ESSENTIAL DIMENSION AND CANONICAL DIMENSION OF GERBES BANDED BY GROUPS OF MULTIPLICATIVE TYPE

ROLAND LÖTSCHER

Abstract. We prove the formula ed(X) = cdim(X) + ed(A) for any gerbe X banded by an algebraic group A which is the kernel of a homomorphism of algebraic tori Q → S with Q invertible and S split. This result is applied to prove new results on the essential dimension of algebraic groups.

1. Introduction

Essential dimension is a notion of complexity of algebraic and geometric objects, introduced by J. Buhler and Z. Reichstein [BR97] around 1995 and in its most general form by A. Merkurjev [BF03].

The essential dimension of an algebraic group G over a field F is defined as the least integer n such that every G-torsor over a field extension K/F is obtained up to isomorphism by scalar extension from a G-torsor over an intermediate field K_{0} of transcendence degree at most n over F. It is denoted by ed(G).

The essential dimension of an algebraic stack X over F is defined similarly (see section 2), where G-torsors are replaced by objects of X. For every algebraic group (and every group algebraic space) G over F there is the classifying stack BG, whose objects are G-torsors. In particular ed(G) = ed(BG).

An algebraic stack X of finite type over F is called a gerbe, if it becomes isomorphic to BG for some group algebraic space G over a finite extension of F (see [Stacks, Definition 06QC and Lemma 06QH]). In particular BG is a gerbe for any algebraic group G.

Let A be an abelian algebraic group over F. If there are natural isomorphisms Aut(x) ≃ A_{U} for x ∈ X (U), U ∈ Sch_{F}, then the gerbe X is said to be banded by A. Gerbes banded by A over F are classified by the second Galois cohomology set H^{2}(F, A), see [Gir71]. A typical example for a gerbe banded by A is the quotient stack [X/G] for some exact sequence of algebraic groups

1 → A → G → H → 1

with A central in G, where X is an H-torsor over F. These gerbes have been used extensively in the computation of the essential dimension of algebraic groups G like finite p-groups, algebraic tori, Spin groups etc. when A is diagonalizable (see [Lö12] for a survey). A key ingredient is the inequality

(1)  ed(G) ≥ ed[X/G] − dim H

2000 Mathematics Subject Classification. 20G15, 14A20, 11E72.

Key words and phrases. essential dimension, canonical dimension, algebraic stack, gerbe, group of multiplicative type.

The author acknowledges support from the Deutsche Forschungsgemeinschaft, GI 706/2-1.

from [Me09, Theorem 4.8], which bounds ed(G) from below in terms of the essential dimension of the gerbe [X/G].

Our aim is to study gerbes whose band comes from a larger class of groups of multiplicative type and to obtain new applications.

Our main result relates essential dimension and canonical dimension of such gerbes X. It is well known that

(2)  cdim(X) ≤ ed(X) ≤ cdim(X) + ed(A),

for any gerbe X banded by an abelian algebraic group A, see [Me08, Proposition 4.9] (for a more general version see [Lö12, Lemma 3.1]). Here cdim(X) stands for the canonical dimension of X, defined as the least integer n such that for every field extension K/F with X (K) ≠ ∅ there exists an intermediate field K_{0} of K/F of transcendence degree at most n such that X (K_{0}) ≠ ∅ (see section 2 for details). The canonical dimension of an algebraic stack X is often more accessible than its essential dimension, since it only depends on the splitting fields of X, i.e., the field extensions K/F such that X (K) ≠ ∅. For instance, if X_{A} is the μ_{n}-gerbe associated with a central simple algebra A of exponent dividing n, then cdim(X_{A}) coincides with the canonical dimension of the Severi Brauer variety of A. This is equal to ind A − 1 when the index of A is a prime power by a result of N. Karpenko [Ka00, Theorem 2.1].

The following question was asked by A. Merkurjev [Me08, Question 4.10]:

Question 1. Do we have the equality ed(X) = cdim(X) + ed(A) for any gerbe X banded by an abelian group A?

Question 1 has a partial positive answer in case A = μ_{n} (where n > 1) [BRV11, Theorem 4.1] and A = (μ_{p})^{s} [KM08, Theorem 3.1]. It clearly also holds when ed(A) = 0 (equivalently A is special, i.e. all A-torsors over field extensions are trivial, see [Me09, Proposition 4.4]). Our main theorem gives an affirmative answer to Question 1 for a larger class of group, which includes all diagonalizable algebraic groups A as well as many non-diagonalizable algebraic groups like norm 1 tori A = R_{L/F}^{(1)} (G_{m}) for an étale F -algebra L:

Theorem 1.1. Let f: Q → S be a homomorphism of algebraic tori over a field F with Q invertible and S split. Let X be a gerbe over F banded by A := ker(f). Then

ed(X) = cdim(X) + ed(A).

Moreover if Q_{d} denotes the largest split subtorus of Q we have

ed(A) = ed(A ∩ Q_{d})

and this is equal to the rank (=minimal number of generators) of the torsion of the character group of the diagonalizable group A ∩ Q_{d}.

Note that in order to prove the theorem we may always assume that f is surjective. So f will be assumed to be surjective in sequel. Moreover A lies between the invertible torus Q and its largest anisotropic subtorus Q_{an}. Conversely every algebraic group, which lies between Q_{an} and Q for an invertible torus Q can be written as the kernel of a homomorphism f: Q → S with S split (take S = Q/A, which is a split torus as a quotient of the split torus Q/Q_{an}).

The rest of the paper is structured as follows. In section 2 we recall some background material on essential dimension and canonical dimension and make some general observations about gerbes banded by groups of multiplicative type. These will be used in section 3, which contains a proof of Theorem 1.1. Finally section 4 contains applications of this theorem.

2. Preliminaries

2.1. Preliminaries on essential dimension and canonical dimension. Let X be a category fibered in groupoids over F (for instance a gerbe over F). For every field extension K/F we have the groupoid X (K) of objects of X over Spec(K). If L/K is a field extension over F and x ∈ X (K) we have a pullback x_{L} ∈ X (L), defined uniquely up to isomorphism.

Let K/F be a field extension, x ∈ X (K) and let K_{0} be an intermediate field of K/F. We say that x is detected over K_{0} if X (K_{0}) ≠ ∅. We say that x is defined over K_{0} if there exists x_{0} ∈ X (K_{0}) with (x_{0})_{K} ≃ x. Note that “being detected” over K_{0} is weaker than “being defined” over K_{0} and depends on x only through the field K (as K_{0} is required to be a subfield of K).

Definition 2.1. The canonical dimension cdim(x) of x ∈ X (K) (resp. the essential dimension ed(x) of x) is defined as the least transcendence degree of an intermediate field K_{0} of K/F over which x is detected (resp. defined).

If p is a prime integer, the canonical p-dimension cdim_{p}(x) and the essential pdimension ed_{p}(x) of x are defined as the minimal value of cdim_{p}(x_{K′}) and ed_{p}(x_{K′}), respectively, where K′/K runs through all extensions of degree prime to p. We will write cdim_{0}(x) = cdim(x), ed_{0}(x) = ed(x) for notational convenience.

The numbers cdim(X), ed(X), cdim_{p}(X) and ed_{p}(X) are defined as the maximal values of cdim(x), ed(x), cdim_{p}(x) and ed_{p}(x), respectively, where x runs through all objects x ∈ X (K) over field extensions K/F.

The following inequalities follow immediately from the definitions: cdim_{p}(x) ≤ ed_{p}(x), cdim_{p}(X) ≤ ed_{p}(X), cdim_{p}(x) ≤ cdim(x), ed_{p}(x) ≤ ed(x), cdim_{p}(X) ≤ cdim(X) and ed_{p}(X) ≤ ed(X) for any prime p (for p = 0 as well).

When G is an algebraic group (not necessarily smooth) we write ed_{p}(G) in place of ed_{p}(BG) for any prime p and for p = 0. Here BG ≃ [Spec F/G] is the classifying stack of G. The groupoid BG(K) is simply the category of G-torsors over Spec(K).

We will use the following result on essential dimension of diagonalizable groups (see [Lö10, Corollary 5.8]):

Lemma 2.1. Let D be a diagonalizable algebraic group. Let D^{∗} denote the character group of D and D_{tors}^{∗} its torsion. Then

ed(D) = rk(D_{tors}^{∗}),

where rk stands for the minimal number of generators.

Moreover we will use the following inequality from [BF03, Theorem 6.19]: If H is a subgroup of an algebraic group H′ then

(3)  ed(H) + dim(H) ≤ ed(H′) + dim(H′).

2.2. Preliminaries on Gerbes. Every gerbe over F banded by the multiplicative group G_{m} is associated to a central simple F -algebra A. The objects X_{A}(K) over a field extension K/F are given by couples (V, α) of a K-vector space V together with an isomorphism α: A_{K} := A ⊗_{F} K →^{≃} End_{K}(V). Let n = deg(A). The gerbe X_{A} can be seen as the quotient stack [X/GL_{n}], where X = Isom(A, M_{n}(F)) is the PGL_{n}-torsor over F corresponding to A.

Alternatively, if L denotes a maximal étale subalgebra of A, the gerbe X_{A} can be represented as the quotient [U/R_{L/F} (G_{m})] for some R_{L/F} (G_{m})/G_{m}-torsor Y over F. Explicitly Y is the scheme of isomorphisms β: A_{K} →^{≃} End_{K}(V) that are compatible with the inclusion of L ⊗_{F} K in A_{K} and End_{K}(V) (by left multiplications), respectively.

The following lemma shows that in fact every gerbe banded by a group scheme of multiplicative type can be realized as the stack quotient of a torsor under a torus by a quasi-trivial torus.

Lemma 2.2. Let X be a gerbe banded by a group A of multiplicative type over F. Let L/F be a Galois extension such that X (L) ≠ ∅ and A_{L} is diagonalizable.

Then there exists a quasi-split algebraic torus P over F containing A and split over L and a P/A-torsor U over F such that X ≃ [U/P].

The splitting fields of X coincide with the splitting fields of U. In particular cdim X = cdim U = cdim(x), for any x ∈ X (F (U)) ≠ ∅.

Proof. Embed A_{L} in G_{m}^{n} for some n ≥ 0. This induces an embedding of R_{L/F} (A_{L}) in the quasi-split torus P = R_{L/F} (G_{m})^{n} split over L. The class of X lies in the kernel of the map H^{2}(F, A) → H^{2}(F, R_{L/F} (A_{L})) = H^{2}(L, A) taking X to X_{L}. Hence it also lies in the kernel of the map H^{2}(F, A) → H^{2}(F, P) induced by the composition A → R_{L/F} (A_{L}) → P, which is injective. Therefore the class of X lies in the image of the connecting map H^{1}(F, P/A) → H^{2}(F, A). In other words, there exists a P/A-torsor U over F such that X ≃ [U/P].

Now a field extension K/F is a splitting field of X if and only if U lifts to a P -torsor over K. This happens if and only if K is a splitting field of U. Hence X and U have the same splitting fields, whence the same canonical dimension. The last equality follows from the fact that U has a smooth projective compactification with the same splitting fields, see [KV84], and the proof of [Me13, Prop. 4.2]. □

Remark 2.1. With notations as in Lemma 2.2 let Ū be a smooth projective compactification of U. Then cdim(X) = cdim(U) is the least dimension of the image of a rational map Ū −−→ Ū by [KM06, Corollary 4.6]. Hence the computation of cdim(X) can be approached with geometric methods. We will not use this description within this paper, however.

We will also be interested in objects of the gerbe X over local F -algebras.

Corollary 2.1. With notations as in Lemma 2.2:

(a) If R is any local F -algebra then X (R) ≠ ∅ if and only U(R) ≠ ∅.

(b) Let R be a local F -algebra which is a valuation ring whose residue field is linearly disjoint from L over F. Let M be the field of fractions of R. Then X (M) ≠ ∅ if and only if X (R) ≠ ∅.

Proof. (a) This follows like in the proof of Lemma 2.2 and the fact that every P -torsor over R is trivial when P is quasi-split and R local.

(b) Clearly X (R) ≠ ∅ implies X (M) ≠ ∅. Conversely assume X (M) ≠ ∅. Let P and U be as above and let T = P/A. Then U(M) ≠ ∅ and so (U/T_{d})(M) ≠ ∅, where T_{d} denotes the largest split subtorus of T over F. Let X be a smooth projective T/T_{d}-equivariant compactification of U/T_{d}. Then X(M) ≠ ∅ and since X is complete, X(R) ≠ ∅.

By assumption R/𝔪 is linearly disjoint from L (which is a Galois splitting field of T/T_{d}) and therefore T/T_{d} is anisotropic over R/𝔪. By [Gi04, Proposition 1.1] X(R/𝔪) = (U/T_{d})(R/𝔪). Therefore the image of X(R) under the map X(R) → X(R/𝔪) is contained in (U/T_{d})(R/𝔪). It follows that (U/T_{d})(R) = X(R) ≠ ∅, i.e. U/T_{d} splits over R. Since R is local U splits over R as well. It follows that X (R) ≠ ∅.

□

Torsors over a local F -algebra under a group of multiplicative type can always be obtained as fibers between morphisms of algebraic tori, see the following well-known lemma.

Lemma 2.3. Let A be a subgroup of an invertible torus Q. If R is a local F -algebra, then every A-torsor over R is the fiber of Q → Q/A over some s ∈ (Q/A)(R).

Proof. The exact sequence 1 → A → Q → Q/A → 1 induces an exact sequence Q(R) → S(R) → H^{1}(R, A) → H^{1}(R, Q) = 1, where the map S(R) → H^{1}(R, A) takes s ∈ S(R) to the fiber of Q → S over s. □

3. Proof of Main Theorem

In this section we will prove Theorem 1.1. We start with a lemma, which will be used in the proof.

Lemma 3.1. Let g: V → W a homomorphism of free abelian groups of finite rank and U a subgroup of V with rk(U) ≤ rk(W). Then rk((V/U)_{tors}) ≤ rk(W/g(U)).

Proof. Let m = rk((V/U)_{tors}) and r = rk(U), r′ = rk(W), s = rk(V). By the elementary divisor theorem, there exist a basis e_{1}, …, e_{s} of the ℤ-module V and integers c_{1}, …, c_{m} > 1 with c_{1}|c_{2}| ⋯ |c_{m} such that the elements

c_{1}e_{1}, …, c_{m}e_{m}, e_{m+1}, …, e_{r}

form a basis of U. Let p be a prime dividing c_{1} (hence dividing every c_{i}). Then W/g(U) surjects onto

W/(g(U) + pW) ≃ (ℤ/pℤ)^{r′}/⟨\overline{g(e_{m+1})}, …, \overline{g(e_{r})}⟩

which has rank ≥ m. Therefore rk(W/g(U)) ≥ m = rk((V/U)_{tors}). □

Proof of Theorem 1.1. The last assertion of the Theorem follows from Lemma 2.1. Moreover recall inequality (2) ed(X) ≤ cdim(X)+ed(A) from the introduction. Let A′ := Q_{d} ∩ A.

We will prove that ed(X) ≥ cdim(X) + ed(A′) and ed(A′) ≥ ed(A). From this the claims follow readily.

For the assertion ed(A′) ≥ ed(A) consider the following diagram with exact rows:

[figure: 1; A′; Q_{d}; S; 1; 1; A; Q; S; 1]

Since Q (which is invertible) and Q_{d} (which is a split torus) are special, we get a surjection in Galois-cohomology H^{1}(−, A′) → H^{1}(−, A) as functors from the category of field extensions of F to the category of sets. Hence the claim follows from [BF03, Lemma 1.9].

Now we will prove the assertion ed(X) ≥ cdim(X)+ed(A′), for which we use the addition BA×X → X , (t, x) ↦ t+x and subtraction X ×X → BA, (x, x′) ↦ x−x′ as in [Me09]. By Lemma 2.2 we can choose a field extension K/F linearly disjoint from F_{alg} and x ∈ X (K) such that cdim(x) = cdim(X).

Let M := K(t_{1}, …, t_{s}), where s := dim S. We will identify S with G_{m}^{s}. We write ⟨t⟩ for the A-torsor over M given by f^{−1}(t_{1}, …, t_{s}). Let x′ := ⟨t⟩ + x_{M} ∈ X (M). We will prove that ed(x′) ≥ cdim(x) + ed(A′).

Let M_{0} be an intermediate field of M/F and y ∈ X (M_{0}) such that y_{M} ≃ x′ and tdeg_{F} (M_{0}) = ed(x′). Let ν be the canonical ℤ^{r}-valued valuation of K_{alg}(t_{1}, …, t_{s}), so that ν(t_{i}) = e_{i} := (0, …, 1, …, 0) ∈ ℤ^{s} (with 1 on place i) and ν|_{K_{alg}} is trivial. Let ν_{0} = ν|_{M_{0}} and R_{0} ⊆ M_{0} the valuation ring of ν_{0} and R the valuation ring of ν|_{M}, whose residue field is K.

Claim: rk(ν_{0}) ≥ ed(A′).

The residue field of R_{0} embeds in K over F, hence it is linearly disjoint from the finite extension L/F. Since y ∈ X (M_{0}) we get X (R_{0}) ≠ ∅ by Corollary 2.1. Choose x_{0} ∈ X (R_{0}). By Lemma 2.3 there exists z = (z_{1}, …, z_{s}) ∈ S(M_{0}) = (M_{0}^{×})^{s} such that y − (x_{0})_{M_{0}} ≃ ⟨z⟩ ∈ BA(M_{0}), where ⟨z⟩ denotes the fiber f^{−1}(z). Since z_{i} ∈ M_{0}^{×} it suffices to show that the subgroup of ℤ^{s} generated by ν(z_{1}), …, ν(z_{r}) has rank ≥ ed(A′). We have

⟨z⟩_{M} ≃ y_{M} − (x_{0})_{M} ≃ x′ − (x_{0})_{M} = (⟨t⟩ + x_{M}) − (x_{0})_{M} ≃ ⟨t⟩ + (x_{M} − (x_{0})_{M}).

Note that x_{M} −(x_{0})_{M} is defined over R (since R contains K and R_{0}). By Lemma 2.3 there exists r = (r_{1}, …, r_{s}) ∈ S(R) = (R^{×})^{s} ⊆ (M^{×})^{s} with x_{M} − (x_{0})_{M} ≃ ⟨r⟩_{M}. Therefore

⟨z⟩_{M} ≃ ⟨t⟩ + ⟨r⟩_{M} ≃ ⟨tr⟩.

It follows that the classes of z and tr are equal in (M^{×})^{s}/f_{M}(Q(M)). Hence there exists w ∈ Q(M) such that z = trf_{M}(w). That yields the relations

ν(z_{j}) = ν(t_{j}) + ν(r_{j}) + ν(f_{M}(w)_{j}) = e_{j} + ν(f_{M}(w)_{j})

for j = 1, …, s. Hence it suffices to show that

rk(ℤ^{s}/⟨ν(f_{M}(w)_{1}), …, ν(f_{M}(w)_{s})⟩) ≥ ed(A′).

We first reduce to the case when Q is quasi-split. Let Q′ be an F -torus such that Q̃ := Q × Q′ is quasi-split and let Ã = A × Q′ and Ã′ = Ã ∩ Q̃_{d} = A′ × Q′_{d}. Then ed(A′) = ed(Ã′), since Q′_{d} is special. We may replace A by Ã and Q by Q̃ and hence assume that Q is quasi-split, Q = ∏_{i=1}^{m} R_{L_{i}/F} (G_{m}) for some separable field extensions L_{1}/F, …, L_{m}/F.

Let C ∈ M_{m,s}(ℤ) be the matrix corresponding to the homomorphism f|_{Q_{d}}: Q_{d} → S. on character groups. Let U denote the span of the rows of C. Then A′ has character lattice ℤ^{m}/U. Hence as observed above

ed(A′) = rk(ℤ^{m}/U)_{tors}.

There exists B ∈ M_{s,m}(ℤ) such that for every field extension F′/F the group homomorphism f_{F′}: ∏_{i=1}^{m}(L_{i} ⊗_{F} F′)^{×} → (F^{′×})^{s} is given by

f_{F′}(x_{1}, …, x_{m}) = (∏_{i=1}^{m} N_{L_{i}⊗_{F} F′/F′}(x_{i})^{b_{ij}})_{1≤j≤s} .

Then c_{ij} = b_{ij}[L_{i} : F] for all i and j.

Since F_{sep} and M = K(t_{1}, ⋯, t_{s}) are linearly disjoint over F we may view F_{sep} ⊗_{F} M as a subfield of K_{alg}(t_{1}, ⋯, t_{s}).

Write w = (w_{1}, …, w_{m}) ∈ Q(M) = ∏_{i=1}^{m}(L_{i} ⊗_{F} M)^{×}. Note that the value ν((σ ⊗ id_{M})(w_{i})) ∈ ℤ^{s} for a F -homomorphism σ: L_{i} → F_{sep} is independent of the particular choice of σ. The product ∏(σ ⊗ id_{M})(w_{i}) taken over all [L_{i} : F] homomorphisms σ is equal to N_{L_{i}⊗_{F} M/M}(w_{i}). Writing ν_{i} := ν((σ ⊗ id_{M})(w_{i})) we get

ν(f_{M}(w)_{j}) = ∑_{i=1}^{m} b_{ij}ν(N_{L_{i}⊗_{F} M/M}(w_{i})) = ∑_{i=1}^{m} b_{ij}[L_{i} : F]ν_{i} = ∑_{i=1}^{m} c_{ij}ν_{i}.

Let g: ℤ^{m} → ℤ^{s} the group homomorphism, which takes the i-th standard basis vector of ℤ^{m} to ν_{i} for all i. Then ⟨ν(f_{M}(w)_{1}), …, ν(f_{M}(w)_{m})⟩ = g(U). Note that rk(U) = s, since f|_{Q_{d}} is surjective. Hence Lemma 3.1 yields:

rk(ℤ^{s}/⟨ν(f_{M}(w)_{1}), …, ν(f_{M}(w)_{m})⟩) = rk(ℤ^{s}/g(U)) ≥ rk((ℤ^{m}/U)_{tors}) = ed(A′).

Therefore rk(ν_{0}) ≥ ed(A′), proving the claim.

Let \overline{M_{0}} denote the residue field of ν_{0}. Note that \overline{M_{0}} can be considered as a subfield of the residue field of ν|_{M}, which is F -isomorphic to K. Since X (R_{0}) ≠ ∅ we have X (\overline{M_{0}}) ≠ ∅, hence

ed(x′) = tdeg_{F} M_{0} ≥ tdeg_{F} \overline{M_{0}} + rk(ν_{0}) ≥ tdeg_{F} \overline{M_{0}} + ed(A′)
≥ cdim(x) + ed(A′).

Therefore ed(X) ≥ cdim(x) + ed(A′) = cdim(X) + ed(A′). This concludes the proof. □

4. Applications

We will discuss three kinds of applications in the following subsections. The first two concern extensions of algebraic groups by abelian algebraic groups. In other words we have a short exact sequence

1 → A → G → H → 1

with A abelian (not necessarily central). We want to use such a sequence to get a lower bound on ed(G), using the inequality (1) for an H-torsor X over a field extension K of F. Note that in general [X/G] is a gerbe banded by the twisted group ^{X}A, but (1) holds regardless of whether A is central in G or not. The twist is isomorphic to A_{K} if A is central in G. When A is not central it may easily happen that A is split and the twist is not split.

Our results on essential dimension of gerbes yield new applications in the following cases

(a) A itself is a (non-split) group of multiplicative type that lies between an invertible torus and its anisotropic part and A is central in G (b) A is (possibly split, but) non-central and the twist ^{X}A lies between an

invertible torus and its anisotropic part. The third application concerns an example of splitting a global problem (i.e. depending on several primes) into two local problems (depending on a single prime).

4.1. Central extensions by non-split groups. A simple example of a torus T for which we can apply Theorem 1.1 is a non-split one-dimensional torus, i.e. T = R_{L/F}^{(1)} (G_{m}), where L/F is a quadratic separable field extension. This group is the center of the unitary group U(B, τ) where B is a central simple L-algebra with F -linear unitary involution τ.

Proposition 4.1. If B is a division algebra and n := deg(B) = 2^{r} for some r ≥ 0 then

ed(U(B, τ)) = n^{2}.

Proof. The inequality ed(U(B, τ)) ≤ n^{2} follows from the fact that U(B, τ) is contained in the special algebraic group R_{L/F} (GL_{1}(B)) with codimension n^{2}.

For the reverse inequality we consider the following commutative diagram with exact rows:

[figure: 1; R_{L/F}^{(1)} (G_{m}); U(B, τ); PGU(B, τ); 1; 1; R_{L/F} (G_{m}); GU(B, τ); PGU(B, τ); 1]

By [Lö11, Theorem 3.2] (and its proof) there exists a field extension K/F linearly disjoint from L/F and a PGU(B, τ)-torsor X over K whose image under the connecting map

H^{1}(K, PGU(B, τ)) → H^{2}(K, R_{L/F} (G_{m})) = Br(L ⊗_{F} K)

is the Brauer class of a central simple L ⊗_{F} K-algebra A of index n^{2} whose norm N_{L⊗_{F} K/K}(A) is split. The gerbe [X/U(B, τ)] banded by R_{L/F}^{(1)} (G_{m}) has the same splitting fields as the K-variety R_{L⊗_{F} K/K}(SB(A)). Therefore by (1) and Theorem 1.1,

ed(U(B, τ)) ≥ ed([X/U(B, τ)]) − dim PGU(B, τ)
= cdim(R_{L⊗_{F} K/K}(SB(A))) + ed (R_{L/F}^{(1)} (G_{m})) − dim PGU(B, τ)
= cdim(R_{L⊗_{F} K/K}(SB(A))) + 1 − (n^{2} − 1).

By [Ka12, Theorem 1.1]

cdim(R_{L⊗_{F} K/K}(SB(A))) = dim(R_{L⊗_{F} K/K}(SB(A))) = 2(n^{2} − 1),

hence the inequality ed(U(B, τ)) ≥ n^{2} follows. □

4.2. Non-central extensions by abelian groups. Let A be a central simple F -algebra of degree d and let n ≥ 1. A simple example of a non-central extension by an abelian group is given by the exact sequence

1 → G_{m}^{n} → GL_{1}(A)^{n} ⋊ S_{n} → PGL_{1}(A)^{n} ⋊ S_{n} → 1,

where the semidirect products are understood with respect to the natural permutation action. The group PGL_{1}(A)^{n} ⋊ S_{n} can be considered as automorphism group scheme of the F -algebra A^{n}. From this observation we get a natural bijection between isomorphism classes of PGL_{1}(A)^{n} ⋊ S_{n}-torsors over a field extension K/F and isomorphism classes of Azumaya algebras of (constant) degree d over n-dimensional étale extensions E/K.

Similarly isomorphism classes of GL_{1}(A)^{n} ⋊ S_{n}-torsors over K can be identified with isomorphism classes of pairs (E, M) consisting of an n-dimensional étale Kalgebra E and an A_{E} := A ⊗_{F} E-module M of reduced dimension d, where a morphism between two pairs (E, M) and (E′, M′) is a pair (ι, φ) consisting of a K-algebra isomorphism ι: E → E′ and an A-linear isomorphism φ: M → M′ satisfying φ(em) = ι(e)φ(m) for all e ∈ E, m ∈ M. In these terms the map

H^{1}(K, GL_{1}(A)^{n} ⋊ S_{n}) → H^{1}(K, PGL_{1}(A)^{n} ⋊ S_{n})

takes the isomorphism class of (E, M) to the isomorphism class of End_{A_{E}}(M).

For a PGL_{1}(A)^{n} ⋊S_{n}-torsor X over K corresponding to an Azumaya algebra B over the n-dimensional étale K-algebra E the gerbe [X/GL_{1}(A)^{n}⋊S_{n}] is banded by R_{E/K}(G_{m}) and its class in H^{2}(K, R_{E/K}(G_{m})) = Br(E) is the class of [B] − [A_{E}].

Now let n = 2 and suppose that A carries an involution σ of the first kind. Let

G := GL_{1}(A) ⋊ S_{2},

where the non-trivial element of S_{2} acts via a ↦ σ(a^{−1}). We have a commutative diagram with exact rows

\xymatrix{1 \ar[r] & R_{F^{2}/F}^{(1)} (G_{m}) \ar@{^{(}->}[d] \ar[r] & G \ar[r] \ar[d] & PGL_{1}(A) ⋊ S_{2} \ar[r] \ar[d] & 1 \\ 1 \ar[r] & R_{F^{2}/F} (G_{m}) \ar[r] & GL_{1}(A)^{2} ⋊ S_{2} \ar[r] & PGL_{1}(A)^{2} ⋊ S_{2} \ar[r] & 1},

where the second and third vertical arrows are induced by the inclusion

GL_{1}(A) → GL_{1}(A)^{2}, a ↦ (a, σ(a^{−1})).

A PGL_{1}(A) ⋊ S_{2}-torsor X over a field extension K/F corresponds to a pair (B, τ) of a central simple K-algebra of degree d with unitary involution, where the trivial torsor corresponds to the pair (A × A, ε) with ε(a, b) = (σ(b), σ(a)). The induced PGL_{1}(A)^{2} ⋊ S_{2}-torsor corresponds to the algebra B. The gerbe [X/G] is then banded by R_{E/K}^{(1)}(G_{m}), where E := Z(B), and its class in

H^{2}(K, R_{E/K}^{(1)}(G_{m})) = ker(cor_{E/K}: Br(E) → Br(K)) ⊆ Br(E)

is again the class of [B] − [A_{E}].

Proposition 4.2. With notations as above:

(a) For any field extension K/F and any pair (B, τ) of a central simple Kalgebra B of degree d with unitary involution τ we have

ed(G) ≥ cdim(R_{E/K}(SB(B ⊗_{F} A^{op}))) + δ_{E} − (d^{2} − 1),

where δ_{E} := ed(R_{E/K}^{(1)}(G_{m})) ∈ {0, 1} is 0 if E is split (i.e., E ≃ K × K) and 1 otherwise (i.e., if E is a field).

(b) Suppose d > 1. Then we have ed(G) ≤ d^{2}.

(c) Suppose that A is a division algebra of index d = 2^{m} with m ≥ 1. Then

ed(G) = d^{2}.

Proof. (a) Let X be the PGL_{1}(A) ⋊ S_{2}-torsor corresponding to (B, τ). Then the gerbe [X/G] has the same splitting fields as R_{E/K}(SB(B⊗_{F} A^{op})), hence the desired inequality follows from inequality (1) and Theorem 1.1.

(b) Define a representation of G on the F -vector space V = A ⊕ A by letting GL_{1}(A) act on V through a · (a_{1}, a_{2}) = (aa_{1}, σ(a^{−1})a_{2}) and S_{2} through permuting coordinates. Clearly dim V = 2d^{2}, hence dim V − dim G = d^{2}. In view of [BF03, Proposition 4.11] it suffices to prove that V is generically free in case d > 1. Let U denote the open subset of the affine space 𝔸(V) consisting of those pairs (a_{1}, a_{2}) where a_{1} and a_{2} are invertible and a_{1}a_{2}^{−1} is not symmetric with respect to σ. It is easy to check that U is G-stable and G acts freely on U. Moreover the symmetric elements Sym(A, σ) form a proper subspace of A in case d > 1, hence U is non-empty. This proves the claim.

(c) By part (b) ed(G) ≤ d^{2}. We will prove ed(G) ≥ d^{2} using part (a). Since dim(R_{E/K}(SB(B ⊗_{F} A^{op}))) = 2(d^{2} − 1) it suffices to construct a field extension K/F and a pair (B, τ) of a central simple K-algebra with unitary involution such that E is a field (so δ_{E} = 1) and

Y := R_{E/K}(SB(B ⊗_{F} A^{op}))

is incompressible, i.e., cdim(Y) = dim(Y). For this by N. Karpenko’s incompressibility result [Ka12, Theorem 1.1] it suffices that B ⊗_{F} A^{op} is a division algebra (note that B and A_{E} both carry unitary involutions, hence N_{E/K}(B ⊗_{F} A^{op}) is split as required). The existence of such a pair (B, τ) was proven in [Lö11, Lemma 2.3].

□

Remark 4.1. (a) In order to prove the correct lower bound on ed(G) in Proposition 4.2 using part (a) we really must choose the pair (B, τ) over a field K such that the center E = Z(B) is non-split. Otherwise B ≃ B_{0} × B_{0}^{op} for some central simple K-algebra B_{0} of degree d and the variety R_{E/K}(SB(B ⊗_{F} A^{op})) has the same splitting fields as SB(B_{0} ⊗_{F} A^{op}), which has strictly smaller dimension. Moreover δ_{E} would be 0 instead of 1.

(b) For m = 0, d = 2^{m} = 1, it is not true that ed(G) = d^{2} = 1, since then G ≃ O_{2} and this group has ed(G) = 2 by [Re99, Theorem 10.3].

4.3. Splitting a global problem into two local problems. In almost all situations, when the (absolute) essential dimension of an algebraic group or algebraic stack is known, it is equal to its essential p-dimension for some prime p. This is a limitation of most methods we have in Galois cohomology and related algebraic areas. For a broad discussion of this phenomenon, see [Re10, §5]. As an example it is well known that ed_{p}(PGL_{p}) = 2 for all primes p, but the question if ed(PGL_{p}) = 2 for all primes p is widely open and a negative answer to this question would disprove the cyclicity conjecture of Albert for central simple algebras of degree p.

When the absolute essential dimension differs from essential p-dimension for all primes p (or is not known to coincide with essential p-dimension for some prime p) it often becomes significantly more difficult to compute it. This usually happens when several primes are immediately tied to the problem. We call such a problem a global problem. A local problem is one, which depends only on one prime.

Theorem 1.1 allows us to attack some of these global problems by decomposing a global problem into two local problems.

We construct the following example: Let K/F be a quadratic separable extension and L/F a qubic separable extension. Let B be a central simple K-algebra of degree 3. Assume that N_{K/F} (B) and B ⊗_{K} (K ⊗_{F} L) ≃ B ⊗_{F} L are split, but B is nonsplit. Let X_{B} be the R_{K/F}^{(1)} (G_{m})-gerbe over F corresponding to the class of B in

H^{2}(F, R_{K/F}^{(1)} (G_{m})) = ker(Br(K) →^{cor} Br(F)).

Proposition 4.3. ed(X_{B}) = 3, ed_{2}(X_{B}) = 1, ed_{3}(X_{B}) = 2 and ed_{p}(X_{B}) = 0 for every prime p ≠ 2, 3.

Proof. By Theorem 1.1

ed(X_{B}) = cdim(X) + ed(R_{K/F}^{(1)} (G_{m})) = cdim(X_{B}) + 1.

Since X_{B} splits over L, which is of degree 3 over F, we have cdim_{p}(X_{B}) = 0 for every prime p ≠ 3, hence for these primes ed_{p}(X_{B}) = ed_{p}(R_{K/F}^{(1)} (G_{m})), which is 1 if p = 2 and 0 if p ≠ 2, 3.

Moreover ed_{3}(X_{B}) ≤ cdim_{3}(X_{B}) + ed_{3}(X_{B}) = cdim_{3}(X_{B}). Since cdim_{3}(X_{B}) ≤ cdim(X_{B}) it remains to show that cdim(X_{B}) ≤ 2 and cdim_{3}(X_{B}) ≥ 2. Note that X_{B} becomes a G_{m}-gerbe over K corresponding to the class of B in Br(K). Since B is non-split of degree 3 we have cdim_{3}(X_{B}) = cdim_{3}((X_{B})_{K}) = 2. To show cdim(X_{B}) ≤ 2 first note that a field extension M/F is a splitting field of X_{B} if and only if B ⊗_{F} M is split. Hence it follows from [Bl10, Corollary 3.6] that X_{B} has the same splitting fields as a Del Pezzo surface S of degree 6 (the one associated to the triple (B, M_{2}(L), KL)). Therefore cdim(X_{B}) ≤ dim S = 2. □

Acknowledgments

I would like to thank Zinovy Reichstein for useful discussions about the topic of this paper. Moreover I am grateful to Philippe Gille for helping me with the proof of Corollary 2.1.

References

[BF03] G. Berhuy, G. Favi, Essential Dimension: A Functorial Point of View (after A. Merkurjev), Doc. Math. 8 (electronic) (2003), 279–330.

[Bl10] M. Blunk, Del Pezzo surfaces of degree 6 over an arbitrary field, J. Algebra 323 (2010), no. 1, 42–58.

[BR97] J. Buhler, Z. Reichstein, On the essential dimension of a finite group, Compositio Mathematica 106 (1997), 159–179.

[BRV11] P. Brosnan, Z. Reichstein, A. Vistoli, Essential dimension of moduli of curves and other algebraic stacks. With an appendix by Najmuddin Fakhruddin. J. Eur. Math. Soc. (JEMS) 13 (2011), no. 4, 1079-1112.

[Gi04] Ph. Gille, Spécialisation de la R-équivalence pour les groupes réductifs, Trans. Amer. Math. Soc. 356 (2004), 4465–4474.

[Gir71] J. Giraud, Cohomologie non abélienne, Springer-Verlag, Berlin, 1971, Die Grundlehren der mathematischen Wissenschaften, Band 179.

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

[Ka12] N. Karpenko, Incompressibility of quadratic Weil transfer of generalized Severi- Brauer varieties., J. Inst. Math. Jussieu 11 (2012), no. 1, 119–131.

[KM06] N. Karpenko, A. Merkurjev, Canonical p-dimension of algebraic groups. Adv. Math. 205, 2 (2006), 410–433.

[KM08] N. Karpenko, A. Merkurjev, Essential dimension of finite p-groups, Inventiones Mathematicae, 172 (2008), 491–508.

[KV84] A. Klyachko, V. Voskresenskii, Toric Fano varieties and systems of roots. (Russian) Izv. Akad. Nauk SSSR Ser. Mat. 48 (1984), no.2, 237–263.

[Lö10] R. Lötscher, Contributions to the essential dimension of finite and algebraic groups,

           Ph.D. thesis (2010), available at edoc.unibas.ch/1147/1/DissertationEdocCC.pdf.

[Lö11] R. Lötscher, Essential dimension of involutions and subalgebras. Israel J. Math.

192(1), (2012), 325–346.

[Lö12] R. Lötscher, A fiber dimension theorem for essential and canonical dimension, Com-

pos. Math. 149(1), (2013), 148–174.

[LMMR13] R. Lötscher, M. MacDonald, A. Meyer, Z. Reichstein, Essential dimension of algebraic tori. Crelle 677 (2013), 1–13.

[Me08] A. Merkurjev, Essential p-dimension of finite groups, Mini Course Lens 2008, (2008), http://www.math.ucla.edu/˜ merkurev/publicat.htm.

[Me09] A. 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.

[Me13] A. Merkurjev, Essential dimension: a survey, Transf. Groups 18, no. 2 (2013), 415–481.

[Re99] Z. Reichstein, On the Notion of Essential Dimension for Algebraic Groups, Transf. Groups 5, no. 3 (2000), 265–304.

[Re10] Z. Reichstein, Essential dimension, Proceedings of the International Congress of Mathematicians, Vol. II, Hindustan Book Agency, New Delhi, 2010, pp. 162–188.

[Stacks] The Stacks Project Authors, Stacks Project, http://math.columbia.edu/algebraic geometry/stacks-git.

Roland Lötscher

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

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