, G):

Lemma 4.1. The set H^{1}(L, G) is naturally (in L/F) isomorphic to the set of conjugacy classes of L-subalgebras E′ of A_{L} such that E ⊗_{F} L_{alg} and E′ ⊗_{L} L_{alg} are conjugate subalgebras of A ⊗_{F} L_{alg} = A_{L} ⊗_{L} L_{alg}.

Proof. Let L/F be a field extension. The set H^{1}(L, Aut_{F} (A, E)) is in one-to-one correspondence with the set of F -isomorphism classes of couples of L-algebras (A′, E′) such that (A′, E′)_{alg} ≃ (A_{L}, E_{L})_{alg}.

We have an exact sequence 1 → G_{m} → G → Aut_{F} (A, E) → 1. The map H^{1}(L, G) → H^{1}(L, Aut_{F} (A, E)) is injective, since H^{1}(L, G_{m}) is trivial and G_{m} lies in the center of G. Hence H^{1}(L, G) is naturally (in L/F) bijective to the kernel of the connecting map H^{1}(L, Aut_{F} (A, E)) → H^{2}(L, G_{m}) = Br(L), which takes a couple (A′, E′) to the Brauer class of A′ ⊗_{F} A^{op}. Thus H^{1}(L, G) ist the set of isomorphism classes of couples of L-algebras (A′, E′) such that A′ ≃ A_{L} and (A′, E′)_{alg} ≃ (A_{L}, E_{L})_{alg}. The map which associates to the conjugacy class of an Lsubalgebra E′ on A_{L} the isomorphism class of the couple (A_{L}, E′) then establishes a (well defined and in L/F natural) bijection between the set of conjugacy classes of L-subalgebras E′ of A_{L} such that E′ ⊗_{L} L_{alg} is conjugate to E ⊗_{F} L_{alg} and the set H^{1}(L, G). □

4.1. Étale subalgebras. In this subsection we try to compute the essential dimension of the functor Fields/F → Sets taking a field L to the set of conjugacy classes of étale subalgebras E of a given type of A_{L}, where A is a fixed central simple F -algebra.

The type of an étale subalgebra E of a central simple algebra B is defined as follows. Denote by L the center of B and let e_{1}, …, e_{d} be the minimal non-zero idempotents of E ⊗_{F} L_{sep}, where d = dim E. Write B_{sep} = B ⊗_{L} L_{sep}. Let r_{i} = rdim e_{i}B_{sep} = \frac{dim e_{i}B_{sep}}{deg B} ∈ ℕ be the reduced dimension of the right ideal e_{i}B_{sep}. Then the multiset t_{E} = [r_{1}, …, r_{d}] is uniquely determined by E and is invariant under conjugation (of étale subalgebras) and scalar extension. The invariant t_{E} is a partition of deg B, i.e., ∑_{i=1}^{d} r_{d} = deg B. It was introduced by D. Krashen [Kr04], who called it the type of the étale subalgebra E of B.

In case A = M_{n}(F) there exists for every partition t = [r_{1}, …, r_{d}] of deg A = n an étale subalgebra E of A of type t_{E} = t. Moreover over a separably closed field F every étale subalgebra of A = M_{n}(F) of given type t = [r_{1}, …, r_{d}] is conjugate to the étale subalgebra E = ⨁_{i=1}^{d} F d_{i} where d_{i} is the diagonal matrix with ones from place s_{i} + 1 := ∑_{j=1}^{i−1} r_{j} + 1 up to place s_{i+1} and zeros on the other places. This follows from the fact that commuting idempotent matrices are simultaneously diagonalizable.

Consider the functor Ét_{A}^{(t)} : Fields/F → Sets, which takes a field extension L/F to the set of conjugacy classes of étale subalgebras of A_{L} of type t.

Example 4.1. Let A be a central simple algebra of degree n and let t = [1, …, 1]

(with n ones). Then the functors Ét_{A}^{(t)} is isomorphic to the functor taking a field L ∈ Fields/F to the set of isomorphism classes of n-dimensional étale L-algebras that embed in A_{L}. In fact two n-dimensional étale subalgebras of A_{L} are conjugate if and only if they are isomorphic.

We will consider the case when A is a division algebra. Every étale subalgebra E of a division algebra is a field and d = dim_{F} E divides n = deg A. Moreover every (étale) subfield of dimension d has type [\frac{n}{d}, …, \frac{n}{d}], since the absolute Galois group of F permutes the idempotents in E_{sep} = ⨁_{i=1}^{d} F_{sep}e_{i} transitively.

Theorem 4.1. Let A be a division F -algebra of degree n = deg A = p^{r} a power of p. Let d = p^{s} divide n and set t = [\frac{n}{d}, …, \frac{n}{d}] (the multiset with d identical elements). Assume that A admits an étale subalgebra of type t. Then

edÉt _{A}^{(t)} = ed_{p} Ét _{A}^{(t)} = n^{2} − \frac{n^{2}}{d} = p^{2r} − p^{2r−s}.

Proof. Let E be an étale subalgebra of A of type t. Let H = Aut_{F} (A, E) and G = Int^{−1}(H) its inverse image under Int: GL_{1}(A) → Aut_{F} (A). It is the normalizer of

the torus R_{E/F} (G_{m}) in GL_{1}(A). By Lemma 4.1 the functors H^{1}(−, G) and Ét_{A}^{(t)} are isomorphic.

We can pass to an algebraic closure in order to compute the dimension of G. Then G is the normalizer of R_{E/F} (G_{m}) in GL_{n} and G^{0} is the centralizer of R_{E/F} (G_{m}) in GL_{n}. We may replace E by the conjugate subalgebra E′ = ∑_{i} F d_{i}, where d_{i} is the diagonal matrix with ones from position (i−1)p^{r−s}+1 to ip^{r−s} and zeros otherwise. Then G^{0} = GL_{p^{r−s}} × ⋯ × GL_{p^{r−s}} ⊆ GL_{n} has dimension p^{s} · p^{2(r−s)} = p^{2r−s}. It follows that dim G = p^{2r−s}.

The image of H^{1}(L, Aut_{F} (A, E)) → H^{1}(L, Aut_{F} (A)) consists of those isomorphism classes of central simple L-algebras that admit an étale subalgebra of type t. In view of Corollary 2.1 it suffices to show that the division F -algebra B = (a_{1}, b_{1})_{L} ⊗_{L} ⋯ ⊗_{L} (a_{r}, b_{r})_{L} from Corollary 2.1 admits an étale subalgebra of type t. This is true, since we can take for E the tensor product E_{1} ⊗_{L} ⋯ ⊗_{L} E_{s} where E_{1}, …, E_{s} are maximal subfields of (a_{1}, b_{1})_{L}, …, (a_{s}, b_{s})_{L} respectively. □

Remark 4.1. In the extreme cases t = [1, …, 1] and t = [n] we can also compute the

essential p-dimension of Ét_{A}^{(t)} for a split algebra A = M_{n}(F) of degree n assuming

char F ≠ p: We have ed_{p} Ét_{M_{n}(F)}^{[n]} = 0 and

ed_{p} Ét_{M_{n}(F)}^{[1,...,1]} = ed_{p} S_{n} = ⌊\frac{n}{p}⌋

The first statement is trivial since the only 1-dimensional étale subalgebra of A is F. The functors Ét_{M_{n}(F)}^{[1,...,1]} and H^{1}(−, S_{n}) are isomorphic by Example 4.1, since every

n-dimensional étale algebra embeds in M_{n}(F). This shows ed_{p} Ét_{M_{n}(F)}^{[1,...,1]} = ed_{p} S_{n}. The equality ed_{p} S_{n} = ⌊\frac{n}{p}⌋ (when char F ≠ p) was first noticed by J.-P. Serre, see [MR08, Corollary 4.2].

In contrast the (absolute) essential dimension of Ét_{M_{n}(F)}^{[1,...,1]}, being equal to the essential dimension of S_{n}, is unknown for n ≥ 8.

4.2. Central simple subalgebras. Let A be a central simple F -algebra and let d be a divisor of deg A. Consider the functor Sub_{A}^{(d)} : Fields/F → Sets, which takes a field extension L/F to the set of isomorphism classes of central simple L-subalgebras of A_{L} of degree d.

Example 4.2. When deg A is divisible by d^{2} and ind A divides \frac{deg A}{d^{2}} the functor Sub_{A}^{(d)} is isomorphic to the functor CSA_{d}, taking a field extension L/F to the set of isomorphism classes of central simple L-algebras of degree d. In fact every central simple algebra B over L of degree d is a subalgebra of B ⊗_{L} B^{op} ≃ M_{d^{2}}(L), hence of A_{L}.

Theorem 4.2. Let n = p^{r} for some r ≥ 0 and let A be a division algebra of degree n. Let d = p^{s} be a divisor of n and assume that A admits a central simple subalgebra of degree d. Then

ed Sub_{A}^{(d)} = ed_{p} Sub_{A}^{(d)} = n^{2} − d^{2} − \frac{n^{2}}{d^{2}} + 1 = p^{2r} − p^{2s} − p^{2(r−s)} + 1.

Proof. Let B be a central simple subalgebra of A of degree d. Set H = Aut_{F} (A, B) and G = Int^{−1}(H) the inverse image of H under Int: GL_{1}(A) → Aut_{F} (A). It is the normalizer of GL_{1}(B) in GL_{1}(A).

I claim that Sub_{A}^{(d)} is isomorphic to H^{1}(−, G). In fact by Lemma 4.1 the set H^{1}(L, G) is in natural one-to-one correspondence with the set of conjugacy classes of central simple subalgebras B′ of A_{L} such that B′ ⊗_{L} L_{sep} and B ⊗_{F} L_{sep} are conjugate subalgebras of A ⊗_{F} L_{sep}. By the Skolem Noether theorem two central simple subalgebras of a central simple algebra are conjugate if and only if they are isomorphic. Hence the claim follows.

Since the division F -algebra B = (a_{1}, b_{1})_{L} ⊗_{L} ⋯ ⊗_{L} (a_{r}, b_{r})_{L} from Corollary 2.1 admits a central simple subalgebra of degree d we obtain

ed G = ed_{p} G = n^{2} − dim G

as in the proof of Theorem 4.1.

Let C denote the centralizer of B in A. Then G fits into an exact sequence

1 → GL_{1}(C) → G → Aut_{F} (B) → 1.

It follows, that G has dimension

dim G = \frac{n^{2}}{d^{2}} + d^{2} − 1.

This concludes the proof. □

5. Essential dimension of related group schemes

In section 3 we have computed the essential dimension of the group scheme Sim(A, σ) for certain algebras with involution (A, σ). The essential dimension of the group scheme Iso(A, σ) of isometries of (A, σ) is closely related:

Proposition 5.1. Let (A, σ) be a central simple F -algebra with involution. Then ed Iso(A, σ) ≤ ed Sim(A, σ) + 1. Moreover if σ is unitary or char F ≠ 2 then ed Iso(A, σ) is either equal to ed Sim(A, σ) or ed Sim(A, σ) + 1.

Furthermore the analogous statements hold for essential p-dimension for every prime p.

Proof. The first inequality follows from [Me09, Corollary 4.3] and the fact that Iso(A, σ) has codimension 1 in Sim(A, σ).

If σ is unitary or if char F ≠ 2 there is a surjection of functors

H^{1}(−, Iso(A, σ)) ↠ H^{1}(−, Sim(A, σ))

(cf. [KMRT98, §29.D and Propositions 2.7 and 2.18]). Hence ed Sim(A, σ) ≤ ed Iso(A, σ) by [BF03, Lemma 1.9] and similarly ed_{p} Sim(A, σ) ≤ ed_{p} Iso(A, σ). □

For division algebras of degree a power of 2 we can say more:

Proposition 5.2. Let n = 2^{r} for some r ≥ 0 and let (A, σ) be a central simple F -algebra with involution, where A is a division algebra of degree n. Then

• If σ is unitary then n^{2} − 1 ≤ ed_{2} U(A, σ) ≤ ed U(A, σ) ≤ n^{2}.
• If σ is orthogonal then ed O(A, σ) = ed_{2} O(A, σ) = \frac{n(n+1)}{2}.
• If σ is symplectic then ed Sp(A, σ) = ed_{2} Sp(A, σ) = \frac{n(n−1)}{2}.
• If (σ, f) is a quadratic pair then ed O(A, σ, f) = ed_{2} O(A, σ, f) = \frac{n(n+1)}{2}.

Moreover in case r ≥ 2 (and char F = 2): ed O^{+}(A, σ, f) = ed_{2} O^{+}(A, σ, f) =

\frac{n(n+1)}{2} and ed GO^{+}(A, σ, f) = ed_{2} GO^{+}(A, σ, f) = \frac{n(n+1)}{2} − 1.

Remark 5.1. The assumption r ≥ 2 in the statements about GO^{+}(A, σ, f) and O^{+}(A, σ, f) is needed. When r = 1 the algebra A is a quaternion division algebra, A = (a, b)_{F}, σ its canonical involution, and I claim that ed GO^{+}(A, σ, f) = 0 < 2 and ed O^{+}(A, σ, f) = 1 < 3. In fact GO^{+}(A, σ, f) = R_{F (u)/F} (G_{m}) and O^{+}(A, σ, f) = R_{F (u)/F}^{(1)} (G_{m}) are tori. Their essential dimension and 2-dimension were computed in [LMMR10] (cf. [BF03, Theorem 2.5]).

Proof. The statements about U(A, σ) follow from Theorem 3.1 and Proposition 5.1.

For orthogonal and symplectic involutions the inequalities ed_{2} O(A, σ) ≤ ed O(A, σ) ≤ \frac{n(n+1)}{2} and ed_{2} Sp(A, σ) ≤ ed Sp(A, σ) ≤ \frac{n(n−1)}{2} are clear. In order to show the reverse inequalities we slightly refine the approach given in section 2. We have a commutative diagram

[figure: 1; μ_{2}; Iso(A, σ); Aut(A, σ); 1; 1; G_{m}; Sim(A, σ); Aut(A, σ); 1]

with exact rows, where the columns are closed embeddings. By [Me09, Example 3.6 and Theorem 4.8] (cf. [BRV08, Corollary 4.2])

ed_{2} Iso(A, σ) ≥ ind(Iso(A, σ), μ_{2}) − dim Iso(A, σ),

where ind(Iso(A, σ), μ_{2}) is the maximal index of the image in H^{2}(L, μ_{2}) = Br_{2}(L) of an H-torsor, taken over all H-torsors over L and all field extensions L/F. From the above diagram it is clear that ind(Iso(A, σ, μ_{2}) = ind(Sim(A, σ), G_{m}). As shown in the proof of Theorem 3.1 the latter equals dim_{F} A. Since dim_{F} A − dim O(A, σ) = \frac{n(n+1)}{2} and dim_{F} A − dim Sp(A, σ) = \frac{n(n−1)}{2} the claim follows.

With a similar argument one gets ed O(A, σ, f) = ed_{2} O(A, σ, f) = \frac{n(n+1)}{2} for a quadratic pair (σ, f). The groups O^{+}(A, σ, f) and GO^{+}(A, σ, f) have finite index in O(A, σ, f) and GO(A, σ, f), respectively. Hence ed O^{+}(A, σ, f) ≤ \frac{n(n+1)}{2} and ed GO^{+}(A, σ, f) ≤ \frac{n(n+1)}{2} − 1. In order to show the reverse inequalities for r ≥ 2 we look at the diagram

[figure: 1; μ_{2}; O^{+}(A, σ, f); PGO^{+}(A, σ, f); 1; 1; G_{m}; GO^{+}(A, σ, f); PGO^{+}(A, σ, f); 1; 1; G_{m}; GL_{1}(A); Int; Aut_{F} (A); 1]

with exact rows. The image of the map H^{1}(L, PGO^{+}(A, σ, f)) → H^{1}(L, Aut_{F} (A)) consists of the classes of central simple L-algebras A′ of degree deg A′ = deg A which admit a quadratic pair (σ′, f′) such that disc(σ_{L}, f_{L}) = disc(σ′, f′). By [Be05, Theorem 2] (note that char L = 2) every central simple L-algebra A′ which is not a quaternion division algebra, and which has a symplectic involution, admits quadratic pairs of arbitrary discriminant. In particular the algebra B = (a_{1}, b_{1}) ⊗_{L} ⋯ ⊗_{L} (a_{n}, b_{n}) from Corollary 2.1 (with p = 2) admits quadratic pairs of arbitrary discriminant. Therefore the inequality ed_{2} GO^{+}(A, σ, f) ≥ \frac{n(n+1)}{2} − 1 is a consequence of Corollary 2.1 and the inequality ed_{2} O^{+}(A, σ, f) ≥ \frac{n(n+1)}{2} follows from [Me09, Example 3.6 and Theorem 4.8]. □

Remark 5.2. Suppose the following statement holds true: The essential dimension ed χ of every gerbe χ banded by R_{K/F}^{(1)} (G_{m}) is equal to cdim χ + 1 (for the notion of gerbes and their essential dimension see [BRV08]). Then one can show that ed U(A, σ) = n^{2} for a central division K-algebra A of degree n a power of 2 and a unitary F -linear involution σ on A.

Acknowledgments

I am grateful to Z. Reichstein for helpful comments.

References

[Be05] G. Berhuy, Erratum to: “On the set of discriminants of quadratic pairs” [JPAA 188 (2004) 33–44], J. Pure Appl. Algebra 195 (2005), 125–126.

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

[BR05] G. Berhuy, Z. Reichstein, On the notion of canonical dimension of algebraic groups, Adv. Math. 198 (2005), 128–171.

[BRV08] P. Brosnan, Z. Reichstein, A. Vistoli, Essential dimension of moduli of curves and other algebraic stacks, arxiv:0907.0924.v2 [math.AG] (October 6, 2009).

[Ka00] N.A. Karpenko, On anisotropy of orthogonal involutions. J. Ramanujan Math. Soc. 15(1) (2000), 1-22.

[Ka09] N.A. Karpenko, Incompressibility of quadratic Weil transfer of generalized Severi-Brauer varieties. Linear Algebraic Groups and Related Structures (preprint server) 362 (2009, Oct 28), 12 pages.

[KMRT98] M.-A. Knus, A. Merkurjev, M. Rost, and J.-P. Tignol, The book of involutions, American Math. Soc. Providence, RI, (1998), With a preface in French by J. Tits.

[Kr04] D. Krashen, Zero cycles on homogeneous varieties, Adv. Math. 223, (2010), no. 6, 2022– 2048.

[LMMR10] R. Lötscher, M. MacDonald, A. Meyer, and Z. Reichstein, Essential dimension of algebraic tori. Linear Algebraic Groups and Related Structures (preprint server) 399 (2010, Aug 17), 12 pages.

[Me09] A.S. Merkurjev, Essential dimension, in Quadratic forms – algebra, arithmetic, and geometry (R. Baeza, W.K. Chan, D.W. Hoffmann, and R. Schulze-Pillot, eds.), Contemporary Mathematics 493 (2009), 299–326.

[JW90] B. Jacob and A. Wadsworth, Division algebras over Henselian fields, J. Algebra 128 (1990), no. 1, 126–179.

[Me10] A.S. Merkurjev Essential dimension of PGL(p^{2}), J. Amer. Math. Soc. 23, (2010), 693–712.

[MR08] A. Meyer, Z. Reichstein The essential dimension of the normalizer of a maximal torus in the projective linear group, Algebra and Number Theory 3 (2009), No. 4, 467–487.

[Ti78] J.-P. Tignol, Sur les classes de similitude de corps à involution de degrée 8, C. R. Acad. Sci. Paris Sér. A-B 286 (1978), no. 20, A875-A876.

[Wa79] W.C. Waterhouse, Introduction to affine group schemes, Springer, New York (1979).

Roland Lötscher

Mathematisches Institut der Ludwig-Maximilians-Universität München, Theresienstraße 39, D-80333 München, Germany

   E-mail address: Roland.Loetscher@mathematik.uni-muenchen.de