Essential dimension of central simple algebras

c

2012 by Institut Mittag-Leffler. All rights reserved

Essential dimension of central simple algebras

by

Sanghoon Baek

Korea Advanced Institute of Science and Technology Daejeon, Republic of Korea

Alexander S. Merkurjev

University of California, Los Angeles Los Angeles, CA, U.S.A.

1. Introduction

LetF:Fields/F!Sets be a functor from the categoryFields/F of field extensions overF to the categorySetsof sets. LetE∈Fields/F andK⊂Ebe a subfield overF. An element α∈F(E) is said to bedefined over K(andK is called afield of definition ofα) if there exists an elementβ∈F(K) such thatαis the image ofβ under the mapF(K)!F(E).

The essential dimension of α, denoted by ed^F(α), is the least transcendence degree tr deg_F(K) over all fields of definitionK ofα. Theessential dimension ofF is

ed(F) = sup{ed^F(α)},

where the supremum is taken over all fieldsE∈Fields/F and all α∈F(E) (see [3, Defi- nition 1.2] or [7,§1]). Informally, the essential dimension ofF is the smallest number of algebraically independent parameters required to defineF and may be thought of as a measure of complexity ofF.

Let p be a prime integer. The essential p-dimension of α, denoted by ed^F_p(α), is defined as the minimum of ed^F(α_E⁰), where E⁰ ranges over all field extensions of E of degree prime top. Theessential p-dimension ofF is

ed_p(F) = sup{ed^F_p(α)},

where the supremum ranges over all fieldsE∈Fields/F and all α∈F(E). By definition, ed(F)>edp(F) for allp.

For every integern>1, a divisormofnand any field extensionE/F, letAlg_n,m(E) denote the set of isomorphism classes of central simpleE-algebras of degree nand ex- ponent dividingm. Equivalently,Alg_n,m(E) is the subset of them-torsion part Brm(E)

The first author was supported by the Beckenbach Dissertation Fellowship at the University of California, Los Angeles. The second author was supported by the NSF grant DMS #0652316.

of the Brauer group ofEconsisting of all elementsasuch that ind(a) dividesn. In par- ticular, if n=m, then Alg_n(E):=Alg_n,n(E) is the set of isomorphism classes of central simpleE-algebras of degreen. We viewAlg_n,mandAlg_n as functorsFields/F!Sets.

In the present paper we give upper and lower bounds for edp(Alg_n,m) for a prime integer pdifferent from char(F). Letp^r (resp.p^s) be the largest power of pdividingn (resp.m). Then edp(Alg_n,m)=edp(Alg_pr,p^s) and edp(Alg_n)=edp(Alg_pr) (see§6). Thus, we may assume thatnandmare thep-powersp^randp^s, respectively.

Using structure theorems on central simple algebras, we can compute the essential (p-)dimension of Alg_pr,p^s for certain small values of r, s and p as follows. Since every central simple algebra A of degree p is cyclic over a finite extension of fields of degree prime to p, A can be given by two parameters (see §2.1). In fact, edp(Alg_p)=2 by [11, Lemma 8.5.7].

By Albert’s theorem, every algebra inAlg4,2 is biquaternion and hence can be given by four parameters. In fact, ed(Alg4,2)=ed2(Alg4,2)=4 (see Remark8.2).

Upper and lower bounds for edp(Alg_pr) can be found in [14] and [9], respectively.

In this paper (see §6 and §7), we establish the following upper and lower bounds for edp(Algp^r,p^s) that match the bounds in the caser=sgiven in [14] and [9].

Theorem 1.1. Let F be a field and p be a prime integer different from char(F).

Then, for any integers r>2 and swith 16s6r, p^2r−2+p^r−s>edp(Alg_pr,p^s)>

(r−1)2^r−1, if p= 2and s= 1, (r−1)p^r+p^r−s, otherwise.

Corollary1.2. (Cf. [8]) Let pbe a prime integer and F be a field of characteristic different from p. Then

edp(Alg_p2) =p²+1.

Corollary 1.3. Let p be an odd prime integer and F be a field of characteristic different from p. Then

ed_p(Alg_p2,p) =p²+p.

The corollary recovers a result in [20] that, for podd, there exists a central simple algebra of degreep²and exponentpover a fieldF which is not decomposable as a tensor product of two algebras of degree pover any finite extension ofF of degree prime top.

Indeed, if every central simple algebra of degreep²and exponentpwere decomposable, then the essentialp-dimension ofAlg_p2,pwould be at most 4.

Corollary 1.4. Let F be a field of characteristic different from 2. Then ed2(Alg_8,2) = ed(Alg_8,2) = 8.

The proof is given in§8. The corollary recovers a result in [1] that there is a central simple algebra of degree 8 and exponent 2 over a fieldF which is not decomposable as a tensor product of three quaternion algebras over any finite extension ofF of degree prime to p. Indeed, if every central simple algebra of degree 8 and exponent 2 were decomposable, then the essential 2-dimension ofAlg8,2 would be at most 6.

The proof of the main theorem splits into two steps. In the first step we relate the essentialp-dimensions ofAlgp^r,p^s and of a certain torusS^Φby means of the iterated degeneration. In the second step, we apply [6, Theorem 1.1] to compute the essential p-dimension ofS^Φ.

2. Character, Brauer group and algebraic tori 2.1. Character and Brauer group

LetF be a field,Fsepbe a separable closure ofF and ΓF=Gal(Fsep/F). For a (discrete) ΓF-moduleM, we writeHⁿ(F, M) for the Galois cohomology group Hⁿ(ΓF, M).

If S is an algebraic group overF, we let H¹(F, S) denote the set H¹(ΓF, S(Fsep)) (see [17]).

Thecharacter group of F is defined by

Ch(F) := Homcont(ΓF,Q/Z) =H¹(F,Q/Z)'H²(F,Z).

Then-torsion character group Chn(F) is identified with H¹(F,Z/nZ). For a character χ∈Ch(F), setF(χ)=(Fsep)^Ker(χ). The field extensionF(χ)/F is cyclic of degree ord(χ).

If Ψ⊂Ch(F) is a finite subgroup, we set

F(Ψ) := (F_sep)^{Tχ∈ΨKer(χ)}.

The Galois group G=Gal(F(Ψ)/F) is abelian and Ψ is canonically isomorphic to the character group Hom(G,Q/Z) ofG. Note that a characterη∈Ch(F) is trivial overF(Ψ) if and only ifη∈Ψ.

IfK/F is a field extension, we writeK(Ψ) forK(Ψ_K), where Ψ_K is the image of Ψ under the natural homomorphism Ch(F)!Ch(K).

We write Br(F) for the Brauer groupH²(F, F_sep^× ) of F. IfL/F is a field extension andα∈Br(F), we letαL denote the image ofαunder the natural map Br(F)!Br(L).

We say thatL is a splitting field of αif αL=0. The index ind(α) ofα is the smallest degree of a splitting field of α. The exponent exp(α) is the order ofα in Br(F). The integer exp(α) divides ind(α).

LetA be a central simpleF-algebra. Thedegree ofAis the square root of dim(A).

We write [A] for the class ofAin Br(F). The index of [A] divides deg(A). If α∈Br(F) and n is a positive multiple of ind(α), then there is a central simple F-algebra A of degreenwith [A]=α.

The cup-product

Ch(F)⊗F^×=H²(F,Z)⊗H⁰(F, F_sep^× )−!H²(F, F_sep^× ) = Br(F)

takesχ⊗bto the class χ∪(b) in Br(F) that is split byF(χ). A classα∈Br(F) is called n-cyclicifα=χ∪(b) for a characterχwithnχ=0. Such classes belong to Br_n(F). Ifnis prime to char(F), then Brn(F)'H²(F, µn), where µn is the ΓF-module of allnth roots of unity inFsep.

Let n be prime to char(F) and suppose that F contains a primitive nth root of unityξ. For anya∈F^×, letχa∈Ch(F) be the unique character with values in

(1/n)Z Z

⊂Q Z such that

γ(a^1/n) =ξ^nχa(γ)a^1/n

for all γ∈Gal(Fsep/F). We write (a, b)n for χa∪(b). The symbol (a, b)n satisfies the following properties (see [16, Chapter XIV, Proposition 4]):

• (a, b)n+(a⁰, b)n=(aa⁰, b)n;

• (a, b)n=−(b, a)n;

• (a,−a)n=0.

For a finite subgroup Φ⊂Ch(F) write Br(F(Φ)/F)dec for thesubgroup of decompos- able elements in Br(F(Φ)/F) generated by the elementsχ∪(a) for allχ∈Φ anda∈F^×. Theindecomposable relative Brauer group Br(F(Φ)/F)_ind is the factor group

Br(F(Φ)/F) Br(F(Φ)/F)dec

.

Similarly, if Φ⊂Chn(F) for somen, then Brn(F(Φ)/F)indis theindecomposablen-torsion relative Brauer group defined as the factor group

Brn(F(Φ)/F) Br(F(Φ)/F)_dec.

LetEbe a complete field with respect to a discrete valuationvandKbe its residue field. Let p be a prime integer different from char(K). There is a natural injective homomorphism Ch(K){p}!Ch(E){p} of the p-primary components of the character

groups that identifies Ch(K){p}with the character group of an unramified field extension of E. For a character χ∈Ch(K){p}, we write χb for the corresponding character in Ch(E){p}.

If in addition E is an extension of a fieldF such thatv is trivial onF, thenK is a field extension ofF and the composition

Ch(F){p} −!Ch(K){p} −!Ch(E){p}

coincides with the canonical homomorphism for the field extensionE/F. By [4,§7.9], there is an exact sequence

0−!Br(K){p}−−ⁱ!Br(E){p}−−−^∂v!Ch(K){p} −!0.

If α∈Br(K){p}, then we write αb for the element i(α) in Br(E){p}. For example, if α=χ∪(¯u) for some χ∈Ch(K){p} and a unit u∈E, then α=b χ∪(u). In the caseb F contains a primitiventh root of unity, where nis a power ofp, ifα=(¯a,¯b)_n witha and bunits inE, thenα=(a, b)b _n.

If β=α+(b χ∪(x)) for an elementb α∈Br(K){p},χ∈Ch(K){p} andx∈E^× such that v(x) is not divisible byp, we have (cf. [18, Proposition 2.4])

ind(β) = ind(α_K(χ)) ord(χ). (2.1)

2.2. Representations of algebraic tori

Let T be an algebraic torus over a field F and L/F be a finite Galois splitting field for T with Galois group G. The group G is called a decomposition group of T. The character group T^∗:=HomL(TL,Gm,L) has the structure of aG-module. The torus T can be reconstructed fromT^∗by

T= Spec(L[T^∗]^G).

A torusP over F split byLis called quasi-split if P^∗ is apermutation G-module, i.e., if there exists aG-invariant Z-basis X forP^∗. The torus P is canonically isomorphic to the group of invertible elements of the ´etaleF-algebraA=Map_G(X, L). The torusP acts linearly by multiplication on the vector spaceAoverF makingAa faithfulP-space (a linear representation ofP) of dimension dim(P). It follows that a homomorphism of algebraic tori ν:T!P, with P being a quasi-split torus, yields a linear representation ofT of dimension dim(P) that is faithful if ν is injective.

LetP be a split torus overF, andP^∗ be its character group. As above, the choice of aZ-basisX forP^∗ allows us to identifyP with the group of invertible elements of a

split ´etaleF-algebra Aand makesA into a faithful P-space over F. Let ν:T!P be a homomorphism of split tori over F. Suppose a finite groupG acts onT andP by tori automorphisms so thatν is aG-equivariant homomorphism. Then the mapν^∗:P^∗!T^∗ is a G-module homomorphism. Suppose that there is aG-invariantZ-basisX forP^∗, i.e., P^∗ is a permutation. Then Gacts on the algebraA byF-algebra automorphisms.

The torus T acts linearly onA viaν. It follows that the semidirect productToGacts linearly onAmakingAinto a (ToG)-space.

LetL be a Galois G-algebra over F (for example, L/F is a Galois field extension with Galois groupG). Thenγ: SpecL!SpecF is aG-torsor. Twisting the split torusT by the torsorγ, we get the torus

T_γ=T×SpecL

G = Spec(L[T^∗]^G), which is split byL, andT_γ^∗ is isomorphic toT^∗ as G-modules.

By [5, Proposition 28.11], the fiber ofH¹(F, ToG)!H¹(F, G) over the class ofγis naturally bijective to the orbit set of the groupGγ(F) inH¹(F, Tγ), i.e.,

H¹(F, ToG)'aH¹(F, T_γ)

Gγ(F) , (2.2)

where the coproduct is taken over all [γ]∈H¹(F, G).

2.3. Generic torsors

LetTbe an algebraic torus split by a finite Galois field extensionL/F withG=Gal(L/F).

Let P be a quasi-split torus split by L and containing T as a subgroup. Set S=P/T. Then the canonical homomorphismγ:P!S is aT-torsor.

Proposition 2.1. The T-torsor γ is generic, i.e., for every field extension K/F with K infinite,every T-torsor γ⁰:E!SpecK and every non-empty open subset W⊂S, there is a morphism s: SpecK!S over F with Im(s)⊂W such that the T-torsors γ⁰ and s^∗(γ)=γ×SSpecK over K are isomorphic.

Proof. AsP is quasi-split, the last term in the exact sequence P(K)−−^γK!S(K)−−^δ!H¹(K, T)−!H¹(K, P)

is trivial. Then there iss∈S(K) withδ(s)=[γ⁰]. AsK is infinite, theK-points ofP are dense in P and we can modify s by an element in the image of γK so that s∈W(K), i.e., Im(s)⊂W. Then the T-torsor γ⁰ over K with the class δ(s) satisfies the required property.

2.4. The algebraic tori P^Φ, S^Φ,T^Φ, U^Φ and V^Φ

Let 16s6r be integers, p be a prime integer, F be a field with char(F)6=p, Φ be a subgroup of Chp(F) of rankrandG=Gal(F(Φ)/F). Choose a basisχ1, χ2, ..., χrfor Φ.

Eachχi can be viewed as a character of G, i.e., as a homomorphism χi:G!Q/Z. Let σ1, σ2, ..., σr be the dual basis forG, i.e.,

χi(σj) =

1/p+Z, ifi=j, 0, otherwise.

Let R be the group ring Z[G]. Consider the surjectiveG-module homomorphism

¯

ε:R!Z/p^sZ, defined by ¯ε(x)=ε(x)+p^sZ, whereε:R!Zis theaugmentation homomor- phism given byε(%)=1 for all%∈G. SetJ:=Ker(¯ε). Thus, we have an exact sequence

0−!J−!R−−^ε¯!Z/p^sZ−!0.

Moreover, theG-moduleJ is generated byIandp^s, whereI:=Ker(ε) is theaugmentation ideal inR.

Consider the G-module homomorphism h:R^r+1!R taking the ith canonical basis elementei toσi−1 for 16i6rander+1 top^s. The image ofhcoincides withJ.

Set N:=Ker(h) and write wi=1+σi+σ²_i+...+σ_i^p−1∈R for 16i6r. Consider the following elements inN:

eij= (σi−1)ej−(σj−1)ei, fi=wiei and gi=−p^sei+(σi−1)er+1

for all 16i, j6r.

Lemma 2.2. The G-module N is generated by eij,fi and gi.

Proof. Consider the surjective morphismk:R^r!Itakingeitoσi−1 for 16i6rand setN⁰:=Ker(k). Then we have the commutative diagram

N _⁰

 //R^r _

k //I _

N

 //R^r+1

h ////J

ε⁰

I  //R ^ε ////Z,

whereR^r+1!Ris the projection morphism to the last coordinate andε⁰:J!Zis given byε⁰(j)=ε(j)/p^s.

By the exactness of the first column of the diagram,N is generated by N⁰ and the liftingsgi ofσi−1 inN. The module N⁰ is generated byeij andfi, by [9, Lemma 3.4].

This completes the proof.

Letε_i:R^r+1!Zbe theith projection followed by the augmentation mapε. It follows from Lemma2.2thatεi(N)=pZfor every 16i6r. Moreover, theG-homomorphism

q:N−!Z^r, x7−!ε1(x)

p , ...,εr(x) p

, is surjective. SetM:=Ker(q) andQ:=R^r+1/M.

LetP^Φ,S^Φ,T^Φ,U^ΦandV^Φbe the algebraic tori overF split by the field extension F(Φ)/F such that

(P^Φ)^∗=R^r+1, (S^Φ)^∗=Q, (T^Φ)^∗=M, (U^Φ)^∗=J and (V^Φ)^∗=N.

The diagram of homomorphisms ofG-modules with exact columns and rows M _

M _

N

q

 //R^r+1

h ////J

Z^r  //Q ////J

(2.3)

yields the following diagram of homomorphisms of the tori T^Φ T^Φ

V^Φ

OOOO

P^Φ

OOOO

oooo γ oo _?U^Φ

G^rm

?

OO

S?^ΦOO

oooo U^Φ._?oo

(2.4)

The commutative diagram

0 //I //

R //Z //

0

0 //J //R //Z/p²Z //0

induces the commutative diagram of homomorphisms of algebraic groups 1 //µp^s //

R_F(Φ)/F(Gm,F(Φ)) //U^Φ //

1

1 //Gm //R_F(Φ)/F(Gm,F(Φ)) //(U⁰)^Φ //1

(2.5)

and then the commutative diagram 0 //H¹(K, U^Φ) //

H²(K, µp^s) //

H²(K⊗F(Φ),Gm)

0 //H¹(K,(U⁰)^Φ) //H²(K,Gm) //H²(K⊗F(Φ),Gm)

(2.6)

for a field extensionK/F. Note that theK-algebraK(Φ) is a direct factor ofK⊗F(Φ).

Hence

Ker(H²(K,Gm)!H²(K⊗F(Φ),Gm)) = Ker(H²(K,Gm)!H²(K(Φ),Gm)).

It follows that

H¹(K,(U⁰)^Φ)'Br(K(Φ)/K) and H¹(K, U^Φ)'Br_ps(K(Φ)/K). (2.7) Lemma 2.3. The map H¹(K, U^Φ)!H¹(K, S^Φ)induces an isomorphism

H¹(K, S^Φ)'Brp^s(K(Φ)/K)ind. Proof. Consider the commutative diagram

1 //U^Φ //

S^Φ //

G^rm //1

1 //(U⁰)^Φ //(S⁰)^Φ //G^rm //1,

where the bottom row is induced by the bottom row of diagram (4) in [9]. This yields a commutative diagram

(K^×)^r //H¹(K, U^Φ) //

H¹(K, S^Φ) //

0

(K^×)^{r λ} //H¹(K,(U⁰)^Φ) //H¹(K,(S⁰)^Φ) //0 with exact rows. The homomorphismλtakes (x1, ..., xr) to

r

X

i=1

((χ_i)_K∪(x_i))

by [9, Lemma 3.6], whence the result.

3. Essential dimension of algebraic tori

Let S be an algebraic group over F and p be a prime integer different from char(F).

The essential dimension ed(S) (resp. essential p-dimension edp(S)) of S is defined to be the essential (p-)dimension of the functor taking a field extensionK/F to the set of isomorphism classesS-torsors(K) ofS-torsors overK. Note that the functorS-torsors is isomorphic to the functor takingKto the set H¹(K, S).

LetS be an algebraic torus overF split by LwithG=Gal(L/F). We assume that Gis a group of orderp^r, wherer>2. LetX be theG-module of characters ofS. Define the group X:=X/(pX+IX), where I is the augmentation ideal in R=Z[G]. For any subgroupH⊂G, consider the compositionX^H,!X!X. For every k, letVk denote the subgroup generated by images of the homomorphismsX^H!Xover all subgroupsH with [G:H]6p^k. We have the sequence of subgroups

0 =V₋₁⊂V₀⊂...⊂V_r=X. (3.1) Ap-presentationofX is aG-homomorphismP!X, withPbeing a permutationG- module, having a finite cokernel of order prime top. Ap-presentation with the smallest rank(P) is calledminimal. The essential p-dimension of algebraic tori was determined in [6, Theorem 1.1] in terms of a minimalp-presentationP!X:

edp(S) = rank(P)−dim(S). (3.2)

We have the following explicit formula for the essential (p-)dimension of S (cf. [9, Theorem 4.3]).

Theorem 3.1. Let S be a torus over a field F and p be a prime integer different from char(F). If the decomposition group Gof S is a p-group, then

ed(S) = edp(S) =

r

X

k=0

(rankVk−rankV_k−1)p^k−dim(S).

Proof. The second equality was proven in [9, Theorem 4.3]. Let ν:P!X be a minimalp-presentation. By definition, the indexm:=[X:Im(ν)] is prime top. LetT be the torus split by L with the character G-module Im(ν). The inclusion of Im(ν) into X yields a homomorphismα:S!T. AsmX⊂Im(ν), there is a homomorphismβ:T!S such that the compositionsαβ andβαare themth power endomorphisms ofT andS, respectively. It follows that for any field extensionK/F, the kernel and cokernel of the induced homomorphism

α∗:H¹(K, S)−!H¹(K, T)

arem-periodic. But both groups arep-groups, sinceS andT are split by ap-extension.

Therefore,α_∗ is an isomorphism.

Thus, the homomorphism α:S!T induces an isomorphism of functors S-torsors−−^∼!T-torsors.

It follows that ed(S)=ed(T). The surjection P!Im(ν) yields a generically free repre- sentation ofT by [10, Lemma 3.3]. Hence, by [3, Proposition 4.11] and (3.2), we have

edp(S)6ed(S) = ed(T)6rank(P)−dim(T) = rank(P)−dim(S) = edp(S), and therefore ed(S)=ed_p(S).

Let F be a field, Φ be a subgroup of Ch_p(F) of rank r>2, L=F(Φ) and G=

Gal(L/F). In this section we compute the essential (p-)dimension of the algebraic tori U^Φ and S^Φ defined by (2.4). For any subgroup H of G, we write n_H:=P

τ∈Hτ in R=Z[G]. An elementx∈Risdecomposable ifx=yzwithy, z∈R, and ε(y), ε(z)∈pZ.

Lemma 3.2. Let H⊂Gbe a non-trivial subgroup and x∈R be such that ε(nHx)∈p²Z.

Then nHxis decomposable.

Proof. If|H|=p, thenε(x)∈pZand hencenHxis decomposable. Otherwise we have H=H⁰×H⁰⁰ for non-trivial subgroups H⁰ and H⁰⁰. AsnH=nH⁰nH⁰⁰, the element nH, and thereforenHx, is decomposable.

Lemma 3.3. If x∈Ris decomposable, then x≡ε(x)modulo pI+I². Proof. Lety=ε(y)+uand z=ε(z)+v for someu, v∈I. Then we have

yz−ε(yz) = (ε(y)v+ε(z)u)+uv∈pI+I².

Consider the sequence of subgroups Vk⊂J¯as in (3.1) with respect to the algebraic torus U^Φ. Ifx∈J, we write ¯xfor the class of xin ¯J. The classesσi−1 and p^s form a basis for ¯J. Hence, rank( ¯J)=r+1.

Lemma 3.4. The group Vk is generated by

(1) the elements nHxsuch that |H|>p^r−k and ε(nHx)∈p^sZif r−k<s;

(2) the elements ¯nH such that |H|>p^r−k if r−k>s.

Proof. The statement follows from the equalityJ^H=R^H∩J=nHR∩J.

Lemma3.5. If k<r−s,then V_k=0.

Proof. By Lemma3.4(2),Vkis generated by ¯nHwith|H|>p^r−k. SincenHis decom- posable and|H|>p^s, in view of Lemma3.3, we have ¯nH=ε(nH)=|H|=0, as|H|∈pJ.

Lemma3.6. If s>2 and r−s6k6r−1,then dim(Vk)=1.

Proof. By Lemma3.4,Vk is generated bynHxwithH non-trivial andε(nHx)∈p^sZ. As s>2, the element nHx is decomposable by Lemma 3.2. In view of Lemma 3.3, nHx=ε(nHx). Hence,Vk is generated byp^s.

Lemma3.7. If s=1and pis odd,thendim(V_r−1)=1.

Proof. We claim thatVr−1is generated by ¯p. By Lemma3.4(2),Vr−1 is generated by ¯nH with|H|>p. If|H|>p²then, by Lemma3.2,nH is decomposable and, in view of Lemma3.3, ¯n_H=ε(n_H)=0.

Suppose that|H|=pand letσ∈H be a generator. We haven_H−p=(σ−1)m, where m=Pp−2

i=0(p−1−i)σⁱ, soε(m)=¹₂p(p−1). Aspis odd,ε(m)∈pZ. Hence,m∈pR+I, and thereforen_H−p∈pI+I²and ¯n_H= ¯pin ¯J.

Lemma3.8. If s=1and p=2,then V_r−1= ¯J.

Proof. By Lemma3.4(2), V_r−1 is generated by ¯n_H with |H|>2. Take non-trivial elementsσ6=τ inG. Then ¯2=1+στ−σ(1+τ)+1+σ∈V_r−1. Also, for anyσ∈G, we have σ−1=1+σ−¯2∈V_r−1. The group ¯J is generated by ¯2 andσ−1 over allσ∈G.

Proposition 3.9. We have ed(U^Φ) = ed_p(U^Φ) =

(r−1)2^r−1, if p= 2and s= 1, (r−1)p^r+p^r−s, otherwise.

Proof. Note thatV_r= ¯J, rank(V_r)=rank( ¯J)=r+1 and dim(U^Φ)=p^r. Case 1. pis odd, orp=2 ands>2. By Lemmas3.5–3.7, we have

rankV_k=







r+1, ifk=r, 1, ifr−s6k < r, 0, if 06k < r−s.

Since the decomposition groupGofU^Φis ap-group, by Theorem 3.1we have ed(U^Φ) = ed_p(U^Φ) =rp^r+p^r−s−dim(U^Φ) =rp^r+p^r−s−p^r= (r−1)p^r+p^r−s. Case 2. p=2 ands=1. By Lemmas3.5and3.8, we have

rankVk=

r+1, ifk=r−1 ork=r, 0, if 06k6r−2.

Again by Theorem3.1,

ed(U^Φ) = ed₂(U^Φ) = (r+1)2^r−1−dim(U^Φ) = (r−1)2^r−1.

Remark 3.10. One can construct a surjective minimal p-presentation %:P⁰!J as follows.

Case 1. p is odd, or p=2 and s>2. Let H be a subgroup of G of order p^s and P⁰:=R^r⊕Z[G/H]. We define%by

ν(x1, ..., xr,y) =¯

r

X

i=1

(σi−1)xi+nHy.

Note that the element nHy is independent of the choice of the representative y∈Z[G]

of ¯y. The image of % containsI and nH. Since nH≡p^s moduloI, we have p^s∈Im(%), and hence%is surjective. Note thateij=(σi−1)ej−(σj−1)ei∈Ker(%). Asσeij6=eij for j6=iand everyσ∈G\{1}, the groupGacts faithfully on Ker(%).

Case 2. p=2 and s=1. Let Hi be the subgroup of G generated by σi and let H=hσ1σ2i. Set

P⁰=

r

a

i=1

Z[G/Hi]⊕Z[G/H].

We define%by

%(¯x1, ...,x¯r,y) =¯

r

X

i=1

(σi+1)xi+(σ1σ2+1)y.

The image of%contains σi+1 and 2=(σ1σ2+1)−σ1(σ2+1)+(σ1+1). Hence,% is sur- jective. Note that we havehij:=(σi+1)ej−(σj+1)ei∈Ker(%). Sinceσhij6=hij fori6=j andσ∈G\hσi, σji, the groupGacts faithfully on Ker(%) ifr>3. In fact,Gacts trivially on Ker(%) ifr=2.

Corollary 3.11. We have ed(S^Φ) = edp(S^Φ) =

(r−1)2^r−1−r, if p= 2and s= 1, (r−1)p^r+p^r−s−r, otherwise.

Proof. By (3.2) and Proposition3.9, there is a minimalp-presentationν:P!J such that

rank(P) =

(r+1)2^r−1, ifp= 2 ands= 1,

rp^r+p^r−s, otherwise. (3.3)

The exact sequence

0−!Z^r−!Q−!J−!0

in the bottom row of (2.3) yields an exact sequence

HomG(P, Q)−!HomG(P, J)−!Ext¹_G(P,Z^r).

As P and Z^r are permutation G-modules, Ext¹_G(P,Z^r)=0. Hence, the homomorphism ν factors through a morphismν⁰:P!Q.

Recall that we write X=X/(pX+IX) for a G-module X. SinceZ^r'(Z/pZ)^r!Q is the zero map, the natural homomorphismQ!J¯is an isomorphism, and hence ν⁰ is a minimal p-presentation of Q. Note that G is a decomposition group of S^Φ and we have dim(S^Φ)=p^r+r. By Theorem3.1, ed(S^Φ)=edp(S^Φ)=rank(P)−dim(S^Φ). Hence, the result follows by (3.3).

4. Degeneration

In this section we relate the essential p-dimensions of Algp^r,p^s and of the torus S^Φ by means of the iterated degeneration (Proposition4.1). The latter is a method of compar- ison of the essentialp-dimension of an object (a central simple algebra in our case) over a complete discrete-valued field and of its specialization over the residue field.

4.1. A simple degeneration

LetF be a field, pbe a prime integer different from char(F) and Φ⊂Ch_p(F) be a finite subgroup. For integersk>0,s>1 and a field extension K/F, let

B_k,s^Φ (K) ={α∈Br(K){p}: ind(αK(Φ))6p^k and exp(α)6p^s}. (4.1) We say that two elementsαandα⁰ inB_k,s^Φ (K) areequivalent ifα−α⁰∈Br(K(Φ)/K)dec. Write Be^Φ_k,s(K) for the set of equivalence classes in B_k,s^Φ (K). To simplify notation, we shall write α for the equivalence class of an element α∈B_k,s^Φ (K) in Be^Φ_k,s(K). We view B^Φ_k,sandBe_k,s^Φ as functors fromFields/F toSets.

In particular, ifk=0, thenB_0,s^Φ (K) andBe^Φ_0,s(K) are bijective to Br_ps(K(Φ)/K) and Br_ps(K(Φ)/K)_ind, respectively. Hence, by (2.7) and Lemma2.3,

B^Φ_0,s'U^Φ-torsors and Be_0,s^Φ 'S^Φ-torsors. (4.2) Moreover, if Φ=0, then

B^Φ_k,s=Be^Φ_k,s'Alg_pk,p^s. (4.3) Let Φ⁰⊂Φ be a subgroup of index p and η∈Φ\Φ⁰. Hence, Φ=hΦ⁰, ηi. Let E/F be a field extension such that ηE∈Φ/ ⁰_E in Ch(E). Choose an element α∈B^Φ_k,s(E), i.e., α∈Br(E){p}such that ind(αE(Φ))6p^k and exp(α)6p^s.

LetE⁰be a field extension ofFwhich is complete with respect to a discrete valuation v⁰ overF with residue fieldE and set

α⁰:=α+(ˆb ηE∪(x))∈Br(E⁰){p}, (4.4) for some x∈(E⁰)^× such that v⁰(x) is prime to p. Since η_E(Φ0)6=0, it follows from (2.1) applied to the element α⁰_E0(Φ⁰) over the complete field E⁰(Φ⁰) with residue field E(Φ⁰) that

ind(α⁰_E0(Φ⁰)) =pind(α_E(Φ))6p^k+1 and exp(α⁰)6lcm(exp(α), p)6p^s. Hence,α⁰∈B_k+1,s^Φ0 (E⁰).

In the case the condition exp(α)6p^s in (4.1) is dropped, the following result was proved in [9, Proposition 5.2].

Proposition 4.1. Suppose that for any finite extension of fields N/E of degree prime to p and any character %∈Ch(N) of order p² such that p%∈ΦN\Φ⁰_N we have ind(α_N(Φ0,%))>p^k. Then

ed^Be

Φ0 k+1,s

p (α⁰)>ed^Be

Φ

pk,s(α)+1.

Proof. The proof of [9, Proposition 5.2] still works with the following modification.

LetM/E⁰ be a finite extension of fields of degree prime top,M0⊂M be a subfield overF andα⁰₀∈B^Φ_k+1,s⁰ (M0) be such that (α⁰₀)M=α⁰_M in Be_k+1,s^Φ0 and

tr deg_F(M0) = ed^Be

Φ0 k+1,s

p (α⁰).

We extend the discrete valuationv⁰ onE⁰ to a (unique) discrete valuationv onM, and let N be its residue field. Let N0 be the residue field of the restriction of v to M0. It was shown in the proof of [9, Proposition 5.2] that there exist α0∈Br(N0){p} with ind(α₀)_N0(Φ)6p^k, a prime elementπ₀ inM₀ andη₀∈Chp(N₀) such that

(α⁰₀)

Mb0=αb₀+(ˆη₀∪(π0)) in Br(Mb₀), (4.5) whereMb0 is the completion of the fieldM0 with respect to the restriction of v onM0, and

αN−(α0)N∈Br(N(Φ)/N)dec. (4.6) By (4.5), we have

exp(α₀) = exp(αb₀)6lcm(exp(α⁰₀)

Mc0, p)6lcm(exp(α⁰₀), p)6p^s,

and hence α₀∈B_k,s^Φ (N₀). Therefore, the class of α_N in Be^Φ_k,s(N) is defined over N₀ by (4.6). It follows that

ed^Be

Φ0 k+1,s

p (α⁰) = tr deg_F(M0)>tr deg_F(N0)+1>ed^Be

Φ k,s

p (α)+1.

4.2. A technical lemma

In this subsection we prove Lemma4.2, which will allow us to apply Proposition4.1.

Until the end of this subsection we assume that the base fieldF contains a primitive p²-th root of unity.

Letχ1, χ2, ..., χr, with r>2, be linearly independent characters in Chp(F) and let Φ=hχ1, χ₂, ..., χ_ri. LetE/F be a field extension such that rank(Φ_E)=randα∈Br(E){p}

be an element that is split byE(Φ) and is such that exp(α)6p^s.

Let E=E₀, E₁, ..., E_r be field extensions of F such that, for anyk=1,2, ..., r, the fieldE_k is complete with respect to a discrete valuationv_k overF andE_k−1is its residue field. For anyk=1,2, ..., r, choose elementsx_k∈E_k^× such thatv_k(x_k) is prime topand define the elementsα_k∈Br(E_k){p}inductively byα₀:=αand

α_k:=αb_k−1+((χb_k)_Ek−1∪(x_k)).

Let Φk be the subgroup of Φ generated by χk+1, ..., χr. Thus, Φ0=Φ, Φr=0 and rank(Φk)=r−k. Note that the character (χk)_Ek−1(Φk) is not trivial. It follows from (2.1) applied to the element (αk)_Ek(Φk)over the complete fieldEk(Φk) with residue field E_k−1(Φk) that

ind(αk)_Ek(Φk)=pind(α_k−1)_{Ek−1(Φk−1)}

for any k=1, ..., r. As ind(α)_E(Φ)=1, we have ind(α_k)_Ek(Φk)=p^k for all k=0,1, ..., r.

Moreover, as exp(α)6p^s, we have exp(α_k)6lcm(exp(α_k−1), p)6p^s. Thus,α_k∈B_k,s^Φk(E_k).

The following lemma assures that under a certain restriction on the elementα, the conditions of Proposition4.1are satisfied for the fields E_k, the groups of characters Φ_k and the elementsα_k. This lemma is similar to [9, Lemma 5.3].

Lemma 4.2. Suppose that for any subgroup Ψ⊂Φwith [Φ:Ψ]=p² and any field ex- tension L/E(Ψ) of degree prime to p, the element αL is not p²-cyclic. Then, for every k=0,1, ..., r−1, any finite extension of fields N/Ek of degree prime to p and any char- acter %∈Ch(N) of order p² such that p%∈(Φk)N\(Φk+1)N,we have

ind(αk)_N(Φk+1,%)>p^k. (4.7) Proof. Let k, N and % satisfy the conditions of the lemma. We construct a new sequence of fields Ee0,Ee1, ...,Eer such that each Eei is a finite extension of Ei of degree prime to p as follows. We set Eek=N. The fields Eej with j <k are constructed by descending induction on j. If we have constructed Eej as a finite extension of Ej of degree prime to p, then we extend the valuation vj to Eej and let Eej−1 be its residue field. The fieldsEemwithm>k are constructed by ascending induction onm. If we have

constructedEe_m as a finite extension ofE_m of degree prime to p, then let Ee_m+1 be an extension ofEm+1of degree [Eem:Em] with residue fieldEem. ReplacingEibyEeiandαi

by (αi)

Ee_i, we may assume thatN=Ek.

We proceed by induction onr. The caser=1 is obvious.

r−1⇒r: First suppose that k<r−1. Consider the fields F⁰=F(χ_r), E⁰=E(χ_r) and E_i⁰=E_i(χ_r), the sequence of characters χ⁰_i=(χ_i)_F⁰ and the sequence of elements α⁰_i:=(α_i)_E⁰

i∈Br(E_i⁰) for 06i6r−1. Let Φ⁰=hχ⁰₁, χ⁰₂, ..., χ⁰_r−1i⊂Ch(F⁰), let Φ⁰_i be the subgroup of Φ⁰ generated byχ⁰_i+1, ..., χ⁰_r−1 and let%⁰=%_E⁰

k.

We check the conditions of the lemma for the new datum. Let Ψ⁰ be a subgroup of Φ⁰ of index p². Then the preimage Ψ of Ψ⁰ under the map Ch(F)!Ch(F⁰) is a subgroup of Φ of indexp²andE⁰(Ψ⁰)=E(Ψ). LetL⁰/E⁰(Ψ⁰) be a field extension of degree prime top. By assumption, the element α⁰_L0=α_L⁰ is not p²-cyclic. We also have that p%⁰=p%_E⁰

k∈(Φk)_E⁰

k=(Φ⁰_k)_E⁰

k. Suppose that p%⁰∈(Φ⁰_k+1)_E⁰

k, i.e.,p%_E⁰

k=p%⁰=η_E⁰

k for some η∈(Φk+1)E_k. It follows that p%−η∈Ker(Ch(Ek)!Ch(E_k⁰))=h(χr)E_ki, and therefore, p%∈(Φk+1)E_k, which is a contradiction. Hencep%⁰∈(Φ⁰_k)_E⁰

k\(Φ⁰_k+1)_E⁰

k. By the induction hypothesis, the inequality (4.7) holds forα⁰_k, i.e,

ind(α⁰_k)E_k⁰(Φ⁰_k+1,%⁰)>p^k. Since (α⁰_k)_E⁰

k(Φ⁰_k+1,%⁰)=(α_k)_Ek(Φk+1,%), inequality (4.7) holds forα_k. Therefore, it remains to show that inequality (4.7) holds in the case k=r−1. Note that in this case p% is a non-zero multiple of (χ_r)_Er−1 and Φ_k+1=Φ_r=0.

Case 1. The character%is unramified with respect tovr−1, i.e.,%= ˆµfor a character µ∈Ch(Er−2) of orderp². Note thatpµis a non-zero multiple of (χr)Er−2.

By (2.1), we have

ind(α_r−2)_{Er−2(χr−1,µ)}=ind(αr−1)E_r−1(%)

p . (4.8)

Consider the fieldsF⁰=F(χ_r−1),E⁰=E(χ_r−1) andE_i⁰=E_i(χ_r−1), the new sequence of charactersχ⁰₁=(χ₁)_F⁰, ..., χ⁰_r−2=(χ_r−2)_F⁰, χ⁰_r−1=(χ_r)_F⁰, the group of characters

Φ⁰=hχ⁰₁, χ⁰₂, ..., χ⁰_r−1i,

the elementsα⁰_i∈Br(E_i⁰), 06i6r−1, defined byα⁰_i=(αi)E_i⁰ fori6r−2 and α⁰_r−1=αb_r−2+(χb_r∪(xr−1))

overE_r−1⁰ , and the character µ. The new datum satisfies the conditions of the lemma.

By the induction hypothesis, the inequality (4.7) holds forα_r−2⁰ , i.e, ind(α⁰_r−2)E_r−2⁰ (µ)>p^r−2.

Since (α⁰_r−2)_E⁰

r−2(µ)=(α_r−2)_{Er−2(χr−1,µ)}, inequality (4.7) holds forα_r−1 in view of equal- ity (4.8).

Case 2. The character%is ramified. Assume that inequality (4.7) does not hold for α_r−1, i.e., we have

ind(α_r−1)_Er−1(%)6p^r−2.

By [9, Lemma 2.3 (2)], there exists a unitu∈Er−1 such thatEr−2(χr)=Er−2(¯u^1/p) and ind(α_r−2−(χ_r−1∪(¯u^1/p)))_Er−2(χr)= ind(α_r−1)_Er−1(%)6p^r−2.

By descending induction on 06j6r−2, we show that there exist an element u_j in E_j^× and a subgroup Ψ_j⊂Φ of rank r−j−2 such that

hχ1, ..., χ_j, χ_r−1, χ_ri∩Ψj= 0, Ej(χr)=Ej(u^1/p_j ) and

ind(α_j−(χ_r−1∪(u^1/p_j )))_Ej(Θj)6p^j, (4.9) where Θj:=hΨj, χri. We set Ψr−2=0 andur−2= ¯u.

j⇒j−1: The field Ej(u^1/p_j )=Ej(χr) is unramified over Ej, and hence vj(uj) is divisible byp. Modifyingu_j by ap²-th power, we may assume thatu_j=vx^mp_j for a unit v∈Ej and an integerm. Then

(αj−(χr−1∪(u^1/p_j )))_Ej(Θj)= ˆβ+(ˆη∪(xj))_Ej(Θj),

whereη=χ_j−mχ_r−1andβ=(α_j−1−(χ_r−1∪(u^1/p_j−1)))_Ej−1(Θj), withu_j−1= ¯v. Asηis not contained in Θ_j, the characterη_Ej−1(Θj)is not trivial. Set Ψ_j−1=hΨ_j, ηi. It follows from (2.1) and the induction hypothesis that

ind(β_{Ej−1(Θj−1)}) = ind(α_j−(χr−1∪(u^1/p_j )))_Ej(Θj)

p 6p^j−1.

This completes the induction step.

Applying inequality (4.9) in the casej=0, we have α_E(Θ0)= (χ_r−1∪(w^1/p))_E(Θ0) for an elementw∈E^× such thatE(w^1/p)=E(χ_r). Hence,

α_E(Ψ

0)(w^1/p2)= (α_E(Θ0))_E(Θ

0)(w^1/p2)= 0 in Br(E(Ψ₀)(w^1/p2)).

Asχr∈Ψ/ 0, the fieldE(Ψ0)(w^1/p)=E(Ψ0)(χr) is a cyclic extension ofE(Ψ0) of degreep.

Hence E(Ψ0)(w^1/p2)/E(Ψ0) is a cyclic extension of degree p². Since α_E(Ψ0) is split by the extensionE(Ψ0)(w^1/p2)/E(Ψ0),α_E(Ψ0) isp²-cyclic. As [Φ:Ψ0]=p², this contradicts the assumption. Hence, the inequality (4.7) holds forα_r−1.

5. Non-cyclicity of the generic element

The aim of this section is the technical Lemma 5.4, which will allow us to later apply Lemma4.2and Proposition4.1.

In this section we assume that the base field F contains a primitive p³-th root of unity. The choice of a primitivep²-th root of unityξallows us to define the symbol (a, b)_p2

as in §2.1. As −1 is a p²-th power in F^×, we have (a,−1)_p2=0. Hence, (a, a)_p2=0 for alla∈F^×. We shall write (a, b)p forp(a, b)_p2=(a^p, b)_p2.

Lemma 5.1. Let E be a field extension of F that is complete with respect to a discrete valuation v with residue field Kand let α∈Br(K). Set β=α+(a, x)b _p for a unit a∈E and x∈E^× such that ¯a /∈(K^×)^p and v(x) is prime to p. If β is p²-cyclic, then α=(¯a, z)_p2 in Br(K)for some z∈K^×.

Proof. Suppose that β=(uπⁱ, tπ^j)_p2 and writex=wπ^k for a prime elementπ, inte- gersi,j andk=v(x), and units u, tandwinE. Then we have

α+(ab ^p, wπ^k)_p2=β= (uπⁱ, tπ^j)_p2= (u, t)_p2+u^j tⁱ, π

p². Applying the residue map∂v, we get ¯a^pk= ¯u^j/¯tⁱ inK^×/(K^×)^p2 and

α= (¯u,¯t)_p2−(¯a,w^p)_p2.

Suppose that i/j is ap-integer (the other case is similar). Ask is not divisible by pand ¯a is not apth power in K^×, j is not divisible byp². It follows that ¯u∈h¯a,¯ti in K^×/(K^×)^p2, and then ¯u∈¯a^r¯t^s(K^×)^p2 for somerands. Hence,α=(¯a,¯t^r/w^p)_p2.

Corollary 5.2. Let x and y be independent variables over F and a, b∈F^×. If (a, b)_p6=0 in Br(F), then, for any field extension M/F(x, y) of degree prime to p, the element (a, x)_p+(b, y)_p in Br(M)is not p²-cyclic.

Proof. LetM/F(x, y) be a field extension of degree prime topandβ=(a, x)p+(b, y)p

overM. As the degree ofM/F(x, y) is prime top, by [7, Lemma 6.1] there exists a field extensionEof the fieldsF((y))((x)) andM overF such that the degree ofE/F((y))((x)) is finite and prime top. The discrete valuationvxon the complete fieldF((y))((x)) extends uniquely to a discrete valuationv ofE. The ramification index of E/F((y))((x)) is prime top, and hencev(x) is prime to p. The residue fieldK of v is an extension ofF((y)) of degree prime top.

Let v⁰ be the valuation on K extending the discrete valuation vy on F((y)). The ramification index e⁰ of K/F((y)) is prime to p. The residue field N of v⁰ is a finite extension ofF of degree prime to p.

Letα=(b, y)_poverK, soβ_E=α+(a, x)b _p. Suppose thatβ isp²-cyclic overM. Then βE is also p²-cyclic. By Lemma5.1, applied toβE overE, we haveα=(a, z)_p2 for some z∈K^×, and hence (b^p, y)_p2=(a, z)_p2. Taking the cup product with (a)_p2∈K^×/(K^×)^p2, we get

(a)_p2∪(b^p, y)_p2= (a)_p2∪(a, z)_p2= (a, a)_p2∪(z)_p2= 0.

Applying the residue map

∂v⁰:H³(K, µ^⊗2_p2)−!H²(N, µ_p2) = Br_p2(N),

we find thate⁰(a, b)_p=e⁰(a, b^p)_p2=0 over N, and hence (a, b)_p=0 in Br(N). Taking the corestriction map Br(N)!Br(F), we see that (a, b)_p=0 in Br(F), a contradiction.

Lemma5.3. For any integer r>2,there exist a field extensionF⁰/F and a subgroup Φ⊂Ch_p(F⁰)of rank rsuch that, for any subgroupΨ⊂Φof index p²,there is an element β∈Br_p(F⁰(Φ)/F⁰) with the property that any field extension M/F⁰(Ψ) of degree prime to p, the element βM is not p²-cyclic.

Proof. Leta1, a2, ..., ar, xandy be independent variables overF and set F⁰:=F(a1, a2, ..., ar, x, y).

For every 16i6r, let χ_i∈Ch_p(F⁰) be a character such that F⁰(χ_i)=F⁰(a^1/p_i ) and set Φ:=hχ₁, χ₂, ..., χ_ri. Let Ψ be a subgroup of Φ of indexp². Choose a basisη₁, η₂, ..., η_rfor Φ such that Ψ=hη₁, η₂, ..., η_r−2i, and elementsb₁, b₂, ..., b_rinF⁰such thatF(η_i)=F(b^1/p_i ) for all 16i6randF(b1, b2, ..., br)=F(a1, a2, ..., ar). Clearly,b1, b2, ..., brare algebraically independent over F and F⁰(Ψ)=L(x, y), where L:=F(b^1/p₁ , ..., b^1/p_r−2, b_r−1, br), with the generators algebraically independent overF.

Let β=(b_r−1, x)p+(br, y)p in Brp(F⁰(Φ)/F⁰) and M/F⁰(Ψ) be a field extension of degree prime to p. Since ∂v((b_r−1, br)p)=¯b_r−1 is non-trivial, where v is the discrete valuation on L associated with br, we have (b_r−1, br)p6=0 in Br(L). The result follows from Corollary5.2.

Let F⁰/F be the field extension and Φ⊂Chp(F⁰) be the subgroup of rank r as in Lemma5.3. Consider the algebraic toriP^Φ,S^Φ,T^Φ,U^ΦandV^ΦoverF⁰defined in§2.4.

The morphismγ:P^Φ!V^Φ in the diagram (2.4) is a U^Φ-torsor. Denote byδ the image of the class ofγunder the composition

H_´et¹(V^Φ, U^Φ)−!H_´et¹(V^Φ,(U⁰)^Φ)−!H_´et²(V^Φ,Gm)

induced by the diagram (2.5). We writeδgenfor the image ofδunder the homomorphism H_´et²(V^Φ,Gm)−!H²(F⁰(V^Φ),Gm) = Br(F⁰(V^Φ))