A FIBER DIMENSION THEOREM FOR ESSENTIAL AND CANONICAL DIMENSION

ROLAND LÖTSCHER

Abstract. The well-known fiber dimension theorem in algebraic geometry says that for every morphism f: X → Y of integral schemes of finite type, the dimension of every fiber of f is at least dim X − dim Y. This has recently been generalized by P. Brosnan, Z. Reichstein and A. Vistoli to certain morphisms of algebraic stacks f: X → Y, where the usual dimension is replaced by essential dimension. We will prove a general version for morphisms of categories fibered in groupoids. Moreover we will prove a variant of this theorem, where essential dimension and canonical dimension are linked.

These results let us relate essential dimension to canonical dimension of algebraic groups. In particular, using the recent computation of the essential dimension of algebraic tori by M. MacDonald, A. Meyer, Z. Reichstein and the author, we establish a lower bound on the canonical dimension of algebraic tori.

Contents

1. Introduction 1 2. Preliminaries 4 2.1. Conventions 4 2.2. Torsors and twists 4 2.3. CFG’s, stacks and gerbes 5 2.4. Essential and canonical dimension of CFG’s 8 3. Fibers for morphisms of CFG’s 10 4. p-exhaustive subgroups 14 5. Canonical dimension of group schemes 22 Acknowledgments 26 References 26

1. Introduction

A category fibered in groupoids (abbreviated CFG) over a field F is roughly a category X equipped with a functor π: X → Sch_{F} to the category Sch_{F} of schemes over F for which pullbacks exist and are unique up to canonical isomorphism. See section 2 for a formal definition.

Date: December 16, 2011. 2000 Mathematics Subject Classification. 20G15, 11E72, 14A20. Key words and phrases. essential dimension, canonical dimension, algebraic groups, fibers,

category fibered in groupoids, algebraic stack, algebraic torus.

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

A typical example of a CFG over F is the quotient [X/G] of a scheme X by the action of an algebraic group G, see Example 2.1. CFG’s of the form [X/G] often arise in moduli problems. Unlike many quotients in geometric invariant theory, they keep a lot of information about the G-equivariant geometry of X.

To every CFG X over F we can attach two numbers, ed X and cdim X (with cdim X ≤ ed X), called essential dimension, resp. canonical dimension of X, see section 2. In the case where X is representable by a scheme X locally of finite type, the essential dimension of X coincides with the usual dimension of X, and cdim X is a number between 0 and dim X, which measures how far X is from having a rational point.

There are versions of essential and canonical dimension relative to a prime p, written ed_{p} X and cdim_{p} X, which basically neglect effects from passing to prime to p field extensions. We will include the case p = 0 for usual dimensions and write ed_{0} X and cdim_{0} X for ed X and cdim X, respectively.

Denote by ℙ = {2, 3, … } the set of all primes. We will prove the following general result on fiber dimensions:

Theorem 1.1. Let f: X → Y be a morphism of CFG’s over F. Then for every p ∈ ℙ ∪ {0}:

ed_{p} X ≤ ed_{p} Y + sup_{y} ed_{p} X_{y}

and

cdim_{p} X ≤ ed_{p} Y + sup_{y} cdim_{p} X_{y},

where the supremum is taken over all field extensions K/F and all morphisms y: Spec K → Y of CFG’s over F.

Here X_{y} (the fiber of f over y) is the 2-fiber product of X and Spec K over Y with respect to f and y, see section 2. It considered as a CFG over K.

The special case of the first inequality, where both CFG’s are represented by schemes locally of finite type over F, is implied by the well-known fiber dimension theorem from algebraic geometry, cf. [Ha77, Exercise II.3.22]. The more general case of the same inequality, when X and Y are algebraic stacks and all fibers X_{y} are representable by quasi-separated algebraic spaces, locally of finite type and of dimension ≤ d for some fixed d ∈ ℕ_{0} is exactly the result of [BRV11, Theorem 3.2].

The second inequality, where canonical and essential dimension are linked, seems to be completely new and is a key ingredient for establishing results on canonical dimension of algebraic groups later on.

Let G be an algebraic group over a field F. The essential p-dimension of G, denoted ed_{p} G, is defined as the essential p-dimension of BG ≃ [Spec F/G], the CFG of G-torsors. It was introduced by J. Buhler and Z. Reichstein in [BR97] and has been object of study for numerous mathematicians since then. See Z. Reichstein’s ICM proceedings [Re10] for a survey on the topic.

The essential dimension of a G-torsor X over a field extension K of F, viewed as object of BG, measures how far X is from being defined (up to isomorphism) over the base field F. On the other hand, the canonical dimension of X, introduced by G. Berhuy and Z. Reichstein in [BR05], is the canonical dimension of the CFG represented by the scheme X and measures how far X is from being split.

Set

cdim G := sup cdim X (and cdim_{p} G := sup cdim_{p} X)

where X runs over all G-torsors over field extensions. Then for G connected and smooth we have ed G = 0 if and only if cdim G = 0 if and only if G is special, i.e., all G-torsors over field extensions of F are split, see [Me09, Proposition 4.4] and recall that a geometrically integral variety X over F has strictly positive canonical dimension unless it has a F -rational point. In general ed G can be much larger than cdim G (e.g. for spin groups, see Corollary 4.10) and vice versa (see Example 5.13).

For split simple (affine) algebraic groups G the value of the canonical p-dimension of G has been computed for every prime p. The case of classical G is due to N. Karpenko and A. Merkurjev [KM06], the case of exceptional G is due to K. Zainoulline [Za07].

The assumption on G being split, i.e., containing a split maximal torus, is essential in their approach. Let B be a Borel subgroup containing the split maximal torus. Then B is special (i.e., has no non-split torsors over field extensions) and therefore, for a G-torsor X the varieties X and X/B have the same splitting fields and in particular the same canonical p-dimension. The variety X/B is smooth, projective and generically split. For these varieties the canonical p-dimension can be expressed through the existence of rational cycles in Chow-groups with 𝔽_{p}-coefficients [KM06, Theorem 5.8]. For a survey on canonical dimension of smooth projective varieties we refer to N. Karpenko’s ICM survey [Ka10].

We will be mainly interested in canonical dimension of tori. In this case all we can do with the above approach is to reduce the study of the canonical p-dimension of torsors of an arbitrary torus to the case of an anisotropic torus (mod out the maximal split subtorus).

Our approach to compute the canonical dimension of tori will be very different from the one above used for split simple algebraic groups. We will use Theorem 1.1 to relate, for an algebraic group G, the essential dimension of suitable subgroups D of G with the canonical dimension of the quotient G/D. This approach produces interesting results for algebraic tori T, which split over a Galois extension of ppower degree, where p is a prime. Here D is any subgroup of T which contains the (unique) largest subgroup C(T) of T of the form μ_{p}^{r}, r ≥ 0. The relation we establish in Corollary 5.5 has the following simple form:

Theorem 1.2. cdim_{p} T/D ≥ dim T/D − ed_{p} D.

Its proof makes full use of the computation of the essential p-dimension of T from [LMMR11]. The general statement for arbitrary G is given in Theorem 5.1. In section 5 we then proceed to find algebraic tori S which can be written as quotients S ≃ T/D with D ⊇ C(T) as in Theorem 1.2 and for which we can show that equality holds. This happens, for instance, for every anisotropic algebraic torus S which splits over a cyclic Galois extension of p-power degree (see Example 5.12) and for every direct product of such tori.

The rest of the paper is structured as follows: In section 2 we recall some basics on torsors, twists, CFG’s, 2-fiber products, stacks, gerbes etc. and define essential and canonical dimension. Section 3 is devoted to the proof of the general fiber dimension results and to applications in basic situations. In section 4 we introduce and study p-exhaustive subgroups. Roughly speaking these are normal subgroups of an algebraic group G for which the essential p-dimension of G can be expressed via the essential p-dimension of gerbes of the form [E/G] for G/C-torsors E. We then apply the fiber dimension results to spin groups. Finally section 5 contains our results on canonical dimension of algebraic groups, in particular of algebraic tori.

2. Preliminaries

2.1. Conventions. We denote by F a field, which serves as our base field.

We will use the “Stacks Project” [Stacks] as our main reference for algebraic spaces, stacks, gerbes etc. All these notions are understood with respect to the fppf-topology. As in [Stacks] (and in contrast to [LMB00], for instance) we try not to ignore any set-theoretical issues. Thus we will work over any big fppf-site Sch_{F} as in [Stacks, Definition 021R]. This site is non-canonical but has the advantage that its class of objects is a set. All schemes over F under consideration are assumed to be objects of Sch_{F}. Note that Sch_{F} contains, among other objects, for every finitely generated F -algebra A and for every finitely generated field extensions K/F some scheme isomorphic to Spec A (resp. Spec K), see [Stacks, Lemma 000R]. For notational convenience we will assume that for every finitely generated field extension K/F there exists a field extension K′/F isomorphic to K such that Spec K′ ∈ Sch_{F}.

All of our group algebraic spaces and group schemes over a field K under consideration are assumed to be locally of finite type over K.

2.2. Torsors and twists. Let G be a group algebraic space (locally of finite type) over a field F in the sense of [Stacks, 043H] (with B = S = Spec F). Usually G will be an affine group scheme of finite type over F for us. However more general group algebraic spaces will appear naturally as automorphism group algebraic spaces of points of algebraic stacks. Since we do not assume algebraic spaces to be quasiseparated, there are group algebraic spaces over a field F which are not group schemes (for an example see [Stacks, Lemma 06E4]).

Let U be an algebraic space over F. A G-torsor over U is an algebraic space E over F with a right action of G (in the sense of [Stacks, Definition 043Q]) and a G-invariant morphism E → U of algebraic spaces which is fppf-locally isomorphic on U to the trivial torsor U × G → U.

A G-torsor over a field extension K/F is a G-torsor E over Spec K. It is trivial if and only if it has a K-rational point. Note that since G is locally of finite type over F every G-torsor E over K becomes trivial over the algebraic closure K_{alg}.

We remark that if G is an affine group scheme (which will usually be the case for us) then every G-torsor over a field extension K/F is representable by a scheme, cf. [Stacks, Remark 049C].

For any G-torsor X over a field extension K/F we can form the twist

XG := Aut_{G}(X),

the group algebraic space over K of G-equivariant automorphisms of X. If X is trivial we have ^{X}G ≃ G_{K}.

More generally if N is a normal subgroup of G we form the twist ^{X}N as follows: First note that for every morphism f: G → H there is an induced H-torsor f_{∗}(X), defined as the quotient

f_{∗}(X) = (X × H)/G,

where G acts by the formula (x, h)g = (xg, f(g)^{−1}h). By descent the quotient exists as an algebraic space and is an H-torsor, see [Stacks, Lemma 04U0].

Now we apply this construction to the canonical morphism π: G → H := G/N. Let Y = π_{∗}(X). Then we get an induced morphism ^{X}G → ^{Y}H of group algebraic spaces. The twist of N by X is defined as the kernel

XN := ker(^{X}G → ^{Y}H)

of this morphism.

If G, N and G/N are smooth affine group schemes over F we associate with a G-torsor X a 1-cocycle z ∈ Z^{1}(Γ, G(F_{sep})) (unique up to the choice of a point x_{0} ∈ X(F_{sep})), where Γ := Gal(F_{sep}/F), and consider the twisted Γ-action on N(F_{sep}) by the cocycle z. This group, denoted _{z}N(F_{sep}) in [Se02] can be identified Γ-equivariantly with the group of F_{sep}-rational points of the twist ^{X}N. Thus our construction of ^{X}N is equivalent to the twist-construction _{z}N(F_{sep}) in Galoiscohomology for smooth affine group schemes.

2.3. CFG’s, stacks and gerbes. Let C be a category. A category fibered in groupoids, abbreviated CFG, over C is a category A equipped with a functor π: A → C subject to the following two conditions

(a) For every morphism ι: U → V in C and object a ∈ A with π(a) = V there exists an object b of A and a morphism f: b → a in A such that π(f) = ι (cf. diagram below).

\xymatrix{∃b \ar@{|->}[d] \ar@{-->}[r]^{f} & a \ar@{|->}[d] \\ U \ar[r]^{ι} & V}

(b) For every pair of morphisms f: b → a and g: c → a in A and every morphism ι: π(c) → π(b) such that π(f) ∘ ι = π(g) there exists a unique morphism h: c → b in A such that π(h) = ι and f ∘ h = g (cf. diagram below).

\xymatrix{c \ar@{|->}[d] \ar[drr]^{g} \ar@{-->}[dr]_{h} \\ π(c) \ar[dr]_{ι} \ar[drr]^{π(g)} & b \ar@{|->}[d] \ar[r]_{f} & a \ar@{|->}[d] \\  & π(b) \ar[r]_{π(f)} & π(a)}

We will call CFG over F a CFG over the category Sch_{F}. Every scheme X ∈ Sch_{F} gives rise to a CFG X̃ over F: Its objects are morphisms T → X, where T ∈ Sch_{F}, its morphisms are morphisms T → S compatible with the morphisms to X and the structure morphism X̃ → Sch_{F} is the projection onto the domain.

Recall that morphisms X → Y of schemes over F are in canonical 1-to-1 correspondence with morphisms X̃ → Ỹ by the Yoneda lemma. In the sequel we will use the notation X for the CFG X̃ associated with a scheme X and make it clear from the context, if the scheme X or the CFG X is meant.

By the term “stack over F ” we mean a CFG X over F satisfying the additional conditions (2) and (3) from [Stacks, Definition 02ZI] on patching isomorphisms and objects of X. Note that all our stacks are fibered in groupoids.

The CFG associated with a scheme X over F is a stack. More generally every algebraic space X over F is a stack. The algebraic spaces over F are precisely those stacks over F whose objects do not have any non-trivial automorphisms lying over the identity of their base, see [Stacks, Proposition 04SZ].

Another type of examples that we will often use in the sequel are quotients of algebraic spaces by group actions:

Example 2.1. Let G be a group algebraic space over F and X be an algebraic space over F on which G acts (from the right). The quotient stack [X/G] of X by the G-action is the CFG over F, whose objects are diagrams

(1)  \xymatrix{E \ar[r]^{φ} \ar[d] & X \\ U} ,

where U ∈ Sch_{F}, E → U is a G-torsor and φ: E → X is a G-equivariant morphism of algebraic spaces over F.

Morphisms between two such objects (1) are pairs consisting of a morphism U → U′ of schemes and a G-equivariant morphism E → E′ of algebraic spaces, such that the diagram

\xymatrix{E \ar[rr] \ar[r]^{φ} \ar[d] & X & E′ \ar[l]_{φ′} \ar[d] \\ U \ar[rr] &  & U′}

commutes. The structure map [X/G] → Sch_{F} is projection onto the bottom row. The quotient stack [X/G] is indeed a stack over F, see [Stacks, Lemma 0370].

In the special case when X = Spec F (with trivial G-action) the quotient stack [X/G] can be canonically identified with BG, the classifying stack of G. An object of BG is simply a G-torsor E → U.

The construction of quotients [X/G] is functorial with respect to G-equivariant morphisms of algebraic spaces. For a G-equivariant morphisms of algebraic spaces X → Y we write f_{∗}^{G}: [X/G] → [Y/G] for the induced morphism of quotient stacks. On objects it is simply given by replacing the morphism E → X by the composition E → X → Y in a diagram (1).

The construction of [X/G] is also functorial with respect to morphisms a: G → H of group algebraic spaces. Let H act on X and let G act on X through a. Then we have a morphism a_{∗}^{X}: [X/G] → [X/H], which takes a diagram (1) to the diagram

\xymatrix{a_{∗}(E) \ar[r]^{ψ} \ar[d] & X \\ U,}

where the H-equivariant map ψ: a_{∗}(E) → X is induced by the G-invariant map E × H → X, (e, h) ↦ φ(e)h.

An algebraic stack over F is a stack X over F whose diagonal X → X × X is representable by algebraic spaces and such that there exists a smooth and surjective morphism U → X for some scheme U ∈ Sch_{F}.

A gerbe over F is an algebraic stack X over F satisfying the additional two conditions (2) and (3) of [Stacks, Definition ZZZ], which say that any two objects of X are locally isomorphic and that objects exist locally. An example of a gerbe is the classifying stack BG for any group algebraic space G over F.

CFG’s (A, π) over F (where π is the structure map π: A → Sch_{F}) form a 2category, in which morphisms (A, π) → (A′, π′) are functors φ: A → A′ such that π′ ∘ φ = π, and in which 2-morphisms φ_{1} → φ_{2} for morphisms φ_{1}, φ_{2} : (A, π) → (A′, π′) are natural transformations t: φ_{1} → φ_{2} such that π′(t_{a}) = id_{π(a)} for all objects a of A.

We will use the notion of 2-fiber product in the 2-category of CFG’s over F. If φ: X → Z and ψ: Y → Z are two morphisms of CFG’s over F a 2-fiber product is a CFG A over F together with morphisms p: A → X and q: A → Y such that the square

\xymatrix{A \ar[r]^{q} \ar[d]_{p} & Y \ar[d]^{ψ} \\ X \ar[r]_{φ} & Z}

2-commutes (i.e. the two compositions A → Z are 2-isomorphic) and is a final object in the 2-category of 2-commutative squares, see [Stacks, Definition 003Q]

for details. In particular for every other 2-commutative square \xymatrix{A′ \ar[r]^{q′} \ar[d]_{p′} & Y \ar[d]^{ψ} \\ X \ar[r]_{φ} & Z} there

exists a morphism α: A′ → A that makes the diagram

\xymatrix{A′ \ar[drr]^{q′} \ar@{-->}[dr]^{α} \ar[ddr]_{p′} \\  & A \ar[r]^{q} \ar[d]_{p} & Y \ar[d]^{ψ} \\  & X \ar[r]_{φ} & Z}

2-commute.

A 2-fiber product is unique up to unique equivalence. A 2-fiber product of φ: X → Z and ψ: Y → Z can be constructed like in [Stacks, Proposition 0040] as a category whose objects are quadruples (U, x, y, f) where U ∈ Sch_{F}, x and y are objects of X and Y, respectively, over U and f: φ(x) →^{∼} ψ(y) is an isomorphism in Z lying over the identity of U.

In some concrete situations like in the following two examples, which will be used later on, 2-fiber products have simpler alternative descriptions:

Example 2.2. Let X, Y and Z be algebraic spaces with a G-action from the right and let f: X → Z and g: Y → Z be G-equivariant morphisms of algebraic spaces. Then G acts diagonally on the (usual) fiber-product X ×_{Z} Y in the category of algebraic spaces and the following diagram is 2-cartesian:

\xymatrix{[(X ×_{Z} Y)/G] \ar[r]^{(π_{Y} )_{∗}^{G}} \ar[d]_{(π_{X})_{∗}^{G}} & [Y/G] \ar[d]^{g_{∗}^{G}} \\ [X/G] \ar[r]_{f_{∗}^{G}} & [Z/G],}

For a proof see e.g. [Wa11, Lemma 2.3.2].

Example 2.3. Let a: G → H be a morphism of group algebraic spaces and f: X → Y an H-equivariant morphism of algebraic spaces. Then the following diagram is 2-cartesian:

(2)  \xymatrix{[X/G] \ar[r]^{f_{∗}^{G}} \ar[d]_{a_{∗}^{X}} & [Y/G] \ar[d]^{a_{∗}^{Y}} \\ [X/H] \ar[r]_{f_{∗}^{H}} & [Y/H].}

This fact is probably well known. By lack of a reference we outline a proof using the construction of the 2-fiber product in [Stacks, Proposition 0040]. Take an object of the 2-fiber product [X/H] ×_{[Y/H]} [Y/G] over U ∈ Sch_{F}. It is given by a G-torsor E over U with a G-equivariant map φ: E → Y, a H-torsor E′ over U with a Hequivariant map ψ: E′ → X and an H-equivariant isomorphism α: E′ → a_{∗}(E) over U fitting into the commutative diagram

\xymatrix{E′ \ar[r]^{α} \ar[d]^{ψ} & a_{∗}(E) \ar[d] \\ X \ar[r]^{f} & Y,}

where the vertical map on the right is induced by the G-invariant map E × H → Y : (e, h) ↦ φ(e)h. We associate with this object the G-torsor E with the Gequivariant map ψ ∘ α^{−1} ∘ ι: E → X, where ι is the map E → a_{∗}(E), e → [e, 1]. This construction yields a morphism [X/H] ×_{[Y/H]} [Y/G] → [X/G] of CFG’s.

On the other hand the 2-commutativity of diagram (2) induces a morphism [X/G] → [X/H] ×_{[Y/H]} [Y/G]. The two morphisms are easily seen to be mutually inverse equivalences. It follows that diagram (2) is 2-cartesian as claimed.

2.4. Essential and canonical dimension of CFG’s. We will define essential and canonical dimension for CFG’s over F, in particular of algebraic stacks. Essential dimension of algebraic stacks has been introduced by P. Brosnan, A. Vistoli and Z. Reichstein in [BRV11] (see also [BRV08]). Since then, several authors have worked on essential dimension of algebraic stacks. The definitions of essential dimension that we give below are equivalent to those in the literature, see e.g. [Me09] or [BRV11]. However the definitions below will be more suitable for our purposes.

Definition 2.4. Let X be a CFG over F. For a finitely generated field extension K/F, a field K_{0} with a morphism Spec K → Spec K_{0} over F and a morphism x: Spec K → X we say that:

• x is defined over K_{0} (or that K_{0} is a field of definition of x) if there exists a morphism x_{0} : Spec K_{0} → X such that the diagram

\xymatrix{Spec K \ar[d] \ar[r]^{x} & X \\ Spec K_{0} \ar[ur]_{x_{0}}}

2-commutes.

• x is detected over K_{0} (or that K_{0} is a detection field of x) if there exists a

morphism x_{0} : Spec K_{0} → X. We define

ed x := min_{K_{0}} tdeg_{F} K_{0} ∈ ℕ_{0}, cdim x := min_{K′_{0}} tdeg_{F} K′_{0} ∈ ℕ_{0}

where the minimum is taken over all fields of definition K_{0} of x, resp. over all detection fields K′_{0} of x. For p ∈ ℙ ∪ {0} we define

ed_{p} x := min ed x_{L} ∈ ℕ_{0}, cdim_{p} x := min cdim x_{L} ∈ ℕ_{0},

where L runs over all prime to p extensions of K such that Spec L ∈ Sch_{F} and x_{L}: Spec L → X is the composite Spec L → Spec K →^{x} X. Here and in the sequel “prime to 0 extension” means “trivial extension”, as usual, so that ed_{0} x = ed x and cdim_{0} x = cdim x.

We set

ed_{p} X := sup_{x} ed_{p} x ∈ ℕ_{0} ∪ {−∞, ∞}, cdim_{p} X := sup_{x} cdim_{p} x ∈ ℕ_{0} ∪ {−∞, ∞},

where the supremum runs over all (finitely generated) field extensions K/F and morphisms x: Spec K → X, and ed X := ed_{0} X, cdim X := cdim_{0} X. We have ed_{p} X = −∞ (or equivalently cdim_{p} X = −∞) if and only if X is empty.

If G is a group algebraic space over F, the essential p-dimension of G for p ∈ ℙ ∪ {0} is defined via its classifying stack BG ≃ [Spec F/G]:

ed_{p} G := ed_{p} BG
Moreover cdim_{p} G := sup cdim_{p} X,

where X runs over all G-torsors over field extensions K of F with Spec K ∈ Sch_{F}.

We can define a functor F_{X} : Fields_{F} → Sets as follows: Choose, for every K ∈ Fields_{F} a field K′ ∈ Fields_{F} and an isomorphism α_{K}: K →^{∼} K′ over F such that Spec K′ belongs to Sch_{F}. Define F_{X} (K) as the set of isomorphism classes in X (K′). For a morphism f: K → L in Fields_{F} define F_{X} (f) as the map between isomorphism classes of objects of X (K′) and X (L′) induced by the field homomorphism α_{L} ∘ f ∘ α_{K}^{−1} : K′ → L′ over F. Then F_{X} is a functor and ed X is easily seen to coincide with ed F_{X} as defined in [BF03]. Similarly cdim X coincides with the essential dimension of the detection functor

D_{X} : Fields_{F} → Sets, K ↦ \begin{cases}\{∅\}, & if X (K′) ≠ ∅ \\ ∅, & otherwise. \end{cases}

We will sometimes tacitly use the following fact:

Lemma 2.5 ([BRV11, Example 2.4]). Let X be a scheme or a quasi-separated algebraic space locally of finite type over F. Then ed_{p} X = dim X for every p ∈ ℙ ∪ {0}.

For every CFG X over F we have cdim_{p} X ≤ ed_{p} X for every p ∈ ℙ ∪ {0}. However note that cdim_{p} G has nothing to do with cdim_{p} BG, which is zero (since F is a detection field for all morphisms x: Spec K → BG), and ed_{p} G = ed_{p} BG has nothing to do with the essential p-dimension of the algebraic space G, which is equal to dim G. Thus there are a-priori no relations between the values of ed_{p} G and cdim_{p} G. However when G is quasi-separated we always have

cdim_{p} G ≤ dim G

since the canonical p-dimension of a G-torsor X is always less or equal to the essential p-dimension of the algebraic space X, which is dim X = dim G by Lemma 2.5.

3. Fibers for morphisms of CFG’s

We start this section by proving our version of the fiber dimension theorem.

Proof of Theorem 1.1. Let x: Spec K → X be a morphism for some finitely generated field extensions K/F. Let y = f ∘ x: Spec K → X → Y. By definition of ed_{p} y there exists a prime to p extension L/K and an intermediate field L_{0} of L/F with tdeg_{F} L_{0} = ed_{p} y together with a 2-commutative diagram:

\xymatrix{Spec L \ar[d] \ar[r] & Spec K \ar[r]^{x} & X \ar[d]^{f} \\ Spec L_{0} \ar[rr]_{y_{0}} &  & Y}

By the universal property of 2-fibered products there exists a morphism z: Spec L → X_{y_{0}} such that the diagram

\xymatrix{Spec L \ar[ddr] \ar[dr]^{z} \ar[r] & Spec K \ar[dr]^{x} \\  & X_{y_{0}} \ar[r] \ar[d] & X \ar[d]^{f} \\  & Spec L_{0} \ar[r]_{y_{0}} & Y}

2-commutes. We will now argue for essential and canonical dimension separately:

• Essential dimension: By the definition of ed_{p} z there exists a prime to p extension M/L and an intermediate field M_{0} of M/L_{0} with tdeg_{L_{0}} M_{0} = ed_{p} z together with a morphism z_{0} : Spec M_{0} → X_{y_{0}} such that the above diagram can be completed to a 2-commutative diagram

\xymatrix{Spec M \ar[d] \ar[r] & Spec L \ar[ddr] \ar[dr]^{z} \ar[r] & Spec K \ar[dr]^{x} \\ Spec M_{0} \ar[rr]^{z_{0}} &  & X_{y_{0}} \ar[r] \ar[d] & X \ar[d]^{f} \\  &  & Spec L_{0} \ar[r]_{y_{0}} & Y}

Therefore x_{M} is defined over M_{0}. It follows that ed_{p} x ≤ tdeg_{F} M_{0} = tdeg_{F} L_{0} + tdeg_{L_{0}} M_{0} = ed_{p} y + ed_{p} z ≤ ed_{p} Y + ed_{p} X_{y_{0}}. Hence the first inequality follows.

• Canonical dimension: By the definition of cdim_{p} z there exists a prime to p extension M′/L and an intermediate field M′_{0} of M′/L_{0} with tdeg_{L_{0}} M′_{0} = cdim_{p} z together with a morphism z′_{0} : Spec M′_{0} → X_{y_{0}}. Hence there exists a morphism Spec M′_{0} → X, which shows that cdim_{p} x ≤ tdeg_{F} M′_{0}. Now the second inequality follows like above.

□

The following lemma on the essential dimension of gerbes X will be useful in the sequel. The case where X is banded by a commutative group scheme is [Me08, Proposition 4.9]. Recall that for any algebraic stack X over F there exists, for every morphism y: Spec K → X, a group algebraic space Aut_{K}(y) over K of automorphisms of y, cf. [Stacks, Lemma 04YP and Lemma 04XR]. Its T -rational points for T ∈ Sch_{K} are the automorphisms y_{T} →^{≃} y_{T} over T.

Lemma 3.1. Let X be a gerbe over F. Then for every p ∈ ℙ ∪ {0},

ed_{p} X ≤ cdim_{p} X + sup ed_{p} Aut_{K}(y),

where the supremum is taken over all field extensions K/F and all morphisms y: Spec K → X.

Proof. Let x: Spec K → X be a morphism. By the definition of cdim_{p} x there exists a prime to p extension L/K, an intermediate extension L_{0}/F of L/F and a morphism x_{0} : Spec L_{0} → X such that tdeg_{F} L_{0} = cdim_{p} x ≤ cdim_{p} X. Then X_{L_{0}} is equivalent to BG, where G := Aut_{L_{0}}(x_{0}), cf. [LMB00, Lemme 3.21]. We get a morphism y: Spec L → BG such that x_{L}: Spec L → X and the composition Spec L →^{y} BG →^{≃} X_{L_{0}} → X are 2-isomorphic. By the definition of ed_{p} y there exists a prime to p extension M/L, an intermediate field M_{0} of the extension M/L_{0} with tdeg_{L_{0}} M_{0} = ed_{p} y and a morphism y_{0} : Spec M_{0} → BG such that the diagram

\xymatrix{Spec M \ar[d] \ar[r] & Spec L \ar[d]^{y} \ar[r] & Spec K \ar[dr]^{x} \\ Spec M_{0} \ar[r]^{y_{0}} & BG \ar[r]^{≃} & X_{L_{0}} \ar[r] & X}

2-commutes. Hence

ed_{p} x ≤ tdeg_{F} M_{0} = tdeg_{F} L_{0} + tdeg_{L_{0}} M_{0} = cdim_{p} x + ed_{p} y
≤ cdim_{p} X + ed_{p} G,

and the claim follows. □

Corollary 3.2. Let 1 → C → G → H → 1 be an exact sequence of group algebraic spaces over F. Let E be an H-torsor over some field extension K/F. Then

ed_{p}[E/G] ≤ cdim_{p}[E/G] + sup ed_{p} ^{X}C,

where X runs over all lifts of E to a G-torsor over field extensions L/K. In particular when C is central in G then

ed_{p}[E/G] ≤ cdim_{p}[E/G] + ed_{p} C_{K}.

Proof. A morphism y: Spec L → [E/G] corresponds to a lifting of E to a G-torsor X and Aut_{L}(y) is isomorphic to the twist ^{X}C. Hence the first inequality follows from Lemma 3.1.

If C is central in G then ^{X}C is isomorphic to C_{L}. Hence the second inequality follows from the first one and the fact sup_{L} ed_{p} C_{L} ≤ ed_{p} C_{K}. □

We will apply Theorem 1.1 in the following cases:

Example 3.3. Let G be a group algebraic space over F and a: X → Y be a Gequivariant morphism of algebraic spaces over F. A morphism y: Spec K → [Y/G] corresponds to a G-torsor E over K with a G-equivariant morphism E → Y. By Example 2.2 the fiber of the morphism a_{∗}^{G}: [X/G] → [Y/G] over y is equivalent to [E ×_{Y} X/G]. Thus, for every p ∈ ℙ ∪ {0},

ed_{p}[X/G] ≤ ed_{p}[Y/G] + sup ed_{p}[E ×_{Y} X/G],

cf. [BRV11, Example 3.1], and

cdim_{p}[X/G] ≤ ed_{p}[Y/G] + sup cdim_{p}[(E ×_{Y} X)/G],

where the supremum is taken over all field extensions K/F and all G-torsors E over K with a G-equivariant morphism E → Y.

Note that [E ×_{Y} X/G] is an algebraic space. Thus if it is quasi-separated or a scheme then we can replace ed_{p}[E×_{Y} X/G] by dim[E×_{Y} X/G] in the first inequality above.

Now we apply this to the following situation: Let g: G → H be a morphism of group schemes over F. Let X be an H-torsor over some field extension L/F. Then G acts on X via g and [(E × X)/G] is an Aut_{H}(X)-torsor over K ∈ Fields_{L}, which is quasi-separated. Thus:

(3)  ed_{p}[X/G] ≤ ed_{p} G + dim H,

cf. [Me09, Theorem 4.8] and [BRV11, Corollary 3.3], and

(4) cdim_{p}[X/G] ≤ ed_{p} G + cdim_{p} Aut_{H}(X).

More generally suppose we are given morphisms g: G → H and h: H → Q of group schemes over F. Let X be an H-torsor over some field extension L/F and let Y = h_{∗}(X) be the induced Q-torsor. Then G acts on X and Y via g and h ∘ g, respectively, and X → Y is G-equivariant. In this situation [(E×_{Y} X)/G] is a torsor over K ∈ Fields_{L} for the group scheme

U := ker(Aut_{H}(X) → Aut_{Q}(Y))

over L (that becomes isomorphic to ker(h: H → Q) over L_{alg}). Thus:

(5)  ed_{p}[X/G] ≤ ed_{p}[Y/G] + dim(ker h),
(6)  cdim_{p}[X/G] ≤ ed_{p}[Y/G] + cdim_{p} U.

Note that in case h is surjective U is simply the twist U = ^{X}C of the kernel C := ker h by X.

Example 3.4. Let f: G → H be a morphism of group algebraic spaces over F and let H act on an algebraic space X. A morphism y: Spec K → [X/H] corresponds to an H-torsor E over K with an H-equivariant morphism E → X. By Example 2.2 the fiber of the morphism f_{∗}^{X}: [X/G] → [X/H] over y is isomorphic to [E/G]. Thus:

ed_{p}[X/G] ≤ ed_{p}[X/H] + sup ed_{p}[E/G],
cdim_{p}[X/G] ≤ ed_{p}[X/H] + sup cdim_{p}[E/G],

where the supremum runs over all field extensions K/F and all H-torsors E over K admitting an H-equivariant morphism E → X.

We have the following interesting special cases:

(a) This case was independently discovered by V. Chernousov and A. Merkurjev and used for split spin groups (cf. section 4). For X = Spec F:

ed_{p} G ≤ ed_{p} H + sup ed_{p}[E/G],

where the supremum is taken over all field extensions K/F and all H-torsors E over K.

When f is surjective [E/G] is a gerbe. Applying Lemma 3.1 yields, with C = ker f:

ed_{p} G ≤ ed_{p} H + sup cdim_{p}[E/G] + sup ed_{p} ^{T}C,

where the suprema are taken over all H-torsors E, resp. all G-torsors T, over field extensions K of F.

(b) For G trivial (and X, H quasi-separated for the first inequality):

dim X ≤ ed_{p}[X/H] + dim H,
cdim_{p} X ≤ ed_{p}[X/H] + sup cdim_{p} E ≤ ed_{p}[X/H] + cdim_{p} H,

where the supremum runs over all field extensions K/F and all H-torsors E over K admitting an H-equivariant morphism E → X.

The following result has been proven by D.-T. Nguyen (for smooth group schemes) and can be seen as a special case of Example 3.4(a). The case where U is commutative is due to D. Tossici and A. Vistoli.

Corollary 3.5 (c.f. [Ng11, Proposition 2.2] and [TV10, Lemma 3.4]). Let 1 → U → G → H → 1 be an exact sequence of group schemes over F with U unipotent. Then

ed_{p} G ≤ ed_{p} H + sup ed_{p} ^{X}U,

where X runs over all G-torsors over field extensions K/F.

In particular when U is central in G then

ed_{p} G ≤ ed_{p} H + ed_{p} U.

Proof. Since U is unipotent every H-torsor lifts to a G-torsor [Oe78] (the reference assumes algebraic groups to be smooth; however the same argument still works in the general case with Galois-cohomology replaced by fppf-cohomology). Therefore every gerbe [E/G] over K from Example 3.4(a) has cdim_{p}[E/G] = 0. The claim follows. □

Note that the same argument works for semi-direct products:

Corollary 3.6. Let G = N ⋊ H be a semidirect product of group schemes N and H over F. Then

ed_{p} H ≤ ed_{p} G ≤ ed_{p} H + sup ed_{p} ^{T}N,

where T runs over all G-torsors over field extensions of F.

Example 3.7. Let f: X → Y be a morphism of stacks over F and I_{X /Y} its relative inertia stack, whose objects are pairs (ξ, α) where ξ ∈ Ob(X) and α is an automorphism of ξ with f(α) = id_{f(ξ)}. The fibers of the canonical morphism I_{X /Y} → X over points x: Spec K → X are the group algebraic spaces given by the kernels of the morphisms Aut_{K}(x) → Aut_{K}(f(x)), see [Stacks, Lemma 050Q] and its proof. We will assume that the morphism I_{X /Y} → X is quasi-separated, so that all the group algebraic spaces ker(Aut_{K}(x) → Aut_{K}(f(x))) are quasi-separated. Then

ed_{p} I_{X /Y} ≤ ed_{p} X + sup dim ker(Aut_{K}(x) → Aut_{K}(f(x))),

where the supremum is taken over all field extensions K/F and all morphisms x: Spec K → X.

We also have

ed_{p} X ≤ ed_{p} I_{X /Y}

since the morphism I_{X /Y} → X is surjective on K-rational points for every K ∈ Fields_{F}. In particular, if X has finite relative inertia over Y, then ed_{p} X = ed_{p} I_{X /Y}. A stack over F for which all automorphism groups Aut_{K}(x) are finite has

ed_{p} X = ed_{p} I_{X} .

Here I_{X} = I_{X / Spec F} denotes the absolute inertia stack.

For a group scheme G and a normal subgroup N the relative inertia stack with respect to the canonical morphism f: BG → B(G/N) is equivalent to [N/G], where G acts by conjugation. The kernels ker(Aut_{K}(x) → Aut_{K}(f(x))) are the twists XN of N by G-torsors X. Therefore:

ed_{p} G ≤ ed_{p}[N/G] ≤ ed_{p} G + dim N.

4. p-exhaustive subgroups

From now on all group schemes under consideration are assumed to be affine.

Definition 4.1. Let p ∈ ℙ ∪ {0}. Let G be a group scheme over F and C be a normal subgroup scheme. Set H = G/C. We say that an H-torsor X over some extension K ∈ Fields_{F} is p-exhaustive (with respect to C and G) if the inequality

ed_{p}[X/G] ≤ ed_{p} G + dim H

from Example 3.3 is an equality.

We say that C is a p-exhaustive (normal) subgroup of G if a p-exhaustive Htorsor X exists.

Clearly G itself is always a p-exhaustive subgroup, for any p ∈ ℙ ∪ {0}. However there may exist smaller p-exhaustive subgroups. We make the following observation:

Lemma 4.2. Let G be a group scheme over F. Let C be a p-exhaustive subgroup of G. Then every normal subgroup D of G containing C is p-exhaustive as well.

Proof. Set H := G/C and Q := G/D. Let X be a p-exhaustive H-torsor, i.e.

ed_{p}[X/G] = ed_{p} G + dim H.

Let h: H → Q the canonical surjective morphism. We will show that the induced Q-torsor Y = h_{∗}(X) is p-exhaustive. By inequality (5) of Example 3.3 we have

ed_{p}[X/G] ≤ ed_{p}[Y/G] + dim H − dim Q.

Therefore

ed_{p}[Y/G] ≥ ed_{p} G + dim Q.

Since the opposite inequality always holds the claim follows. □

If C is a central subgroup of G isomorphic to μ_{p}^{r} for some r ≥ 0 we can use a result of N. Karpenko and A. Merkurjev [KM08] to compute, at least in principle, the essential p-dimension of [E/G] for every H = G/C-torsor E over some field extension K. Denote by β^{E}: Hom(C, μ_{p}) → Br(K) the group homomorphism, which takes a character χ to the image of the class of E under the map

H^{1}(K, H) → H^{2}(K, C(C)) →^{χ_{∗}} H^{2}(K, μ_{p}) = Br_{p}(K).

Then by [KM08] (cf. [Me09, Example 3.7])

ed_{p}[E/G] = min {∑_{χ∈B} ind β^{E}(χ)} ,

where the minimum runs over all bases B of Hom(C, μ_{p}) ≃ (ℤ/pℤ)^{r}.

The case r = 1 is due to [BRV11]. Note that in this case, ed_{p}[E/G] is the index of β^{E}(χ) for any generator χ of Hom(C, μ_{p}) ≃ ℤ/pℤ.

We remark that by [KM08, Theorem 4.4 and Remark 4.5] the indices arising in these formulas can be expressed in representation theoretic terms. Namely, ind β^{E}(χ) is the greatest common divisor gcd dim ρ taken over all (irreducible) representations ρ of G for which C acts via multiplication by χ. However we will not use this description in the sequel.

In several recent papers about essential dimension p-exhaustive central subgroups of the form μ_{p}^{r} have been used (implicitly) to compute the exact value of the essential p-dimension ed_{p} G for some classes of group schemes G whose center is of multiplicative type. Recall from [LMMR11, p.4] that we can associate with G a subgroup C(G) which is the (uniquely determined) largest central subgroup of G of the form μ_{p}^{r}, r ≥ 0. The center Z(G) of G or even the subgroup C(G) is p-exhaustive in several cases, summarized in the following list:

Example 4.3. Let p be a prime. For the following group schemes G the subgroup C(G) (exists and) is p-exhaustive:

(a) G is a group scheme of multiplicative type over a field F which splits over a Galois extension of p-power degree. See [LMMR11, Theorem 1.1] and its proof.

(b) G is a p-group over a field F of characteristic ≠ p, containing a primitive pth root of unity such that G becomes constant over a Galois field extension of p-power degree. See [LMMR11, Theorem 7.1] and its proof. The case where G is constant is contained in [KM08].

(c) G = Spin_{n} (for p = 2) over a field of characteristic 0, where n ≥ 15. This is due to [BRV10, Me09] and a result of V. Chernousov and A. Merkurjev. We will give more details below.

(d) G = HSpin_{n} (for p = 2) over a field of characteristic 0, where n ≥ 20 is divisible by 4. See [BRV10].

Let A be a division-algebra of p-power degree over its center. For the following group schemes G the center Z(G) is p-exhaustive:

(e) G is the normalizer G = N_{GL_{1}(A)}(GL_{1}(B)) where B is a separable subalgebra of A and Z(A) = F. See [Lo11]. Here Z(G) ≃ G_{m}.

(f) G = Sim(A, σ) (with p = 2), where σ is an involution on A with Z(A)^{σ} = F. See [Lo11]. Here Z(G) ≃ G_{m} if σ is of the first kind and Z(G) ≃ R_{K/F} (G_{m}) with K = Z(A) separable of degree 2 over F if σ is of the second kind.

(g) G = Iso(A, σ) (with p = 2), where Z(A) = F and σ is an involution of the first kind on A. See [Lo11].

(h) G = GO(A, σ, f), O(A, σ, f) or, if r ≥ 2, GO^{+}(A, σ, f) or O^{+}(A, σ, f), where (σ, f) is a quadratic pair on A and Z(A) = F, p = 2, char F = 2.

Remark 4.4. For a general normal subgroup C of G we do not know if the maximal value of ed_{p}[E/G] is reached for a versal H = G/C-torsor (in the sense of [BF03]). However if C is central and C ≃ μ_{p}^{r} for a prime p this is true. In particular C is pexhaustive if and only if for any versal torsor E we have ed_{p}[E/G] = ed_{p} G+dim G. Let us prove this. By the above formula it suffices to show that for E versal

ind β^{E}(χ) ≥ ind β^{E′}(χ)

for every χ ∈ Hom(C, μ_{p}) and every other H-torsor E′ over some field extension of F.

Recall that a versal torsor E over some field extension K is by definition the generic fiber of a classifying H-torsor π: X → Y (here Y is an irreducible scheme over F). There is an Azumaya algebra A over Y such that for K ∈ Fields_{F} and y ∈ Y (K), E′ = X_{y}, the class of β^{E′}(χ) is represented by the Azumaya K-algebra A_{y}, see [KM08, Lemma 4.3] and its proof. In particular β^{E}(χ) is represented by the generic fiber of A.

Now let D be a central division F (Y)-algebra representing the class of β^{E}(χ). Then we can lift D to some Azumaya-algebra B of constant degree equal to deg D = ind β^{E}(χ) over some non-empty open subset U of Y. Shrinking U if necessary allows us to assume that B is Brauer-equivalent to A_{U}.

Let E′ be another H-torsor over some extension K ∈ Fields_{F}. In order to prove the claim we can replace E′ by E′ ×Spec K(T) if necessary and thus assume that K is infinite. Since π is classifying there exists a point y ∈ U(K) such that the fiber of π over y is isomorphic to E′. Therefore B_{y} is Brauer equivalent to A_{y}, which represents the class β^{E′}(χ) in Br(K). This implies that ind β^{E′}(χ) ≤ deg B_{y} = ind β^{E}(χ). The claim follows.

We will prove two general lemmas on the behaviour of p-exhaustive central subgroups of the form μ_{p}^{r}. The first one generalizes the additivity theorem from [LMMR09, Theorem 8.1].

Lemma 4.5. Let p be a prime and G_{1}, G_{2} be group schemes over F. Let C_{1} ≃ μ_{p}^{r_{1}} and C_{2} ≃ μ_{p}^{r_{2}} be central subgroups of G_{1} and G_{2}, respectively. Set H_{1} = G_{1}/C_{1}, H_{2} = G_{2}/C_{2}. Let E be a versal H_{1} × H_{2}-torsor over some extension K ∈ Fields_{F}. Write E ≃ E_{1} × E_{2}, where E_{i} is an H_{i}-torsor, for i = 1, 2. Then

ed_{p}[E/(G_{1} × G_{2})] = ed_{p}[E_{1}/G_{1}] + ed_{p}[E_{2}/G_{2}].

In particular, if C_{1} and C_{2} are p-exhaustive, then C_{1} × C_{2} is p-exhaustive as well and

ed_{p} G_{1} × G_{2} = ed_{p} G_{1} + ed_{p} G_{2}.

Proof. Set G = G_{1} × G_{2}, H = H_{1} × H_{2}, C = C_{1} × C_{2}. Choose a basis B of Hom(C, μ_{p}) such that

ed_{p}[E/G] = ∑_{χ∈B} ind β^{E}(χ).

By elementary linear algebra there exists a partition B = B_{1} ∐ B_{2} such that the image of B_{j} under the projection π_{j}: Hom(C, μ_{p}) = Hom(C_{1}, μ_{p})×Hom(C_{2}, μ_{p}) → Hom(C_{j}, μ_{p}) is a basis of Hom(C_{j}, μ_{p}), for both j = 1, 2.

Let T_{1} denote the trivial H_{1}-torsor. Then β^{T_{1}×E_{2}}(χ) = β^{E_{2}}(π_{2}(χ)). Therefore by Remark 4.4, ind β^{E}(χ) ≥ ind β^{E_{2}}(π_{2}(χ)), for every χ ∈ Hom(C, μ_{p}). Similarly ind β^{E}(χ) ≥ ind β^{E_{1}}(π_{1}(χ)). We conclude:

ed_{p}[E/G] ≥ ∑_{φ∈π_{1}(B)} ind β^{E_{1}}(φ) + ∑_{ψ∈π_{2}(B)} ind β^{E_{2}}(ψ)

≥ ed_{p}[E_{1}/G_{1}] + ed_{p}[E_{2}/G_{2}] ≥ ed_{p}([E_{1}/G_{2}] × [E_{2}/G_{2}]) = ed_{p}[E/G].

Therefore ed_{p}[E/G] = ed_{p}[E_{1}/G_{1}] + ed_{p}[E_{2}/G_{2}] as claimed.

Now assume that C_{j} is p-exhaustive in G_{j}, for j = 1, 2. It is easy to see that that E_{j} is a versal H_{j}-torsor, for j = 1, 2. Therefore in view of Remark 4.4, ed_{p}[E_{j}/G_{j}] = ed_{p} G_{j} + dim H_{j}. Hence

ed_{p}[E/G] = ed_{p} G_{1} + ed_{p} G_{2} + dim H ≥ ed_{p} G + dim H.

It follows that C is p-exhaustive and ed_{p} G_{1} + ed_{p} G_{2} = ed_{p} G. □

Lemma 4.6. Let G be a group scheme over F and C ≃ μ_{p}^{r} a central subgroup of rank r ≥ 1. Assume that for all but at most r − 1 index p subgroups D of C the subgroup C/D of G/D is p-exhaustive. Then C is a p-exhaustive subgroup of G.

Proof. Set H = G/C. Let E be a versal H = G/C-torsor over some extension K ∈ Fields_{F}. Choose a basis B of Hom(C, μ_{p}) ≃ (ℤ/pℤ)^{r} such that

(7)  ed_{p}[E/G] = ∑_{χ∈B} ind β^{E}(χ)

We will first show that for any D = ker χ_{0} with χ_{0} ∈ B: (8) ed_{p}[E/G] = ed_{p}[E/(G/D)] + sup_{XliftingE} ed_{p}[X/G] = ed_{p}[E/(G/D)] + sup_{X} ed_{p}[X/G],

where on the right X runs over all G/D-torsors over field extensions of K and on the left X runs only over all G/D-torsors over field extensions of K that lift E.

Note that the left most expression in equation (8) is ≤ the middle expression by Example 3.4 and therefore ≤ the right most expression.

For every field extension L/F we have a commutative diagram

[figure: H^{1}(L, G/D); H^{2}(L, D); H^{1}(L, H); H^{2}(L, C); (χ_{0})_{∗}; H^{2}(L, μ_{p}); H^{1}(L, H); H^{2}(L, C/D); ≃; H^{2}(L, μ_{p}).]

In particular it follows that

(9)  ed_{p}[E/(G/D)] = ind β^{E}(χ_{0}).

Let X be a G/D-torsor over L ∈ Fields_{K} and let X̄ be the induced H-torsor. For χ ∈ Hom(C, μ_{p}) the image of χ|_{D} under β^{X}: Hom(D, μ_{p}) → Br(L) coincides with βX̄(χ). Since the characters χ|_{D} with χ ∈ B \ {χ_{0}} form a basis of Hom(D, μ_{p}),

(10)  ed_{p}[X/G] ≤ ∑_{χ∈B\{χ_{0}}} ind βX̄(χ) ≤ ∑_{χ∈B\{χ_{0}}} ind β^{E}(χ).

Combination of (7), (9) and (10) implies that the right most expression in (8) is ≤ the left most expression. Therefore we have proven (8).

By assumption there is at least one subgroup D = ker χ_{0} with χ_{0} ∈ B such that the subgroup C/D of G/D is p-exhaustive. For such D we get with Remark 4.4,

(11)  ed_{p}[E/(G/D)] = ed_{p} G/D + dim G/D = ed_{p} G/D + dim H.

Example 3.4(a) implies,

(12)  ed_{p} G ≤ ed_{p} G/D + sup ed_{p}[X/G],

where the supremum is taken over all G/D-torsors X over field extensions of K. Combining (8), (11) and (12) shows

ed_{p} G + dim H ≤ ed_{p}[E/G].

Hence the claim follows. □

We will now consider spin groups for application. Essential dimension of spin groups has been subject of investigation in several articles, including [Ro99], [CS06],

[BRV11] and [Me09]. Assume char F ≠ 2. Let Spin_{n} denote the spin group for a maximally isotropic non-degenerate quadratic form of dimension n. The essential dimension of Spin_{n} for n ≤ 14 has been computed by M. Rost [Ro99], see also [Ga09]. Then came P. Brosnan, A. Vistoli and Z. Reichstein [BRV11] who established a strong lower bound on Spin_{n} for any n ≥ 15 using essential dimension of algebraic stacks, basically applying inequality (3) from Example 3.3 to the surjective homomorphism Spin_{n} → O_{n}^{+} with kernel μ_{2}. For fields of characteristic 0 they also proved an upper bound using generically free representations. In case n ≢ 0 (mod 4) their lower bound matched the upper bound.

Then came A. Merkurjev [Me09], who improved the lower bound in case n ≡ 0 (mod 4), by considering the surjective homomorphism Spin_{n} → PGO_{n}^{+} with kernel μ_{2} × μ_{2} instead. This bound matched the upper bound from [BRV11] when n is a power of 2. At the RAGE conference in Atlanta 2011 Merkurjev also showed how to improve the upper bound in case n ≡ 0 (mod 4) by relating the essential dimension of Spin_{n} with the essential dimension of the semi-spinor group HSpin_{n}. As Merkurjev communicated to the author, this result will appear in a joint preprint with V. Chernousov. This upper bound can be seen as a special case of Example 3.4(a) for the morphism f: Spin_{n} → HSpin_{n}. Again the two bounds match. Thus ed Spin_{n} is known for any field of characteristic 0. We refer to [Me09, §4.3] for the list of values.

Since the new upper bound of Chernousov and Merkurjev for ed Spin_{n}, n ≥ 20 divisible by 4, is such a natural application of Theorem 1.1 we will reproduce it below. Also we feel that non-split spin groups have been excluded unnecessarily for investigation so far, so we would like to fill this gap.

We will entirely focus on the case n ≡ 0 (mod 4), since the other cases can be treated with published results. Moreover we will always assume that (σ, f) has trivial discriminant. The case where n ≡ 0 (mod 4) and (σ, f) has non-trivial discriminant looks more difficult.

Let (A, σ, f) be a quadratic pair over F with n := deg A divisible by 4. We assume that (σ, f) has trivial disciminant. In other words the center Z = Z(C(A, σ, f)) of the Clifford algebra of (A, σ, f) is isomorphic to F × F. We have an inclusion

Spin(A, σ, f) ⊆ R_{Z/F} (GL_{1}(C(A, σ, f))) = GL_{1}(C^{+}(A, σ, f)) × GL_{1}(C^{−}(A, σ, f)).

The center of Spin(A, σ, f) is μ_{2} × μ_{2}. We denote the image of Spin(A, σ, f) in the first (resp. second) component by Spin^{+}(A, σ, f) (resp. Spin^{−}(A, σ, f)). In other words Spin^{+}(A, σ, f) is the quotient of Spin(A, σ, f) by the central subgroup {1}×μ_{2}. Similarly Spin^{−}(A, σ, f) is the quotient of Spin(A, σ, f) by μ_{2} ×{1}. Note that unlike the split case, these two groups do not need to be isomorphic.

The quotient of Spin(A, σ, f) by the diagonal subgroup of μ_{2}×μ_{2} is O^{+}(A, σ, f). The quotient of Spin(A, σ, f) by the full center μ_{2} × μ_{2} is PGO^{+}(A, σ, f).

Proposition 4.7. Assume char F ≠ 2. Then

d^{+} := sup ed_{2}[E/Spin^{+}(A, σ, f)] = 2^{\frac{n−2}{2}} ind C^{+}(A, σ, f)
d^{−} := sup ed_{2}[E/Spin^{−}(A, σ, f)] = 2^{\frac{n−2}{2}} ind C^{−}(A, σ, f)
d := sup ed_{2}[E/O^{+}(A, σ, f)] = 2^{ν_{2}(n)+ν_{2}(ind A)},
sup ed_{2}[E/Spin(A, σ, f)] = min{d + d^{+}, d + d^{−}, d^{+} + d^{−}},

where E runs over all PGO^{+}(A, σ, f)-torsors E over field extensions of F. These values are attained for a versal PGO^{+}(A, σ, f)-torsor E.

Furthermore if n ≥ 20 then

sup ed_{2}[E/Spin(A, σ, f)] = min{d^{+}, d^{−}} + d.

Proof. For a field extension K/F the fppf-cohomology set H^{1}(K, PGO^{+}(A, σ, f)) is in natural bijection with isomorphism classes of quadruples (B, τ, g, φ) where B is a central simple K-algebra of degree deg B = deg A, (τ, g) is a quadratic pair on B and φ is an isomorphism Z(C(B, τ, g)) →^{∼} K ×K. The connecting map associated with the exact sequence 1 → μ_{2} ×μ_{2} → Spin(A, σ, f) → PGO^{+}(A, σ, f) → 1 takes the isomorphism class of (B, τ, g, φ) to the element

([C^{+}(B, τ, g)] − [C^{+}(A, σ, f)_{K}], [C^{−}(B, τ, g)] − [C^{−}(A, σ, f)_{K}])

in Br_{2}(K) × Br_{2}(K) = H^{2}(K, μ_{2} × μ_{2}), where C^{+}(B, τ, g) = C(B, τ, g)φ^{−1}(1, 0) and C^{−}(B, τ, g) = C(B, τ, g)φ^{−1}(0, 1) are the two components of C(B, τ, g) labeled with respect to φ, cf. [KMRT98, Exercise VII.15].

Similarly, the connecting maps associated with the exact sequences

1 → μ_{2} → G → PGO^{+}(A, σ, f) → 1

for G = Spin^{+}(A, σ, f), Spin^{−}(A, σ, f), O^{+}(A, σ, f) takes the class of (B, τ, g, φ) to [C^{+}(B, τ, g)] − [C^{+}(A, σ, f)_{K}], [C^{−}(B, τ, g)] − [C^{−}(A, σ, f)_{K}] and [B] − [A_{K}], respectively.

We always have ind B ≤ 2^{ν_{2}(n)} and ind C^{δ}(B, τ, g) ≤ 2^{\frac{n−2}{2}} . By [MPW96, (5.49)] (here we use the assumption char F ≠ 2) there exists a quadruple (B, τ, g, φ) as above such that for every central simple F -algebra D:

ind(D ⊗_{F} C^{δ}(B, τ, g)) = 2^{\frac{n−2}{2}} ind D, ∀δ ∈ \{+, −\},
ind(D ⊗_{F} B) = 2^{ν_{2}(n)} ind(D).

In particular:

d^{δ} = ind(C^{δ}(A, σ, f)^{op} ⊗_{F} C^{δ}(B, τ, g)) = 2^{\frac{n−2}{2}} ind(C^{δ}(A, σ, f)), ∀δ ∈ \{+, −\},
d = ind(A^{op} ⊗_{F} B) = 2^{ν_{2}(n)+ν_{2}(ind A)}.

Moreover it follows that

sup ed_{2}[E/Spin(A, σ, f)] = min{d + d^{+}, d + d^{−}, d^{+} + d^{−}}.

Now assume n ≥ 20. Then 4ν_{2}(n) ≤ n − 2, hence

d ≤ 2^{2ν_{2}(n)} ≤ 2^{\frac{n−2}{2}} ≤ min\{d^{+}, d^{−}\}.

Thus min{d + d^{+}, d + d^{−}, d^{+} + d^{−}} = min{d^{+}, d^{−}} + d. □

Proposition 4.8. Assume char F ≠ 2.

(a) Let δ ∈ {+, −} and let m = ind(C^{δ}(A, σ, f)). Suppose that the m-fold direct sum of the canonical representation of HSpin_{n} is generically free. Then the center μ_{2} of Spin^{δ}(A, σ, f) is a 2-exhaustive subgroup of Spin^{δ}(A, σ, f). Moreover

ed_{2} Spin^{δ}(A, σ, f) = 2^{\frac{n−2}{2}} m − \frac{n(n − 1)}{2}.

(b) Suppose that A is division. Then the center μ_{2} of O^{+}(A, σ, f) is a 2exhaustive subgroup of O^{+}(A, σ, f). Moreover

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

Proof. (a) Let D be a division F -algebra, representing the Brauer class of C^{δ}(A, σ, f). We have a representation arising from the composition

ρ: Spin^{δ}(A, σ, f) ↪ GL_{1}(C^{δ}(A, σ, f)) ↪ GL_{1}(C^{δ}(A, σ, f) ⊗_{F} D^{op}) →^{≃} GL_{N}

with N = 2^{\frac{n−2}{2}} m. Over F_{sep} this representation decomposes as the mfold direct sum of the canonical HSpin_{n}-representation, which is generically free by assumption. Hence ρ is generically free as well. Therefore ed_{2} Spin^{δ}(A, σ, f) ≤ N − dim Spin^{δ}(A, σ, f) by [BF03, Proposition 4.11]. Combining this inequality with Proposition 4.7 shows that the center of Spin^{δ}(A, σ, f) is 2-exhaustive and gives us the value of ed_{2} Spin^{δ}(A, σ, f).

(b) Since O^{+}(A, σ, f) is a subgroup of the group GL_{1}(A) of essential dimension 0 we have ed_{2} O^{+}(A, σ, f) ≤ dim GL_{1}(A) − dim O^{+}(A, σ, f) = \frac{n(n+1)}{2} (by [BF03, Theorem 6.19] or Example 3.4(a)). Now the claim follows again from Proposition 4.7.

□

Remark 4.9. Let m ≥ 1. In the following cases the m-fold direct sum of the canonical representation of HSpin_{n} is generically free:

(a) m ≥ 2^{\frac{n−2}{2}} , which is the dimension of the canonical representation of HSpin_{n}.

(b) char F =0 and m ≥ 8.

(c) char F =0 and m ≥ 2 if n ≥ 16.

(d) char F =0 and m arbitrary if n ≥ 20. The first case is obvious, since GL_{m} acts generically freely on the m-fold direct sum of its canonical m-dimensional representation. The other cases follow from [PV94, Theorem 8.8 and Theorem 8.9]. We do not know if the assumption char F = 0 can be dropped or not.

Combining Proposition 4.8 and Remark 4.9 with Lemma 4.6 we can compute the essential 2-dimension of Spin(A, σ, f) in many cases. In particular we get the following result:

Corollary 4.10. Assume char F ≠ 2. Set d = 2^{ν_{2}(n)+ν_{2}(ind(A))}, d^{+} = 2^{\frac{n−2}{2}} ind C^{+}(A, σ, f) and d^{−} = 2^{\frac{n−2}{2}} ind C^{+}(A, σ, f) like in Proposition 4.7. In the following cases the center μ_{2} × μ_{2} of Spin(A, σ, f) is 2-exhaustive and

ed_{2} Spin(A, σ, f) = min\{d + d^{+}, d + d^{−}, d^{+} + d^{−}\} − \frac{n(n − 1)}{2} :

(a) At least two of the algebras A, C^{+}(A, σ, f) and C^{−}(A, σ, f) are division.

(b) char F = 0 and n ≥ 20. Here the formula simplifies to

ed_{2} Spin(A, σ, f) = min\{d^{+}, d^{−}\} + d − \frac{n(n − 1)}{2} :

(c) char F = 0 and both C^{+}(A, σ, f) and C^{−}(A, σ, f) have index at least 8.

(d) char F = 0, n ≥ 16, and A is division or none of C^{+}(A, σ, f) and C^{−}(A, σ, f) is split.

Remark 4.11. All results from Proposition 4.8 and Corollary 4.10 hold with ed_{2} replaced by ed. For the lower bounds this is clear and for the upper bound only very slight modifications in the proofs are needed.

Remark 4.12. In case n = 8 the result of Corollary 4.10(a) can be improved. It suffices that two of the three algebras (all of degree 8) have index ≥ 4. This follows from the fact that the 4-fold direct sum of the representation

Spin_{8} ↪ O_{8}^{+} × O_{8}^{+} ↪ GL_{16}

is generically free, which can easily be checked.

Moreover in case char F = 0, n = 16 no assumptions on the indices are really needed. This follows from the fact that the representation

Spin_{16} ↪ HSpin_{16} × O_{16}^{+} ↪ GL_{128} × GL_{16} ↪ GL_{144}

is already generically free, see [BRV10, p.5].

5. Canonical dimension of group schemes

In the following theorem, we reveal a relation between canonical and essential dimension of group schemes for p-exhaustive subgroups, introduced in section 4.

Theorem 5.1. Let p ∈ ℙ ∪ {0}. Let G be a group scheme over F and let C be a p-exhaustive subgroup of G. Let H = G/C and X be a p-exhaustive H-torsor over some field extension K/F. Then

cdim_{p} Aut_{H}(X) ≥ dim H − sup ed_{p} ^{Z}C,

where the supremum is taken over all field extensions L/K and all lifts of X to a G-torsor Z over L.

In particular, if C is central then

cdim_{p} Aut_{H}(X) ≥ dim H − ed_{p} C,

and if H is abelian then

cdim_{p} H ≥ dim H − ed_{p} ^{X}C,

and if C is central and H abelian then

cdim_{p} H ≥ dim H − ed_{p} C.

Proof. Since X is p-exhaustive we have

(13)  ed_{p}[X/G] = ed_{p} G + dim H.

By inequality (4) of Example 3.3,

(14)  cdim_{p}[X/G] ≤ ed_{p} G + cdim_{p} Aut_{H}(X).

Corollary 3.2 yields the inequality

(15)  ed_{p}[X/G] ≤ cdim_{p}[X/G] + sup ed_{p} ^{Z}C.

Combining (13), (14) and (15) yields the desired inequality.

□

Remark 5.2. Suppose, given a group scheme G over F and a prime p, we want to study the question if the subgroup C(G) ≃ μ_{p}^{r} (from above) is p-exhaustive. Theorem 5.1 gives an obstruction to an affirmative answer to this question. Namely C(G) can only be p-exhaustive if one of the twisted inner forms H′ = Aut_{H}(X) of H has cdim_{p} H′ ≥ dim H − r.

Combing Theorem 5.1 with items (c), (e) and (f) of Example 4.3 we get the following results:

Corollary 5.3. (a) Let n ≥ 15 with n ≢ 0 (mod 4). Assume char F = 0. Then there exists an n-dimensional quadratic form q of trivial discriminant over some field extension of F such that

cdim_{2} O^{+}(q) ≥ dim O^{+}(q) − 1 = \frac{n(n − 1)}{2} − 1.

(b) Let n ≥ 15 with n ≡ 0 (mod 4). Assume char F = 0. Then there exists a central simple algebra of degree n over some field extension of F and an orthogonal involution σ on A such that

cdim_{2} PGO^{+}(A, σ) ≥ dim PGO^{+}(A, σ) − 2 = \frac{n(n − 1)}{2} − 2.

(c) Let p be a prime and let a, b, n ≥ 0 be integers with a + b ≤ n. Then there exists a central simple algebra A of degree p^{n} over some field extension K of F and a separable subalgebra B of A such that B ⊗_{K} K_{sep} ≃ M_{p^{a}}(K_{sep}) × ⋯ × M_{p^{a}}(K_{sep}), C_{A}(B) ⊗_{K} K_{sep} ≃ M_{p^{b}}(K_{sep}) × ⋯ × M_{p^{b}}(K_{sep}) (both p^{n−a−b} times) and

cdim_{p} Aut_{K}(A, B) = dim Aut_{K}(A, B) = p^{n+a−b} + p^{n−a+b} − p^{n−a−b} − 1.

(d) Let n = 2^{r} for some r ≥ 1. Then there exists a central simple algebra A of degree n over some field extension K of F and an involution σ on A of orthogonal (resp. symplectic) type on A such that

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

(e) Let n = 2^{r} for some r ≥ 0 and let K/F be a separable quadratic extension. Then there exists a field extension L/F linearly disjoint from K/F, a central simple M := L⊗_{F} K-algebra A of degree n and a unitary L-linear involution σ on A such that

cdim_{2} Aut_{M}(A, σ) = dim Aut_{M}(A, σ) = n^{2} − 1.

Remark 5.4. The split forms of the groups appearing in Corollary 5.3 usually have clearly lower canonical p-dimension. For example for the special orthogonal groups

cdim_{2} O_{2n+1}^{+} = cdim_{2} O_{2n+2}^{+} = \frac{n(n + 1)}{2},

which was conjectured in [BR05] and proven independently in [Ka05] and [Vi05]. This value is to compare with the values n(2n + 1) − 1 (resp. (n + 1)(2n + 1) − 1) for the quadratic forms q of dimension 2n + 1 and 2n + 2, respectively, from part (a) of the corollary.

Another example is the group Aut_{K}(A, B) from part (c) of the corollary, where A is a central division K-algebra of degree d = p^{n}, B is a maximal étale subalgebra of A and cdim_{p} Aut_{K}(A, B) = dim Aut_{K}(A, B) = d − 1. Here the split form Aut_{K}(M_{d}(K), K^{d}) ≃ G_{m}^{d}/G_{m} ⋊ S_{d} has canonical p-dimension equal to 0 (this follows from [KM06, Remark 3.7], since the maps H^{1}(−, G_{m}^{d}/G_{m} ⋊ S_{d}) → H^{1}(−, S_{d}) of pointed sets have trivial kernel and since cdim_{p} S_{d} = 0).

Now we turn our attention to the case of groups of multiplicative type.

Corollary 5.5. Let G be a group scheme of multiplicative type which splits over a Galois extension of p-power degree. Let C be any subgroup of G containing C(G) and set H = G/C. Then, for every p ∈ ℙ ∪ {0},

cdim_{p} H ≥ dim H − ed_{p} C.

Proof. As recorded in Example 4.3(a) the subgroup C(G) of G is p-exhaustive. Hence by Lemma 4.2 C is p-exhaustive as well. The claim now follows from Theorem 5.1. This proves Corollary 5.5 and hence Theorem 1.2 from the introduction. □

Example 5.6. Let L/F be a field extension such that the normal closure of L has p-power degree over F. Let K be an intermediate field of the extension L/F. Let T := R_{L/F} (G_{m})/R_{K/F} (G_{m}). Then

cdim_{p} T = dim T = [L : F] − [K : F].

Proof. Apply Corollary 5.5 to G = R_{L/F} (G_{m}) and C = R_{K/F} (G_{m}) and note that ed_{p} C = 0. □

A famous and often applied result of N. Karpenko says that Severi-Brauer varieties SB(A) of p-power degree central division algebras A are p-incompressible, i.e. have cdim_{p} SB(A) = dim SB(A), [Ka00, Theorem 2.1] (see also [Ka10, Proposition 2.2]). Karpenko more recently proved that Weil transfers R_{K/F} (SB(A)) of “suitably generic” central simple K-algebras A of 2-power degree are 2-incompressible, when K/F is a quadratic separable extension [Ka11]. The following Corollary 5.7 basically tells us that the same happens for Weil restrictions with respect to separable field extensions of higher degree.

Corollary 5.7. Under the assumptions of Example 5.6 there exists a field extension M/F and an Azumaya M ⊗_{F} K-algebra A of degree [L : K] over M and split over M ⊗_{F} L such that the Weil restriction R_{M⊗_{F} K/M}(SB(A)) of the Severi-Brauer variety SB(A) is p-incompressible.

Proof. There is a natural isomorphism

H^{1}(M, T) ≃ ker(Br(K ⊗_{F} M) → Br(L ⊗_{F} M)).

Let a be a T -torsor over some field extension M/F with maximal canonical pdimension. Let A be an Azumaya K ⊗_{F} M-algebra (split by L ⊗_{F} M) of degree [L : K] corresponding to a. Then the splitting fields of t are precisely the splitting fields of R_{M⊗_{F} K/M}(SB(A)). Therefore

cdim_{p} R_{M⊗_{F} K/M}(SB(A)) = cdim_{p} a = [L : F] − [K : F] = [K : F]([L : K] − 1)
= dim R_{M⊗_{F} K/M}(SB(A)),

which proves the claim. □

Our goal is now to find a condition on an algebraic torus T which ensures that the lower bound from Corollary 5.5 is an equality.

Corollary 5.8. Let T be an algebraic torus over a field F, p a prime, T_{d} the largest split subtorus of T. Let K/F be a splitting field of T. Make the following two assumptions:

(a) [K : F] is a power of p.

(b) The Tate cohomology group Ĥ^{−1}(Gal(K/F), X(T)) is trivial. Then for every diagonalizable subgroup C of T containing the p-torsion of T_{d}:

cdim T/C = cdim_{p} T/C = dim T/T_{d} = dim T/C − ed_{p} C.

Proof. Assumption (b) implies C ⊆ T_{d}. Moreover T_{d}/C is a split torus and therefore special. Thus cdim T/C ≤ cdim T/T_{d} by [KM06, Lemma 6.5]. The inequalities cdim_{p} T/C ≤ cdim T/C ≤ dim T/T_{d} follow.

We have C(T) = T_{d}[p] and therefore ed_{p} C = dim T_{d}/C. Applying Corollary 5.5 we immediately get the inequality cdim_{p} T/C ≥ dim T/C −ed_{p} C = dim T/T_{d}. This concludes the proof. □

Definition 5.9. Let r ∈ ℕ_{0}, p a prime. Define C_{p}^{(r)} to be the class of all F -tori of the form T/C, where:

(a) T is a torus admitting a Galois-splitting field K/F of p-power degree such that Ĥ^{−1}(Gal(K/F), X(T)) = 0,

(b) C is a diagonalizable subgroup of T which contains T_{d}[p] and which has ed_{p} C = r.

Some properties of this construction are listed in the following lemma:

Lemma 5.10. (a) For S in C_{p}^{(r)}: cdim_{p} S = dim S − r.

(b) S_{1} ∈ C_{p}^{(r_{1})}, S_{2} ∈ C_{p}^{(r_{2})} ⇒ S_{1} × S_{2} ∈ C_{p}^{(r_{1}+r_{2})}.

(c) S ∈ C_{p}^{(r)}, S′ ⊆ S subtorus with S/S′ anisotropic. Then S′ ∈ C_{p}^{(r)}

Proof. (a) This is a reformulation of Corollary 5.8.

(b) The simple proof is left to the reader.

(c) Write S = T/C, with T, C (and K) as in the definition of C_{p}^{(r)}. Let T′ be the preimage of S′ under the canonical projection T → S. Clearly T′ is split over K as well and C contains T′_{d}[p] ⊆ T_{d}[p]. Moreover S′ ≃ T′/C. Note that S′ contains the image of T_{d}, since S/S′ is anisotropic. Hence T′ contains T_{d}, which in turn contains C. Thus T′/T_{d} is an epimorphic image of S′, hence a torus. It follows that T′ is a torus as well.

It remains to verify the condition Ĥ^{−1}(Gal(K/F), X(T′)) = 0. We have a short exact sequence 1 → X(S/S′) → X(T) → X(T′) → 1. Since S/S′ is anisotropic X(S/S′) has trivial fixed point set under Gal(K/F). In particular Ĥ^{0}(Gal(K/F), X(S/S′)) = 0. We also have Ĥ^{−1}(Gal(K/F), X(T)) = 0, hence the claim follows from the (standard) long exact sequence in Tate cohomology.

□

Example 5.11. Let L_{1}, …, L_{n} be separable field extensions whose normal closures K_{1}, …, K_{n} have p-power degree over F. Then any subtorus T of the product ∏_{i=1}^{n}(R_{L_{i}/F} (G_{m})/G_{m}) belongs to C_{p}^{(0)}, hence has cdim T = cdim_{p} T = dim T.

Proof. For every i the torus T_{i} = R_{L_{i}/F} (G_{m}) is split by K_{i} and satisfies the condition Ĥ^{−1}(Gal(K_{i}/F), X(T_{i})) = 0. The subgroup C_{i} = G_{m} coincides with (T_{i})_{d}, hence contains (T_{i})_{d}[p], and has ed_{p} C_{i} = 0. Therefore T_{i}/C_{i} ∈ C_{p}^{(0)}. Lemma 5.10(b) implies that the torus S := ∏_{i=1}^{n}(R_{L_{i}/F} (G_{m})/G_{m}) lies in C_{p}^{(0)}, too. Since S is anisotropic (and therefore S/T as well) Lemma 5.10(c) implies that T also belongs to C_{p}^{(0)}. □

Example 5.12. Let p be a prime and S an anisotropic algebraic torus over F whose minimal Galois splitting field is cyclic of p-power degree over F. Then S belongs to the class C_{p}^{(0)}. In particular

cdim_{p} S = cdim S = dim S.

Proof. Let K/F be the minimal Galois splitting field of S. Embed S in R_{K/F} (G_{m})^{N} for some N ≫ 0. Since S is anisotropic, it lies in the subtorus (R_{K/F}^{(1)} (G_{m}))^{N}, which is isomorphic to S′ := (R_{K/F} (G_{m})/G_{m})^{N} since K/F is cyclic. Thus we are in the situation of Example 5.11 and the claim follows. □

Example 5.13. Let L/F be a cyclic Galois extension of degree p^{r} > 1 and let T = R_{L/F}^{(1)} (G_{m}) the corresponding norm 1 torus. Then ed T = ed_{p} T = 1, but cdim T = cdim_{p} T = dim T = p^{r} − 1 can be arbitrarily large.

Example 5.14. Let T be an algebraic torus over F. Assume that there exists an element τ of Gal(F_{sep}/F) which acts as −1 on X(T). Then cdim T = cdim_{2} T = dim T.

Proof. Let F′ be the fixed field F_{sep}^{τ}. Then T_{F′} is of the form (T′)^{dim T}, where T′ is a non-split 1-dimensional torus. By Example 5.12 T_{F′} has canonical 2-dimension equal to dim T. Since dim T ≥ cdim_{2} T ≥ cdim T ≥ cdim T_{F′} the claim follows. □

Example 5.15. Let T be a 2-dimensional algebraic torus. Then (a) cdim T = 0 if and only if T is quasi-split.

(b) cdim T = 1 if and only if T ≃ G_{m} × T′ where T′ is a non-split onedimensional torus.

(c) cdim T = 2, otherwise.

Proof. In every case it is clear that cdim T cannot be larger than the claimed value. Moreover the equality cdim G_{m} ×T′ = 1 for a non-split one-dimensional torus T′ is contained in Example 5.14. It remains to show that if T is neither quasi-split, nor of the form G_{m} × T′ with T′ non-split, then cdim T ≥ 2. Let L/F be the minimal Galois splitting field of T. Then Gal(L/F) is a finite group embedding in GL_{2}(ℤ).

First assume that there exists an element σ of order 3 in Gal(L/F). Let F′ = L^{σ}. Then T_{F′} is isomorphic to R_{L/F′}(G_{m})/G_{m}, which has canonical dimension 2 by Example 5.12. Hence T has canonical dimension 2 as well.

Now assume that Gal(L/F) does not contain elements of order 3. Then Gal(L/F) embeds in the (unique up to conjugacy) maximal 2-subgroup D_{8} of GL_{2}(ℤ). Since T is neither quasi-split, nor of the form G_{m}×T′ with T′ one-dimensional, one easily sees that Gal(L/F) contains an element which acts as −1 on X(T). Now the claim follows from Example 5.14. □

Acknowledgments

I would like to thank Zinovy Reichstein for helpful remarks and for raising the problem of computing the canonical dimension of algebraic tori. Also I would like to thank Alexander Merkurjev and Duy Tan Nguyen for their comments. Moreover I am grateful to Stefan Gille and Fabien Morel for useful discussions.

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

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