Essential Dimension of Separable Algebras

Embedding in a Fixed Central Simple Algebra

To Alexander Merkurjev on His 60th Birthday

Roland Lötscher^{1}

Received: March 20, 2014

Abstract. In this paper we fix a central simple F -algebra A of prime power degree and consider separable algebras over extensions K/F, which embed in A_{K}. We study the minimal number of independent parameters, called essential dimension, needed to define these separable algebras. In case the index of A does not exceed a certain bound, the task is equivalent to the problem of computing the essential dimension of the algebraic groups (PGL_{d})^{m} ⋊ S_{m}, which is extremely difficult in general. In the other case, however, we manage to compute the exact value of the essential dimension of the given class of separable algebras, except in one case for A of index 2, which we study in greater detail.

2010 Mathematics Subject Classification: 16W10, 16K20 Keywords and Phrases: essential dimension, central simple algebras, separable algebra, étale algebras, non-split algebraic group

1. Introduction

Central simple algebras over fields are at the core of non-commutative algebra. Their history is rooted in the middle of the 19th century, when W. Hamilton discovered the quaternions over the real numbers. In the early 20th century J. Wedderburn gave a classification of finite dimensional semisimple algebras by means of division rings and subsequently R. Brauer introduced the Brauer group of a field, which lead to diverse research in algebra and number theory. Moreover central simple algebras and the Brauer group arise naturally in Galois cohomology and are therefore central for the theory of algebraic groups over fields. We refer to [2, 1] for surveys on these topics, including discussion of open problems.

1Partially supported by the Deutsche Forschungsgemeinschaft, GI 706/2-1

Documenta Mathematica · Extra Volume Merkurjev (2015) 443–459

Essential dimension is a more recent topic, introduced around 1995 by J. Buhler and Z. Reichstein [4] and in full generality by A. Merkurjev [3]. The essential dimension of a functor F: Fields_{F} → Sets from the category of field extensions of a fixed base field F to the category of sets is defined as the least integer n, such that every object a ∈ F(K) over a field extension K/F is defined over a subextension K_{0}/F of transcendence degree at most n. Here a ∈ F(K) is said to be defined over K_{0} if it lies in the image of the map F(K_{0}) → F(K) induced by the inclusion K_{0} → K. The functors F we are mostly interested in take a field extension K/F to the set of isomorphism classes of algebraic objects over K of some kind. The essential dimension of F is then roughly the number of independent paramters needed to define these objects. The essential dimension of an algebraic group G over a field F is defined as the essential dimension of the Galois cohomology functor

H^{1}(−, G): Fields_{F} → Sets, K ↦ H^{1}(K, G).

It is denoted by ed(G) and measures the complexity of G-torsors up to isomorphism, and hence of isomorphism classes of certain objects such as central simple algebras (for projective linear groups), quadratic forms (for orthogonal groups), étale algebras (for symmetric groups) etc. See [21, 17] for recent surveys on the topic.

Two of the motivating problems in essential dimension are the computation of the essential dimension of the projective linear group PGL_{d} and the symmetric group S_{n}, since they provide insight to the structure of central simple algebras (of degree d) and étale algebras (of dimension n), respectively. The first problem goes back to C. Procesi [19], who asked for fields of definition of the universal division algebra and discovered, in modern terms, that ed(PGL_{d}) ≤ d^{2}. This upper bound has been improved after the introduction of essential dimension, but it is still quadratic in d. See Remark 4.5 for details. A recent breakthrough has been made by A. Merkurjev [16] for a lower bound on ed(PGL_{d}). Namely, if d = p^{a} for some prime p different from char(F), he showed that ed(PGL_{d}) ≥ (a−1)p^{a}+1. In fact he established this lower bound for the essential p-dimension of PGL_{d}, denoted ed_{p}(PGL_{d}), which measures complexity of degree d central simple algebras up to prime to p field extensions, and showed in particular that ed_{p}(PGL_{p^{2}}) = p^{2} + 1 when char(F) ≠ p [15]. For exponent a ≥ 3 the problem of computing ed_{p}(PGL_{p^{a}}) is still wide open. Moreover even the value of ed(PGL_{p}) is unknown for any prime p ≥ 5 and related to the long-standing cyclicity-conjecture of degree p division algebras due to Albert.

The second problem is related to classical work of F. Klein, C. Hermite and F. Joubert on simplifying minimal polynomials of generators of separable field extensions (of degree n = 5 and 6) by means of Tschirnhaus-transformations, and was the main inspiration of [4]. In our language Hermite and Joubert showed that ed(S_{5}) ≤ 2 and ed(S_{6}) ≤ 3 (over a field F of characteristic zero), and Klein proved that ed(S_{5}) > 1, hence ed(S_{5}) = 2. The gap between the best lower bound (roughly \frac{n}{2}) and the best upper bound n − 3 on ed(S_{n}) for n ≥ 5 is still quite large in general. See [7], where it is also proven that ed(S_{7}) = 4 in characteristic zero.

In this paper we study separable algebras B. A (finite-dimensional) algebra B over a field is called separable, if it is semisimple (i.e., its Jacobson radical is trivial) and remains semisimple over every field extension. This includes both the case of central simple algebras and étale algebras. We restrict our attention to those separable K-algebras which embed in A_{K} = A⊗_{F} K for a fixed central simple F -algebra A. Here F is our base field and K/F a field extension. This originates in my earlier paper [13], which covers the case where A is a division algebra. The aim in this paper is to prove results for lower index of A.

Throughout A is a central simple algebra over a field F and B ⊆ A a separable subalgebra. The type of B in A is defined as the multiset θ_{B} = [(r_{1}, d_{1}), …, (r_{m}, d_{m})] such that the algebra B and its centralizer C = C_{A}(B) have the form

B_{sep} ≃ M_{d_{1}}(F_{sep}) × ⋯ × M_{d_{m}}(F_{sep}), C_{sep} ≃ M_{r_{1}}(F_{sep}) × ⋯ × M_{r_{m}}(F_{sep})

over a separable closure F_{sep}. Note that central simple and étale subalgebras are those of type θ_{B} = [(d, r)] (with d = deg(B)) and θ_{B} = [(1, r_{1}), …, (1, r_{m})] (with m = dim(B)), respectively. We will assume throughout that the type θ_{B} of B is constant, i.e. θ_{B} = [(d, r), …, (d, r)] (m-times) for some r, d, m ≥ 1. This assumption is automatically satisfied if A is a division algebra. By [13, Lemma 4.2(a)] the product drm is the degree of A.

Denote by Forms(B): Fields_{F} → Sets the functor that takes a field extension K/F to the set of isomorphism classes of K-algebras B′ which become isomorphic to B over a separable closure of K and by Forms_{A}^{θ}(B) the subfunctor of Forms(B) formed by those isomorphism classes B′ of forms of B which admit an embedding in A of type θ_{B}. We are interested in ed(Forms_{A}^{θ}(B)). By [13, Lemma 4.6] we have a natural isomorphism

Forms_{A}^{θ}(B) ≃ H^{1}(−, G),

of functors Fields_{F} → Sets, where G is the normalizer

G := N_{GL_{1}(A)}(GL_{1}(B)).

Our main result is the following theorem, which shows an interesting dichotomy between the case where the index of A exceeds the bound \frac{r}{d} and when it does not. The case where A is a division algebra is [13, Theorem 4.10]. As there we get examples of algebraic groups, where ed(G) is determined explicitly, but ed(G_{alg}) is unknown. Here we see that the mystery starts exactly once ind(A) ≤ \frac{r}{d}.

Theorem 1.1. Let G = N_{GL_{1}(A)}(GL_{1}(B)) with A central simple and B ⊆ A a separable subalgebra of type θ_{B} = [(d, r), …, (d, r)] (m-times). Suppose that deg(A) = drm is a power of a prime p and that d ≤ r, so that d|r. Then exatly one of the following cases occurs:

(a) ind(A) ≤ \frac{r}{d}: Then Forms_{A}^{θ}(B) = Forms(B) and the three functors H^{1}(−, G), Forms(B) and H^{1}(−, (PGL_{d})^{m} ⋊ S_{m}) are naturally isomorphic. In particular

ed(G) = ed(Forms(B)) = ed((PGL_{d})^{m} ⋊ S_{m}).

(b) ind(A) > \frac{r}{d}: Then

ed(G) = ed(Forms_{A}^{θ}(B)) = deg(A) ind(A) − dim(G),
= drm ind(A) − m(r^{2} + d^{2} − 1).

except possibly when d = r > 1 and ind(A) = 2.

Note that the assumption r ≤ d is harmless. Indeed since

N_{GL_{1}(A)}(GL_{1}(B)) ⊆ N_{GL_{1}(A)}(GL_{1}(C_{A}(B))) ⊆ N_{GL_{1}(A)}(GL_{1}(C_{A}(C_{A}(B))))

and C_{A}(C_{A}(B)) = B by the double centralizer property of semisimple subalgebras [8, Theorem 4.10] we can always replace B by its centralizer (which amounts to switching r and d) without changing ed(G).

There is a big contrast between the two cases in Theorem 1.1. In case (a) the computation of ed(G) = ed((PGL_{d})^{m} ⋊ S_{m}) = ed(Forms(B)) is very hard in general. For instance when B is central simple (i.e., m = 1), we have ed(G) = ed(PGL_{d}) with d = deg(B), and in case B is étale (i.e., d = 1), ed(G) = ed(S_{m}) where m = dim(B). In contrast the above theorem gives the precise value of ed(G) in case (b) with only a small exception. The exception occurs when d = r > 1 and ind(A) = 2, i.e., when A ≃ M_{d/2}(Q) for a non-split quaternion F -algebra Q and B and the centralizer C = C_{A}(B) become isomorphic to (M_{d}(F_{sep}))^{m} over F_{sep}. Note that we then automatically have p = 2, so r = d and m are 2-primary. This special case will be treated separately. We will provide lower bounds and upper bounds on ed(G). When m = 1 the set H^{1}(K, G) then classifies central simple K-algebras B′ of degree d, whose tensor product with a fixed quaternion algebra over F is not division (see Example 4.1). In particular we will prove that ed(G) is either 2 or 3 when r = d = 2 and m = 1 (see Corollary 4.6).

The rest of the paper is structured as follows. In section 2 we study representations of G = N_{GL_{1}(A)}(GL_{1}(B)) with respect to generic freeness. This is used in section 3 to prove that ed(G) does not exceed the value suggested in Theorem 1.1(b). We will conclude the proof of the whole theorem in that section. It remains to study the case excluded from Theorem 1.1, where A has index 2 and r = d > 1. This is finally done in section 4.

2. Results on the Canonical Representation

The group G = N_{GL_{1}(A)}(GL_{1}(B)), as every subgroup of GL_{1}(A), has a canonical representation defined as follows:

Definition 2.1. Let H be a subgroup of GL_{1}(A) for a central simple algebra A. Let D be a division F -algebra representing the Brauer class of A. Fix an isomorphism A ⊗_{F} D^{op} ≃ End(V) for an F -vector space V. We call the representation

H ↪ GL_{1}(A) ↪ GL_{1}(A ⊗_{F} D^{op}) ≃ GL(V)

the canonical representation of H, denoted ρ_{can}^{H}: H → GL(V).

Clearly ρ_{can}^{H} is faithful of dimension deg(A) ind(A) and its equivalence class does not depend on the chosen isomorphism A ⊗_{F} D^{op} ≃ End(V). Strictly speaking ρ_{can}^{H} depends on the embedding of H in GL_{1}(A). However it will always be clear from the context, which embedding is meant.

Recall that a representation H → GL(W) of an algebraic group H over F in a F -vector space W is called generically free, if the affine space 𝔸(W) contains a non-empty H-invariant open subset U on which H acts freely, i.e., any u ∈ U(F_{alg}) has trivial stabilizer in H_{alg} := H_{F_{alg}}. By stabilizer we will always mean the scheme-theoretic stabilizer (whose group of R-rational points for any commutative F_{alg}-algebra R is the subgroup of H(R) = H_{alg}(R) formed by those h ∈ H(R) satisfying hu = u). Generic freeness of W can be tested over a separable or algebraic closure. In fact if U ⊆ 𝔸(W)_{F_{alg}} is an H_{alg}-invariant nonempty open subset with free H_{alg}-action then the union of all Gal(F_{alg}/F)- translates of U descends to a nonempty H-invariant open subset with free H-action, see [23, Prop. 11.2.8]. Every generically free representation is faithful, but the converse need not be true. In particular, every generically free representation V of H has dimension dim(V) ≥ dim(H) and when ed(H) > 0 this inequality is strict by [3, Proposition 4.11]. The main result of this section is the following Theorem:

Theorem 2.2. Assume that d divides r. Then the canonical representation of G = N_{GL_{1}(A)}(GL_{1}(B)) is generically free if and only if the index of A satisfies

ind(A) ≥ \begin{cases}2, & if d = r = 1, m > 1, \\ 3, & if d = r > 1, \\ r, & if d = m = 1 \\ \frac{r}{d} + 1, & if d < r and (d > 1 or m > 1). \end{cases}

In order to prove Theorem 2.2 we start with a couple of intermediate results. We will need the notion of stabilizer in general position, abbreviated SGP. An SGP for an action of an algebraic group H (over a field F) on a geometrically irreducible F -variety X is a subgroup S of H with the property that there exists a non-empty open subscheme U of X such that all points u ∈ U(F_{alg}) have (scheme-theoretic) stabilizers conjugate to S_{alg} = S_{F_{alg}}. We can always make such a subscheme U invariant under H as follows: Consider U′ := ⋃_{h∈H(F_{alg})} hU_{alg}, which is a nonempty H_{alg}-invariant open subscheme of X_{alg}. By construction the stabilizer of every u ∈ U′(F_{alg}) is conjugate to S_{alg}.

Now U′(F_{alg}) is also invariant under the action of the absolute Galois group of F. Therefore, by [23, Prop. 11.2.8] it descends to an H-invariant open subset of X with the same properties as U. Clearly a representation of H is generically free, if and only if it the trivial subgroup of H is an SGP for that action. Moreover if H acts on X with kernel N, then S is an SGP for the H-action on X if and only if S contains N and S/N is an SGP for the (faithful) H/N-action on X. The following lemma is well known for algebraically closed fields of characteristic 0. We adapt the proof of [18, Proposition 8] to our more general situation, when F is an arbitrary field.

Lemma 2.3. Let H act on two geometrically irreducible F -varieties X and Y. Suppose that S_{1} is an SGP for the H-action on X and S_{2} is an SGP for the S_{1}-action on Y. Then S_{2} is an SGP for the H-action on X × Y.

Proof. First by replacing X with a suitable non-empty H-invariant open subvariety we may assume that every x ∈ X(F_{alg}) has stabilizer conjugate to (S_{1})_{alg} in H_{alg}. Let U_{Y} be a non-empty S_{1}-invariant open subset of Y such that all u ∈ U_{Y} (F_{alg}) have stabilizer conjugate to (S_{2})_{alg} in (S_{1})_{alg}. Let C = H · (X^{S_{1}} × U_{Y}^{S_{2}}) denote the set-theoretical image of (X^{S_{1}} × U_{Y}^{S_{2}}) in X ×Y under the action map H ×(X ×Y) → X ×Y. Endow the closure Z := C̄ with the reduced scheme structure and consider the morphism p_{X}: Z → X of schemes given by the composition Z ↪ X × Y →^{π_{X}} X. The fiber of p_{X} over any x ∈ X(F_{alg}) has dimension equal to dim Y. In fact if h_{x} ∈ H(F_{alg}) is such that (H_{alg})_{x} = h_{x}(S_{1})_{alg}h_{x}^{−1} then p_{X}^{−1}(x)(F_{alg}) contains {x} × h_{x}U_{Y} (F_{alg}), as one easily checks. Therefore by the fiber dimension theorem dim Z = dim X+dim Y and it follows that C is dense in X × Y. Since C is constructible (by Chevalley’s Theorem) there exists a non-empty open subset U ⊂ X × Y contained in C. The stabilizer of every u ∈ U(F_{alg}) is conjugate to (S_{2})_{alg}, since this is obviously true for elements of (X^{S_{1}} × U_{Y}^{S_{2}})(F_{alg}). Therefore S_{2} is an SGP for the H-action on X × Y. □

The following proposition will be the key step in order to establish the case of Theorem 2.2, where m = 1.

Proposition 2.4. Let V be a vector space over a field F, whose dual we denote by V^{∗}, and let

H = GL(V^{∗}) × GL(V).

For any commutative F -algebra R and φ ∈ End(V_{R}) denote by φ^{∗} ∈ End(V_{R}^{∗}) the dual endomorphism (given by the formula (φ^{∗}(f))(v) = f(φ(v)) for v ∈ V_{R}, f ∈ V_{R}^{∗} = Hom_{R}(V_{R}, R)).

(a) The image S ≃ GL(V) of the homomorphism

GL(V) → H, φ ↦ ((φ^{∗})^{−1}, φ)

is an SGP for the natural H-action on V^{∗} ⊗_{F} V.

(b) Let E be a maximal étale subalgebra of End(V). Then the image T ≃ GL_{1}(E) of the homomorphism

GL_{1}(E) → H, φ ↦ ((φ^{∗})^{−1}, φ)

is an SGP for the natural H-action on (V^{∗} ⊗_{F} V)^{⊕2}.

(c) Let Z(H) ≃ G_{m} × G_{m} denote the center of H. The image of the homomorphism

G_{m} → Z(H) ⊆ H, λ ↦ (λ^{−1}, λ)

is an SGP for the natural H-action on (V^{∗} ⊗_{F} V)^{⊕3}.

(d) Suppose V = V_{1} ⊗_{F} V_{2} and consider the subgroup

H′ = GL(V_{1}^{∗}) × GL(V)

of H = GL(V^{∗}) × GL(V). Let t = dim(V_{2}). Then the image S′ of the homomorphism

GL(V_{1}) → H′, φ ↦ ((φ^{∗})^{−1}, φ)

is an SGP for the natural H′-action on (V_{1}^{∗} ⊗_{F} V)^{⊕t}.

Moreover if t > 1, the image of the homomorphism

G_{m} → Z(H′) ⊆ H′, λ ↦ (λ^{−1}, λ)

is an SGP for the natural H′-action on (V_{1}^{∗} ⊗_{F} V)^{⊕(t+1)}.

Proof. (a) We use the canonical identification of V^{∗}⊗_{F} V with the underlying F -vector space of the F -algebra End_{F} (V^{∗}), where a pure tensor f⊗v corresponds to the endomorphism of V^{∗} defined by f′ ↦ f′(v)f. The H-action on (the affine space associated with) V^{∗} ⊗_{F} V = End_{F} (V^{∗}) is then given by the formula

(ψ, φ) · ρ = ψρφ^{∗}.

Let U = GL(V^{∗}) ⊆ 𝔸(End(V^{∗})), which is a non-empty and H-invariant open subset. The stabilizer of ρ ∈ U(F_{alg}) in H_{alg} is given by the image of the homomorphism

GL(V)_{alg} → H_{alg}, φ ↦ (ρ(φ^{∗})^{−1}ρ^{−1}, φ)

which is a conjugate of S_{alg} over F_{alg}. This shows the claim.

(b) Let S be the subgroup of H from part (a). By Lemma 2.3 it suffices to show that T is an SGP for the S-action on V^{∗} ⊗_{F} V. Let U ⊆ 𝔸(V^{∗} ⊗_{F} V) = 𝔸(End(V^{∗})) be as in part (a). Identify (V^{∗})^{∗} with V in the usual way, so that ψ^{∗} ∈ End(V) for ψ ∈ End(V^{∗}). For any ρ ∈ U(F_{alg}) the stabilizer of ρ in S_{alg} is the image of the centralizer C_{GL(V)_{alg}}(ρ^{∗}) under the homomorphism GL(V)_{alg} → S_{alg}, φ ↦ ((φ^{∗})^{−1}, φ). When ρ^{∗} is semisimple regular C_{GL(V)_{alg}}(ρ^{∗}) is a maximal torus of GL(V)_{alg}. Now the claim follows from the well known facts that all maximal tori of GL(V)_{alg} are conjugate and the semisimple regular elements in 𝔸(End(V^{∗})) form a non-empty open subset.

(c) By part (b) T ≃ GL_{1}(E) is an SGP for the H-action on two copies of V^{∗} ⊗_{F} V. The kernel of the T -action on V^{∗} ⊗_{F} V is the image of G_{m} in H and coincides with the SGP for this action, since T is a torus, see e.g. [12, Proposition 3.7(A)]. Now the claim follows with Lemma 2.3.

(d) Note that (V_{1}^{∗} ⊗_{F} V)^{⊕t} is H-equivariantly isomorphic to V_{1}^{∗} ⊗_{F} V_{2}^{∗} ⊗_{F} V ≃ V^{∗} ⊗_{F} V. Define the open subset U ⊆ 𝔸(V^{∗} ⊗_{F} V) like in part (a). Then every ρ ∈ U(F_{alg}) has stabilizer in (H′)_{alg} given by the image of the homomorphism

GL(V_{1})_{alg} → (H′)_{alg}, α ↦ ((α^{∗})^{−1}, ρ^{∗}α(ρ^{∗})^{−1})

which is conjugate to (S′)_{alg} over F_{alg}. This shows the first claim.

As an S′-representation V_{1}^{∗} ⊗_{F} V is isomorphic to the t-fold direct sum of W = End(V_{1}^{∗}) where S′ acts through the formula

((φ^{∗})^{−1}, φ) · ρ = (φ^{∗})^{−1}ρφ^{∗}.

As in the proof of part (b) and (c) the S′-action on W has SGP isomorphic to GL_{1}(E′) for a maximal étale subalgebra E′ of GL(V_{1}) and the S′-action on W^{⊕2} and, since t > 1, also on W^{⊕t} ≃ V_{1}^{∗} ⊗_{F} V, has as SGP the kernel of this action, which is the image of G_{m} in H′ by the given homomorphism. Now the claim follows from Lemma 2.3.

□

The next lemma will allow a reduction to the case m = 1 in Theorem 2.2 when d ≠ 1.

Lemma 2.5. (a) Let m ≥ 1. A representation of an algebraic group H on a vector space V of dimension dim(V) > dim(H) is generically free if and only if the associated representation of the wreath product H^{m} ⋊S_{m} on V^{⊕m} is generically free.

(b) Suppose A is split and d ≠ 1. Then for any t ≥ 1 generic freeness of (ρ_{can}^{G})^{⊕t} depends only on r and d, not on m.

Proof. (a) If H^{m} ⋊ S_{m} acts generically freely on V^{⊕m} then so does the subgroup H^{m}. Let

U ⊆ 𝔸(V^{⊕m}) = \underbrace{𝔸(V) × ⋯ × 𝔸(V)}_{m times}

be a non-empty H^{m}-invariant open subset where H^{m} acts freely. Then the projection π_{1}(U) ⊆ 𝔸(V) is non-empty open and H-invariant with free H-action. Hence H acts generically freely on V.

Conversely suppose that H acts generically freely on V. Let U_{0} ⊆ 𝔸(V) a friendly open subset, i.e., an H-invariant non-empty open subset such that there exists an H-torsor π: U_{0} → Y for some irreducible F - scheme Y (which we will fix). Existence of U_{0} is granted by a Theorem of P. Gabriel, see [3, Theorem 4.7] or [22, Exposé V, Théoréme 10.3.1]. Since dim(U_{0}) = dim(V) > dim(H) we have dim(Y) > 0. Hence the open subset Y^{(m)} of Y^{m} where the m coordinates are different, is nonempty open with free natural S_{m}-action on it. Now the inverse image of Y^{(m)} in U_{0}^{m} under the morphism π^{m}: U_{0}^{m} → Y^{m} is H^{m} ⋊S_{m}-invariant, nonempty and open with H^{m} ⋊ S_{m} acting freely on it.

(b) Since the property of being generically free can be checked over an algebraic closure F_{alg} and (ρ_{can}^{G})_{F_{alg}} = ρ_{can}^{G_{alg}} we may assume without loss of generality that F is algebraically closed. Let

H = (GL(V_{1}) × GL(V_{2}))/G_{m},

where V_{1} and V_{2} are vector spaces of dimension dim(V_{1}) = d, dim(V_{2}) = r and G_{m} is embedded in the center of GL(V_{1}) × GL(V_{2}) through λ ↦ (λ, λ^{−1}). Then

G ≃ H^{m} ⋊ S_{m}.

In particular for m = 1 the two groups H and G are isomorphic. Moreover, in general, ρ_{can}^{G} is given by the obvious homomorphism

G → GL((V_{1} ⊗_{F} V_{2})^{⊕m}).

In order to establish the claim, it suffices to show that the representation of H on V := (V_{1} ⊗_{F} V_{2})^{⊕t} is generically free if and only if the associated representation (ρ_{can}^{G})^{⊕t} of G on V^{⊕m} is generically free. When dim(V) > dim(H) the claim follows from part (a). On the other hand when dim(V) ≤ dim(H) or equivalently dim(V^{⊕m}) ≤ dim G the two representations of G and H, respectively, are both not generically free, since otherwise the respective group would have essential dimension 0. This is both excluded by the assumption d ≠ 1, since B ≃ M_{d}(F)^{m} and M_{d}(F) have nontrivial forms over some field extension K/F which embed in A ⊗_{F} K ≃ M_{drm}(K). Correspondingly there is a non-trivial G-torsor (resp. H-torsor) over K. This torsor cannot be defined over any subfield of transcendence degree 0 over F, since F is algebraically closed.

□

The following lemma tells us how ρ_{can}^{H} looks over F_{sep}, for any subgroup H of GL_{1}(A).

Lemma 2.6. Over F_{sep} the representation ρ_{can}^{H} decomposes as a direct sum of ind(A) copies of the canonical representation of H_{sep} = H_{F_{sep}}.

Proof. Fix isomorphisms A_{sep} →^{∼} End(V), (D^{op})_{sep} →^{∼} End(W) with F_{sep}- vector spaces V and W. Let w_{1}, …, w_{a} be a basis of W, with a = dim(W) = ind(A). Then (ρ_{can}^{H})_{F_{sep}} is equivalent to the composition H_{sep} ↪ GL(V) ↪ GL(V ⊗_{F_{sep}} W), whilst ρ_{can}^{H_{sep}} is equivalent to the inclusion H_{sep} ↪ GL(V). Since the subspaces V ⊗_{F_{sep}} F_{sep}w_{i} of V ⊗_{F_{sep}} W are GL(V)-invariant and GL(V)-equivariantly (and therefore H_{sep}-equivariantly) isomorphic to V, the claim follows. □

We are now ready to prove our main result from this section.

Proof of Theorem 2.2. In view of Lemma 2.6 it suffices to show that the least integer t ≥ 1 such that the t-fold direct sum of ρ_{can}^{G_{sep}} is generically free, is given by the lower bound on the index in the statement of the theorem.

(a) Case d = r = 1, m > 1: Here B is a maximal étale subalgebra of A of dimension deg(A) = m > 1. The canonical representation of G_{sep} is given by the natural action of (G_{m})^{m} ⋊ S_{m} on V = F^{m}. Let U ⊆ 𝔸(V) = 𝔸^{m} denote the open subset where all coordinates are non-zero. The group G_{sep} operates transitively on U. Therefore the stabilizer of any u ∈ U(F_{alg}) is conjugate to the stabilizer of (1, …, 1) in G_{sep}, which is S_{m}. Therefore S_{m} is an SGP for the canonical representation of G_{sep}. Moreover S_{m} acts freely on the S_{m}-invariant open subset of U, where all coordinates are different. Thus the canonical representation of G_{sep} is not generically free, but two copies of it are, by Lemma 2.3.

(b) Case d = r > 1: We must show that two copies of the canonical representation of G_{sep} are not generically free, but three copies are. By Lemma 2.5, since d > 1, we may assume that m = 1. Let V be an F_{sep}-vector space of dimension d = r. Identify B_{sep} with End(V^{∗}) and its centralizer in A_{sep} with End(V). This identifies G_{sep} with (GL(V^{∗}) × GL(V))/G_{m}, where G_{m} is embedded in the center of GL(V^{∗}) × GL(V) via λ ↦ (λ^{−1}, λ). Its canonical representation is given by the natural action on V^{∗} ⊗_{F} V. By Proposition 2.4(b) the sum of two copies of this representation has an SGP in general position of the form G_{m}^{d}/G_{m}, hence it is not generically free. Moreover Proposition 2.4(c) shows that the sum of three copies of that representation is generically free.

(c) Case d = m = 1: Here G = GL_{1}(A) with A of degree drm = r. By dimension reasons we need at least r copies of the canonical representation of G_{sep} (whose dimension is r) in order to get a generically free representation. On the other hand r copies are clearly enough.

(d) Case d < r and (d > 1 or m > 1):

First assume d > 1. This case is similar to case (b). We must show that \frac{r}{d} + 1 copies of the canonical representation of G_{sep} are generically free, but \frac{r}{d} copies are not. By Lemma 2.5 we may assume that m = 1. Let V_{1} and V_{2} be F_{sep}-vector spaces of dimension d and \frac{r}{d}, respectively, and set V = V_{1} ⊗_{F_{sep}} V_{2}, which is of dimension r. Identify B_{sep} with End(V_{1}^{∗}) and its centralizer in A_{sep} with End(V), so that G_{sep} = (GL(V_{1}^{∗}) × GL(V))/G_{m}. Its canonical representation is given by the natural action on V_{1}^{∗} ⊗_{F_{sep}} V. By Proposition 2.4(c) exactly dim(V_{2}) + 1 = \frac{r}{d} + 1 copies of this representation are needed in order to get a generically free representation. This establishes the claim in case d > 1.

Now assume d = 1 < r and m > 1. Here B is étale of dimension m with 1 < m < rm = deg(A). Let V denote an r-dimensional F_{sep}-vector space. Then G_{sep} ≃ (GL(V))^{m} ⋊ S_{m} and its canonical representation is given by the natural action on V^{⊕m}. We have dim G = r^{2}m = r · dim(V^{⊕m}). Since G_{sep} is not connected it has ed(G_{sep}) > 0, see [11, Lemma 10.1], hence we need at least r+1 copies of V^{⊕m} in order to get a generically free representation. On the other hand the connected component G_{sep}^{0} ≃ (GL(V))^{m} acts generically freely on r copies of V^{⊕m} and S_{m} acts generically freely on V^{⊕m}, which implies that G_{sep} acts generically freely on r + 1 copies of V^{⊕m}. This concludes the proof.

□

3. Proof of Theorem 1.1

The purpose of this section consists in proving the results on ed(G) as formulated in our main theorem.

Proof of Theorem 1.1. (a) The inequality ind(A) ≤ \frac{r}{d} implies that r is divisible by d ind(A), since ind A, r and d are powers of p. In this case natural isomorphism between the functors of H^{1}(−, G) and Forms(B) was established in [13, Remark 4.8]. In fact when r is divisible by d ind(A) every form B′ of B over a field extension K/F can be embedded in A ⊗_{F} K with type [(d, r), …, (d, r)].

Now for every F -form B′ of B the functors Forms(B) and Forms(B′) are equivalent as functors to the category of sets. The split form of B over F is M_{d}(F)^{m} and its automorphism group scheme is (PGL_{d})^{m} ⋊ S_{m}. This shows that Forms(B) is naturally isomorphic to the Galois cohomology functor H^{1}(−, (PGL_{d})^{m} ⋊ S_{m}).

(b) Assume ind(A) > \frac{r}{d}. For any algebraic group H over F we have the standard inequality

ed(H) ≤ dim(ρ) − dim(H)

for any generically free representation ρ of H, see [3, Proposition 4.11]. The canonical representation of the group G = N_{GL_{1}(A)}(GL_{1}(B)) has dimension deg(A) ind(A). Therefore Theorem 2.2 yields the inequality

(1)  ed(G) ≤ deg(A) ind(A) − dim(G)

in case

ind(A) ≥ \begin{cases}2, & if d = r = 1, m > 1, \\ 3, & if d = r > 1, \\ r, & if d = m = 1 \\ \frac{r}{d} + 1, & if d < r and (d > 1 or m > 1). \end{cases}

Combining this with the assumption ind(A) > \frac{r}{d} shows that inequality (1) is always satisfied, except possibly when d = r > 1 and ind(A) = 2.

Now we show the converse to inequality (1). We follow the approach given in [13]. Let Aut_{F} (A, B) denote the group scheme of automorphisms of B-preserving automorphisms of A. We have an exact sequence

1 → G_{m} → G →^{Int} Aut_{F} (A, B) → 1,

where Int: G = N_{GL_{1}(A)}(GL_{1}(B)) → Aut_{F} (A, B) takes, for every commutative F -algebra R, the element g ∈ G(R) ⊆ (A ⊗_{F} R)^{×} to the inner automorphism of A ⊗_{F} R given by conjugation by g. The connection map

H^{1}(K, Aut_{F} (A, B)) → H^{2}(K, G_{m}) = Br(K)

sends the isomorphism class of a K-form (A′, B′) of (A, B) to the Brauer class [A′]−[A⊗_{F} K] = [A′ ⊗_{F} A^{op}]. Write deg(A) = p^{s}. By [13, Lemma 2.3] there exists a field extension K/F and a central simple K-algebra A′ of the form A′ = D_{1}⊗_{K} ⋯⊗_{K} D_{s} for division K-algebras D_{1}, …, D_{s} of degree p, such that

ind(A′ ⊗_{F} A^{op}) = p^{s} ind(A) = deg(A) ind(A).

Write d = p^{a}, r = p^{b}, m = p^{c}, so that a + b + c = s. Choose a maximal étale K-subalgebra L_{i} of D_{a+i} for i ∈ {1, …, c}. Then

B′ := D_{1} ⊗_{K} ⋯ ⊗_{K} D_{a} ⊗_{K} L_{1} ⊗_{K} ⋯ ⊗_{K} L_{c}

is a separable K-subalgebra of A′ of type [(d, r), …, (d, r)] (like B in A). This implies that (A′, B′) is a K-form of (A, B) by [13, Lemma 4.2(d)]. Therefore the maximal index of a Brauer class contained in the image of a connection map H^{1}(K, Aut_{F} (A, B)) → Br(K) for a field extension K/F is precisely deg(A) ind(A). Now the inequality

ed(G) ≥ deg(A) ind(A) − dim(G)

follows from [5, Corollary 4.2].

□

Remark 3.1. Theorem 1.1 holds with essential dimension replaced by essential p-dimension. For definition of ed_{p}(G) see [14] or [21]. In fact part (a) follows from the description of the Galois cohomology functor H^{1}(−, G) like for essential dimension. Moreover we always have ed_{p}(G) ≤ ed(G) and the lower bounds given in part (b) are actually lower bounds on ed_{p}(G). This follows from the p-incompressibility of Severi-Brauer varieties of division algebras of p-power degree [9, Theorem 2.1] and [14, Theorem 4.6].

4. The Special Case

In this section we consider the case, which was not resolved by Theorem 1.1. Hence we assume throughout this section that

A = M_{2^{n}}(Q)

for some integer n ≥ 0 and a non-split quaternion F -algebra Q, and B ⊆ A is a separable subalgebra with

B_{sep} ≃ (M_{2^{a}}(F_{sep}))^{2^{c}} ≃ C_{sep},

where C ⊆ A is the centralizer of B in A and a, c are integers with a ≥ 1, c ≥ 0. Note that the relation drm = deg(A) implies 2a + c = n + 1. Recall that G = N_{GL_{1}(A)}(GL_{1}(B)).

Example 4.1. Suppose m = 1, which means that B is central simple of degree d = 2^{a}. Then the functor H^{1}(−, G) ≃ Forms_{A}^{θ}(B) (with θ_{B} = [(d, d)]) classifies central simple algebras B′ of degree d over field extensions K/F such that B′ ⊗_{F} Q is not a division algebra. This is shown as follows: B′ embeds in A_{K} if and only if B′ ⊗_{F} Q embeds in A_{K} ⊗_{F} Q ≃ M_{2d^{2}}(K). If this is the case, the centralizer of B′ ⊗_{F} Q in M_{2d^{2}}(K) has degree d and has opposite Brauer class to B′ ⊗_{F} Q. Therefore the index of B′ ⊗_{F} Q divides d, i.e. B′ ⊗_{F} Q is not a division algebra. Conversely, if the index of B′ ⊗_{F} Q divides d, then the opposite algebra is Brauer equivalent to a degree d algebra, so B′ ⊗_{F} Q embeds in M_{2d^{2}}(K).

Let L/F be a maximal separable subfield of Q (of dimension 2 over F). The algebra A splits over L. In particular we get the lower bound

ed(Forms(M_{d}(L)^{m})) = ed(Forms(B_{L})) = ed(G_{L}) ≤ ed(G)

on ed(G) by Theorem 1.1(a) and [3, Proposition 1.5]. Moroever we have the upper bound

ed(G) ≤ 4 deg(A) − dim(G) = 4 · 2^{2a+c} − 2^{c}((2^{a})^{2} + (2^{a})^{2} − 1)
= 2^{2a+c+2} − 2^{2a+c+1} + 2^{c}
= 2^{c}(2^{2a+1} + 1),

since 2 copies of ρ_{can}^{G} are generically free by Theorem 2.2 and Lemma 2.6. The main effort in this section will go into proving a better upper bound on ed(G).

For this purpose we will show that the canonical representation of the normalizer of a maximal torus (and even of some larger subgroup) of G is generically free. The following lemma reveals that this will improve the above upper bound on ed(G).

Lemma 4.2. Let T be a maximal torus of G and H a subgroup of G containing the normalizer N_{G}(T). Suppose that ρ_{can}^{H} is generically free. Then

ed(G) ≤ ed(H) ≤ 2 deg(A) − dim H
= 2^{c+2a+1} − dim H.

Proof. The connected component G_{alg}^{0} ≃ ((GL_{2^{a}} × GL_{2^{a}})/G_{m})^{2^{c}} of G_{alg} is reductive. Therefore the inclusion ι: N_{G}(T) ↪ G induces a surjection of functors

ι_{∗}: H^{1}(−, N_{G}(T)) ↠ H^{1}(−, G),

see e.g. [6, Corollary 5.3]. Note that G is supposed to be connected reductive there, but the proof goes through if only G^{0} is reductive (over F_{alg}). Since ι factors through H, the map ι_{∗} factors through H^{1}(−, H). By [3, Lemma 1.9] this proves the first inequality. The second inequality follows from dim(ρ_{can}^{H}) = 2 deg(A) and [3, Proposition 4.11]. □

In order to make use of Lemma 4.2 we will need the following result:

Lemma 4.3. Let R be a connected reductive algebraic group over F. Let T be a maximal torus of R and let T_{R} ≃ R/N_{R}(T) denote the variety of maximal tori in R. Assume that R/Z(R) is simple, i.e., has no nontrivial normal subgroups. Then there exists a non-empty open subscheme U of T_{R} such that every maximal torus of R_{alg} contained in U(F_{alg}) intersects (N_{R}(T))_{alg} exactly in Z(R)_{alg}.

Proof. First note that T contains Z(R), since R is reductive. If T is central in R, then T = Z(R), and the claim easily follows. Hence we may assume that T is non-central. We let R act on T_{R} through conjugation. The kernel of this action is a proper normal subgroup of R containing Z(R). Hence it is equal to Z(R). Therefore the kernel of the T -action on T_{R} obtained by restriction is also Z(R). By [12, Proposition 3.7] the induced T/Z(R)-action on T_{R} is generically free. Hence there exists a non-empty open subscheme Ũ of T_{R} such that every S ∈ Ũ(F_{alg}) has stabilizer in T_{alg} equal to Z(R)_{alg}, i.e. N_{R_{alg}}(S)∩T_{alg} = Z(R)_{alg}. For a ∈ R(F_{alg}) with S = aT_{alg}a^{−1} this is equivalent to (N_{R}(T))_{alg} ∩ (a^{−1}T_{alg}a) = Z(R)_{alg}. Denote by π: R → T_{R}, a ↦ aT a^{−1} the projection map and by ι: R → R, a ↦ a^{−1} the inversion map. Then U := (π∘ι)(π^{−1}(Ũ)) has the desired property. □

Proposition 4.4. With the standing assumptions r = d = 2^{a} > 1, m = 2^{c} ≥ 1 and ind(A) = 2:

ed(G) ≤ 2^{c+2a+1} − 2^{c}(2^{2a} + 2^{a} − 1)
= 2^{c}(2^{2a} − 2^{a} + 1).

Proof. We first consider the case m = 1 (i.e., c = 0): Let E be a maximal étale subalgebra of the centralizer C = C_{A}(B) and let

H = (GL_{1}(B) × N_{GL_{1}(C)}(GL_{1}(E)))/G_{m} ⊆ G.

We will show that ρ_{can}^{H} is generically free. Since dim(H) = 2^{2a} + 2^{a} − 1 this would establish the claim in case m = 1 in view of Lemma 4.2. Over F_{alg} we may identify H_{alg} with (GL(V^{∗}) × N_{GL(V)}(T))/G_{m} where V is an F_{alg}-vector space of dimension d = 2^{a} and T is a maximal torus of GL(V). Moreover ρ_{can}^{H} becomes a direct sum of two copies of the natural representation

H_{alg} → GL(V^{∗} ⊗_{F_{alg}} V) = GL(End(V^{∗}))

over F_{alg}. Hence it suffices to show that G_{m} is an SGP for the natural action of H′ := GL(V^{∗}) × N_{GL(V)}(T) on two copies of W := End(V^{∗}). Identify N := N_{GL(V)}(T) with its image in H′ under the map φ ↦ ((φ^{∗})^{−1}, φ). The proof of Proposition 2.4(b) shows that N is an SGP for the H′ action on one copy of W. Moreover the stabilizer of any ρ ∈ End(V^{∗}) in N is given by the intersection of N with the centralizer C_{GL(V)}(ρ^{∗}). When ρ is semisimple regular, C_{GL(V)}(ρ^{∗}) is a maximal torus of GL(V). It can be considered as a rational point of the variety of maximal tori T_{GL(V)} of GL(V). By Lemma 4.3 there exists a non-empty open subscheme U of T_{GL(V)} such that N∩S = G_{m} for every S ∈ U(F_{alg}). Let GL(V^{∗})^{ss,reg} ⊂ 𝔸(W) denote the open subset given by the regular semisimple elements. We have a morphism GL(V^{∗})^{ss,reg} → T_{GL(V)}, sending a semisimple regular element ρ to the centralizer C_{GL(V)}(ρ^{∗}). The preimage P of U in GL(V^{∗})^{ss,reg} is a non-empty open subset of 𝔸(W) such that every ρ ∈ P (F_{alg}) has stabilizer in N equal to G_{m}. By Lemma 2.3 this implies the claim.

Now let m = 2^{c} be arbitrary. Since the functor H^{1}(−, G): Fields_{F} → Sets depends only on the type of B, we may replace B by any subalgebra of A of the same type as B without changing ed(G). As

A = M_{2^{n}}(Q) = M_{m}(B_{0} ⊗_{F} C_{0}),

with B_{0} = M_{2^{a}}(F) and C_{0} = M_{2^{a−1}}(Q), we may take for B the m×m diagonalmatrices with entries in B_{0}. Its centralizer C is the set of m×m diagonalmatrices with entries in C_{0}. Therefore

G = (G_{0})^{m} ⋊ S_{m}

where

G_{0} = (GL_{1}(B_{0}) × GL_{1}(C_{0})) /G_{m} = N_{GL_{1}(B_{0}⊗_{F} C_{0})}(GL_{1}(B_{0}))

has ed(G_{0}) ≤ 2^{2a} − 2^{a} + 1 by the case m = 1. By [13, Lemma 4.13] we have ed(G) ≤ m ed(G_{0}) and the claim follows. □

Remark 4.5. Consider the case m = 1. Since ed((PGL_{2^{a}})_{sep}) = ed(G_{sep}) ≤ ed(G) the upper bound

ed(G) ≤ 2^{2a} − 2^{a} + 1

should be compared with the best existing upper bound on the essential dimension of (PGL_{2^{a}})_{sep}, namely

ed((PGL_{2^{a}})_{sep}) ≤ 2^{2a} − 3 · 2^{a} + 1

by [10, Proposition 1.6] (which assumes char(F) = 0).

Corollary 4.6. Suppose B is central simple (i.e., m = 1) and char(F) ≠ 2. Then

max{2, (a − 1)2^{a} + 1} ≤ ed(G) ≤ 2^{2a} − 2^{a} + 1.

In particular when B has degree 2 we have

ed(G) ∈ {2, 3}

and when B has degree 4 we have

ed(G) ∈ {5, 6, …, 13}.

If B is central simple of degree 2 and char(F) = 2 we still have ed(G) ∈ {2, 3}.

Proof. The upper bound on ed(G) is contained in Proposition 4.4. By Theorem 1.1(a) we have ed(Forms(M_{2^{a}}(F_{sep}))) = ed(G_{sep}) ≤ ed(G) . Hence the lower bound

(a − 1)2^{a} + 1 ≤ ed(G)

follows from [16, Theorem 6.1] (which assumes char(F) ≠ 2) and the lower bound 2 ≤ ed(G) follows from [20, Lemma 9.4(a)] (the paper assumes characteristic 0, but the proof works in arbitrary characteristic). □

References

[1] Auel, A., E. Brussel, S. Garibaldi and U. Vishne. “Open problems on Central Simple Algebras.” Transformation Groups 16, no. 1, (2011): 219–264.

[2] Amitsur, S.A. “Division algebras. A survey.” Algebraists homage: papers in ring theory and related topics (New Haven, Conn., 1981), Contemporary Mathematics 13, Amer. Math. Soc., Providence, R.I., (1982): 3–26.

[3] Berhuy, G., and G. Favi. “Essential Dimension: A Functorial Point of View (after A. Merkurjev).” Documenta Mathematica 8 (electronic) (2003): 279– 330.

[4] Buhler, J., and Z. Reichstein. “On the essential dimension of a finite group.” Compositio Mathematica 106 (1997): 159–179.

[5] Brosnan, P., Z. Reichstein, and A. Vistoli. “Essential dimension of moduli of curves and other algebraic stacks.” Journal of the European Mathematical Society 13, no. 4, (2011): 1079–1112. With an appendix by N. Fakhruddin.

[6] Chernousov, V., Ph. Gille, and Z. Reichstein. “Reduction of structure for torsors over semilocal rings.” Manuscripta Mathematica 126 (2008): 465– 480.

[7] Duncan, A. “Essential dimensions of A_{7} and S_{7}.” Mathematical Research Letters 17, no. 2 (2010): 263–266.

[8] Jacobson, J. Basic algebra II, 2nd ed., New York, NY: W. H. Freeman and Company. xvii, 686 p. (1989).

[9] Karpenko, N. “On anisotropy of orthogonal involutions.” Journal of the Ramanujan Mathematical Society 15, no. 1, (2000): 1– 22.

[10] Lemire, N. “Essential dimension of algebraic groups and integral representations of Weyl groups”, Transformation Groups 9, no. 4, (2004): 337–379.

[11] Lötscher R., M. MacDonald, A. Meyer, and Z. Reichstein. “Essential pdimension of algebraic groups whose connected component is a torus.” Algebra & Number Theory 7, no. 8 (2013): 1817–1840.

[12] R. Lötscher. “Contributions to the essential dimension of finite and algebraic groups.” (PhD diss., University of Basel, 2010). Electronic: http://edoc.unibas.ch/1147/1/DissertationEdocCC.pdf.

[13] Lötscher, R. “Essential dimension of involutions and subalgebras.” Israel Journal of Mathematics 192, Part A, (2012): 325–346.

[14] Merkurjev, A.S. “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.

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

[16] Merkurjev, A.S. “A lower bound on the essential dimension of simple algebras.” Algebra & Number Theory 4, no.8, (2010): 1055–1076.

[17] Merkurjev, A.S. “Essential dimension: a survey.” Transformation Groups 18, no. 2, (2013): 415–481.

[18] Popov, V.L. “Closed orbits of Borel subgroups.” Matematicheskii Sbornik. Novaya Seriya 135, no. 3, (1988): 385–402. English transl.: Math. USSR, Sb. 63, 375–392 (1989).

[19] Procesi, C. “Non-commutative affine rings.” Atti della Accademia Nazionale dei Lincei. Memorie. Classe di Scienze Fisiche, Matematiche e Naturali. Serie VIII. Sezione I 8 (1967): 237–255.

[20] Reichstein, Z. “On the notion of essential dimension for algebraic groups.” Transformation Groups 5, no. 3, (2000): 265–304.

[21] Reichstein, Z. “Essential dimension.” Proceedings of the International Congress of Mathematicians, Volume II, 162–188, Hindustan Book Agency, New Delhi, 2010.

[22] Gabriel, P. “Construction de schémas quotients”, pp. 249–289 in Schémas en groupes (SGA 3), tome I: Propriétés générales des schémas en groupes, edited by P. Gille and P. Polo, Documents Mathématiques 7, Société Mathématique de France, Paris, 2011.

[23] Springer, T.A. Linear algebraic groups, 2nd ed., Progress in Mathematics (Boston, MA, 1998), Birkhäuser, Boston, 1998.

Roland Lötscher

Mathematisches Institut

Ludwig-Maximilians-Universität Theresienstraße 39 D-80333 München Germany roland.loetscher@math.lmu.de 460

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