ESSENTIAL p-DIMENSION OF ALGEBRAIC GROUPS WHOSE CONNECTED COMPONENT IS A TORUS

ROLAND LÖTSCHER^{(1)}, MARK MACDONALD^{(2)}, AUREL MEYER^{(3)}, AND ZINOVY REICHSTEIN^{(4)}

Abstract. Following up on our earlier work and the work of N. Karpenko and A. Merkurjev, we study the essential p-dimension of linear algebraic groups G whose connected component G^{0} is a torus.

Contents

1. Introduction  1
2. The p-special closure of a field  4
3. p-isogenies  6
4. Proof of the Main Theorem: an overview  7
5. Dimensions of irreducible representations  9
6. Proof of Proposition 4.3  11
7. Faithful versus generically free  12
8. Proof of Theorem 1.3  14
9. Additivity  15
10. Groups of low essential p-dimension  18
References  21

1. Introduction

Let p be a prime integer and k a base field of characteristic ≠ p. In this paper we will study the essential dimension of linear algebraic k-groups G whose connected component G^{0} is an algebraic torus. This is a natural class of groups; for example, normalizers of maximal tori in reductive linear algebraic groups are of this form. We will use the notational conventions of

2000 Mathematics Subject Classification. 20G15, 11E72. Key words and phrases. Essential dimension, algebraic torus, twisted finite group,

generically free representation.

(1) Supported by the Deutsche Forschungsgemeinschaft, GI 706/2-1.

(2) Partially supported by a postdoctoral fellowship from the Canadian National Science and Engineering Research Council (NSERC).

(3) Supported by a postdoctoral fellowship from the Swiss National Science Foundation.

(4) Partially supported by a Discovery grant from the Canadian National Science and Engineering Research Council (NSERC).

[LMMR]. For background material and further references on the notion of essential dimension, see [Re_{2}].

For the purpose of computing ed(G; p) we may replace the base field k by a p-special closure k^{(p)}. A p-special closure of k is a maximal directed limit of finite prime to p extensions over k; for details see Section 2. The resulting field k = k^{(p)} is then p-special, i.e. every field extension of k has ppower degree. (Some authors use the term “p-closed” in place of “p-special”.) Furthermore, the finite group G/G^{0} has a Sylow p-subgroup F defined over k = k^{(p)}; see [LMMR, Remark 7.2]. Since G is smooth we may replace G by the preimage of F without changing the essential p-dimension; see [LMMR, Lemma 4.1]. It is thus natural to restrict our attention to the case where G/G^{0} is a finite p-group, i.e., to those G which fit into an exact sequence of k-groups of the form

(1.1)  1 → T → G ⟶^{π} F → 1 ,

where T is a torus and F is a smooth finite p-group. Note that F may be twisted (i.e. non-constant) and T may be non-split over k. Moreover, the extension (1.1) is not assumed to be split either.

To state our main result, recall that a linear representation ρ: G → GL(V) is called generically free if there exists a G-invariant dense open subset U ⊂ V such that the scheme-theoretic stabilizer of every point of U is trivial. A generically free representation is clearly faithful but the converse does not always hold; see below. We will say that ρ is p-generically free (respectively, p-faithful) if ker ρ is finite of order prime to p, and ρ descends to a generically free (respectively, faithful) representation of G/ ker ρ.

Theorem 1.1. Let G be an extension of a (possibly twisted) finite p-group F by an algebraic torus T defined over a p-special field k of characteristic ≠ p. Then

min dim ρ − dim G ≤ ed(G; p) ≤ min dim μ − dim G ,

where the minima are taken over all p-faithful linear representations ρ of G and p-generically free representations μ of G, respectively.

If G = G^{0} is a torus or G = F is a finite p-group then the upper and lower bounds of Theorem 1.1 coincide (see, [LMMR, Lemma 2.5] and [MR_{1}, Remark 2.1], respectively). In these cases Theorem 1.1 reduces to [LMMR, Theorems 1.1 and 7.1], respectively. In the case of constant finite p-groups this result is due to N. Karpenko and A. Merkurjev, whose work [KM] was the starting point for both [LMMR] and the present paper. We will show that the upper and lower bounds of Theorem 1.1 coincide for a larger class of groups, which we call tame; see Definition 1.2 and Theorem 1.3(b) below.

ESSENTIAL p-DIMENSION 3

In general, groups G of the form (1.1) may have faithful (respectively, p-faithful) representations which are not generically free (respectively, pgenerically free). This phenomenon is not well understood; there is no classification of such representations, and we do not even know for which groups G they occur ^{1}. It is, however, the source of many of the subtleties we will encounter.

To give the reader a better feel for this phenomenon, let us briefly consider the following “toy” example. Let k = ℂ, p = 2 and G = O_{2}, the group of 2 × 2 orthogonal matrices. It is well known that G ≃ 𝔾_{m} ⋊ ℤ/2ℤ, where G^{0} = SO_{2} = 𝔾_{m} is a 1-dimensional torus. It is easy to see that the natural representation i: G ↪ GL_{2} is the unique 2-dimensional faithful representation of G. However this representation is not generically free: the stabilizer Stab_{G}(v) of every anisotropic vector v = (a, b) ∈ ℂ^{2} is a subgroup of G = O_{2} of order 2 generated by the reflection in the line spanned by v. Here “anisotropic” means a^{2} + b^{2} ≠ 0; anisotropic vectors are clearly Zariski dense in ℂ^{2}. On the other hand, the 3-dimensional representation i ⊕ det is easily seen to be generically free. Here det denotes the one-dimensional representation det: O_{2} → GL_{1}. Since dim G = 1, Theorem 1.1 yields 1 ≤ ed(O_{2}; 2) ≤ 2. The true value of ed(O_{2}; 2) is 2; see [Re_{1}, Theorem 10.3].

We now proceed to state our result about the gap between the upper and lower bounds of Theorem 1.1. The group C(F) which appears in the definition below is the maximal split p-torsion subgroup of the center of F; for a precise definition, see Section 4.

Definition 1.2. Let G, T := G^{0} and π: G → F := G/T be as in (1.1). Consider the natural (conjugation) action of F on T. We say that G is tame if C(F) lies in the kernel of this action. Equivalently, G is tame if T is central in π^{−1}(C(F)).

For any group G as in (1.1) over a p-special field k, we define gap(G; p) to be the difference between the dimensions of a minimal p-faithful Grepresentation, and a minimal p-generically free G-representation. This is precisely the “gap” between the upper and lower bounds in Theorem 1.1.

Theorem 1.3. Let k be a p-special field of characteristic ≠ p and G be as in (1.1). Then

(a) gap(G; p) ≤ dim T − dim T^{C(F)}.

(b) If G is tame then gap(G; p) = 0, i.e.

ed(G; p) = min dim ρ − dim G,

where the minimum is taken over all p-faithful k-representations of G.

In many cases the lower bound of Theorem 1.1 is much larger than dim T, so Theorem 1.3(a) may be interpreted as saying that the gap between the lower and upper bounds of Theorem 1.1 is not too wide, even if G is not tame.

As a consequence of Theorem 1.1, we will also prove the following “Additivity Theorem”.

1More is known about faithful representations which are not generically free in the case where G is connected semisimple. For an overview of this topic, see [PV, Section 7].

Theorem 1.4. Let k a p-special field of characteristic ≠ p and G_{1}, G_{2} be groups such that gap(G_{1}; p) = gap(G_{2}; p) = 0. Then gap(G_{1} × G_{2}; p) = 0 and ed(G_{1} × G_{2}; p) = ed(G_{1}; p) + ed(G_{2}; p).

The rest of this paper is structured as follows. In Section 2 we discuss the notion of p-special closure k^{(p)} of a field k and show that passing from k to k^{(p)} does not change the essential p-dimension of any k-group. In Section 3 we show that if G → Q is an isogeny of degree prime to p then the essential pdimensions of G and Q coincide. Sections 4, 5 and 6 are devoted to the proof of our Main Theorem 1.1. In Sections 7 and 8 we prove Theorem 1.3 and in Section 9 we prove the Additivity Theorem 1.4. In Section 10 we classify central extensions G of p-groups by tori of small essential p-dimension.

2. The p-special closure of a field

A field L is called p-special if every finite extension of L has degree a power of p. By [EKM, Proposition 101.16] there exists for every field K an algebraic field extension L/K such that L is p-special and every finite sub-extension of L/F has degree prime to p. Such a field L is called a pspecial closure of K and will be denoted by K^{(p)}. The following properties of p-special closures will be important for us in the sequel.

Lemma 2.1. Let K be a field with char K ≠ p and let K_{alg} be an algebraic closure of K containing K^{(p)}.

(a) K^{(p)} is a direct limit of finite extensions K_{i}/K of degree prime to p.

(b) The field K^{(p)} is perfect.

(c) The cohomological q-dimension of Ψ = Gal(K_{alg}/K^{(p)}) is cd_{q}(Ψ) = 0 for any prime q ≠ p.

Proof. (a)The finite sub-extensions K′/K of K^{(p)}/K form a direct system with limit K^{(p)}. Moreover the degrees [K′ : K] are all prime to p. (b) Every finite extension of K^{(p)} has p-power degree. Since char K ≠ p it is separable.

(c) By construction Ψ is a profinite p-group. The result follows from [Se, Cor. 2, I. 3]. □

We call a covariant functor F: Fields /k → Sets limit-preserving if for any directed system of fields {K_{i}}, F(lim_{→} K_{i}) = lim_{→} F(K_{i}). For example if

G is an algebraic group, the functor F_{G} = H^{1}(∗, G) is limit-preserving; see [Ma, 2.1].

Lemma 2.2. Let F be limit-preserving and α ∈ F(K) an object. Denote the image of α in F(K^{(p)}) by α_{K^{(p)}}.

(a) ed_{F}(α; p) = ed_{F}(α_{K^{(p)}}; p) = ed_{F}(α_{K^{(p)}}).

(b) ed(F; p) = ed(F_{k^{(p)}}; p), where F_{k^{(p)}} : Fields /k^{(p)} → Sets denotes the restriction of F to Fields /k^{(p)}.

Proof. (a) The inequalities ed_{F}(α; p) ≥ ed_{F}(α_{K^{(p)}}; p) = ed_{F}(α_{K^{(p)}}) are clear from the definition and since K^{(p)} has no finite extensions of degree prime to p. It remains to prove ed_{F}(α; p) ≤ ed_{F}(α_{K^{(p)}}). If L/K is finite of degree prime to p,

(2.1)  ed_{F}(α; p) = ed_{F}(α_{L}; p),

cf. [Me_{1}, Proposition 1.5] and its proof. For the p-special closure K^{(p)} this is similar and uses (2.1) repeatedly:

Suppose there is a subfield K_{0} ⊂ K^{(p)} and α_{K^{(p)}} comes from an element β ∈ F(K_{0}), so that β_{K^{(p)}} = α_{K^{(p)}}. Write K^{(p)} = lim L, where L is a direct system of finite prime to p extensions of K. Then K_{0} = lim L_{0} with L_{0} = {L∩K_{0} | L ∈ L} and by assumption on F, F(K_{0}) = lim_{L′∈L_{0}} F(L′). Thus there

is a field L′ = L ∩ K_{0} (L ∈ L) and γ ∈ F(L′) such that γ_{K_{0}} = β. Since α_{L} and γ_{L} become equal over K^{(p)}, after possibly passing to a finite extension, we may assume they are equal over L which is finite of degree prime to p over K. Combining these constructions with (2.1) we see that

ed_{F}(α; p) = ed_{F}(α_{L}; p) = ed_{F}(γ_{L}; p) ≤ ed_{F}(γ_{L}) ≤ ed_{F}(α_{K^{(p)}}).

(b) This follows immediately from (a), taking α of maximal essential pdimension. □

Proposition 2.3. Let F, G: Fields /k → Sets be limit-preserving functors and F → G a natural transformation. If the map

F(K) → G(K)

is bijective (resp. surjective) for any p-special field containing k then

ed(F; p) = ed(G; p) (resp. ed(F; p) ≥ ed(G; p)).

Proof. Assume the maps are surjective. By Lemma 2.1(a), the natural transformation is p-surjective, in the terminology of [Me_{1}], so we can apply [Me_{1}, Prop. 1.5] to conclude ed(F; p) ≥ ed(G; p).

Now assume the maps are bijective. Let α be in F(K) for some K/k and β its image in G(K). We claim that ed(α; p) = ed(β; p). First, by Lemma 2.2 we can assume that K is p-special and it is enough to prove that ed(α) = ed(β).

Assume that β comes from β_{0} ∈ G(K_{0}) for some field K_{0} ⊂ K. Any finite prime to p extension of K_{0} is isomorphic to a subfield of K (cf. [Me_{1}, Lemma 6.1]) and so also any p-special closure of K_{0} (which has the same transcendence degree over k). We may therefore assume that K_{0} is p-special. By assumption F(K_{0}) → G(K_{0}) and F(K) → G(K) are bijective. The unique element α_{0} ∈ F(K_{0}) which maps to β_{0} must therefore map to α under the natural restriction map. This shows that ed(α) ≤ ed(β). The other inequality always holds and the claim follows.

Taking α of maximal essential dimension, we obtain ed(F; p) = ed(α; p) = ed(β; p) ≤ ed(G; p). □

3. p-isogenies

An isogeny of algebraic groups is a surjective morphism G → Q with finite kernel. If the kernel is of order prime to p we say that the isogeny is a p-isogeny. In this section we will prove Proposition 3.1 which says that p-isogenous groups have the same essential p-dimension.

Proposition 3.1. Suppose G → Q is a p-isogeny of algebraic groups over a field k of characteristic ≠ p. Then

(a) For any p-special field K containing k the natural map H^{1}(K, G) → H^{1}(K, Q) is bijective.

(b) ed(G; p) = ed(Q; p).

Example 3.2. Let E_{6}^{sc}, E_{7}^{sc} be simply connected simple groups of type E_{6}, E_{7} respectively. In [GR, 9.4, 9.6] it is shown that if k is an algebraically closed field of characteristic ≠ 2 and 3 respectively, then

ed(E_{6}^{sc}; 2) = 3 and ed(E_{7}^{sc}; 3) = 3.

For the adjoint groups E_{6}^{ad} = E_{6}^{sc}/μ_{3}, E_{7}^{ad} = E_{7}^{sc}/μ_{2} we therefore have

ed(E_{6}^{ad}; 2) = 3 and ed(E_{7}^{ad}; 3) = 3.

For the proof of Proposition 3.1 will need two lemmas.

Lemma 3.3. Let N be a finite algebraic group over a field k of characteristic ≠ p. The following are equivalent:

(a) p does not divide the order of N.

(b) p does not divide the order of N(k_{alg}). If N is also assumed to be abelian, denote by N[p] the p-torsion subgroup of N. The following are equivalent to the above conditions.

(a′) N[p](k_{alg}) = {1}.

(b′) N[p](k^{(p)}) = {1}.

Proof. (a) ⟺ (b): Let N^{0} be the connected component of N and N^{et} = N/N^{0} the étale quotient. Recall that the order of a finite algebraic group N over k is defined as |N| = dim_{k} k[N] and |N| = |N^{0}||N^{et}|, see, for example, [Ta]. If char k = 0, N^{0} is trivial, if char k = q ≠ p is positive, |N^{0}| is a power of q. Hence N is of order prime to p if and only if the étale algebraic group N^{et} is. Since N^{0} is connected and finite, N^{0}(k_{alg}) = {1} and so N(k_{alg}) is of order prime to p if and only if the group N^{et}(k_{alg}) is. Then |N^{et}| = dim_{k} k[N^{et}] = |N^{et}(k_{alg})|, cf. [Bou, V.29 Corollary].

(b) ⟺ (a′) ⇒ (b′) are clear.

(a′) ⇐ (b′): Suppose N[p](k_{alg}) is nontrivial. The Galois group Γ = Gal(k_{alg}/k^{(p)}) is a pro-p group and acts on the p-group N[p](k_{alg}). The image of Γ in Aut(N[p](k_{alg})) is again a (finite) p-group and the size of every Γ-orbit in N[p](k_{alg}) is a power of p. Since Γ fixes the identity in N[p](k_{alg}), this is only possible if it also fixes at least p − 1 more elements. It follows that N[p](k^{(p)}) is non-trivial. □

Remark 3.4. Part (b′) could be replaced by the slightly stronger statement that N[p](k^{(p)} ∩ k_{sep}) = {1}, but we will not need this in the sequel.

Proof of Proposition 3.1. (a) Let N be the kernel of G → Q and K be a p-special field over k. Since K_{sep} = K_{alg} (see Lemma 2.1(b)), the sequence of K_{sep}-points 1 → N(K_{sep}) → G(K_{sep}) → Q(K_{sep}) → 1 is exact. By Lemma 3.3, the order of N(K_{sep}) is not divisible by p and therefore coprime to the order of any finite quotient of Ψ = Gal(K_{sep}/K). By [Se, I.5, ex. 2] this implies that H^{1}(K, N) = {1}. Similarly, if _{c}N is the group N twisted by a cocycle c : Ψ → G, _{c}N(K_{sep}) = N(K_{sep}) is of order prime to p and H^{1}(K, _{c}N) = {1}. It follows that H^{1}(K, G) → H^{1}(K, Q) is injective, cf. [Se, I.5.5].

Surjectivity is a consequence of [Se, I. Proposition 46] and the fact that the q-cohomological dimension of Ψ is 0 for any divisor q of |N(K_{sep})| (Lemma 2.1(c)).

This concludes the proof of part (a). Part (b) follows from part (a) and Proposition 2.3. □

4. Proof of the Main Theorem: an overview

We will assume throughout this section that the field k is p-special of char k ≠ p, and G is a smooth affine k-group, such that G^{0} = T is a torus and G/G^{0} = F is a finite p-group, as in (1.1).

The upper bound in Theorem 1.1 is an easy consequence of Proposition 3.1. Indeed, suppose μ: G → GL(V) is a p-generically free representation. That is, Ker μ is a finite group of order prime to p and μ descends to a generically free representation of G′ := G/ Ker μ. By Proposition 3.1 ed(G; p) = ed(G′; p). On the other hand,

ed(G′; p) ≤ ed(G′) ≤ dim μ − dim G′ = dim μ − dim G;

see [BF, Lemma 4.11] or [Me_{1}, Corollary 4.2]. This completes the proof of the upper bound in Theorem 1.1.

The rest of this section will be devoted to outlining a proof of the lower bound of Theorem 1.1. The details (namely, the proofs of Propositions 4.2 and 4.3) will be supplied in the next two sections. The starting point of our argument is [LMMR, Theorem 3.1], which we reproduce as Theorem 4.1 below for the reader’s convenience.

Theorem 4.1. Consider an exact sequence of algebraic groups over k

1 → C → G → Q → 1

such that C is central in G and is isomorphic to μ_{p}^{r} for some r ≥ 0. Given a character χ: C → μ_{p} denote by Rep^{χ} the class of irreducible representations ϕ: G → GL(V), such that ϕ(c) = χ(c) Id for every c ∈ C.

Assume further that

(4.1)  gcd{dim ϕ | ϕ ∈ Rep^{χ}} = min{dim ϕ | ϕ ∈ Rep^{χ}}

for every character χ: C → μ_{p}. Then

ed(G; p) ≥ min dim ψ − dim G ,

where the minimum is taken over all finite-dimensional representations ψ of G such that ψ_{| C} is faithful.

To prove the lower bound of Theorem 1.1, we will apply Theorem 4.1 to the exact sequence

(4.2)  1 → C(G) → G → Q → 1 .

where C(G) is a central subgroup of G defined as follows. Recall from [LMMR, Section 2] that if A is a k-group of multiplicative type, Split_{k}(A) is defined as the maximal split k-subgroup of A. That is, if X(A) is the character Gal(k_{sep}/k)-lattice of A then the character lattice of Split_{k}(A) is defined as the largest quotient of X(A) with trivial Gal(k_{sep}/k)-action.

If G be an extension of a finite p-group by a torus, as in (1.1), denote by Z(G)[p] the p-torsion subgroup of the center Z(G). Note that Z(G) is a commutative group, which is an extension of a p-group by a torus. Since char k ≠ p it is smooth. Moreover Z(G)(k_{alg}) consists of semi-simple elements. It follows that Z(G) is of multiplicative type. We now define C(G) := Split_{k}(Z(G)[p]).

In order to show that Theorem 4.1 can be aplied to the sequence (4.2), we need to check that condition (4.1) is satisfied. This is a consequence of the following proposition, which will be proved in the next section.

Proposition 4.2. Let

1 → T → G → F → 1

be an exact sequence of (linear) algebraic k-groups, where F is a finite pgroup and T a torus. Then the dimension of every irreducible representation of G over k is a power of p.

Applying Theorem 4.1 to the exact sequence (4.2) now yields

ed(G; p) ≥ min dim ρ − dim G ,

where the minimum is taken over all representations ρ: G → GL(V) such that ρ_{| C(G)} is faithful. This resembles the lower bound of Theorem 1.1; the only difference is that in the statement of Theorem 1.1 we take the minimum over p-faithful representations ρ, and here we ask that ρ_{| C(G)} should be faithful. The following proposition shows that the two bounds are, in fact, the same, thus completing the proof of Theorem 1.1.

Proposition 4.3. A finite-dimensional representation ρ of G is p-faithful if and only if ρ_{| C(G)} is faithful.

We will prove Proposition 4.3 in Section 6.

Remark 4.4. The inequality min dim ρ − dim G ≤ ed(G; p) of Theorem 1.1, where ρ ranges over all p-faithful representations of G, fails if we take the minimum over just the faithful (rather than p-faithful) representations, even in the case where G = T is a torus.

Indeed, choose T so that the Gal(k_{sep}/k)-character lattice X(T) of T is a direct summand of a permutation lattice, but X(T) itself is not permutation (see [CTS_{1}, 8A.] for an example of such a lattice).

In other words, there exists a k-torus T′ such that T × T′ is quasi-split. This implies that H^{1}(K, T × T′) = {1} and thus H^{1}(K, T) = {1} for any field extension K/k. Consequently ed(T; p) = 0 for every prime p.

On the other hand, we claim that the dimension of the minimal faithful representation of T is strictly bigger than dim T. Assume the contrary. Then by [LMMR, Lemma 2.6] there exists a surjective homomorphism f: P → X(T) of Gal(k_{sep}/k)-lattices, where P is permutation and rank P = dim T. This implies that f has finite kernel and hence, is injective. We conclude that f is an isomorphism and thus X(T) is a permutation Gal(k_{sep}/k)-lattice, a contradiction. □

5. Dimensions of irreducible representations

The purpose of this section is to prove Proposition 4.2. Recall that G^{0} is a torus, which we denote by T and G/G^{0} is a finite p-group which we denote by F. The case, where T = {1}, i.e., G = F is an arbitrary (possibly twisted) finite p-group, is established in the course of the proof of [LMMR, Theorem 7.1]. Our proof of Proposition 4.2 below is based on leveraging this case as follows.

Lemma 5.1. Let G be an algebraic group defined over a field k and

F_{1} ⊆ F_{2} ⊆ ⋯ ⊂ G

be an ascending sequence of finite k-subgroups whose union ∪_{n≥1}F_{n} is Zariski dense in G. If ρ: G → GL(V) is an irreducible representation of G then ρ_{| F_{i}} is irreducible for sufficiently large integers i.

Proof. For each d = 1, ..., dim V −1 consider the G-action on the Grassmannian Gr(d, V) of d-dimensional subspaces of V. Let X^{(d)} = Gr(d, V)^{G} and X_{i}^{(d)} = Gr(d, V)^{F_{i}} be the subvariety of d-dimensional G- (resp. F_{i}-)invariant subspaces of V. Then X_{1}^{(d)} ⊇ X_{2}^{(d)} ⊇ … and since the union of the groups F_{i} is dense in G,

X^{(d)} = ∩_{i≥0}X_{i}^{(d)} .

By the Noetherian property of Gr(d, V), we have X^{(d)} = X_{m_{d}}^{(d)} for some m_{d} ≥ 0.

Since V does not have any G-invariant d-dimensional k-subspaces, we know that X^{(d)}(k) = ∅. Thus, X_{m_{d}}^{(d)}(k) = ∅, i.e., V does not have any F_{m_{d}}- invariant d-dimensional k-subspaces. Setting m := max{m_{1}, …, m_{dim V −1}}, we see that ρ_{| F_{m}} is irreducible. □

We now proceed with the proof of Proposition 4.2. By Lemma 5.1, it suffices to construct a sequence of finite p-subgroups

F_{1} ⊆ F_{2} ⊆ ⋯ ⊂ G

defined over k whose union ∪_{n≥1}F_{n} is Zariski dense in G.

In fact, it suffices to construct one p-subgroup F′ ⊂ G, defined over k such that F′ surjects onto F. Indeed, once F′ is constructed, we can define F_{i} ⊂ G as the subgroup generated by F′ and T [p^{i}], for every i ≥ 0. Since ∪_{n≥1}F_{n} contains both F′ and T [p^{i}], for every i ≥ 0 it is Zariski dense in G, as desired.

The following lemma, which establishes the existence of F′, is thus the final step in our proof of Proposition 4.2.

Lemma 5.2. Let 1 → T → G ⟶^{π} F → 1 be an extension of a p-group F by a torus T over a field k, as in (1.1). Then G has a p-subgroup F′ with π(F′) = F.

In the case where F is split and k is algebraically closed this is proved in [CGR, p. 564]; cf. also the proof of [BS, Lemme 5.11].

Proof. Denote by Ẽx^{1}(F, T) the group of equivalence classes of extensions of F by T. We claim that Ẽx^{1}(F, T) is torsion. Let Ex^{1}(F, T) ⊂ Ẽx^{1}(F, T) be the classes of extensions which have a scheme-theoretic section (i.e. G(K) → F (K) is surjective for all K/k). There is a natural isomorphism Ex^{1}(F, T) ≃ H^{2}(F, T), where the latter one denotes Hochschild cohomology, see [DG, III. 6.2, Proposition]. By [Sch_{3}] the usual restriction-corestriction arguments can be applied in Hochschild cohomology and in particular, m · H^{2}(F, T) = 0 where m is the order of F. Now recall that M ↦ Ẽx^{i}(F, M) and M ↦ Ex^{i}(F, M) are both derived functors of the crossed homomorphisms M ↦ Ex^{0}(F, M), where in the first case M is in the category of F -module sheaves and in the second, F -module functors, cf. [DG, III. 6.2]. Since F is finite and T an affine scheme, by [Sch_{1}, Satz 1.2 & Satz 3.3] there is an exact sequence of F -module schemes 1 → T → M_{1} → M_{2} → 1 and an exact sequence Ex^{0}(F, M_{1}) → Ex^{0}(F, M_{2}) → Ẽx^{1}(F, T) → H^{2}(F, M_{1}) ≃ Ex^{1}(F, M_{1}). The F -module sequence also induces a long exact sequence on Ex(F, ∗) and we have a diagram

\xymatrix{ &  & Ẽx^{1}(F, T) \ar[dr] \\ Ex^{0}(F, M_{1}) \ar[r] & Ex^{0}(F, M_{2}) \ar[ur] \ar[dr] &  & Ex^{1}(F, M_{1}) \\  &  & Ex^{1}(F, T) \ar[ur] \ar@{^{(}->}[uu]}

An element in Ẽx^{1}(F, T) can thus be killed first in Ex^{1}(F, M_{1}) so it comes from Ex^{0}(F, M_{2}). Then kill its image in Ex^{1}(F, T) ≃ H^{2}(F, T), so it comes from Ex^{0}(F, M_{1}), hence is 0 in Ẽx^{1}(F, T). In particular, multiplying twice by the order m of F, we see that m^{2} · Ẽx^{1}(F, T) = 0. This proves the claim.

Now let us consider the exact sequence 1 → N → T ⟶^{×m^{2}} T → 1, where N is the kernel of multiplication by m^{2}. Clearly N is finite and we have an induced exact sequence

Ẽx^{1}(F, N) → Ẽx^{1}(F, T) ⟶^{×m^{2}} Ẽx^{1}(F, T)

which shows that the given extension G comes from an extension F′ of F by N. Then G is the pushout of F′ by N → T and we can identify F′ with a subgroup of G. □

6. Proof of Proposition 4.3

We will prove Proposition 6.1 below; Proposition 4.3 is an immediate consequence, with N = Ker ρ. Recall that G is an algebraic group over a p-special field k of characteristic ≠ p such that G^{0} is a torus, G/G^{0} is a finite p-group, as in (1.1), and C(G) is the split central p-subgroup of G defined in Section 4.

Proposition 6.1. Let N be a normal k-subgroup of G. Then the following conditions are equivalent:

(i) N is finite of order prime to p,

(ii) N ∩ C(G) = {1},

(iii) N ∩ Z(G)[p] = {1},

In particular, taking N = G, we see that C(G) ≠ {1} if G ≠ {1}.

Proof. (i)⟹ (ii) is obvious, since C(G) is a p-group.

(ii) ⟹ (iii). Assume the contrary: A := N ∩ Z(G)[p] ≠ {1}. By [LMMR, Section 2]

{1} ≠ C(A) ⊂ N ∩ C(G) ,

contradicting (ii).

Our proof of the implication (iii) ⟹ (i) will rely on the following assertion:

Claim: Let M be a non-trivial normal finite p-subgroup of G such that the commutator (G^{0}, M) is trivial. Then M ∩ Z(G)[p] ≠ {1}.

To prove the claim, note that M(k_{sep}) is non-trivial and the conjugation action of G(k_{sep}) on M(k_{sep}) factors through an action of the p-group (G/G^{0})(k_{sep}). Thus each orbit has p^{n} elements for some n ≥ 0; consequently, the number of fixed points is divisible by p. The intersection (M∩Z(G))(k_{sep}) is precisely the fixed point set for this action; hence, M ∩ Z(G)[p] ≠ {1}. This proves the claim.

We now continue with the proof of the implication (iii) ⟹ (i). For notational convenience, set T := G^{0}. Assume that N ◁G and N ∩Z(G)[p] = {1}. Applying the claim to the normal subgroup M := (N ∩ T)[p] of G, we see that (N ∩ T)[p] = {1}, i.e., N ∩ T is a finite group of order prime to p. The exact sequence

(6.1)  1 → N ∩ T → N → \overline{N} → 1 ,

where \overline{N} is the image of N in G/T, shows that N is finite. Now observe that for every r ≥ 1, the commutator (N, T [p^{r}]) is a p-subgroup of N ∩ T. Thus (N, T [p^{r}]) = {1} for every r ≥ 1. We claim that this implies (N, T) = {1}. If N is smooth, this is straightforward; see [Bo, Proposition 2.4, p. 59]. If N is not smooth, note that the map c: N × T → G sending (n, t) to the commutator ntn^{−1}t^{−1} descends to \overline{c}: \overline{N} × T → G (indeed, N ∩ T clearly commutes with T). Since |\overline{N}| is a power of p and char(k) ≠ p, \overline{N} is smooth over k, and we can pass to the separable closure k_{sep} and apply the usual Zariski density argument to show that the image of \overline{c} is trivial.

We thus conclude that N ∩ T is central in N. Since gcd(|N ∩ T |, \overline{N}) = 1, by [Sch_{2}, Corollary 5.4] the extension (6.1) splits, i.e., N ≃ (N ∩ T) × \overline{N}. This turns \overline{N} into a finite p-subgroup of G with (T, \overline{N}) = \{1\}. The claim implies that \overline{N} is trivial. Hence N = N ∩ T is a finite group of order prime to p, as claimed.

This completes the proof of Proposition 6.1 and thus of Theorem 1.1. □

7. Faithful versus generically free

Throughout this section we will assume that k is a p-special field of characteristic ≠ p and G is a (smooth) algebraic k-group such that T := G^{0} is a torus and F := G/G^{0} is a finite p-group, as in (1.1). As we have seen in the introduction, some groups of this type have faithful linear representations that are not generically free. In this section we take a closer look at this phenomenon.

If F′ is a subgroup of F then we will use the notation G_{F′} to denote the subgroup π^{−1}(F′) of G. Here π is the natural projection G → G/T = F.

Lemma 7.1. Suppose T is central in G. Then (a) G has only finitely many k-subgroups S such that S ∩ T = {1}.

(b) Every faithful action of G on a geometrically irreducible variety X is generically free.

Proof. After replacing k by its algebraic closure k_{alg} we may assume without loss of generality that k is algebraically closed.

(a) Since F has finitely many subgroups, it suffices to show that for every subgroup F_{0} ⊂ F, there are only finitely many S ⊂ G such that π(S) = F_{0} and S ∩ T = {1}.

After replacing G by G_{F_{0}}, we may assume that F_{0} = F. In other words, we will show that π has at most finitely many sections s: F → G. Fix one such section, s_{0} : F → G. Denote the exponent of F by e. Suppose s: F → G is another section. Then for every f ∈ F (k), we can write s(f) = s_{0}(f)t for some t ∈ T. Since T is central in G, t and s_{0}(f) commute. Since s(f)^{e} = s_{0}(f)^{e} = 1, we see that t^{e} = 1. In other words, t ∈ T (k) is an e-torsion element, and there are only finitely many e-torsion elements in T. We conclude that there are only finitely many choices of s(f) for each f ∈ F. Hence, there are only finitely many sections F → G, as claimed.

(b) The restriction of the G-action on X to T is faithful and hence, generically free; cf., e.g., [Lo, Proposition 3.7(A)]. Hence, there exists a dense open T -invariant subset U ⊂ X such that Stab_{T} (u) = {1} for all u ∈ U. In other words, if S = Stab_{G}(u) then S ∩ T = {1}. By part (a) G has finitely many non-trivial subgroups S with this property. Denote them by S_{1}, …, S_{n}. Since G acts faithfully, X^{S_{i}} is a proper closed subvariety of X for any i = 1, …, n. Since X is irreducible,

U′ = U \ (X^{S_{1}} ∪ ⋯ ∪ X^{S_{n}})

is a dense open T -invariant subset of X, and the stabilizer Stab_{G}(u) is trivial for every u ∈ U′. Replacing U′ by the intersection of its (finitely many) G(k_{alg})-translates, we may assume that U′ is G-invariant.This shows that the G-action on X is generically free. □

Proposition 7.2. (a) A faithful action of G on a geometrically irreducible variety X is generically free if and only if the action of the subgroup G_{C(F)} ⊆ G on X is generically free.

(b) A p-faithful action of G on a geometrically irreducible variety X is p-generically free if and only if the action of the subgroup G_{C(F)} ⊆ G on X is p-generically free.

Proof. (a) The (faithful) T -action on X is necessarily generically free cf. [Lo, Proposition 3.7(A)]. Thus by [Ga, Exposé V, Théorème 10.3.1] (or [BF, Theorem 4.7]) X has a dense open T -invariant subvariety U defined over k, which is the total space of a T -torsor, U → Y := U/G, where Y is also smooth and geometrically irreducible. Since G/T is finite, after replacing U by the intersection of its (finitely many) G(k_{alg})-translates, we may assume that U is G-invariant.

The G-action on U descends to an F -action on Y (by descent). Now it is easy to see that the following conditions are equivalent:

(i) the G-action on X is generically free, and

(ii) the F -action on Y is generically free; cf. [LR, Lemma 2.1]. Since F is finite, (ii) is equivalent to

(iii) F acts faithfully on Y. Since k is p-special, Proposition 6.1 tells us that the kernel of the F -action on Y is trivial iff the kernel of the C(F)-action on Y is trivial. In other words, (iii) is equivalent to

(iv) C(F) acts faithfully (or equivalently, generically freely) on Y and consequently, to

(v) the G_{C(F)}-action on U (or, equivalently, on X) is generically free. Note that (iv) and (v) are the same as (ii) and (i), respectively, except that F is replaced by C(F) and G by G_{C(F)}. Thus the equivalence of (iv) and (v) follows from the equivalence of (i) and (ii). We conclude that (i) and (v) are equivalent, as desired.

(b) Let K be the kernel of the G-action on X, which is contained in T by assumption. Notice that (G/K)/(T/K) = G/T = F. So part (a) says the G/K-action on X is generically free if and only if the G_{C(F)}/K-action on X is generically free, and part (b) follows. □

Corollary 7.3. Consider an action of a tame k-group G (see Definition 1.2) on a geometrically irreducible k-variety X.

(a) If this action is faithful then it is generically free.

(b) If this action is p-faithful then it is is p-generically free.

Proof. (a) Since G is tame, T is central in G_{C(F)}. Hence, the G_{C(F)}-action on X is generically free by Lemma 7.1(b). By Proposition 7.2(a), the G-action on X is generically free.

(b) Let K be the kernel of the action. Note that G/K is also tame. Now apply part (a) to G/K. □

8. Proof of Theorem 1.3

In this section we will prove the following proposition, which implies Theorem 1.3(a). Theorem 1.3(b) is an immediate consequence of part (a) and Theorem 1.1 (or alternatively, of Corollary 7.3(b) and Theorem 1.1).

We continue to use the notational conventions and assumptions on k and G described at the beginning of Section 7.

Proposition 8.1. Let ρ: G → GL(V) be a linear representation of G.

(a) If ρ is faithful then G has a generically free representation of dimension dim ρ + dim T − dim T^{C(F)}.

(b) If ρ is p-faithful, then G has a p-generically free representation of dimension dim ρ + dim T − dim T^{C(F)}.

Proof. (a) T^{C(F)} is preserved by the conjugation action of G, so the adjoint representation of G decomposes as Lie(T) = Lie(T^{C(F)}) ⊕ W for some G-representation W. Since the G-action on Lie(T) factors through F, the existence of W follows from Maschke’s Theorem. Let μ be the G-representation on V ⊕ W. Then dim μ = dim ρ + dim T − dim T^{C(G)}. It thus remains to show that μ is a generically free representation of G.

Let K be the kernel of the G_{C(F)} action on Lie(T). We claim T is central in K. The finite p-group K/T acts on T (by conjugation), and it fixes the identity. By construction K/T acts trivially on the tangent space at the identity, which implies K/T acts trivially on T, since the characteristic is ≠ p; cf. [GR, Proof of 4.1]. This proves the claim.

By Lemma 7.1, the K-action on V is generically free. Now G_{C(F)} acts trivially on Lie(T^{C(F)}), so G_{C(F)}/K acts faithfully on W. Since G_{C(F)}/K is finite, this action is also generically free. Therefore G_{C(F)} acts generically freely on V ⊕ W [MR_{1}, Lemma 3.2]. Finally, by Proposition 7.2(a), G acts generically freely on V ⊕ W, as desired.

(b) By our assumption ker ρ ⊂ T. Set \overline{T} := T/ ker ρ. It is easy to see that dim T^{C(F)} ≤ dim \overline{T}^{C(F)}. Hence, by part (a), there exists a generically free representation of G/ ker ρ of dimension

dim \overline{T} − dim \overline{T}^{C(F)} ≤ dim T − dim T^{C(F)} .

We may now view this representation as a p-generically free representation of G. This completes the proof of Theorem 1.3. □

Remark 8.2. A similar argument shows that for any tame normal subgroup H ⊂ G over a p-special field k, gap(G; p) ≤ ed(G/H; p).

9. Additivity

Our proof of the Additivity Theorem 1.4 relies on the following lemma. Let G be an algebraic group defined over k and C be a k-subgroup of G. Denote the minimal dimension of a representation ρ of G such that ρ_{| C} is faithful by f(G, C).

Lemma 9.1. For i = 1, 2 let G_{i} be an algebraic group defined over a field k and C_{i} be a central k-subgroup of G_{i}. Assume that C_{i} is isomorphic to μ_{p}^{r_{i}} over k for some r_{1}, r_{2} ≥ 0. Then

f(G_{1} × G_{2}; C_{1} × C_{2}) = f(G_{1}; C_{1}) + f(G_{2}; C_{2}) .

Our argument below is a variant of the proof of [KM, Theorem 5.1], where G is assumed to be a (constant) finite p-group and C = C(G) (recall that C(G) is defined at the beginning of Section 4).

Proof. For i = 1, 2 let π_{i}: G_{1} × G_{2} → G_{i} be the natural projection and ϵ_{i}: G_{i} → G_{1} × G_{2} be the natural inclusion.

If ρ_{i} is a d_{i}-dimensional representation of G_{i} whose restriction to C_{i} is faithful, then clearly ρ_{1} ∘ π_{1} ⊕ ρ_{2} ∘ π_{2} is a d_{1} + d_{2}-dimensional representation of G_{1} × G_{2} whose restriction to C_{1} × C_{2} is faithful. This shows that

f(G_{1} × G_{1}; C_{1} × C_{2}) ≤ f(G_{1}; C_{1}) + f(G_{2}; C_{2}) .

To prove the opposite inequality, let ρ: G_{1} × G_{2} → GL(V) be a representation such that ρ_{| C_{1}×C_{2}} is faithful, and of minimal dimension

d = f(G_{1} × G_{1}; C_{1} × C_{2})

with this property. Let ρ_{1}, ρ_{2}, …, ρ_{n} denote the irreducible decomposition factors in a decomposition series of ρ. Since C_{1} × C_{2} is central in G_{1} × G_{2}, each ρ_{i} restricts to a multiplicative character of C_{1}×C_{2} which we will denote by χ_{i}. Moreover since C_{1} × C_{2} ≃ μ_{p}^{r_{1}+r_{2}} is linearly reductive ρ_{| C_{1}×C_{2}} is a direct sum χ_{1}^{⊕d_{1}} ⊕ ⋯ ⊕ χ_{n}^{⊕d_{n}} where d_{i} = dim V_{i}. It is easy to see that the following conditions are equivalent:

(i) ρ_{| C_{1}×C_{2}} is faithful,

(ii) χ_{1}, …, χ_{n} generate (C_{1} × C_{2})^{∗} as an abelian group. In particular we may assume that ρ = ρ_{1} ⊕ ⋯ ⊕ ρ_{n}. Since C_{i} is isomorphic to μ_{p}^{r_{i}}, we will think of (C_{1} × C_{2})^{∗} as a 𝔽_{p}-vector space of dimension r_{1} + r_{2}. Since (i) ⇔ (ii) above, we know that χ_{1}, …, χ_{n} span (C_{1} × C_{2})^{∗}. In fact, they form a basis of (C_{1} × C_{2})^{∗}, i.e., n = r_{1} + r_{2}. Indeed, if they were not linearly independent we would be able to drop some of the terms in the irreducible decomposition ρ_{1} ⊕ ⋯ ⊕ ρ_{n}, so that the restriction of the resulting representation to C_{1} × C_{2} would still be faithful, contradicting the minimality of dim ρ.

We claim that it is always possible to replace each ρ_{j} by ρ′_{j}, where ρ′_{j} is either ρ_{j} ∘ ϵ_{1} ∘ π_{1} or ρ_{j} ∘ ϵ_{2} ∘ π_{2} such that the restriction of the resulting representation ρ′ = ρ′_{1} ⊕ ⋯ ⊕ ρ′_{n} to C_{1} × C_{2} remains faithful. Since dim ρ_{i} = dim ρ′_{i}, we see that dim ρ′ = dim ρ. Moreover, ρ′ will then be of the form α_{1} ∘ π_{1} ⊕ α_{2} ∘ π_{2}, where α_{i} is a representation of G_{i} whose restriction to C_{i} is faithful. Thus, if we can prove the above claim, we will have

f(G_{1} × G_{1}; C_{1} × C_{2}) = dim ρ = dim ρ′ = dim α_{1} + dim α_{2}
≥ f(G_{1}, C_{1}) + f(G_{2}, C_{2}) ,

as desired.

To prove the claim, we will define ρ′_{j} recursively for j = 1, …, n. Suppose ρ′_{1}, …, ρ′_{j−1} have already be defined, so that the restriction of

ρ′_{1} ⊕ ⋯ ⊕ ρ′_{j−1} ⊕ ρ_{j} ⋯ ⊕ ρ_{n}

to C_{1} × C_{2} is faithful. For notational simplicity, we will assume that ρ_{1} = ρ′_{1}, …, ρ_{j−1} = ρ′_{j−1}. Note that

χ_{j} = (χ_{j} ∘ ϵ_{1} ∘ π_{1}) + (χ_{j} ∘ ϵ_{2} ∘ π_{2}) .

Since χ_{1}, …, χ_{n} form a basis (C_{1}×C_{2})^{∗} as an 𝔽_{p}-vector space, we see that (a) χ_{j} ∘ϵ_{1}∘π_{1} or (b) χ_{j} ∘ϵ_{2}∘π_{2} does not lie in Span_{𝔽_{p}}(χ_{1}, …, χ_{j−1}, χ_{j+1}, …, χ_{n}). Set

ρ′_{j} := \begin{cases}ρ_{j} ∘ ϵ_{1} ∘ π_{1} in case (a), and \\ ρ_{j} ∘ ϵ_{2} ∘ π_{2}, otherwise. \end{cases}

Using the equivalence of (i) and (ii) above, we see that the restriction of

ρ_{1} ⊕ ⋯ ⊕ ρ_{j−1} ⊕ ρ′_{j} ⊕ ρ_{j+1}, ⋯ ⊕ ρ_{n}

to C is faithful. This completes the proof of the claim and thus of Lemma 9.1. □ Proof of Theorem 1.4. Clearly G_{1} × G_{2} is again a p-group extended by a torus.

Reall that C(G) is defined as the maximal split p-torsion subgroup of the center of G; see Section 4. It follows from this definition that

C(G_{1} × G_{2}) = C(G_{1}) × C(G_{2}) ;

cf. [LMMR, Lemma 2.1]. Lemma 9.1 and Proposition 4.3 show that the minimal dimension of a p-faithful representation is f(G, C(G)) = f(G_{1}, C(G_{1}))+ f(G_{2}, C(G_{2})), which is the sum of the minimal dimensions of p-faithful representations of G_{1} and G_{2}. For i ∈ {1, 2}, since gap(G_{i}; p) = 0, there exists a p-generically free representation ρ_{i} of G_{i} of dimension f(G_{i}, C(G_{i})). The direct sum ρ_{1} ⊕ρ_{2} is a p-generically free representation of G and its dimension is f(G, C(G)). It follows that gap(G_{1} × G_{2}; p) = 0. By Theorem 1.1

ed(G; p) = f(G, C(G)) − dim G;

cf. Proposition 4.3, and similarly for G_{1} and G_{2}. Hence the equality ed(G; p) = ed(G_{1}; p) + ed(G_{2}; p) follows. This concludes the proof. □

Example 9.2. Let T be a torus over a field k. Suppose there exists an element τ in the absolute Galois group Gal(k_{sep}/k) which acts on the character lattice X(T) via multiplication by −1. Then ed(T; 2) ≥ dim T.

Proof. Let n := dim T. Over the fixed field K := (k_{sep})^{τ} the torus T becomes isomorphic to a direct product of n copies of a non-split one-dimensional torus T_{1}. Using [LMMR, Theorem 1.1] it is easy to see that ed(T_{1}; 2) = 1. By the Additivity Theorem 1.4 we conclude that ed(T; 2) ≥ ed(T_{K}; 2) = ed((T_{1})^{n}; 2) = n ed(T_{1}; 2) = dim T. □

We conclude this section with an example which shows, in particular, that the property gap(G; p) = 0 is not preserved under base field extensions. Let T be an algebraic torus over a field p-special field k of characteristic ≠ p and F a non-trivial p-subgroup of the constant group S_{n}. Then we can form the wreath product

T ≀ F := T^{n} ⋊ F,

where F acts on T^{n} by permutations.

Example 9.3. gap(T ≀ F; p) = 0 if and only if ed(T; p) > 0. Moreover,

ed(T ≀ F; p) = \begin{cases}ed(T^{n}; p) = n ed(T; p), & if ed(T; p) > 0, \\ ed(F; p), & otherwise. \end{cases}

Proof. Let W be a p-faithful T -representation of minimal dimension. By [LMMR, Theorems 1.1], ed(T; p) = dim W − dim T.

Then W^{⊕n} is naturally a p-faithful T ≀ F -representation. Lemma 9.1 and Proposition 4.3 applied to T^{n} tell us that W^{⊕n} has minimal dimension among all p-faithful representations of T ≀ F.

Suppose ed(T; p) > 0, i.e., dim W > dim T. The group F acts faithfully on the rational quotient W^{⊕n}/T^{n} = (W/T)^{n}, since dim W/T = dim W − dim T > 0. It is easy to see that the T ≀ F -action on W^{⊕n} is p-generically free; cf. e.g, [MR_{1}, Lemma 3.3]. In particular, gap(T ≀ F; p) = 0 and

ed(T ≀F; p) = dim W^{⊕n}−dim(T ≀F) = n(dim W −dim T) = n ed(T; p) = ed(T^{n}; p) ,

where the last equality follows from Theorem 1.4.

Now assume that ed(T; p) = 0, i.e., dim W = dim T. The group T ≀ F cannot have a p-generically free representation V of dimension dim W^{⊕n} = dim T ≀ F, since T^{n} would then have a dense orbit in V. It follows that gap(T ≀ F; p) > 0. In order to compute its essential p-dimension of T ≀ F we use the fact that the natural projection T ≀ F → F has a section. Hence, the map H^{1}(−, T ≀ F) → H^{1}(−, F) also has a section and is, consequently, a surjection. This implies that ed(T ≀ F; p) ≥ ed(F; p). Let W′ be a faithful F - representation of dimension ed(F; p). The direct sum W^{⊕n} ⊕ W′ considered as a T ≀F representation is p-generically free. So, ed(T ≀F; p) = ed(F; p). □

10. Groups of low essential p-dimension

In [LMMR] we have identified tori of essential dimension 0 as those tori, whose character lattice is invertible, i.e. a direct summand of a permutation module, see [LMMR, Example 5.4]. The following lemma shows that among the algebraic groups G studied in this paper, i.e. extensions of p-groups by tori, there are no other examples of groups of ed(G; p) = 0.

Lemma 10.1. Let G be an algebraic group over a field k such that G/G^{0} is a p-group. If ed(G; p) = 0, then G is connected.

Proof. Assume the contrary: F := G/G^{0} ≠ {1}. Let X be an irreducible G-torsor over some field K/k. For example, we can construct X as follows. Start with a faithful linear representation G ↪ GL_{n} for some n ≥ 0. The natural projection GL_{n} → GL_{n} /G is a G-torsor. Pulling back to the generic point Spec(K) → GL_{n} /G, we obtain an irreducible G-torsor over K.

Now X/G^{0} → Spec(K) is an irreducible F -torsor. Since F ≠ {1} is not connected, this torsor is non-split. As F is a p-group, X/G^{0} remains nonsplit over every prime to p extension L/K. It follows that the degree of every closed point of X is divisible by p, hence p is a torsion prime of G. Therefore ed(G; p) > 0 by [Me_{1}, Proposition 4.4]. This contradicts the assumption ed(G; p) = 0 and so F must be trivial. □

For the remainder of this section we will assume the base field k is of characteristic ≠ p.

Proposition 10.2. Let G be a central extension of a p-group F by a torus T. If ed(G; p) ≤ p − 2, then G is of multiplicative type.

Proof. Without loss of generality, assume k = k_{alg}. By Theorem 1.1, there is a p-faithful representation V of G, with dim V ≤ dim T + p − 2.

First consider the case when V is faithful. By Nagata’s theorem [Na] G is linearly reductive, hence we can write V = ⨁_{i=1}^{r} V_{i} for some non-trivial irreducible G-representations V_{i}. Since T is central and diagonalizable it acts by a fixed character on V_{i}, for every i. Hence r ≥ dim T by faithfulness of V. It follows that 1 ≤ dim V_{i} ≤ p − 1 for each i. But every irreducible Grepresentation has dimension a power of p (Proposition 4.2), so each V_{i} is one-dimensional. In other words, G is of multiplicative type.

Now consider the general case, where V is only p-faithful, and let K ⊂ G be the kernel of that representation. Then G/K is of multiplicative type, so it embeds into a torus T_{1}. Since T is central in G, a subgroup F′ as in Lemma 5.2 is normal, so let T_{2} = G/F′, which is also a torus. The kernel of the natural map G → T_{1} × T_{2} is contained in K ∩ F′, which is trivial since p does not divide the order of K. In other words, G embeds into a torus, and hence is of multiplicative type. □

Example 10.3. Proposition 10.2 does not generalize to tame groups. For a counter-example, assume the p-special field k contains a primitive p^{2}-root of unity, and consider the group G = 𝔾_{m}^{p} ⋊ℤ/p^{2}, where a generator in ℤ/p^{2} acts by cyclically permuting the p copies of 𝔾_{m}. The group G is tame, because C(ℤ/p^{2}) = ℤ/p = μ_{p} acts trivially on 𝔾_{m}^{p}. On the other hand, G is not abelian and hence, is not of multiplicative type.

We claim that ed(G; p) = 1 and hence, ≤ p − 2 for every odd prime p. There is a natural p-dimensional faithful representation ρ of G; ρ embeds 𝔾_{m}^{p} into GL_{p} diagonally, in the standard basis e_{1}, …, e_{p}, and ℤ/p^{2} cyclically permutes e_{1}, …, e_{p}. Taking the direct sum of ρ with the 1-dimensional representation χ: G → ℤ/p^{2} = μ_{p^{2}} ↪ 𝔾_{m} = GL_{1}, we obtain a faithful p + 1-dimensional representation ρ ⊕ χ which is therefore generically free by Corollary 7.3 (this can also be verified directly). Hence, ed(G; p) ≤ (p + 1) − dim(G) = 1. On the other hand, by Lemma 10.1, we see that ed(G; p) ≥ 1 and thus = 1, as claimed. □

Let Γ_{p} a finite p-group, and let ϕ : P → X be a map of ℤΓ_{p}-modules. As in [LMMR], we will call ϕ a p-presentation if P is permutation, and the cokernel is finite of order prime to p. We will denote by I the augmentation ideal of ℤ[Γ_{p}], and by \overline{X} := X/(pX +IX) the largest p-torsion quotient with trivial Γ_{p}-action. The induced map on quotient modules will be denoted ϕ̄: \overline{P} → \overline{X}.

In the sequel for G a group of multiplicative type over k, the group Γ_{p} in the definition of “p-presentation” is understood to be a Sylow p-subgroup of Γ = Gal(ℓ/k), where ℓ/k is a Galois splitting field of G.

Lemma 10.4. Let ϕ : P → X be a map of ℤΓ_{p}-modules. Then the cokernel of ϕ is finite of order prime to p if and only if ϕ̄ is surjective.

Proof. This is shown in [Me_{2}, proof of Theorem 4.3], and from a different perspective in [LMMR, Lemma 2.2]. □

Proposition 10.5. Let G be a central extension of a p-group F by a torus T, and let 0 ≤ r ≤ p − 2. The following statements are equivalent.

(a) ed(G; p) ≤ r (b) G is of multiplicative type and there is a p-presentation P → X(G)

whose kernel is isomorphic to the trivial ℤΓ_{p}-module ℤ^{r}.

Proof. Assuming (a), by Proposition 10.2 G is of multiplicative type. By [LMMR, Corollary 5.1] we know there is a p-presentation P → X(G), whose kernel L is of free of rank ed(G; p) ≤ p−2. By [AP, Satz], Γ_{p} must act trivially on L.

(b) ⇒ (a) follows from [LMMR, Corollary 5.1]. □

Proposition 10.6. Assume G is multiplicative type, with a p-presentation ϕ: P → X(G) whose kernel is isomorphic to the trivial ℤΓ_{p}-module ℤ^{r} for some r ≥ 0. Then ed(G; p) ≤ r and the following conditions are equivalent.

(a) ed(G; p) = r,

(b) ker ϕ is contained in pP + IP,

(c) ker ϕ is contained in

{∑_{λ∈Λ} a_{λ}λ ∈ P | a_{λ} ≡ 0 (mod p) ∀λ ∈ Λ^{Γ_{p}}} .

Here I denotes the augmentation ideal in ℤ[Γ_{p}] and Λ is a Γ_{p}-invariant basis of P.

Proof. (a) ⇔ (b): We have a commutative diagram:

\xymatrix{1 \ar[r] & ℤ^{r} \ar[r] \ar[d] & P \ar[r]^{ϕ} \ar[d] \ar[dr] & X(G) \ar[d] \\  & (ℤ/pℤ)^{r} \ar[r] & \overline{P} \ar[r]^{ϕ̄} & \overline{X(G)}}

with exact rows. By Lemma 10.4, ϕ̄ is a surjection. Therefore ker ϕ ⊆ pP +IP if and only if ϕ̄ is an isomorphism.

Write P as a direct sum P ≃ ⨁_{j=1}^{m} P_{j} of transitive permutation ℤΓ_{p}- modules P_{1}, …, P_{m}. Then P/(pP +IP) ≃ ⨁_{j=1}^{m} P_{j}/(pP_{j} +IP_{j}) ≃ (ℤ/pℤ)^{m}. If ϕ̄ is not an isomorphism we can replace P by the direct sum P̂ of only m−1 P_{j}’s without losing surjectivity of ϕ̄. The composition P̂ ↪ P → X(G) is then still a p-presentation of X(G), by Lemma 10.4. Hence ed(G; p) ≤ rank P̂ − dim G < rank P − dim G = r.

Conversely assume that ϕ̄ is an isomorphism. Let ψ: P′ → X(G) be a p-presentation such that ed(G; p) = rank ker ψ. Let d be the index [X(G) : ϕ(P)], which is finite and prime to p. Since the map X(G) → d·X(G), x ↦ dx is an isomorphism we may assume that the image of ψ is contained in ϕ(P). We have an exact sequence Hom_{ℤΓ_{p}}(P′, P) → Hom_{ℤΓ_{p}}(P′, ϕ(P)) → Ext_{ℤΓ_{p}}^{1}(P′, ℤ^{r}) and the last group is zero by [Lor, Lemma 2.5.1]. Therefore ψ = ϕ ∘ ψ′ for some map ψ′: P′ → P of ℤΓ_{p}-modules. Since ϕ̄ is an isomorphism and ψ is a p-presentation it follows from Lemma 10.4 that ψ′ is a p-presentation as well, and in particular that rank P′ ≥ rank P. Thus ed(G; p) = rank ker ψ ≥ rank ker ϕ = r.

(b) ⇔ (c): It suffices to show that P^{Γ_{p}} ∩ (pP + IP) consists precisely of the elements of P^{Γ_{p}} of the form ∑_{λ∈Λ} a_{λ}λ with a_{λ} ≡ 0 (mod p) for all λ ∈ Λ^{Γ_{p}}, for any permutation ℤΓ_{p}-module P. One easily reduces to the case where P is a transitive permutation module. Then P^{Γ_{p}} consists precisely of the ℤ-multiples of ∑_{λ∈Λ} λ and pP + IP are the elements ∑_{λ∈Λ} a_{λ}λ with ∑_{λ∈Λ} a_{λ} ≡ 0 (mod p). Thus for n ∈ ℤ the element n ∑_{λ∈Λ} λ lies in pP +IP iff n · |Λ| ≡ 0 (mod p), iff n ≡ 0 (mod p) or |Λ| ≡ 0 (mod p). Since |Λ| is a power of p the claim follows. □

Example 10.7. Let E be an étale algebra over a field k. It can be written as E = ℓ_{1}×⋯×ℓ_{m} with some separable field extensions ℓ_{i}/k. The kernel of the norm map n: R_{E/k}(𝔾_{m}) → 𝔾_{m} is denoted by R_{E/k}^{(1)}(𝔾_{m}). Let G = n^{−1}(μ_{p^{r}}) for some r ≥ 0. It is a group of multiplicative type fitting into an exact sequence

1 → R_{E/k}^{(1)}(𝔾_{m}) → G → μ_{p^{r}} → 1.

Let ℓ be a finite Galois extension of k containing ℓ_{1}, …, ℓ_{m} (so ℓ splits G), Γ = Gal(ℓ/k), Γ_{ℓ_{i}} = Gal(ℓ/ℓ_{i}) and Γ_{p} a p-Sylow subgroup of Γ. The character module of G has a p-presentation

P := ⨁_{i=1}^{m} ℤ[Γ/Γ_{ℓ_{i}}] → X(G),

with kernel generated by the element (p^{r}, ⋯, p^{r}) ∈ P. This element is fixed by Γ_{p}, so ed(G; p) ≤ 1. It satisfies condition (c) of Proposition 10.6 iff r > 0 or every Γ_{p}-set Γ/Γ_{ℓ_{i}} is fixed-point free. Note that Γ/Γ_{ℓ_{i}} has Γ_{p}-fixed points if and only if [ℓ_{i} : k] = |Γ/Γ_{ℓ_{i}}| is prime to p. We thus have

ed(G; p) = \{ \begin{matrix}0, & if r = 0 and [ℓ_{i} : k] is prime to p for some i, \\ 1, & otherwise. \end{matrix}

References

[AP] H. Abold, W. Plesken, Ein Sylowsatz für endliche p-Untergruppen von GL(n, ℤ), Math. Ann. 232 (1978), no. 2, 183–186.

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

[Bo] A. Borel Linear Algebraic Groups, Benjamin (1969).

[BS] A. Borel, J.-P. Serre, Théorèmes de finitude en cohomologie galoisienne, Comment. Math. Helv. 39 (1964), 111–164.

[Bou] N. Bourbaki, Algebra. II. Chapters 4–7. Translated from the French by P. M. Cohn and J. Howie. Elements of Mathematics. Springer-Verlag, Berlin, (1990).

[CGR] V. Chernousov, Ph. Gille, Z. Reichstein, Resolving G-torsors by abelian base extensions, J. Algebra 296 (2006), no. 2, 561–581.

[CTS_{1}] J.-L. Colliot-Thélène, J. J. Sansuc, R-équvalence sur les tores, Ann. scient. ENS

4, 10 (1977), 175–230.

[DG] M. Demazure, P. Gabriel, Groupes algébriques. Tome I, Masson & Cie, Paris; North-Holland Publishing Co., Amsterdam, 1970.

[EKM] R. Elman, N. Karpenko, A. Merkurjev: The algebraic and geometric theory of. quadratic forms. Amer. Math. Soc. Coll. Publ. 56, Providence, RI: Amer. Math. Soc. (2008).

[Ga] P. Gabriel, Construction de préschémas quotient, SGA 3, Exposé V (1963), in Réédition de SGA 3, available at

http://people.math.jussieu.fr/~polo/SGA3/.

[GR] Ph. Gille, Z. Reichstein, A lower bound on the essential dimension of a connected linear group, Comment. Math. Helv. 4, no. 1 (2009), 189–212.

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

[Lo] 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.

[LMMR] R. Lötscher, M. MacDonald, A. Meyer, Z. Reichstein, Essential dimension of algebraic tori, to appear in Crelle.

[Lor] M. Lorenz, Multiplicative Invariant Theory, Encyclopaedia of Mathematical Sciences, Springer (2005).

[LR] M. Lorenz, Z. Reichstein Lattices and parameter reduction in division algebras, MSRI preprint 2000-001, (2000), arXiv:math/0001026

[Ma] B. Margaux, Passage to the limit in non-abelian Čech cohomology. J. Lie Theory 17, no. 3 (2007), 591–596.

[Me_{1}] 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.

[Me_{2}] A. Merkurjev, A lower bound on the essential dimension of simple algebras,

Algebra and Number Theory, 4, no. 8, (2010), 1055–1076.

[MR_{1}] A. Meyer, Z. Reichstein, The essential dimension of the normalizer of a maximal torus in the projective linear group, Algebra and Number Theory, 3, no. 4 (2009), 467–487.

[Na] M. Nagata, Complete reducibility of rational representations of a matric group, J. Math. Kyoto Univ. 1 (1961–1962), 87–99.

[PV] V. L. Popov, E. B. Vinberg, Invariant Theory, in: Algebraic Geometry IV, Encyclopaedia of Mathematical Sciences, Vol. 55, Springer-Verlag, Berlin, 1994, pp. 123–284.

[Re_{1}] Z. Reichstein, On the notion of essential dimension for algebraic groups, Trans-

formation Groups, 5, 3 (2000), 265-304.

[Re_{2}] Z. Reichstein, Essential Dimension, Proceedings of the International Congress

of Mathematicians, Vol. II, 162–188, Hindustan Book Agency, New Delhi, 2010.

[Sch_{1}] H.-J. Schneider, Zerlegbare Erweiterungen affiner Gruppen J. Algebra 66, no. 2

(1980), 569–593.

[Sch_{2}] H.-J. Schneider, Decomposable Extensions of Affine Groups, in Lecture Notes in

Mathematics 795, Springer Berlin/Heidelberg (1980), 98–115.

[Sch_{3}] H.-J. Schneider, Restriktion und Corestriktion für algebraische Gruppen J. Al-

gebra, 68, no. 1 (1981), 177–189.

[Se] J.-P. Serre, Galois cohomology. Springer Monographs in Mathematics. Springer-Verlag, Berlin, 2002.

[Ta] J. Tate, Finite flat group schemes. Modular forms and Fermat’s last theorem (Boston, MA, 1995), 121–154, Springer, New York, 1997.

Mathematisches Institut, Universität München, 80333 München, Germany

E-mail address: roland.loetscher@mathematik.uni-muenchen.de

Department of Mathematics, University of British Columbia, Vancouver V6T1Z2, Canada

E-mail address: mlm@math.ubc.ca

Départment de Mathématiques, Université Paris-Sud 11, 91405 Orsay, France

E-mail address: aurel.meyer@math.u-psud.fr

Department of Mathematics, University of British Columbia, Vancouver V6T1Z2, Canada

E-mail address: reichst@math.ubc.ca