ESSENTIAL p-DIMENSION OF PGL_{n}

ANTHONY RUOZZI

1. Introduction

k a fixed base field of characteristic p. All fields mentioned in this paper are assumed to contain k.

Consider any functor F : Fields/k → Set. We say that an element a ∈ F(K) is defined over k ⊂ K_{0} ⊂ K if it is in the image of the map F (K_{0}) → F (K). The essential dimension, ed_{k}(a), is the least transcendence degree/k of a field of definition for a. The essential dimension of F, ed_{k}(F) = sup{ed_{k}(a)} where the supremum is taken over all a ∈ K for all field extensions K/k. The basic properties of this definition are outlined in [BF].

In this paper, we will be interested in the slightly simpler computation of essential p-dimension. Here, we are allowed some extra flexibility: ed_{k}(a; p) is the minimum essential dimension of the image of a in F (L) over all L/K finite prime to p extensions. As above, we define ed_{k}(F; p) as the supremum of the essential p-dimensions over all elements and fields, a ∈ K.

A field F/k is called p-closed if every finite extension of F has degree prime to p. For limit-preserving functors, the essential p-dimension of any element can be computed over a p-closure [LMMR Lemma 3.3], so we can always assume that our fields F are p-closed.

A natural transformation of functors F → G will be called psurjective if for any K/k, there is a finite extension L/K of degree prime to p such that F (L) G(L). More specifically, if F/k is p-closed and F (F) G(F), then the map is p-surjective and ed_{k}(F; p) ≥ ed_{k}(G; p) [LMMR Prop 3.4].

Of particular interest are the functors H^{1}(−, G) for an algebraic group G/k. For ease of notation, the essential dimension of such functors will be denoted ed(G). We will be studying the functor H^{1}(F, PGL_{n}) which classifies central simple algebras/F of degree n. In what follows, this functor will be denoted by Alg_{n}(−). Its essential dimension gives the least number of parameters needed to define a “generic” central simple algebra of degree n.

Because we can apply prmiary decomposition to central simple algebras, the computation of ed(Alg_{n}(−); p) reduces to a computation of ed(Alg_{p^{s}}(−); p) where p^{s} is the largest power of p dividing n. It is well-known that ed(Alg_{p}; p) = 2; cf. [R2]. Recently, Meyer and Reichstein gave an upper bound for s ≥ 2 [MR2 Theorem 1.1]:

ed(Alg_{p^{s}}; p) ≤ 2p^{2s−2} − p^{s} + 1

and conjectured that this bound is sharp. The goal of this paper is to further strengthen this result.

Theorem 1.1. ed(Alg_{p^{s}}) ≤ p^{2s−2} + 1 for s ≥ 2.

For s = 2, Merkurjev showed that this bound is sharp [M], so we can consider s ≥ 3.

2. Essential Dimension and Tori

Throughout this section, let F/k be an arbitrary field extension. Fix a finite group G and a finite G-set, X, of n elements. Consider the augmentation exact sequence of G-modules:

0 → I → [X] → → 0.

Construct any resolution of I

0 → M → P → I → 0

where P is a permutation module. Fixing bases and using the usual anti-equivalence of categories, this sequence corresponds to an exact sequence of algebraic tori split over F:

1 → T → U → S → 1

where U = GL_{1}(E) is the diagonal subgroup in GL_{F}(E) for the split étale algebra E/F corresponding to P. Thus we have a faithful representation T ↪ GL_{F}(E).

Since T acts on E via this representation, G acts on E by algebra automorphisms, and the above representation is G-equivariant, this extends to a representation T G → GL_{F}(E). We will construct an upper bound for the essential dimension using the following important result:

Theorem 2.1. If G acts faithfully on M in the resolution of I constructed above, then ed(T G) ≤ rank(P) − rank(I) = rank(M).

Proof.This result is a combination of [R Theorem 3.4] and [MR1 Lemma 3.3].

Since we a interested in essential p-dimension, the following result will also be useful:

Proposition 2.2. Let H = Syl_{p}(G). Then ed(T G; p) = ed(T H; p).

Proof.This is just a special case of [MR1 Lemma 4.1].

3. Division Algebras

In what follows we are interested in computing the essential pdimension, so we can assume that F ⊃ k is p-closed. Let D/F be a central division algebra of degree n = p^{s} with s ≥ 2. The following results will be useful:

Proposition 3.1. D contains degree p cyclic extension of F.

Proof.See [RS Prop 1.1].

Theorem 3.2. Let L_{1} ⊂ D be a degree p cyclic extension/F. Then there is another degree p cyclic exentsion L_{2}/F contained in D such that L_{1}L_{2} ⊂ D is a bicyclic extension.

Proof.Fix a generator < σ >= Gal(L_{1}/F). The Skolem-Noether theorem gives an element y ∈ D^{×} such that yxy^{−1} = σ(x) for all x ∈ L_{1}. There are two possibilities:

y^{p} ∈ F: In this case, y^{p} = a defines a cyclic algebra/F B = (L_{1}, a). By the double centralizer theorem C_{D}B is division algebra/F of degree ≥ p. Thus, C_{D}B ⊂ D has a cyclic subfield L_{2}. Since L_{1}⊗L_{2} ⊂ B⊗C_{D}B ≃ D is a subfield, L_{1} ⊗ L_{2} ≃ L_{1}L_{2} is bicyclic as desired.

y^{p} F: Let K = F(y^{p}). By the proposition, K contains a cyclic subfield L_{2}. Any element of L_{2} commutes with F(y). Then x ∈ L_{1} ∩ L_{2} commuting with y implies that σ acts trivially, so L_{1} and L_{2} are disjoint and give L_{1}L_{2} bicyclic.

Corollary 3.3. For A any central simple algebra/F of degree p^{s} ≥ p^{2} and L_{1} ⊂ A an étale sub-algebra of degree p, there is a maximal étale sub-algebra K ⊂ A that can be written as K = L_{1} ⊗_{F} L_{2} for L_{2}/F an étale algebra of dimension p^{s−1}.

Proof.First, consider the case where A is a division algebra. By the theorem, A contains two distinct degree p cyclic extensions L_{1} and L over F. Proceed by induction on s.

If s = 2 , then taking L_{2} = L we are done. Otherwise, assume we have the result for any division algebra of degree p^{s−1} and any degree p subfield L′. The centralizer C_{A}L is a division algebra over L_{1} of degree p^{s−1}. By definition it contains the degree p subfield L_{1}L/L, so by induction hypothesis, C_{A}L has subfield L_{2}/L disjoint from L_{1}L of degree p^{s−2}.

Since L_{2} ∩ L_{1}L = L and L is disjoint from L_{1}, L_{2} ∩ L_{1} = F. It follows that L_{2}/F is a degree p^{s−1} extension disjoint from L_{1} and L_{1} ⊗ L_{2} ⊂ A.

Suppose A is not a division algebra. If A is split, the result is immediate. Otherwise, choose a division algebra D ∼ A. deg(D) = p^{t} ≤ p^{s} = deg(A). If t = 1, then any maximal subfield K ⊂ D gives the desired étale algebra K ⊗ F^{×p^{s−1}} ⊂ A, so suppose t ≥ 2. By the above argument, D has a subfield K = L_{1}L_{2} ≃ L_{1} ⊗ L_{2} where [L_{1} : F] = p and [L_{2} : F] = p^{t−1}. Set L = K^{×p^{s−t}} an étale sub-algebra of B. Since L has dimension p^{s}, it is maximal. But L ≃ L_{1} ⊗ L_{2}^{×p^{s−t}}, so L_{1} and L_{2}^{×p^{s−t}} are étale algebras of the desired degree.

Consider the functor H^{1}(F, S_{n}) which classifies n-dimensional étale algebras up to isomorphism [KMRT 29.9]. Let G = S_{p}×S_{p^{s−1}}. We have the usual isomorphism

H^{1}(F, S_{p}) × H^{1}(F, S_{p^{s−1}}) → H^{1}(F, G).

Converting this to the laguange of algebras, H^{1}(F, G) can be identified as pairs (L_{1}, L_{2}) ofétale algebras of dimensions p and p^{s−1}, respectively. Under the natural inclusion

H^{1}(F, G) → H^{1}(F, S_{p^{s}}),

the image of such a pair is theétale algebra L_{1}⊗L_{2}. Let K = L_{1}⊗L_{2} ⊂ A be given for a central simple algebra A as in the corollary. Using these identifications, any such K ⊂ A can be viewed as an element of H^{1}(F, G).

Recall the notation of section 2. Consider the split torus T for G as above and X a G-set of p^{s} elements. For its cohomology group, we have a disjoint union of fibers

H^{1}(F, T G) = ∐_{γ∈H^{1}(F,G)}H^{1}(F, T_{γ})/G_{γ}^{Γ}

where T_{γ} denotes T with action twisted by the cocycle γ and Γ = Gal(F^{sep}/F) [KMRT 28.C]. T_{γ} is also a torus with character module I. In particular, if X corresponds to a p^{s}-dimensional étale algebra N representated by γ then the usual anti-equivalence of categories gives an exact sequence

1 → _{m} → R_{N/F}( _{m,N}) → T_{γ} → 1.

Passing to cohomology and applying Hilbert theorem 90 gives

1 → H^{1}(F, T_{γ}) → H^{2}(F, _{m}) → H^{2}(F, R_{N/F}( _{m}))

showing that H^{1}(F, T_{γ}) ≃ Br(N/F). Now, G_{γ}^{Γ} acts on G_{m} trivially, so given g ∈ G_{γ}^{Γ}, we get a diagram:

1 −−⟶ H^{1}(F, T_{γ}) −−⟶ H^{2}(F, _{m})
↓^{g}  ∥
1 −−⟶ H^{1}(F, T_{γ}) −−⟶ H^{2}(F, _{m})

and the commutativity implies that the action of g must be trivial for all g ∈ G_{γ}^{Γ}. Combining these observations with the above,

H^{1}(F, T G) = ∐_{N} Br(N/F),

and thus we can define a map ϕ_{G}(F) : H^{1}(F, T G) → Alg_{p^{s}}(F) which sends [B] ∈ Br(N/F) to the unique (up to isomorphism) C ∼ B with degree C = p^{s}. Since any A ∈ Alg_{p^{s}} is split over K = L_{1} ⊗ L_{2}, it is in the image of this map. We have proven:

Proposition 3.4. ϕ_{G} is p-surjective.

Let G_{s} = Syl_{p}(S_{p} × S_{p^{s−1}}) = Syl_{p}(S_{p}) × Syl_{p}(S_{p^{s−1}}) := Σ_{1} × Σ_{s−1}. Using Proposition 2.2 and the property of p-surjective maps stated in the introduction,

ed_{k}(T G_{s}; p) = ed_{k}(T (S_{p} × S_{p^{s−1}}); p) ≥ ed_{k}(Alg_{p^{s}}(−); p).

We can therefore reduce to the computation of the essential dimension of T G_{s}. The calculations for this case will be done in the next section.

4. Construction

We will produce an upper bound for the essential dimension of T G_{s}. By section 2, this requires finding a faithful G_{s}-module, M, in a resolution of I of smallest rank. In what follows, we denote G_{s} = Σ_{s−1} × /p = Σ_{s−1}× < σ >, where as above Σ_{s} = Syl_{p}(S_{p^{s}}).

First observe that G_{s} ⊂ S_{p^{s}} acts by permutation on a set X_{s} of p^{s} elements. This action can be descrbibed as an action on p blocks of p^{s−1} elements where σ cyclically permutes the blocks and Σ_{s−1} acts as usual on a block of p^{s−1} elements. In particular, the action is transitive, so if H_{s} = Stab(x) for some x ∈ X_{s}, then X_{s} ≃ G_{s}/H_{s} as G_{s}-sets.

Begin with the case s = 2. G_{2} = Σ_{1} × Σ_{1} =< τ_{1} > × < σ >. No non-trivial element of G_{2} fixes x ∈ X_{2}, so H_{2} = 1. Identifying X_{2} ≃ G_{2}, I is generated by σ − 1 and τ_{1} − 1 as a G_{2}-module. Since G_{2} is abelian, the map

[G_{2}] ⊕ [G_{2}] → I

defined by sending a generator of the first term to σ−1 and a generator of the second to τ_{1} − 1 is a well-defined G_{2}-module homomorphism. It is surjective by definition. The kernel of this map, M, has rank:

rank(M) = 2 rank( [G_{2}]) − rank(I) = 2p^{2} − (p^{2} − 1) = p^{2} + 1.

For the general case, Σ_{s−1} = (Σ_{s−2})^{p} /p = (Σ_{s−2})^{p} < τ_{s−1} >. We will show inductively that H_{s} = H_{s−1} × (Σ_{s−2})^{p−1} and G_{s} =< σ, τ_{s−1}, H_{s} >.

The case s = 2 was verified above. Suppose the formula holds for s−1 ≥ 2. Σ_{s} acts on a set X_{s} of p^{s} elements, and H_{s} = Stab(x). As above, this action can be thought of as an action on p blocks of p^{s−1} elements. On any of these blocks, Σ_{s−1} ⊂ G_{s} acts as usual by considering it as a collection of p blocks of p^{s−2} elements. Therefore, we see that to be the stabilizer of x is to stabilize the block containing x and allow the others to be permuted freely. That is, H_{s} = H_{s−1} × (Σ_{s−2})^{p−1}.

Now, τ_{s−2} ∈< τ_{s−1}, (Σ_{s−2})^{p−1} > and since < τ_{s−2}, H_{s−1} >= Σ_{s−2} by assumption, we can conclude that G_{s} =< σ, τ_{s−1}, H_{s} >. By an easy argument, I is then generated by σx − x and τ_{s−1}x − x; cf. [MR2 proof of Theorem 4.1]. Setting H′_{s} = τ_{s−1}Hτ_{s−1}^{−1} ∩ H ≃ H_{s−1} × H_{s−1} × (Σ_{s−2})^{p−2}, we can define a map as above

[G_{s}/H_{s}] ⊕ [G_{s}/H′_{s}] → I

by sending a generator of the first to σx − x and a generator of the second to τ_{s−1}x − x. This is well-defined since H′_{s}is exactly the subset of G_{s} that fixes τ_{s−1}x − x. We then have constructed a surjective G_{s}-module map with rank(M) =

rank( [G_{s}/H_{s}])+rank( [G_{s}/H′_{s}])−rank(I) = p^{s}+p^{s+(s−2)}−(p^{s}−1) = p^{2s−2}+1.

5. Conclusions

We now complete the proof of the theorem stated in the introduction. Recall that we are assuming that the base field k has characteristic p.

Theorem 5.1. ed(Alg_{p^{s}}; p) ≤ p^{2s−2} + 1 for s ≥ 2.

Proof.For F p-closed, the construction in the previous section produced a G_{s}-module M of rank p^{2s−2} + 1. By section 2, it remains to show that the G-action on M is faithful (cf. [MR Lemma 3.2] for a more general argument). Faithfulness can be checked over , so we have the split exact sequences:

0 → I ⊗ → [G_{s}/H_{s}] → → 0
0 → N → [G_{s}/H′_{s}] → [G_{s}/H_{s}] → 0
0 → M ⊗ → [G_{s}/H_{s}] ⊕ [G_{s}/H′_{s}] → I ⊗ → 0

Combining these together,

(M ⊗ ) ⊕ (I ⊗ ) ≃ [G_{s}/H_{s}] ⊕ [G_{s}/H′_{s}]
≃ [G_{s}/H_{s}] ⊕ N ⊕ [G_{s}/H_{s}]
≃ [G_{s}/H_{s}] ⊕ N ⊕ ⊕ (I ⊗ ).

Therefore, [G_{s}/H_{s}] is a direct summand of M ⊗ , so it suffices to check that the G_{s} action on [G_{s}/H_{s}] is faithful. However, if the coset gH_{s} is fixed by every element in G_{s}, then g ∈ ⋂ _{g_{s}∈G_{s}} g_{s}H_{s}g_{s}^{−1}. A quick induction argument shows that for all s ≥ 2 this group is trivial.

Since then the action is faithful, we have the bound:

p^{2s−2} + 1 ≥ ed(T G_{s}; p) ≥ ed(Alg_{p^{s}}; p).

References

[A] A. Albert, Structure of algebras, Colluquim Publications, Vol. 24, Amer. Math. Soc., Providence, 1961.

[BF]G. Berhuy, G. Favi, Essential dimension: a functorial point of view (after A.

Merkurjev), Doc. Math. 8 (2003), 279-330. [KMRT]M. Knus, A. Merkurjev, M. Rost, J.P. Tignol, The book of involutions, Col-

luquim Publications, Vol. 44, Amer. Math. Soc., Providence, 1998.

[LRRS] M. Lorenz, Z. Reichstein, L.H. Rowen, D.J. Saltman, Fields of definition for division algebras, J. London Math. Soc. (2) 68 (2003), no. 3, 651-670.

[LMMR] . Lotscher, M. MacDonald, A. Meyer, Z. Riechstein, Essential p-dimension of algebraic tori, preprint, www.mathematik.uni-bielefeld.de/LAG/man/363.ps.gz.

[M] A. Merkurjev, Essential p-dimension of PGL_{p^{2}}, J. Amer. Math. Soc. 23 (2010), 693-712.

[MR1] A. Meyer, Z. Reichstein, The essential dimension of the normalizer of a maximal torus in the projective linear group, Algebra and Number Theory, 3 (2009), no.4, 467-487.

[MR2] A. Meyer, Z. Reichstein, An upper bound on the essential dimension of a central simple algebra, to appear in the issue of Journal of Algebra dedicated to Corrado de Concini’s 60th birthday.

[R] Z. Reichstein, On the notion of essential dimension for algebraic groups, Transform. Groups 5 (2000), no. 3, 265-304.

[R2] Z. Reichstein, Essential dimension: a survey, lecture notes for the workshop on essential and canonical dimension in Lens, France, http://www.math.ubc.ca/ reichst/lens-notes6-27-8.pdf.

[RS] L.H. Rowen, D.J. Saltman, Prime to p extensions of division algebras, Israel J. math. 78 (1992), no. 2-3, 197-207.

[S] D.J. Saltman, Lectures on division algebras, CMBS Regional Conference Series in Mathematics, No. 94, Amer. Math. Soc., Providence, 1999.

Department of Mathematics, University of California, Los Angeles, CA 90095-1555, USA

  E-mail address: aruozzi@math.ucla.edu