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Essential dimension and linear codes Cernele, Shane 2014

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ESSENTIAL DIMENSION AND LINEAR CODESbySHANE CERNELEA THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THEREQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinTHE FACULTY OF GRADUATE AND POSTDOCTORAL STUDIES(Mathematics)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)April 2014c©Shane Cernele, 2014AbstractThe essential dimension of an algebraic group G is a measure of the complex-ity of G-torsors. One of the central open problems in the theory of essentialdimension is to compute the essential dimension of PGLn, whose torsors cor-respond to central simple algebras up to isomorphism. In this thesis, westudy the essential dimension of groups of the form G/µ, where G is a reduc-tive algebraic group satisfying certain properties, and µ is a central subgroupof G. In particular, we consider the caseG = GLn1 × · · · ×GLnrwhere each ni a power of a single prime p, which is a generalization of thegroup PGLpa = GLpa /Gm. We will see that torsors for G/µ correspond totuples of central simple algebras satisfying certain properties. Surprisingly,computing the essential dimension of G/µ becomes easier when r ≥ 3.Using techniques from Galois cohomology, representation theory and theessential dimension of stacks, we give upper and lower bounds for the essentialdimension of G/µ. To do this, we first attach a linear ‘code’ Cµ to the centralsubgroup µ, and define a weight function on Cµ. Our upper and lower boundsare given in terms of a minimal weight generator matrix for Cµ. In some caseswe can determine the exact value of the essential dimension of G/µ.iiPrefaceThis dissertation is original, unpublished, independent work by the Author,S. Cernele.iiiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . vii1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.1 Essential Dimension and Canonical Dimension . . . . . . . . . 132.2 Quotient Stacks . . . . . . . . . . . . . . . . . . . . . . . . . . 153 Galois Cohomology of G/µ . . . . . . . . . . . . . . . . . . . . . 184 On Minimal Generator Matrices . . . . . . . . . . . . . . . . . 245 Codes and the Brauer Group . . . . . . . . . . . . . . . . . . . 286 An Upper Bound . . . . . . . . . . . . . . . . . . . . . . . . . . 367 Central Simple Algebras with Tensor Product of BoundedIndex. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417.1 General Results . . . . . . . . . . . . . . . . . . . . . . . . . . 417.2 Small Cases in Theorem 7.2 . . . . . . . . . . . . . . . . . . . 448 Examples of Linear Error-Correcting Codes . . . . . . . . . . 50iv9 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56Appendix A Disjoint Central Simple Algebras . . . . . . . . . . 60Appendix B Quotient Stacks . . . . . . . . . . . . . . . . . . . . . 66Appendix C Products of Groups with p 6= 2 . . . . . . . . . . . . 71vList of Tables1 General Notation . . . . . . . . . . . . . . . . . . . . . . . . . 32 Notation for the Groups Gi . . . . . . . . . . . . . . . . . . . 53 Notation Related to G and µ . . . . . . . . . . . . . . . . . . 8viAcknowledgementsFirst, I would like to thank my supervisor Zinovy Reichstein. His guidance,suggestions and feedback on early versions of this thesis have been invaluable.I would also like to thank Zinovy Reichstein and Roland Lo¨tscher forsuggesting the methods used to prove Corollary B.4, and Zinovy Reichsteinfor showing me the proof of Theorem 7.9.I would like to thank my committee members Julia Gordon and LiorSilberman for their helpful questions and conversations. Also, thanks toAthena Nguyen for showing me the results of her Master’s thesis, and to myfellow graduate students Mario Garcia Armas and Jerome Lefebvre for thecountless mathematical discussions we’ve had together.vii1 Introduction1.1 BackgroundInformally, the essential dimension of an object is the minimum number ofalgebraically independent variables required to define that object. Essentialdimension was introduced by Buhler and Reichstein ([BR97]) in 1997, andthe definition has since been generalized by Reichstein ([R00]) and Merkurjev([BF03]). For the definition of essential dimension see §2, and for recentsurveys see [R10] and [M13].Let k be a base field of characteristic zero. The essential dimension of analgebraic group G/k is the maximum essential dimension of an element of thefirst Galois cohomology set H1(K,G), over all field extensions K/k. For ex-ample, H1(K,PGLn) can be identified with central simple algebras of degreen over K (up to isomorphism), and so edk(PGLn) is the minimum number ofalgebraically independent variables needed to define a central simple algebraof degree n over any field extension of k. For n ≥ 5 and odd, from [LRRS03]we haveedk(PGLn) ≤12(n− 1)(n− 2)and if ab | n for some a > 1, from [R00, Theorem 9.3 & Proposition 9.8a] wehaveedk(PGLn) ≥ 2bStronger results are known for a ‘local version’ of essential dimension ata prime p, called essential p-dimension. We denote essential p-dimension byedk(G; p), and by definition edk(G; p) ≤ edk(G).In the case where G = PGLn, if n = pab with (p, b) = 1 thenedk(PGLn; p) = edk(PGLpa ; p)and so we can reduce to studying only central simple algebras of p-primary1degree.Every central simple algebra of index p becomes a cyclic algebra after aprime-to-p extension of the base field; from this one can deduceedk(PGLp; p) = 2(see [RY00, Lemma 8.5.7]). When a ≥ 2 we have the following result:(a− 1)pa + 1 ≤ edk(PGLpa ; p) ≤ p2a−2 + 1The lower bound is from [M10] and the upper bound is from [Ru11].The set H1(K,G) where G = GLpa /µps (s < a) corresponds to centralsimple algebras of degree pa and exponent dividing ps. The essential p-dimension of GLpa /µps was studied in [BM12], where the authors show:edk(GLpa /µps ; p) ≤ 2p2a−2 − pa + pa−sedk(GLpa /µps ; p) ≥(a− 1)2a−1 if p = 2 and s = 1(a− 1)pa + pa−s otherwiseIn this paper we will study the essential dimension of a certain class ofgroups, including the groups G/µ where G = GLpa1 × . . . × GLpar for someprime p, and µ ≤ Z(G). The Galois cohomology of G/µ is related to r-tuples of central simple algebras; see Section 3. Surprisingly, computing theessential dimension becomes easier when r ≥ 3.Let Cµ be the submodule of Zr consisting of all r-tuples (x1, . . . , xr) ∈ Zrsuch that λx11 · . . . · λxrr = 1 for all (λ1, . . . , λr) ∈ µ. Let Cµ be the (finite)image of Cµ under the natural surjectionZr → Z /pa1 Z× · · · × Z /par ZIf (c1, . . . , cr) ∈ Cµ then write c = (u1pk1 , . . . , urpkr) with ui ∈ (Z / pai Z)∗2and 0 ≤ ki ≤ ai, and define the ‘weight of c’, denoted w(c), byw(c) =r∑i=1(ai − ki).The main result of this paper is the following. Let (Y1, . . . , Yt) be agenerating set of Cµ such thatt∑i=1w(Yi) is minimal. Let M =t∑i=1pw(Yi).Thenedk(G/µ) ≥M + (r − t)− p2a1 − . . .− p2arand for many choices of µ (when r ≥ 3), equality holds.1.2 NotationWe will now introduce some notation before stating our main results. Themajor notation is summarized in the tables throughout this section. Webegin by defining some standard notation in the table below.Table 1: General NotationNotation Definition and Assumptionsk Base field of characteristic zero.p Positive prime integer.Z(B) Center of the group B.X(A) Character lattice of the diagonalizable group A.Br(K) Brauer group of the field K.jth Galois cohomology group of B over K.Hj(K,B) We assume B is abelian for j > 1.3Good references for more information on the Brauer group include [GS06]and [BO13]. For the the definition and properties of Galois cohomology,consult [S97].We now proceed to define the groups and maps we will study in thispaper. For i = 1, . . . , r, let Gi be a reductive linear algebraic group withZ(Gi) ≤ Gm. In other words, we are once and for all identifying Z(Gi) witha subgroup of Gm, so that we may have the identity character: Z(Gi) ↪→ Gm.Denote Gi = Gi/Z(Gi) and consider δiK : H1(K,Gi) → H2(K,Z(Gi)) ≤Br(K) for any K/k. Here δiK is the coboundary map induced from thesequence1→ Z(Gi)→ Gi → Gi → 1for any K/k, see [S97, Section I.5].We will assume that the image of δiK consists of elements of p-primaryorder for some prime p. Let pai be the maximal index of δiK(E) (over all K/kand E ∈ H1(K,Gi)), and let pbi be the maximal exponent of δiK(E) (over allK/k and E ∈ H1(K,Gi)). We make the following additional assumptions:i) For each i, bi ∈ {1, ai}.ii) Either Z(Gi) = Gm for all i, or Z(Gi) = µpbi for all i. In particular,Z(G1)× · · · × Z(Gr) is either connected or finite.Let ni = pai .Example 1.1. Examples of groups Gi satisfying bi ∈ {1, ai} and havingZ(Gi) ∈ {Gm, µpbi} include:a) GLni and SLni for p arbitrary, ai ≥ 1 (bi = ai). In this caseGi = PGLni ,and H1(K,PGLni) classifies central simple algebras of degree n over K.The coboundary map sends a central simple algebra to its Brauer classin Br(K).b) For p = 2, GOni , Oni , GSPni , and SPni when ai ≥ 1 (bi = 1), and GO+niand SOni when ai ≥ 2 (bi = 1). In these cases H1(K,Gi) classifies4central simple algebras of degree ni with involution (of the first kind)satisfying certain properties. The coboundary map sends a centralsimple algebra A with involution to the Brauer class of A in Br(K)(see [KMRT98, Section 29]). Note that the groups GOn, On and SOnwhen p 6= 2 have a trivial coboundary map; we discuss these groupsfurther in Appendix C.c) E6 (simply connected) for p = 3, ai = 3 (bi = 1).d) Non-abelian finite p-groups Gi where Z(Gi) ∼= µp and the dimensionof a minimal faithful representation of Gi is ni (see [KM08, Theorem4.4]). In this case, ai ≥ 1 and bi = 1.We summarize the notation related to the groups Gi in the table below.Table 2: Notation for the Groups GiNotation Definition and Assumptionsr Positive integer.Gi(i = 1 . . . r)Reductive linear algebraic group.Z(Gi) ≤ Gm.Gi(i = 1, . . . , r)Gi/Z(Gi).δiKThe coboundary map H1(K,Gi)→ H2(K,Z(Gi))induced from the sequence 1 → Z(Gi) → Gi →Gi → 1.We assume every element of im(δiK) has p-primaryorder.aiMaximum index of an element in im(δiK) over allK/k.Continued on next page...5Table 2: Notation for the Groups Gi (continued)Notation Definition and AssumptionsbiMaximum exponent of an element in im(δiK) overall K/k.We assume bi ∈ {1, ai}.If |Z(Gi)| <∞ then we assume Z(Gi) ∼= µpbi .ni pai .We now proceed to define the groups whose essential dimension we areinterested in, and some of their related structures. LetG = G1 × · · · ×Gr, and G = G/Z(G) =r∏i=1Gi.Let µ be a subgroup of Z(G) (in particular, µ ≤ Grm), and letδK : H1(K,G)→ H2(K,Z(G)/µ)be the coboundary map induced from the sequence 1→ Z(G)/µ→ G/µ→G→ 1. We will compute bounds on the essential dimension of G/µ over k.From µ ≤ Z(G) we get a surjective map X(Z(G)) → X(µ) given byrestricting a character of Z(G) to µ. We defineCµ = ker (X(Z(G))  X(µ))and observe that Cµ ∼= X(Z(G)/µ).Note that X(Z(G)) is a Z-module of rank r, and comes with a canonicalcoordinate system. This coordinate system is determined by r generators,which are the r maps Z(G) → Z(Gi) ↪→ Gm. Thus we can think of an6element of Cµ as an r-tuple (z1, . . . , zr), wherezi ∈Z if Z(Gi) ∼= GmZ / pbi Z if Z(Gi) ∼= µpbiand we can write Cµ explicitly as:Cµ = {(c1, . . . , cr) ∈ X(Z(G)) | λc11 · . . . · λcrr = 1 for all (λ1, . . . , λr) ∈ µ} .Set F = µpb1 ×· · ·×µpbr ≤ Z(G), and for any subgroup τ of Z(G), defineτf = τ ∩F . Given µ ≤ Z(G), we define the code associated to µ, denoted Cµ,to be the image of Cµ under the natural surjection X(Z(G))  X(Z(G)f ).In other words, Cµ is the code given by reducing the ith coordinate in eachelement of Cµ modulo pbi . Note that this construction is trivial in the casewhere |Z(G)| < ∞, since in this case we assumed Z(G) = F and henceCµ = Cµ.Remark 1.2. Cµ can be identified with X(Z(G)f/µf ), and thus if α and βare subgroups of Z(G), then αf = βf if and only if Cα = Cβ.We will now assign ‘weights’ to the elements of our code. Let µ ≤ Z(G)with associated code Cµ. Define a map vi : Z /pbi Z → Z as follows. Forz ∈ Z /pbi Z, if z = 0 then define vi(z) = ai. Otherwise, write z = upk withu invertible in Z /pbi Z and 0 ≤ k < bi, and define vi(z) = k.For an element z = (z1, . . . , zr) ∈ Cµ, we define the weight of z, denotedw(z) to be:w(z) =r∑i=1(ai − vi(zi))Remark 1.3. In the case where ai = bi = 1 for all i, w(z) is the usualHamming weight of z.Remark 1.4. Since we assumed bi ∈ {1, ai}, our weight function has thefollowing important property. Suppose that for i = 1, . . . , r, Ei is a central7simple algebra with index pai and exponent pbi , and zi ∈ Z /pbi Z. Thenind(E⊗zii ) = pai−vi(zi), and further if z = (z1, . . . , zr) ∈ Cµ thenind(E⊗z11 ⊗ · · · ⊗ E⊗zrr ) ≤r∏i=1ind(E⊗zii ) = pw(z)We summarize the notation related to the group G/µ and the code asso-ciated to µ in the following table.Table 3: Notation Related to G and µNotation Definition and AssumptionsGG1 × · · · ×Gr.We assume Z(G) is finite or connected.G G/Z(G).µ Subgroup of Z(G).CµThe Z-module given by ker (X(Z(G))→ X(µ)).Explicitly, Cµ is given by:{(c1, . . . , cr) ∈ X(Z(G)) | λc11 · . . . · λcrr = 1,∀ (λ1, . . . , λr) ∈ µ}Continued on next page...8Table 3: Notation Related to G and µ (continued)Notation Definition and AssumptionsCµImage of Cµ under the mapX(Z(G))→ X(µpb1 × · · · × µpbr )induced from µpb1 × · · · × µpbr ↪→ Z(G).Equivalently, Cµ is the Z-module given byreducing the ith coordinate of Cµ modulo pbi , fori = 1, . . . , r.Cµ is called the code associated to µ.ww : Cµ → Z(u1pk1 , . . . , urpkr) 7→∑uipki 6=0(ai − ki)Here, ui ∈ (Z / pbi Z)∗ and 0 ≤ ki ≤ bi.δKThe coboundary mapH1(K,G)→ H2(K,Z(G)/µ)induced from the sequence1→ Z(G)/µ→ G/µ→ G→ 1.91.3 Main ResultsWe will now state the main results of this paper. We begin with a resultexplaining the significance of the code associated to µ.Theorem 1.5. Let µ, τ ≤ Z(G) and K/k. If Cµ = Cτ (or equivalently, ifµf = τf) then there is an isomorphism of functors H1(−, G/µ)→ H1(−, G/τ).In particular, the essential dimension and essential p-dimension of G/µ de-pend only on the code Cµ.A generator matrix Y of a code is a matrix whose rows generate the code,and if Y is a generator matrix, then Yi denotes the ith row of the matrix andyij denotes the entry in the ith row and jth column.Definition 1.6. Let Y be a generator matrix for Cµ, with rows Y1, . . . , Yt.We say that Y is minimal if, for any other generator matrix Z of Cµ withrows Z1, . . . , Zl,t∑i=1w(Yi) ≤l∑i=1w(Zi).We can now state upper and lower bounds on the essential dimension ofG/µ, where µ ≤ Z(G). Recall that, by Theorem 1.5, we are free to replaceµ with any τ ≤ Z(G) such that Cµ = Cτ .Theorem 1.7. Let µ ≤ Z(G) and let Y be a minimal generator matrix forCµ with rows Y1, . . . , Yt.1. edk(G/µ; p) ≥(t∑i=1pw(Yi))− d− dim(G)2. edk(G/µ) ≤(t∑i=1pw(Yi))− d+ edk(G)where d =t, if Z(G) is connected;0, if Z(G) is finite.10Although the upper and lower bounds in Theorem 1.7 never meet, formany families of subgroups µ the termt∑i=1pw(Yi) appearing in both the upperand lower bound is much larger than any of the other terms in either formula.Definition 1.8. Let Y be a generator matrix for Cµ. We say that Y is veryacceptable if:1. Each yij equals −1, 0 or 1 in Z /pbi Z.2. Y contains no column of all zeroes.3. For each i, the Hamming weight of Yi is at least f(p), wheref(p) =5, if p = 24, if p = 33, otherwiseWhen Gi has a faithful representation of dimension ni such that Z(Gi)acts by the identity character (for example, every Gi in Example 1.1), we cansometimes use very acceptable generator matrices to find a stronger upperbound.Theorem 1.9. Suppose that Gi has a faithful representation of dimension nisuch that Z(Gi) acts by the identity character, for all i. Let µ ≤ Z(G) and letY be a very acceptable minimal generator matrix for Cµ with rows Y1, . . . , Yt.Suppose additionally that for all 1 ≤ i, j, k ≤ r, we have ai < aj + ak (orequivalently, ni < nj · nk). Thenedk(G/µ) = edk(G/µ; p) =(t∑i=1pw(Yi))− d− dim(G)where d =t, if Z(G) is connected;0, if Z(G) is finite.11Remark 1.10. The conditions in Theorem 1.9 that for all 1 ≤ i, j, k ≤ r,ai < aj + ak, and that Y is very acceptable can be replaced by assuming Yis an acceptable generator matrix. The definition of an acceptable generatormatrix is more complicated to describe; see Section 6.Example 1.11. A motivating example to keep in mind is the following. Letn1 = n2 = · · · = nr = pa for r ≥ 5 and a ≥ 1, and let Gi ∼= GLni for1 ≤ i ≤ r. Let µ < Z(G) be defined by:µ := {(λ1, . . . , λr) ∈ Z(G) | λ1 · . . . · λr = 1}Thus Cµ = 〈(1, 1, . . . , 1)〉 ≤ X(Z(G)) = Zr. By Theorem 1.9, we haveed(G/µ) = ed(G/µ; p) = pra − rp2 + r − 1See Section 7 for a generalization of this example.The rest of this report is structured as follows. In §2 we discuss somepreliminaries on essential dimension, including the definition. Then in §3we study the Galois cohomology of G/µ and prove Theorem 1.5. In §4, wewill discuss codes and minimal generator matrices, and in §5 we discuss therelationship between codes and subgroups of the Brauer group and proveTheorem 1.7. In §6 we prove Theorem 1.9. In §7 we discuss an interestingexample, and in §8 we look at a class of codes where we can find acceptableminimal generator matrices. Throughout, all diagrams are commutative, Gand G are groups of the form described in this section, and µ denotes asubgroup of Z(G).122 Preliminaries2.1 Essential Dimension and Canonical DimensionIn this section we will give the definition of essential dimension and essentialp-dimension. All fields are assumed to contain our base field k. Let F be acovariant functor from Fieldsk to Sets. If we have a field extension givenby Li↪→ K and α ∈ F(L), then we write (α)K for F(i)(α) ∈ F(K).Let K/k be a field extension and α ∈ F(K). If K/L is a field extension,then we say α descends to L if there exists α0 ∈ F(L) that that (α0)K = α.The essential dimension of α (over k), denoted edk(α), is defined to be theminimum value of trdegk(L) over all fields L such that such that α descendsto L.The essential dimension of F (over k), denoted edk(F) is defined to bethe maximum value of edk(α), where K runs over all field extensions of kand α ∈ F(K).Essential p-dimension is defined similarly, for a covariant functor F fromFieldsk to Sets. If K/L and α ∈ F(K) then we say α p-descends to L ifthere exists α0 ∈ F(L) and a finite exenstion K ′/K of degree prime-to-p,such that (α0)K′ = (α)K′ . The essential p-dimension of α (over k), denotededk(α; p), is defined to be the minimum value of trdegk(L) over all fieldsL such that such that α p-descends to L. The essential p-dimension of F ,denoted edk(F ; p), is defined to be the maximum value of edk(α; p), whereK runs over all field extensions of k and α ∈ F(K). We have from thedefinitions that edk(F) ≥ edk(F ; p).For an algebraic group G, the essential dimension (resp. p-dimension) ofG is defined to be the essential dimension (resp. p-dimension) of the Galoiscohomology functor edk(H1(−, G)). For example, by Hilbert’s Theorem 90H1(K,GLn) = 0 for all K, and hence edk(GLn) = edk(GLn; p) = 0 (for anyp).It is clear from the definitions that for any group G, edk(G) ≥ edk(G; p).13However, for some groups edk(G) is strictly greater than edk(G; p) for anyprime p; for an example, see [D10].We now recall some results from the theory of essential dimension.Theorem 2.1. [BF03, Lemma 1.9] Let F , T be functors from Fieldsk toSets. If there is a surjective morphism of functors F  T thenedk(T ) ≤ edk(F).Theorem 2.2. [S97, III.4.3, Lemma 6] Let G be a reductive linear algebraicgroup, and N be the normalizer of the maximal torus in G. Then the inducedmap in cohomolgy H1(−, N) → H1(−, G) is surjective. In particular, byTheorem 2.1,edk(G) ≤ edk(N).Let G be a linear algebraic group.Definition 2.3. A representation ρ : G → GL(V ) is called generically freeif there exists a dense open subset U ⊂ V such that the geometric stabilizerof every point x of U is trivial.Remark 2.4. Note that if V is a faithful representation of G, then G actsgenerically freely on the vector space End(V ). This is because, with U =Aut(V ) ⊂ End(V ), it is easy to see that the geometric stabilizer of everypoint in U is trivial. In particular, generically free representations alwaysexist for linear algebraic groups.Theorem 2.5. [BF03, Proposition 4.11] Suppose G has a generically freerepesentation ρ : G→ GL(V ). Thenedk(G) ≤ dim(V )− dim(G).A special case of essential dimension is canonical dimension. For a functorF : Fieldsk → Sets, we define the detection functor DF : Fieldsk → Setsas follows. For a field K,14DF(K) ={∗}, if F(K) 6= ∅;∅, if F(K) = ∅.We define the canonical dimension (resp. p-dimension) of F to be theessential dimension (resp. p-dimension) of DF , and denote it by cdimk(F)(resp. cdimk(F ; p)). For more on canonical dimension, including the defini-tion of the canonical dimension of an algebraic group (which will not be usedin this paper), see [M13, Section 4].2.2 Quotient StacksA good reference for background on quotient stacks is [M13, Section 5].Let 1 → D → H → H → 1 be an exact sequence of algebraic groupsover k, where D is diagonalizable and central in H. For any K/k, letdK : H1(K,H) → H2(K,D) be the coboundary map. Let K/k and E ∈H1(K,H), and view E as an H-torsor over K. In particular, E is an H-scheme via the map H → H. We define a fibered category over the categoryof schemes over K (called the quotient stack for the action of H on E) anddenote it by [E/H]. The objects over a scheme X are diagrams (T, pi, φ)given by:Tφ //piEXwhere φ is H-equivariant and pi is an H-torsor. A morphism between objects(T, pi, φ) and (T ′, pi′, φ′) over X is a G-equivariant morphism from T to T ′satisfying the obvious commuting relationships over E and X.Remark 2.6. (See [M13, Section 5c]) If L/K is a field extension, then anobject (T, pi, φ) of [E/H](L) induces an H-equivariant map from T to ELover L, which in particular implies that EL is the image of T under the15induced map in cohomology H1(L,H) → H1(L,H). It follows that for anyfield L/K,[E/H](L) 6= ∅ ⇐⇒ EL is in the image of H1(L,H)→ H1(L,H)⇐⇒ dL(EL) = 0 ∈ H2(L,D)We now have a functor F : FieldsK → Sets given byF(L) = ([E/H](L)) / ≈for any L/K. This allows us to define edK([E/H]) = edK(F), edK([E/H]; p) =edK(F ; p), cdimK([E/H]) = cdimK(F), and cdimK([E/H]; p) = cdimK(F ; p).The following theorem provides a relationship between the essential di-mension of an algebraic group and the essential dimension of certain quotientstacks.Theorem 2.7. (See [M13, Section 5]) Let 1 → D → H → H → 1 be anexact sequence of algebraic groups, with D central and diagonalizable. Then1. edk(H; p) ≥ maxA,L(cdimL([A/H]; p)) + edk(D; p)− dim(H)2. edk(H) ≤ maxA,L(cdimL([A/H])) + edk(D) + edk(H)Here, L runs over all field extensions of k, A ∈ H1(L,H).Proof. Choose K/k and E ∈ H1(K,H). Then edk(H; p) ≥ edK(H; p) andfrom [BRV11, Corollary 3.3] (see also [M13, Corollary 5.7]), we have:edK(H; p) ≥ edK([E/H]; p)− dim(H).To complete the proof of the lower bound, we must show edK([E/H]; p) =cdimK([E/H]; p) + edk(D; p). If D ∼= (µp)t for some t then the result followsfrom [M13, Theorem 5.11]. One can check that their proof holds with only16trivial modifications in the general case when D is only assumed to be centraland diagonalizable. Alternatively, if we assume the image of H1(−, H) →H2(−, D) consists only of elements of p-primary order (which is true in allof our applications), then using [M13, Theorem 5.11] and [KM08, Theorem4.4 & Remark 4.5] we can deduce the result for a particular (E,K) withcdimK([E/H]; p) = maxA,L(cdimL([A/H]; p)); see Corollary B.4 in AppendixB.The upper bound can be deduced from [M13, Corollary 5.8 & Proposition5.10]. For completeness we provide a direct proof in Appendix B.173 Galois Cohomology of G/µIn this section we will discuss the Galois Cohomology of G/µ, and prove astronger version of Theorem 1.5. The following theorem is a more generalversion of [N11, Theorem 5.1.3].Theorem 3.1. Suppose Z(G) is a torus and let µ ≤ Z(G). From µ ↪→ Z(G)we get G/µ G, and hence an induced map in cohomology H1(K,G/µ)→H1(K,G) for any K/k. ThenH1(K,G/µ)→ H1(K,G)is injective, and identifies H1(K,G/µ) with r-tuples (E1, . . . , Er), with Ei ∈H1(K,Gi), such that for all χ = (c1, . . . , cr) ∈ Cµ,δ1K(E1)⊗c1 ⊗ · · · ⊗ δrK(Er)⊗cr = 0 ∈ Br(K)Note that δiK(Ei)⊗ci is always well-defined since exp(δiK(Ei)) | pbi.Corollary 3.2. From µf ↪→ µ we get G/µf  G/µ, and hence an inducedmap in cohomology γ : H1(K,G/µf )→ H1(K,G/µ) for any K/k. The mapγ is a bijection.We can use these two results to prove Theorem 3.4, which is a strongerversion of Theorem 1.5.Definition 3.3. Two codes are called (linearly) equivalent if one can beobtained from the other by repeatedly performing the following operations:1. Permuting entries i and j in every vector of the code, for any i, j withGni ∼= Gnj .2. Multiplying the ith entry in every vector of the code by any λ ∈(Z /pbi Z)∗, for any i with Gi ∼= GLni .18Theorem 3.4. Let µ, τ ≤ Z(G) and K/k. If Cµ is equivalent to Cτ , thenthere is an isomorphism of functors H1(−, G/µ) → H1(−, G/τ). In partic-ular, the essential dimension and essential p-dimension of G/µ depend onlyon Cµ up to equivalence.Proof of Theorem 3.4. In the case where Cµ = Cτ , the result is immediatefrom Corollary 3.2 and Remark 1.2. Now using the definition of equivalenceand induction, we may assume Cµ is obtained from Cτ by either permutingentries i and j where Gi ∼= Gj, or by multiplying the ith entry in every vectorof Cτ by some λ ∈ (Z /pbi Z)∗, for some i with Gi ∼= GLni . In the formercase, the automorphism of G which swaps Gi with Gj sets up an isomorphismG/µ ∼= G/τ and the result follows. In the latter case we must have that Z(G)is a torus, and using the description of H1(K,G/µ) given by Theorem 3.1,it is easy to check thatH1(K,G/µ) → H1(K,G/τ)(E1, . . . , Er) 7→ (E1, . . . , Ei−1, [E⊗λi ], Ei+1, . . . , Er)is an isomorphism. Here, [E⊗λi ] means the algebra of degree ni which isBrauer equivalent to E⊗λi (such an algebra is unique up to isomorphism).We will now prove Theorem 3.1 and Corollary 3.2, beginning with anumber of elementary results from Galois cohomology. Throughout, we willidentify H2(K,Grm) withr∏i=1H2(K,Gm) in the usual way.Remark 3.5. Let a = (a1, . . . , ar) ∈ X(Grm). Then the induced map incohomology is easily seen to be given by:a∗ :r∏i=1H2(K,Gm)→ H2(K,Gm)(x1, . . . , xr) 7→ xa11 · xa22 · . . . · xarr19Proposition 3.6. (See [N11, Lemma 5.1.2]) Let K/k be a field extension.1. If i : A ↪→ B is injective, where A and B are diagonalizable groups,then i∗ : H2(K,A) → H2(K,B) is injective. In particular, we canidentify H2(K,A) as a subgroup of H2(K,B).2. Suppose A ≤ Grm, thus giving X(Grm/A) a coordinate system. Then theimage of the map H2(K,A)→ H2(K,Grm) identifies H2(K,A) with thesubgroup of H2(K,Gm)r consisting of r-tuples (A1, . . . Ar) such that forall χ = (c1, . . . , cr) ∈ X(Grm/A),Ac11 ⊗ · · · ⊗ Acrr = 0 ∈ H2(K,Gm)Proof. 1. Put A into any exact sequence 1 → A → Glm → Glm /A → 1.Then since Glm /A is a split torus, we use the long exact sequence incohomology and Hilbert’s Theorem 90 to get:0→ H2(K,A)→ H2(K,Glm)It follows that H2(K,A) → H2(K,Grm) is injective. Now, choose anyembedding j : B ↪→ Gqm. Then Ai↪→ Bj↪→ Gqm and we get the followingdiagram:H2(K,A)i∗ //(j◦i)∗H2(K,B)j∗wwH2(K,Gqm)From our previous argument we know that (j ◦ i)∗ is injective, andhence so is i∗.2. From 1 → A → Grmp→ Grm /A → 1 we get the induced sequence incohomologyH2(K,A)→ H2(K,Grm)p∗→ H2(K,Grm /A)20By part (1), we can identify H2(K,A) with its image in H1(K,Grm),and by exactness this image equals ker(p∗). Let Y = (Y1, . . . , Yr) ∈H2(K,Grm). Since Grm /A is diagonalizable, p∗(Y ) = 0 if and only ifχ∗(p∗(Y )) = 0 for all χ ∈ X(Grm /A) ≤ X(Grm). If χ = (a1, . . . , ar) ∈X(Grm/A) then by Remark 3.5, χ∗(p∗(Y )) = Y⊗a11 ⊗ · · · ⊗ Y⊗arr , andthe result follows.Proof of Theorem 3.1. We argue as in [N11, §5.1]. Consider the followingdiagram:1 −−−→ Z(G) −−−→ G −−−→r∏i=1Gi −−−→ 1τyy∥∥∥1 −−−→ Z(G)/µ −−−→ G/µpi−−−→r∏i=1Gi −−−→ 1Since Z(G)/µ is a split torus, we can use the long exact sequences incohomology and Hilbert’s Theorem 90 to get the following diagram withexact rows:H1(K,r∏i=1Gi)(δ1K ,...,δrK)−−−−−−→ H2(K,Z(G))∥∥∥ τ∗y0 −−−→ H1(K,G/µ)pi∗−−−→ H1(K,r∏i=1Gi)δK−−−→ H2(K,Z(G)/µ)By [S97, I.5, Proposition 42], pi∗ is injective. Thus we may identifyH1(K,G/µ) with the set of r-tuples (E1, . . . , Er), Ei ∈ H1(K,Gi), suchthat (δ1K(E1), . . . , δrK(Er)) ∈ ker τ∗. From the exact sequence1→ µ→ Z(G)τ→ Z(G)/µ→ 121ker(τ∗) equalsim(H2(K,µ)→ H2(K,Z(G)))Viewed inside Br(K)r, this is the same as the image of the map H2(K,µ)→H2(K,Grm). Since the exponent of δiK(Ei) divides pbi and Cµ is obtainedfrom Cµ by reducing the ith coordinate modulo pbi , the result follows fromProposition 3.6.2.Proof of Corollary 3.2. The result is trivial if Z(G) is finite since in this casewe assumed µ = µf . Thus assume Z(G) is a torus. Consider the followingdiagram:H1(K,G/µ)γ //pi∗H1(K,G/µf )pif∗vvH1(K,G)By Theorem 3.1, pi∗ and pif∗ are injective. Since exp(δiK(Ei)) | pbi , then alsoby Theorem 3.1 pi∗ and pif∗ have the same image. It follows that γ is abijection.In the sequel, δK will continue to denote the coboundary mapH1(K,G)→H2(K,Z(G)/µ).Definition 3.7. For any K/k and E ∈ H1(K,G) we have a mapΨE,K : Cµ → Br(K)χ 7→ χ∗ ◦ δK(E)Clearly ΨE,K factors through ΨE,K : Cµ → Br(K) by definition of Cµ. Inthe sequel, if (c1, . . . , cr) ∈ Cµ (or Cµ) and Ei ∈ H1(K,Gi) for all i, then wewill write[δ1K(E1)⊗c1 ⊗ · · · ⊗ δrK(Er)⊗cr ]22to mean the Brauer class of[δ1K(E1)⊗c′1 ⊗ · · · ⊗ δrK(Er)⊗c′r ]where (c′1, . . . , c′r) is any set of integer representatives for (c1, . . . , cr) respec-tively.We end this section with the following result, which follows easily fromthe proof of Theorem 3.1.Lemma 3.8. Let K/k and c = (c1, . . . , cr) ∈ Cµ. Let δiK : H1(K,Gi) →H2(K,Z(Gi)), and view H2(K,Z(Gi)) ≤ Br(K) via the inclusion Z(Gi) ↪→Gm. If E = (E1, . . . , Er) ∈ H1(K,G) with each Ei ∈ H1(K,Gi), thenΨE,K(c) = [δ1K(E1)⊗c1 ⊗ · · · ⊗ δrK(Er)⊗cr ]234 On Minimal Generator MatricesThe Z-module Cµ can be thought of as a Z /pb Z-module, where b := max{b1,. . . , br}. In this section we will develop some preliminary algebraic resultsfor this context. In particular, we show in Example 4.8 that if we replacedour weight function w with the weight function pw, then the set of minimalgenerator matrices would remain unchanged. We assume for simplicity thatgenerating sets are ordered and do not contain 0.Let R be a local ring, I the unique maximal ideal of R, and let M bea finitely generated R-module. For m ∈ M , let m denote the image of min M/IM . The following lemma can be deduced from Nakayama’s Lemma,and is an immediate consequence of [AM69, Proposition 2.8].Lemma 4.1. The set {m1, . . . ,mt} is a generating set of minimal size forM as an R-module if and only if {m1, . . . ,mt} is a basis for M/IM as anR/I-vector space.Let w : M → Z≥0 be a function with w(m) 6= 0 if m 6= 0. For eachgenerating set B = {m1, . . . ,mt} of M , we definew(B) := (w(m1), . . . , w(mt), 0, 0, . . . ) ∈ ZNWe define word(B) to be the element of ZN obtained by rearranging the entriesof w(B) in decreasing order, and we call word(B) the w-profile of B.Remark 4.2. If B is arranged in weight-decreasing order, then word(B) =w(B).If γ is a w-profile of M , we call a generating set Bγ = {β1, . . . , βl} arepresentative generating set for γ if the w-profile of Bγ equals γ.We put a partial order ≤ on ZN as follows. For γ, β ∈ ZN, γ ≤ β if γi ≤ βifor all i ≥ 1, where γi denotes the ith component of γ ∈ ZN. Let Prof(M)(or, Profw(M)) denote the set of w-profiles of generating sets of M .24Theorem 4.3. (Prof(M),≤) has a unique minimal element, and this ele-ment is comparable to every other element.Proof. Prof(M) has no infinite descending totally ordered chain, so it sufficesto show that there is a unique minimal element. Towards a contradiction,suppose X and Y are representative generating sets for distinct minimalelements of Prof(M). By Lemma 4.1, both X and Y must have the samesize, say t. Thus write X = {x1, . . . , xt} and Y = {y1, . . . , yt} with w(x1) ≥· · · ≥ w(xt) and w(y1) ≥ · · · ≥ w(yt). Suppose s is minimal such thatw(xi) = w(yi) for all i > s. Since by assumption the w-profiles of X and Yare distinct, s ≥ 1. Without loss of generality, assume w(xs) < w(ys).We can extend the set {xs, . . . , xt} to a minimal generating set of Mby adding elements of Y . That is, for some J = {j1, . . . , js−1} ⊂ Y withw(j1) ≥ w(j2) ≥ · · · ≥ w(js−1), we have that{j1, . . . , js−1, xs, . . . , xt}is a basis for M/IM as an R/I-vector space. By Lemma 4.1,Γ := {j1, . . . , js−1, xs, . . . , xt}generates M as an R-module.We will now compare the weights of the elements of Γ with the weightsof the elements of Y . By construction, w(xi) = w(yi) for s + 1 ≤ i ≤ t,and w(xs) < w(ys) by assumption. Since J is an ordered subset of Y andw(y1), . . . , w(ys−1) are the largest s − 1 weights of elements in Y , we havew(ji) ≤ w(yi) for 1 ≤ i ≤ s − 1. Thus we have w(Γ) < w(Y ) = word(Y ).It remains to show that word(Γ) < word(Y ), since this would contradict theminimality of Y .Let ji = xi for s ≤ i ≤ t so that we may writeΓ := {j1, . . . , js−1, js, . . . , jt}.25If there exists a, b with a < b such that w(ja) < w(jb), then we swap these twoelements to get a new generating set Γ′. We must show that w(Γ′) < word(Y ),since then after finitely many such swaps we obtain a generating set Γ′′, whichis just Γ rearranged into weight-decreasing order. Thus inductively we wouldhave w(Γ′′) < word(Y ), and hence:word(Γ) = word(Γ′′) = w(Γ′′) < word(Y ).Note that the first inequality is true because Γ and Γ′′ contain the sameelements, and the second equality follows from Remark 4.2.Since Γ only changes in positions a and b, it is enough to show thatw(jb) ≤ w(ya) and w(ja) < w(yb) (note that the second inquality is automat-ically strict). Since w(Γ) < w(Y ), we have w(ja) ≤ w(ya) and w(jb) ≤ w(yb),and since Y is in decreasing order, we have w(yb) ≤ w(ya). Thus the firstinequality follows from w(jb) ≤ w(yb) ≤ w(ya), and the second inequalityfollows from w(ja) < w(jb) ≤ w(yb).Corollary 4.4. A generating set B = {m1, . . . ,ml} of M minimizesl∑i=1w(mi)if and only if word(B) is the minimal element of Prof(M).Corollary 4.5. Suppose we form a generating set of M inductively as fol-lows: Select m1 ∈ M with m1 6= 0 such that w(m1) is minimal. Supposem1, . . . ,mi have been selected. Then select mi+1 such that w(mi+1) is mini-mal among all elements m ∈M such that m /∈ 〈m1, . . . ,mi〉. Continue untilit is not possible to select another element. Then the w-profile of the resultinggenerating set is the minimal element of Prof(M).Example 4.6. [KM08, Remark 4.7] Take R = Fp, G a finite p-group, Dto be the elements of exponent at most p in Z(G) and M = X(D), whereX(D) is the group of characters of D. For x ∈M , define w(x) to be the leastdimension of a representation of G, say Vx, such that D acts by x. Then if26{x1, . . . , xt} is the basis provided by the greedy algorithm in Corollary 4.5,then Vx1 ⊕ · · · ⊕ Vxt is a faithful representation of G of minimal dimension.Corollary 4.7. Suppose τ : M → Z is a function such that τ(m1) ≥τ(m2) iff w(m1) ≥ w(m2). Then the generating sets {b1, . . . , bt} that mini-mizet∑i=1τ(bt) are precisely those whose w-profile is the minimal element ofProf(M).Example 4.8. Take R = Z /pb Z, with M and w to be arbitrary. Takingτ to be the function pw and applying Corollary 4.4 and Corollary 4.7 showsthat choosing a generating set {m1, . . . ,mt} of M that minimizest∑i=1w(mi)is the same as choosing a generating set {m′1, . . . ,m′l} of M that minimizesl∑i=1pw(m′i).275 Codes and the Brauer GroupIn this section we will prove Theorem 1.7, which gives formulas for boundson the essential dimension of G/µ involving the weights of elements of theassociated code Cµ. We do this by establishing a relationship between Cµand the image of the coboundary map δK : H1(k,G)→ H2(K,Z(G)/µ), andusing this relationship to construct our bounds.Recall from Definition 3.7 that for any K/k and E ∈ H1(K,G) we havea map ΨE,K : Cµ → Br(K) given byΨE,K : Cµ → Br(K)χ 7→ χ∗ ◦ δK(E)where δK denotes the coboundary map H1(k,G) → H2(K,Z(G)/µ). De-fine TE,K ≤ Br(K) to be the (finite) image of ΨE,K . Let {t1, . . . , tl} be agenerating set of TE,K withl∑i=1ind(ti) minimal, and defineind(E,K) =l∑i=1(ind(ti)− 1)We will prove the following theorem.Theorem 5.1. Let µ ≤ Z(G), and let Y be a minimal generator matrix forCµ with t rows. ThenmaxE,K(ind(E,K)) =(t∑i=1pw(Yi))− tWe can use this to prove Theorem 1.7 as follows. Since TE,K is a p-groupfor any K/k and E ∈ H1(K,G), by [KM08, Theorem 2.1 & Remark 2.9]28(and applying Remark 2.6 and Corollary 4.5) we have:cdimK([E/G]; p) = cdimK([E/G]) = ind(E,K).If rank(Cµ) = t then we can find a subgroup τ ≤ Z(G) with rank(Cτ ) = tand Cτ = Cµ. Thus by Theorem 1.5 we may assume rank(Cµ) = t, and henceedk(Z(G)/µ) = edk(Z(G)/µ; p) = t− d, whered =t, if Z(G) is connected;0, if Z(G) is finite.Theorem 1.7 is now an immediate consequence of Theorem 5.1 and Theorem2.7. Thus, it suffices to prove Theorem 5.1, which is the content of theremainder of this section.As in [KM08], if 1→ D → H → H → 1 is an exact sequence of algebraicgroups with D central and diagonalizable, and χ ∈ X(D), let Rep(χ)(H)denote the category of all finite dimensional representations ρ of H such thatρ(z) is scalar multiplication by χ(z) for all z ∈ D. In particular, we have thecategories:1. Repχ(G) corresponding to the exact sequence 1→ Z(G)→ G→ G→1, where χ ∈ X(Z(G)).2. Repχ(Gi) corresponding to the exact sequence 1 → Z(Gi) → Gi →Gi → 1, where χ ∈ X(Z(Gi)).3. Repχ(G/µ) corresponding to the exact sequence 1→ Z(G)/µ→ G/µ→G→ 1, where χ ∈ X(Z(G)/µ) ∼= Cµ.Let dK : H1(K,H)→ H2(K,D) be the coboundary map. If K/k is a fieldextension and E ∈ H1(K,H) then for any χ ∈ X(D) and V ∈ Rep(χ)(H)we have that ind(χ∗ ◦ dK(E)) divides dim(V ) (see [M13, Theorem 6.1.1]).Indeed, we have the diagram:291 −−−→ Gm −−−→ GL(V ) −−−→ PGL(V ) −−−→ 1χxxx1 −−−→ D −−−→ H −−−→ H −−−→ 1which gives the following in cohomology:H1(K,PGL(V )) −−−→ Br(K)x χ∗xH1(K,H)dK−−−→ H2(K,D)Since the image of H1(K,PGL(V )) → Br(K) consists of classes of algebrasof index dividing dim(V ), we see ind(χ∗ ◦ dK(E)) | dim(V ). Thus for anyχ ∈ X(D),ind(χ∗ ◦ dK(E)) | gcd{dim(V ) | V ∈ Rep(χ)(H)}Theorem 5.2. [KM08, Theorem 4.4 & Remark 4.5] Let 1 → D → H →H → 1 as above. Then there exists a field K/k and E ∈ H1(K,H) such thatfor any χ ∈ X(D) we haveind(χ∗ ◦ dK(E)) = gcd{dim(V ) | V ∈ Rep(χ)(H)}Since we are studying only reductive groups in characteristic zero, thiscan be reduced toind(χ∗ ◦ dK(E)) = gcd{dim(V ) | V irreducible, V ∈ Rep(χ)(H)}Before using this to prove Theorem 5.1, we need one more preliminaryresult. Recall that pai and pbi were defined to be the maximum index andexponent respectively of δiK(E) over all E ∈ H1(K,Gi) and K/k. We nowprove a lemma which says that they can both be attained by the same torsor.30Lemma 5.3. For each i, there exists K/k and E ∈ H1(K,Gi) such ind(δiK(E)) =pai and exp(δiK(E)) = pbi.Proof. Let V be a generically free representation of Gi. Then there existsa ‘friendly’ subset U ⊂ V (see [BF03, Theorem 4.7]), ie a dense open Gi-invariant subset U ⊂ V such that the categorical quotient U/Gi exists andU → U/Gi is a Gi-torsor. Then the generic fiber of this Gi-torsor givesa Gi torsor E with base K = k(U/Gi) (ie E ∈ H1(K,Gi)). By [GMS03,Example 5.4], E is versal. By [KM08, Theorem 4.4 & Remark 4.5] and thediscussion preceding Theorem 5.2 above, ind(δiK(E)) is the maximum valueof ind(δiL(A)) over all L/k and A ∈ H1(L,Gi), ie. ind(δiK(E)) = pai .Let pci be the exponent of δiK(E) ∈ Br(K). Consider the natural trans-formationH1(−, Gi)δi−→ H2(−, Z(Gi))P→ H2(−, Z(Gi))where P is the map sending A to A⊗pci for any A ∈ H2(L,Z(Gi)) and anyL/k. This natural transformation is a cohomological invariant of Gi, andin fact lands in H2(−, µpbi ) ⊂ H2(−, Z(Gi)). By construction, this invariantevaluates to the class of zero when applied to the versal torsor E ∈ H1(K,Gi)and hence by [GMS03, Theorem 12.3], the invariant is identically zero. Inparticular, δiL(E) has maximal exponent over all L/k and A ∈ H1(L,Gi),and hence bi = ci as required.Recall that for the exact sequence 1 → Z(G)/µ → G/µ → G → 1, wehave the notation d = δ and χ∗ ◦ dK(E) = ΨE,K(χ).Proof of Theorem 5.1. Recall that {Y1, . . . , Yr} is assumed to be a minimalgenerating set of Cµ. If K/k, E = (E1, . . . , Er) ∈ H1(K,G), and χ =(c1, . . . , cr) ∈ Cµ, then it follows from Remark 1.4 and Lemma 3.8 thatind(ΨE,K(χ)) = ind([E⊗c11 ⊗ · · · ⊗ E⊗crr ]) ≤ pw(χ). Since {ΨE,K(Y1), . . . ,31ΨE,K(Yr)} generate TE,K , the inequalitymaxE,K(ind(E,K)) ≤(t∑i=1pw(Yi))− tfollows immediately.It remains to provemaxE,K(ind(E,K)) ≥(t∑i=1pw(Yi))− tWe may assume that k is algebraically closed. It suffices to find K/k andE ∈ H1(K,H) such that ind(ΨE,K(χ)) ≥ pw(χ) for all χ ∈ Cµ, or equivalentlythat ind(ΨE,K(χ)) = pw(χ) for all χ ∈ Cµ (here χ means the image of χ inCµ). Indeed, thenΨE,K : Cµ → TE,Kwill be an isomorphism, and ind(E,K) will be the minimum value ofl∑i=1(pw(χi))− lover all generating sets χ1, . . . , χl of Cµ. By Example 4.8 this value ist∑i=1(pw(Yi))− t.By Theorem 5.2, we can find K/k and E ∈ H1(K,G) such that for allχ ∈ Cµ,ind(ΨE,K(χ)) = gcd(dim(V ) | V irreducible, V ∈ Rep(χ)(G/µ))If χ ∈ Cµ, then via the inclusion Cµ ↪→ X(Z(G)) we can view χ ∈32X(Z(G)). We can view a representation of G/µ as a representation of G viathe morphism G G/µ. If V is a representation of G such that Z(G) actsby τ ∈ X(Z(G)), then it is easy to see that V is a well-defined representationof G/µ precisely when τ ∈ Cµ. It follows that for any χ ∈ Cµ, the functorF : Rep(χ)(G/µ) → Rep(χ)(G)V 7→ Vis an isomorphism of categories. Thusind(ΨE,K(χ)) = gcd{dim(V ) | V irreducible, V ∈ Rep(χ)(G)}(1)Since k is algebraically closed, a representation V of G decomposes asV = V1 ⊗ · · · ⊗ Vr, where Vi is an irreducible representation of Gi for i =1, . . . , r. If χ = (c1, . . . , cr) ∈ Cµ then Z(Gi) acts on Vi by the character(ci) ∈ X(Z(Gi)).If Ji is any set of integers for 1 ≤ i ≤ r, then one can easily check thefollowing gcd result:gcdji∈Ji,i=1,...,r{j1 · . . . · jr} = gcdj1∈J1{j1} · . . . · gcdjr∈Jr{jr}Applying this result with Ji = {dim(W ) | W ∈ Rep(ci)(G)}, (1) reduces to:ind(ΨE,K(χ)) =r∏i=1(gcd{dim(Vi) | Vi irreducible, Vi ∈ Rep(ci)(Gi)}).By Lemma 5.3, there existsK/k and Ti ∈ H1(K,Gi) such that ind(δiK(Ti)) =pai and exp(δiK(Ti)) = pbi . By Remark 1.4, if ci is the reduction of ci modpbi , then ind(δiK(Ti)⊗ci) = pai−vi(ci). Note that, by Remark 3.5, δiK(Ti)⊗ci =ci∗ ◦ δiK(Ti), and so by the discussion preceding Theorem 5.2 applied to the33exact sequence 1→ Z(Gi)→ Gi → Gi → 1,gcd{dim(Vi) | Vi irreducible, Vi ∈ Rep(ci)(Gi)}is at least as large as ind(δiK(Ti)⊗ci). Thus we havegcd{dim(Vi) | Vi irreducible, Vi ∈ Rep(ci)(Gi)}≥ pai−vi(ci)and hence,ind(ΨE,K(χ)) ≥r∏i=1pai−vi(ci) = p∑ri=1(ai−vi(ci)) = pw(χ)as required.Remark 5.4. In the case where the image of the coboundary map δK is wellunderstood, one can prove Theorem 5.1 using the theory of central simplealgebras; see Appendix A.Remark 5.5. An alternate method to prove the lower bound on ed(G/µ; p)in Theorem 1.7 would be to find a finite p-subgroup Y of G/µ and apply thebounded(G/µ; p) ≥ ed(Y ; p)− dim(G/µ).Suppose that for 1 ≤ i ≤ r, one can find a finite p-subgroup Hi ≤ Gi withZ(Hi) = µpbi ≤ Z(Gi), and such that the maximal index and exponentof the coboundary map H1(−, Hi/Z(Hi)) → H2(−, Z(Hi)) are pai and pbirespectively. Then set H = H1 × · · · × Hr, so that we have H/µf ≤ G/µ.Theorem 1.7 applies, and gives ed(H/µf ; p) ≥t∑i=1pw(Yi). Combining thiswith the bound for the essential dimension of a subgroup above yields:34ed(G/µ; p) ≥ ed(H/µf ; p)− dim(G/µ)≥t∑i=1pw(Yi) − d− dim(G)where d =t, if Z(G) is connected;0, if Z(G) is finite.This shows that, if one found such subgroups Hi ≤ Gi, then the lowerbound on ed(G/µ; p) provided by computing the essential p-dimension ofH/µf would be the same as the lower bound in Theorem 1.7.Computing the essential p-dimension of H/µf can be done using [KM08,Theorem 4.1], which says that the essential p-dimension of a finite p-groupover k equals the minimal dimension of a faithful representation of that group.This is used in [MR10, Theorem 1.2] to give a formula for the essential p-dimension of a finite p-group purely in terms of its group structure. Forexample, in the case Gi = GLp, one can take the group Hi to be any finitenon-abelian group of order p3, and the inclusion Hi ↪→ Gi given by anyfaithful irreducible representation of Hi; see the group Γ defined in the proofof [MR10, Theorem 1.5].356 An Upper BoundIn this section we will prove Theorem 1.9. Let H = GL(V1) × · · · × GL(Vr)and H ′ = SL(V1) × · · · × SL(Vr) where Vi = kni . Then both H and H ′ actnaturally on the vector spaceVc1,...,cr = V⊗c11 ⊗ · · · ⊗ V⊗crrwhere c1, . . . , cr ∈ {±1} (here, V −1 denotes the dual of V ). We denote sucha representation by ρ(c1,...,cr) : H → GL(Vc1,...,cr).Theorem 6.1. Suppose r ≥ 3, 2 ≤ n1 ≤ . . . ≤ nr and nr ≤n1·...·nr−12 . Thenthe kernel of ρ(c1,...,cr) is central in H, and the action of H/ ker(ρ(c1,...,cr)) onVc1,...,cr is generically free in all but the following exceptional cases:1. r = 3, n1 = 2, n2 = n3.2. r = 4, n1 = n2 = n3 = n4 = 2.3. r = 3, n1 = n2 = n3 = 3.Proof. We first reduce to the case where (c1, . . . , cr) = (1, . . . , 1). Supposethe theorem is true in this case, and let (c1, . . . , cr) ∈ {±1}r. By choosingbases of V1, . . . , Vr we can identify Vi with V⊗cii (we can take the identitymap if ci = 1). Define an automorphism:σ : H → H(h1, . . . , hr) 7→ (h∗1, . . . , h∗r)whereh∗i =hi if ci = 1;(h−1i )T if ci = −1.36Now ρ(c1,...,cr) is isomorphic to the representation ρ(1,...,1) ◦σ. Since Z(H) is acharacteristic subgroup, we see that the theorem holds for ρ(c1,...,cr) as well.Denote ρ(1,...,1) and V(1,...,1) by ρ and V repectively. It remains to provethe theorem is true for the representation ρ.By [P87, Theorem 2], with the conditions in our theorem, the H ′/Z(H ′)action on P(V ) is generically free. Thus the stabilizer in general positionfor the H ′-action on V is central. Since a central element of H ′ either actstrivially on V or non-trivially on all non-zero elements of V , we see that thestabilizer in general position for the H ′-action on V is equal to the (central)kernel of this action. It remains to extend this result to the H-action on V .We may assume k = k for the purposes of checking whether a represen-tation is generically free. Suppose v ∈ V is in general position and h ∈ Hstabilizes v. Write h = λ ·h′ with λ ∈ (k∗)r and h′ ∈ H ′. Then we must haveh′ acting by scalar multiplication on v, and hence h′ (mod Z(H ′))stabilizesthe image of v in P(V ). Thus h′ ∈ Z(H ′), and hence h ∈ Z(H). As before, acentral element of H either acts trivially on V or acts non-trivially on everynon-zero element of V , and so the stabilizer of a point v ∈ V in generalposition equals the (central) kernel of ρ. Thus the H/ ker(ρ)-action on V isgenerically free, as required.We can now apply this to the essential dimension of G/µ, where µ is asubgroup of Z(G) and Gi ≤ GL(Vi) is a faithful representation of dimensionni whose central character is the identity character. In other words, with Has above we have G ≤ H.Let χ = (c1, . . . , cr) ∈ Cµ. For 1 ≤ j ≤ r, define cˆj to be the uniqueinteger such that cˆj ≡ cj mod pbj and −pbj/2 < cˆj ≤ pbj/2. Define arepresentation ρχ of G byVχ =r⊗i=0V ⊗cˆjiwhere V ⊗1i is the standard representation, V⊗0i is the trivial representation,37and V ⊗−1i is the dual of Vi.We define the set m(χ) bym(χ) = {i | ci 6= 0}Definition 6.2. We say that χ = (c1, . . . , cr) ∈ Cµ is acceptable if the fol-lowing conditions hold:1. −1 ≤ cˆj ≤ 1 for 1 ≤ j ≤ r.2. maxi∈m(χ){ai} <12∑j∈m(χ)aj (note this implies |m(χ)| ≥ 3).3. {ni}i∈m(χ) 6= {2, n, n}, {2, 2, 2, 2} or {3, 3, 3}, for any positive integer n.By Theorem 6.1, if χ is acceptable then the stabilizer in general positionfor ρχ equals ker ρχ, and if (g1, . . . , gr) ∈ ker ρχ then gi ∈ Z(Gi) for alli ∈ m(χ).Remark 6.3. The first condition in the definition of acceptable impliesdim(Vχ) = pw(χ) for any acceptable χ.Definition 6.4. Let Y be a generator matrix for Cµ with rows Y1, . . . , Ym.We say that Y is acceptable if for each j, 1 ≤ j ≤ r, there exists i such thatyij 6= 0 and Yi is acceptable.Theorem 6.5. Suppose µ ≤ Z(G) and Cµ has an acceptable generator matrixY with rows Y1, . . . , Yt. Thened(G/µ) ≤t∑i=1dim(VYi)− dim(G)− dwhere d =t, if Z(G) is connected;0, if Z(G) is finite.38Proof. Let zi = (yˆi1, . . . , yˆir) ∈ X(Z(G)) for 1 ≤ i ≤ t and let τ be the sub-group of Z(G) such that Cτ is generated by z1, . . . , zt. Then by constructionCµ = Cτ .To each Yi, we have the associated representation ρYi : G→ GL(VYi). If Yiis acceptable we have that the stabilizer in general position for ρYi is ker ρYi .Let ρ =⊕i ρYi and let (v1, . . . , vt) be in general position in V =⊕i VYi . Inparticular, vi is in general position in VYi for all i. Then it follows from thecomments after Definition 6.2 thatStabρ(v) =r⋂i=1StabρYi vi≤⋂i | Yi acceptableStabρYi vi=⋂i | Yi acceptableker ρYi≤ Z(G)where for the last containment we use the property that for each j thereexists i such that yij 6= 0 and Yi is acceptable. In particular, ker ρ ≤ Z(G).Thus by construction we have ker ρ = τ , and hence the stabilizer in generalposition for ρ : G → GL(W ) equals τ . It follows that ρ is a generically freerepresentation of G/τ , and henceed(G/τ) ≤t∑i=1dim(VYi)− dim(G/τ)By observing dim(G/τ) = dim(G) + d and applying Theorem 1.5 we get thedesired result.Notice that a very acceptable generator matrix is acceptable. Theorem1.9 follows from Theorem 6.5 by applying Remark 6.3 and the lower bound39in Theorem 1.7.407 Central Simple Algebras with Tensor Prod-uct of Bounded Index.7.1 General ResultsSuppose p is a prime, r ≥ 1, a1, . . . , ar ∈ Z≥1, and z ∈ Z≥0. Consider thefunctor F (a1,...,ar);z : Fieldsk → Sets given byF (a1,...,ar);z(K) =r-tuples (A1, . . . , Ar) of central simple K-algebrasup to isomorphism, such that deg(Ai) = pai ∀i,and ind(A1 ⊗ . . .⊗ Ar) | pz.This functor places a restriction on the index of a certain algebra, andis reminiscent of the functor H1(−,GLpa /µps) discussed in the Introduction,which places a restriction on the exponent of a certain algebra:H1(K,GLpa /µps) ={central simple K-algebras A up to isomorphismsuch that deg(A) = pa and exp(A) | ps}Projection to the first r algebras sets up an isomorphism of functors:F (a1,...,ar,z);0 → F (a1,...,ar);zand thus we may assume z = 0. We will also assume a1 ≤ a2 ≤ · · · ≤ ar.The functor F (a1,...,ar);0 classifies r-tuples (A1, . . . , Ar) of central simplealgebras of specified degrees satisfying the splitting condition A1⊗· · ·⊗Ar = 1in Br(K). If we ignored the condition that the tensor product is split, wewould be left with the functor T = H1(−,PGLa1) × · · · × H1(−,PGLpar ).The essential dimension of this function is at most quadratic in the pai , thatisedk(T ) < p2a1 + . . .+ p2arWe will see in Theorem 7.2 below that, unless ar ≥ a1+· · ·+ar−1 or r ≤ 2, the41leading term in the essential dimension of F (a1,...,ar);0 is pa1+···+ar . In otherwords, when trying to descend a tuple of algebras satisfying the splittingcondition, enforcing the splitting condition may require significantly morevariables than would be needed to just define the algebras individually.If we set Gi = GLni (ni = pai), G = G1 × · · · ×Gr, andµ = {(λ1, . . . , λr) ∈ Z(G) | λ1 · . . . · λr = 1}then Cµ = [1, . . . , 1] and by Theorem 3.1 we haveF (a1,...,ar);0 ∼= H1(−, G/µ).Lemma 7.1. Let K/k be a field extension.a) If r = 1 then Fa1;0(K) = {pt}.b) If r ≥ 2 and ar ≥r−1∑i=1ai (which is automatic if r = 2), then projection tothe first r − 1 algebras gives an isomorphismγ : F (a1,...,ar);0 →r−1∏i=1H1(−,PGLpai ).Proof. Part a) is obvious. For part b), a tuple (A1, . . . , Ar) in F (a1,...,ar);0(K)for some field K is uniquely determined by A1, . . . , Ar−1, since Ar is theunique central simple algebra of degree par whose Brauer class is(A1 ⊗ · · · ⊗ Ar−1)opIt follows that γ is injective. To see that γ is surjective, observe that for an42arbitrary tuple (A1, . . . , Ar−1) inr−1∏i=1H1(−,PGLpai ), we haveind(A1 ⊗ · · · ⊗ Ar−1)op ≤r−1∏i=1ind(Ai) ≤ pa1+···+ar−1 .Thus the condition on the ai’s guarantees that the Brauer class(A1 ⊗ · · · ⊗ Ar−1)opwill in fact have a representative central simple algebra Ar of degree par , andthus (A1, . . . , Ar−1) is equal to γ(A1, . . . , Ar).Theorem 7.2. If r ≥ 3, ar <r−1∑i=1ai, and (pa1 , . . . , par) /∈ {(2, n, n)n∈Z,(2, 2, 2, 2), (3, 3, 3)}, thenedk(F (a1,...,ar);0) = edk(F (a1,...,ar);0; p) = p∑ri=1 ai −r∑i=1p2ai + r − 1Proof. The matrix [1, . . . , 1] is an acceptable and minimal generator matrixfor Cµ. Since F (a1,...,ar);0 ∼= H1(−, G/µ) we haveedk(F (a1,...,ar);0) = edk(G/µ)edk(F (a1,...,ar);0; p) = edk(G/µ; p)and so the result follows from Theorem 1.9.Now let r ≥ 3, and a1 ≤ a2 ≤ · · · ≤ ar−1 such that (pa1 , . . . , par−1) /∈{(2, n)n∈Z, (3, 3)}. Let ar =(r−1∑i=1ai)− 1. Then by Theorem 3.143F (a1,...,ar−1);ar(K) =m-tuples (A1, . . . , Ar−1) of central simple k-algebrasup to isomorphism, such that deg(Ai) = pai ∀i andA1 ⊗ . . .⊗ Ar−1 is not a division algebra.Corollary 7.3. Let r ≥ 3, and a1 ≤ a2 ≤ · · · ≤ ar−1 such that (pa1 , . . . , par−1) /∈{(2, n)n∈Z, (3, 3)}. Let ar =(r−1∑i=1ai)− 1. Thenedk(F (a1,...,ar−1);ar) = edk(F (a1,...,ar−1);ar ; p) = p2ar+1 −r∑i=1p2ai + r − 1Proof. Theorem 7.2 applies, and gives:edk(F (a1,...,ar−1);ar) = edk(F (a1,...,ar−1);ar ; p)= edk(F (a1,...,ar);0; p)= pa1+···+ar −r∑i=1p2ai + r − 1= p2ar+1 −r∑i=1p2ai + r − 17.2 Small Cases in Theorem 7.2We now turn to the special cases from Theorem 7.2. Recall that r ≥ 3,a1 ≤ a2 ≤ · · · ≤ ar. If we set Gi = GLni (ni = pai) for 1 ≤ i ≤ r then thisfunctor is isomorphic to H1(−, G/µ), where Cµ = [1, 1, . . . , 1].Remark 7.4. Note that in all cases, projection to the (r− 1)st algebra givesa surjective morphism of functors from F (a1,...,ar−1);ar to H1(−,PGLpar−1 ).44By Theorem 2.1, we get the lower bounded(F (a1,...,ar−1);ar ; p) ≥ edk(PGLpar−1 ; p)We have a natural representation of G/µ given by W = V1 ⊗ · · · ⊗ Vr,where each Vi is a vector space of dimension ni, and Gi acts on Vi. Outsideof the exceptional cases (pa1 , . . . , par) ∈ {(2, n, n)n∈Z, (2, 2, 2, 2), (3, 3, 3)}, thisrepresentation is generically free. For the exceptional cases, one could tryreplacing the representation W := V1⊗· · ·⊗Vr with W⊕W and checking if itis generically free. However, we can find a better upper bound using normal-izers of maximal tori, by instead finding an upper bound on this normalizerand applying Theorem 2.2.A maximal torus M in GLn is the set of diagonal matrices, and thenormalizer of M is M o Sn where Sn acts by permutation. Let T be amaximal torus in GLn1 × · · · ×GLnr . Then T/µ is a maximal torus of G/µ.One can check that the normalizer of T/µ has the form T/µ o S, whereS = Sn1 × · · · × Snr and each Sni acts by permutation on Gi⋂T/µ.Since T  T/µ we can identify X(T/µ) with a subgroup of X(T ). RecallX(T ) = Zn1 × . . . × Znr . Then X(T/µ) is the set of characters in X(T )which are trivial on µ. Let Ci = Zni so that X(T ) = C1 × . . .× Cr, and letγi : Ci → Z be the augmentation map. Take t = (t1, . . . , tr) ∈ µ, so thatti ∈ Z(Gi) and t1 · . . . · tr = 1. For χ = (c1, . . . , cr) ∈ X(T ) with ci ∈ Ci, wehave:χ(t) = tγ1(c1)1 · . . . · tγr(cr)rIt is now easy to see (for example, by writing tr = (t1 · . . . · tr−1)−1) thatχ(t) = 1 for all t ∈ µ precisely whenγ1(c1) = γ2(c2) = . . . = γr(cr)and this is the condition that describes X(T/µ) as a submodule of X(T ).45We have the induced action of S on X(T ), where each Sni acts by permu-tation on Ci. To any S-invariant generating set Λ ⊂ X(T ), [MR09, Section 3]describes a method to construct a representation VΛ of T/µoS, of dimension|Λ|. To use this to give an upper bound on essential dimension, we requirethis representation VΛ to be generically free, and the following lemma tellsus how to check this.Lemma 7.5. ([MR09, Lemma 3.3]) Let R be the kernel of the natural mapof Z[S]-modules Z[Λ] → X(T ). Then the representation VΛ is genericallyfree precisely when the S-action on R is faithful.We construct an S-invariant generating set Λ of X(T/µ) as follows. Letcimi , where 1 ≤ i ≤ r and 1 ≤ mi ≤ ni be the vector in Ci which has a 1 inthe mthi position and a zero in all other positions. Then defineΛ = {(c1m1 , . . . , crmr) | 1 ≤ mi ≤ ni (1 ≤ i ≤ r)}Then Λ clearly generates X(T/µ) and is S-invariant. It remains toverify the condition in the lemma that S acts faithfully on R. Suppose1 6= (σ1, . . . , σr) ∈ S. Without loss of generality, we may assume σ1 6= 1 andσ1(1) = j 6= 1. Consider the elements r1, r2, r3, r4 ∈ R:r1 = (c11, c21, . . . , cr1)r2 = (c11, c22, c31, c41, . . . , cr1)r3 = (c11, c21, c32, c41, c51, . . . , cr1)r4 = (c11, c22, c32, c41, c51, . . . , cr1)Setting r = r1 − r2 − r3 + r4 ∈ Z[Λ] we see r ∈ R, but each σ1(ri) willhave the form (c1j , . . . ). Thus σ(r) 6= r, and the result follows.46Remark 7.6. Note that this argument depended crucially on r ≥ 3, but noton p or n1, . . . , nr.Since |Λ| =r∏i=1ni, and dim(T/µ o S) = dim(T/µ) =(r∑i=1ni)− r + 1,we get the following corollary by Theorem 2.5.Corollary 7.7. We have ed(T/µ o S) ≤r∏i=1ni −(r∑i=1ni)+ r − 1. Inparticular, by Theorem 2.2:ed(G/µ) ≤r∏i=1ni −(r∑i=1ni)+ r − 1.Theorem 7.8. 1. p = 2, a > 1:(a− 1)2a + 1 ≤ edk(F (1,a,a);0) ≤ 22a+1 − 2a+12. p = 2: 4 ≤ edk(F (1,1,1,1);0) ≤ 113. p = 3: 2 ≤ edk(F (1,1,1);0) ≤ 20The bounds are also valid for essential p-dimension in all four cases.Proof. The upper bounds all follow from Corollary 7.7. The lower boundsin (1) and (3) follow from Remark 7.4 and [R00, Theorem 9.3 & Proposition9.8a]. For the lower bound in part (2), observe that projection the the first2 algebras gives a surjective morphism of functorsF (1,1,1,1);0 → H1(−,PGL2)×H1(−,PGL2).It follows from [RY00, Section 8] that PGL2×PGL2 has a self-centralizingfinite 2-subgroup of rank 4, and hence by [RY00, Theorem 7.8.1] and [RY00,47Lemma 8.5.7]edk(PGL2×PGL2; p) = 4.Thus the lower bound follows from Theorem 2.1.We could use the methods of the above theorem to prove that for p = 2we have2 ≤ edk(F (1,1,1);0) ≤ 4but in this case we can determine the essential dimension exactly.Theorem 7.9. For p = 2,edk(F (1,1,1);0) = edk(F (1,1,1);0; 2) = 3.Proof. We begin with the upper bound. Recall that F (1,1,1);0(L) classifiestriples of quaternion algebras (Q1, Q2, Q3) (up to isomorphism over L) suchthatQ1⊗Q2⊗Q3 is split. By a theorem of Albert [L05, Theorem III.4.8], sinceQ1 ⊗Q2 is not a division algebra, we may write Q1 = (a, b) and Q2 = (a, c).Thus Q3 ∼= Q1 ⊗ Q2 ∼= (a, bc). Hence the triple (Q1, Q2, Q3) descends tothe field K = k(a, b, c) while still satisfying the splitting property. Thusedk(F (1,1,1);0) ≤ 3.To prove the lower bound, consider the mapΓ : F (1,1,1);0 → H1(−, SO4)(Q1, Q2, Q3) 7→ αHere α is defined to be the quadratic form such that α ⊕ H⊕H ∼=N(Q1)⊕−N(Q2) where H = 〈1,−1〉 is the 2-dimensionsal hyperbolic form.(Equivalently, using the definition of the Albert form given in [L05, p.69], αis the quadratic form such that α ⊕ H ∼= AQ1.Q2 where AQ1,Q2 is the Albertform of Q1 and Q2.) By the Witt cancellation theorem, α is unique up to48isomorphism. We can explicitly compute α as follows, for arbitrary K/k.Suppose Q1 = (a, b) and Q2 = (a, c) as above. ThenN(Q1) = 〈〈−a,−b〉〉 = 〈1,−a,−b, ab〉N(Q2) = 〈1,−a,−c, ac〉and soN(Q1)⊕−N(Q2) = 〈1,−1,−a, a,−b, c, ab,−ac〉.This is isomorphic to〈−b, c, ab,−ac〉 ⊕H⊕H .Thus α ∼= 〈−b, c, ab,−ac〉. SinceH1(K, SO4) classifies 4-dimensional quadraticforms over K of discriminant 1, it is clear from specializing the values a, band c in our expression for α that Γ is surjective. Since edk(SO4; 2) = 3 (see[RY00, Theorem 8.1 & Remark 8.2]), by Theorem 2.1 we haveedk(F (1,1,1);0; 2) ≥ 3.498 Examples of Linear Error-Correcting CodesGiven a code Cµ, the two questions we must determine are:1. What does a minimal generator matrix look like?2. Can a minimal generator matrix be chosen such that all the coefficientsare 0,−1 or 1?Example 4.8 and Corollary 4.5 provide a partial answer to the first ques-tion: the greedy algorithm will always result in a minimal generator matrix.In the case bi = 1 for all i, the code Cµ is a linear error-correcting code overFp in the traditional sense. If a1 = a2 = · · · = ar then the weight on Cµ willjust be a1 times the usual Hamming weight. Of note though is that unlessGi = GLp for all i, our notion of equivalence of codes does not coincide withthe usual notion of linear equivalence of linear error-correcting codes.Example 8.1. Cµ is a traditional code, and the weight is a scaling of theHamming weight, when:1. p arbitrary: Gi ∈ {GLp, SLp}, for all i.2. p = 2: Gi ∈ {GL2,GO2,GSP2, SL2 = SP2,O2}, for all i.3. p = 2: Gi ∈ {GOn,GSPn, SPn,On,GO+n , SOn} where n = 2a > 2, forall i.4. p = 3: Gi = E6 ≤ GL27 for all i.In this case, if a code can be generated by its minimum (Hamming) weightvectors then this completely answers the first question above. In this section50we recall a class of traditional codes (called generalized Reed-Muller codes)that are generated by their minimum weight vectors, and where the secondquestion sometimes has an affirmative answer. The primary references are[DK00] and [AK92, Section 5].Generalized Reed-Muller codes are a family of codes that is closed undertaking dual codes and contains, for example, all extended Hamming codes.We recall the definition. Let q be a power of a prime p, m ≥ 1, r ≥ 1such that r ≤ m(q − 1), and V = Fmq with standard basis e1, . . . , em. Theunderlying vector space for the generalized Reed-Muller code RFq(r,m) is thevector space W of all functions from V to Fq. We have dim(W ) = qm, andour distinguished basis for W is the set of characteristic functions of vectorsin V .Any monomial n(x1, . . . , xm) defines an element of W , since we can eval-uate n(v) by writing v =∑mi=1 ziei, with zi ∈ Fq, and defining n(v) =n(z1, . . . , zm). Of course, xqi = xi as elements of W by Fermat’s little theo-rem, and it follows that we can identify W with the underlying vector spaceof the ringFq[x1, . . . , xm]/(xq1 − x1, . . . , xqm − xm)For any monomial n = xi11 · . . . · ximm , we define degxj(n) = ij and deg(n) =i1 + · · · + im. The reduced monic monomials (that is, monic monomials nwith degxi n < q ∀i) give us a new basis of W . We now define RFq(r,m) tobe the span in W of all reduced monic monomials n with deg(n) ≤ r.Theorem 8.2. Write r = t(q − 1) + s with 0 ≤ s < q − 1.1. [AK92, Theorem 5.5.3] The minimum weight codewords of RFq(r,m)have weight (q − s)qm−t−1.2. [DK00, Theorem 1] If q = p then RFq(r,m) is generated by its minimumweight codewords.3. [DGM70, Theorem 2.6.3] If s = 0 and q = p then RFq(r,m) is generatedby minimum weight codewords whose entries are all 0 or 1.51Using this theorem and Theorems 1.7 and 1.9, we conclude the following.Corollary 8.3. Suppose Cµ = RFp(r,m) (up to equivalence). Write r =t(p− 1) + s with 0 ≤ s < p− 1. Let D = dimRFp(r,m).1.edk(G/µ; p) ≥ Dp(p−s)pm−t−1 − d− dim(G)edk(G/µ) ≤ Dp(p−s)pm−t−1 − d+ edk(G)2. Suppose (p− 1) | r.edk(G/µ; p) = Dp(p−s)pm−t−1 − d− dim(G)where d =D, if Z(G) is connected;0, if Z(G) is finite.Remark 8.4. [AK92, Theorem 5.4.1] The dimension of RFq(r,m) is givenby:dimRFq(r,m) =r∑i=0m∑k=0(−1)k(mk)(i− kq +m− 1i− kq)[DGM70, Theorem 2.6.3] gives us an explicit description of the minimumweight codewords, which we recall here in the s = 0 case. Let w1, . . . , wt ∈ Fq.Consider the codeword:P (x1, . . . , xm) =t∏i=1(1− (xi − wi)q−1)We see that the degree of P is precisely t(q−1) = r, which is the maximumallowable degree of a polynomial defining an element of RFq(r,m). It is alsoclear the codeword corresponding to P contains entries 0 and 1. The entryin the codeword associated to the vector v ∈ V is equal to 1 precisely whenxi(v) = wi for 1 ≤ i ≤ t. Recall that a k-flat in V is a subset of V of the52form v0 + U where vo ∈ V and U is a k-dimensional subspace of V . Thenwe have that the codeword corresponding to P is the incidence vector of the(m − t)-flat given by (w1, . . . , wt, ∗, . . . , ∗). Thus the weight of P is qm−t,and P corresponds to a minimum weight vector. From [DGM70, Theorem2.6.3], all minimal weight codewords in W can be obtained from codewordsof the form P , along with scalar multiplication and replacing x1, . . . , xt withany other set of t linearly independent linear polynomials. In particular,all minimal weight codewords lie in the subspace generated by the minimalweight codewords whose entries are all 0 and 1.Remark 8.5. The question of whether a code has a generator matrix whereeach element has minimal weight has been studied for other classes of codesas well, see [KL06, Section 1] for an overview. In particular, the authorsshow that certain extended binary BCH codes are always generated by theirminimum weight vectors.539 ConclusionIn this report we have computed bounds on the essential dimension of certainfamilies of reductive algebraic groups. One of the motivating examples wasthe group G/µ, whereG = GLpa1 × · · · ×GLpar ,p is a prime and µ is a central subgroup of G. This example was particularlyinteresting because we interepreted the Galois cohomology of this group astuples of central simple algebras satisfying relations in the Brauer group.Surprisingly, this problem became easier for r ≥ 3, and we were able togive asymptotically sharp bounds (or even exact values) for many familiesof central subgroups. We also looked at one particular family of centralsubgroups where ed(G/µ) grew ‘exponentially in r’, or informally:ed(G/µ) = pa1+···+ar − smaller order terms.This is in contrast to the group PGLpa1 × · · ·×PGLpar , whose cohomologyclassifies tuples of central simple algebras without any additional conditions.In this case, the essential dimension grows much more slowly in r:ed(PGLpa1 × · · · × PGLpar ) < p2a1 + · · ·+ p2ar .Our bounds for the essential dimension of G/µ were given in terms ofa ‘code’ Cµ and a weight function on this code. Specifically, computingthe upper and lower bounds depend on finding a minimal weight generatormatrix for Cµ. For some families of codes (for example, see Sections 7 and8) we could determine a minimal weight generator matrix. For other, morecomplicated codes, it may be more difficult to determine the structure of aminimal weight generator matrix. This is related to the general notion ofweight distribution in codes.54In section 7 we studied an interesting family of groups where Cµ wasparticularly simple, and the cohomology could be interpreted as tuples ofcentral simple algebras satisfying certain index conditions. One example ofthis was the functor of pairs of central simple algebras (A,B) of degree pa,where A⊗B is not a division algebra. This functor was of particular interestboth because of its connection to linkages of cyclic algebras (Theorem 7.9),and because the problem of determining a structural condition for when thetensor product of two central simple algebras is not a division algebra is anopen problem ([ABGV12, Problem 9.1]). Both of these connections could beareas for future research.One of the primary limitations of this research was the requirement thateach ni be a power of the same prime. This requirement was not needed toconclude that the essential dimension of G/µ depended only on Cµ (Theorem1.5), and the upper bound (Theorem 6.5) can also be formulated without thisassumption. However, it was needed to deduce the lower bound. Specifically,we appealed to [KM08] regarding a formula for the canonical dimension ofa finite p-subgroup of the Brauer group. Although one can find a formulafor the canonical p-dimension of a finite subgroup of the Brauer group (see[KM08, Remark 2.10]), it is unclear what the formula for absolute canonicaldimension might look like when the finite subgroup is not a p-group (see[M13, Conjecture 4.23] for a related conjecture), and this could also be thesubject of future research.55Bibliography[ABGV12] Auel, Asher; Brussel, Eric; Garibaldi, Skip; Vishne, Uzi. Openproblems on central simple algebras. Transform. Groups 16 (2011), no.1, 219-264.[AK92] Assmus, E. F., Jr.; Key, J. D. Designs and their codes. CambridgeTracts in Mathematics, 103. 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Soc. 333 (1992), no. 2,729-739.59Appendix A Disjoint Central Simple AlgebrasDefinition A.1. [IK99, Definitsubion 2.5] A collection of central simple alge-bras {A1, . . . , Ar} over a field K is called disjoint if for all m1, . . . ,mr ∈ Z≥0,we haveind(A⊗m11 ⊗ . . .⊗ A⊗mrr ) = ind(A⊗m11 ) · . . . · ind(A⊗mrr )As a consequence of the latter part of the proof of Theorem 5.1 (in the caseµ = {1}), we have that there exists K/k and Ei ∈ H1(K,Gi), 1 ≤ i ≤ r, suchthat the underlying division algebras of the set {δiK(Ei)}i=1,...,r form a disjointset of division algebras over K, with ind(δiK(Ei)) = pai and exp(δiK(Ei)) =pbi . In the case Gi ∈ {PGLni ,PGSp2ai ,PGO2ai ,PGO+2ai (ai ≥ 2)} we canprove this directly; see Corollary A.6 below. When combined with Remark1.4 this provides an alternate proof of Theorem 5.1 in these cases.Theorem A.2. Let a1, . . . , ar and b1, . . . , br be positive integers with bi ≤ai. Then there exists a finitely generated field extension K/k and a disjointcollection {Z1, . . . , Zr} of central division algebras over K with ind(Zi) = pai,exp(Zi) = pbi, and ind(Z⊗pdi ) = pai−d for any 1 ≤ d < bi.We begin with a weaker existence result.Lemma A.3. Let a1, . . . , ar be positive integers. Then there exists a finitelygenerated field extension K/k and a disjoint collection {A1, . . . , Ar} of centraldivision algebras over K with ind(Ai) = exp(Ai) = pai.We require some preliminary results before the proof. We may assumek contains all primitive pdthroots of unity for all d, and choose a sequence{1 = ζ1, ζp, ζp2 , . . . } ⊂ k such that, for all b ≥ 1, ζpb is a primitive pbthroot ofunity, and ζppb+1 = ζpb . Recall that if u,w ∈ K and a ∈ Z≥0, then the symbolalgebra (u,w)pa is the (central simple) K-algebra generated by x and y suchthat xpa= u, ypa= w and uw = ζpawu.60Proposition A.4. (see [D83, Chapter 11], or [R88, Proposition 7.1.17]) Letc, d ∈ K, a ∈ Z≥1i) (c, d)⊗ppa is Brauer equivalent to (c, d)pa−1 (Here, (c, d)1 = K is split).ii) (c, d)npa is Brauer equivalent to (c, dn)pa.Lemma A.5. Let a1, . . . , ar be non-negative integers, u1, . . . , ur, w1, . . . , wrbe commuting indeterminates and K = k(u1, . . . , ur, w1, . . . , wr). Then thecentral simple K-algebraA = (u1, w1)pa1 ⊗ · · · ⊗ (ur, wr)paris a division algebra.Proof. Let {x1, . . . , xr, y1, . . . , yr} be the elements in A such that by defi-nition xpaii = ui and ypaii = wi. Consider the k-algebra R generated byx1, . . . , xr, y1, . . . , yr. Each element of R has a unique expression as a finitesum of the form∑0≤i1,...,ir,j1,...,jrλi1,...,jrxi11 . . . xirr yj11 . . . yjrrwith each λi1,...,jr ∈ k.A standard leading monomial trick shows that R is a domain, and itis easy to see that Z(R) = k[u1, . . . , ur, w1, . . . , wr]. Thus A is the centrallocalization of R, and since R is a domain, so is A.Proof of Lemma A.3. Take K = k(u1, . . . , ur, w1, . . . , wr) as in the previouslemma. Let Ai = (ui, wi)pai . Since deg(Ai) = pai , to see that Ai has exponent(and thus also index) pai it suffices to show that Apai−1i is not split. UsingProposition A.4i) repeatedly, it is equivalent to show that (ui, wi)p is notsplit, and this follows from Lemma A.5 with ai = 1 and aj = 0 for j 6= i.61Let m1, . . . ,mr ∈ Z≥0. We wish to showind(A⊗m11 ⊗ . . .⊗ A⊗mrr ) = ind(A⊗m11 ) · . . . · ind(A⊗mrr )Observe that replacing each A⊗mii with a Brauer-equivalent element does notchange either side. Thus first, if A⊗mii is split for any i, then we remove itfrom both sides. Next, we apply Proposition A.4i) repeatedly. Thus we canremove factors of p from each of the mi (and consequently reduce the ai),and so we can assume without loss of generality that (mi, p) = 1 for all i.By Proposition A.4ii), A⊗miiBr∼ (ui, wmii )pai . Let Di = (ui, wmii )pai . Let K′be the subfield of K given by k(u1, . . . , ur, wm11 , . . . , wmrr ). Then each D⊗miiis defined over K ′. By Lemma A.5, D⊗m11 ⊗ . . .⊗D⊗mrr is a division algebraover K ′, and so in particularind(D⊗m11 ⊗ . . .⊗D⊗mrr ) = ind(D⊗m11 ) · . . . · ind(D⊗mrr )over K ′. Since (mi, p) = 1 we have that K/K ′ is an extension of degree primeto p. Since each Di has index a power of p, by [GS06, Corollary 4.5.11b] wehave that the equality still holds after scalar extension to K, which was thedesired result.Proof of Theorem A.2. Using the previous lemma, for some field K/k, thereis a disjoint collection B1, . . . , Br of central division K-algebras such thatind(Bi) = exp(Bi) = pai . Let t be maximal, with 1 ≤ t ≤ r + 1, such thatthere exists a field L/K with the following properties:1. {Bi ⊗K L}i=1,...,r is a disjoint collection of central division E-algebras.2. If i < t then exp(Bi ⊗K L) = pbi and ind((Bi ⊗ L)⊗pd) = pai−d for any0 ≤ d < bi.3. If t ≤ i ≤ r then ind(Bi ⊗K L) = exp(Bi ⊗K L) = pai .62If t = r+1 then we are done, so towards a contradiction suppose t < r+1.Let L/K be a field extension satisfying the three properties, and set Ai =Bi⊗K L for all i. Let Lt be the function field of the Brauer Severi variety ofA⊗pbtt . Now consider the r-tuple (A1 ⊗L Lt, . . . , Ar ⊗L Lt) of central simplealgebras over Lt. Using the fact that ind((Aj ⊗L Lt)⊗m) = ind(A⊗mj ⊗L Lt),the index reduction formula of Schofield and Van Den Bergh [SV92, Theorem1.3], and the properties above, we get the following formula for any positiveinteger d:ind((Ai ⊗L Lt)⊗pd)= minz∈Z(ind(A⊗pdi ⊗L A⊗zpbtt ))=ind(A⊗pdi ), if t 6= iind(A⊗pdi ) if t = i, d < bt1, if t = i, d ≥ btIn particular, for i 6= t we have that the exponent and index of Ai ⊗L Ltare the same as the exponent and index respectively of Ai. For i = t, if d < btthen from ind(At) = exp(At) = pat , we get:ind((At ⊗L Lt)⊗pd)= ind(A⊗pdi ) = pai−dand we also have exp(Ai ⊗L Lt) = pbt . Thus conditions ii) and iii) aboveare satisfied for the algebras {A1 ⊗L Lt, . . . , Ar ⊗L Lt} with t + 1 replacingt. Since Ai ⊗L Lt = Bi ⊗K Lt for i ≤ i ≤ r, to arrive at a contradiction itsuffices to verify condition i) for the algebras {A1 ⊗L Lt, . . . , Ar ⊗L Lt}.We will prove this using our inductive hypothesis that A1, . . . , Ar aredisjoint, and the index reduction formula used above. Let m1, . . . ,mr ∈ Z.Then63indr⊗i=1(Ai ⊗L Lt)⊗mi = ind((r⊗i=1A⊗mii)⊗L Lt)= minz∈Zind((r⊗i=1A⊗mii)⊗L Azpbtt)=∏1≤i≤ri 6=tind(A⊗mii ) ·minz∈Zind(A⊗mtt ⊗L Azpbt )t )=∏1≤i≤ri 6=tind((Ai ⊗L Lt)⊗mi) · ind(A⊗mtt ⊗L Lt)=r∏1=1ind((Ai ⊗L Lt)⊗mi)as required.Corollary A.6. Let p be prime, with integers a1, . . . , ar and groups G1, . . . , Gras in the Introduction. Suppose Gi ∈ {PGLni, PGSp2ai , PGO2ai , PGO+2ai(ai ≥ 2)}. Then there exists K/k and Ei ∈ H1(K,Gi) such that the underly-ing division algebras of the set {δiK(Ei)}i=1,...,r form a disjoint set of divisionalgebras over K, with ind(δiK(Ei)) = pai and exp(δiK(Ei)) = pbi.Proof. From [KMRT98, Section 29], the image of H1(K,Gi) in Br(K) underthe boundary map is given by the following (modulo Brauer equivalence):δiK(H1(K,PGLni)) ={divisional algebras over K of index dividing ni}64δiK(H1(K,PGSp2ai )) ={divisional algebras over K of index dividing2ai and exponent dividing 2}δiK(H1(K,PGO2ai )) ={divisional algebras over K of index dividing2ai and exponent dividing 2}δiK(H1(K,PGO+2ai )) =divisional algebras A over K of index dividing2ai , exponent dividing 2, and for which thereexists an involution σ on A with trivialdiscriminantNote that δiK(H1(K,PGO+2ai )) contains the class of any division algebraover K of index dividing 2ai and exponent dividing 2 which can be properlydecomposed as the product of division algebras of exponent dividing 2. Theresult is now an easy application of Theorem A.2.65Appendix B Quotient StacksIn this section we will further discuss the proof of Theorem 2.7, beginningwith the lower bound. Recall that we have an exact sequence 1 → D →H → H/D → 1 with D central and diagonalizable, and such that the imageof the coboundary map dK : H1(K,H/D) → H2(K,D) consists of only p-primary elements for any K/k. We need to show that there exists K/k andE ∈ H1(K,H/D) with cdimK([E/H]; p) = maxA,L(cdimL([A/H]; p)) (over allL/k and A ∈ H1(K,H/D)) such thatedk(H; p) ≥ cdimK([E/H]; p) + edk(D; p)− dim(H).This result follows from [M13, Theorem 5.11], but the result was only statedin the case D = µtp for some t. We will prove the result instead (CorollaryB.4) using [M13, Theorem 5.11] and [KM08, Theorem 4.4 & Remark 4.5].Remark B.1. For any K/k and E ∈ H1(K,H/D), by [KM08, Theorem 2.1& Remark 2.9] we can writecdim([E/H]; p) = minχ1,...,χtt∑i=1ind(χi∗ ◦ dK(E))where the minimum is taken over all generating sets of X(D). By Theorem5.2, cdimK([E/H]; p) can be maximized by choosing K and E such that forany χ ∈ X(D) we haveind(χ∗ ◦ dL(A)) = gcd{dim(V ) | V ∈ Rep(χ)(H)}.Let Dp be the p-torsion subgroup of D, so that we have an exact sequence1 → Dp → H → H/Dp, and let d′L : H1(L,H/Dp) → H2(L,Dp) be thecoboundary map for any L/k.For χ ∈ X(D) and χ′ ∈ X(Dp), let Rep(χ)D (H) denote the category ofall finite dimensional representations ρ of H such that ρ(z) is multiplication66by χ(z) for all z ∈ D, and let Rep(χ′)Dp (H) denote the category of all finitedimensional representations ρ′ of H such that ρ′(z) is multiplication by χ′(z)for all z ∈ Dp.Choose K/k, K ′/k, E ∈ H1(K,H/D) and E ′ ∈ H1(K,H/Dp) so that forall χ ∈ X(D), χ′ ∈ X(Dp):ind(χ∗ ◦ dK(E)) = gcd{dim(V ) | V ∈ Rep(χ)D (H)}ind(χ′∗ ◦ d′K′(E′)) = gcd{dim(V ) | V ∈ Rep(χ′)Dp (H)}Lemma B.2. Let χ′ ∈ X(Dp) and let χi (i ∈ I) be the preimages of χ′ underthe natural map X(D)→ X(Dp). In other words, (χi)|Dp = χ′ ∀i. Thenind(χ′∗ ◦ d′K′(E)) = mini∈I(ind(χi∗ ◦ dK(E)))Proof. By our choice of E and E ′, it is equivalent to showgcd{dim(V ) | V ∈ Rep(χ′)Dp (H)}= mini∈I(gcd{dim(V ) | V ∈ Rep(χi)D (H)})Since Rep(χ′)Dp (H) =⋃i∈IRep(χi)D (H), using general properties of gcd we havegcd{dim(V ) | V ∈ Rep(χ′)Dp (H)}= gcdi∈I(gcd{dim(V ) | V ∈ Rep(χi)D (H)})Since gcd{dim(V ) | V ∈ Rep(χi)D (H)}is a power of p for all i by assump-tion, we can replace gcdi∈Iby mini∈Iand the result follows.Theorem B.3. Let K/k, K ′/k, E ∈ H1(K,H/D) and E ′ ∈ H1(K ′, H/Dp)be as chosen above. Then cdimK(E/H) = cdimK′(E ′/H) = cdimK(E/H; p) =cdimK′(E ′/H; p).67Proof. DefineTE,K = 〈χ∗ ◦ dK(E))〉χ∈X(D)andTE′,K′ = 〈χ′∗ ◦ d′K(E′)〉χ′∈X(Dp).Then, by assumption, both TE,K and TE′,K′ are (finite) p-groups. As inSection 5, let b1, . . . , bl be a generating set of TE,K withl∑i=1ind(bi) minimal,and defineind(E,K) =l∑i=1(ind(bi)− 1)Similarly, let b′1, . . . , b′t be a generating set of TE′,K′ witht∑i=1ind(bi) minimal,and defineind(E ′, K ′) =l∑i=1(ind(b′i)− 1)By applying [KM08, Theorem 2.1 & Remark 2.9] it is equivalent to showind(E,K) = ind(E ′, K ′).We will first show ind(E,K) ≤ ind(E ′, K ′). Choose a generating setχ′1, . . . , χ′m ∈ X(Dp) such that χ′i∗ ◦ d′K′(E′) = b′i for i ≤ t and χ′i∗ ◦d′K′(E′) = 0 for i > t. Using Lemma B.2, choose χ1, . . . , χm ∈ X(D)such that χi|Dp = χ′i and ind(χ′i∗ ◦ d′K′(E′)) = ind(χi∗ ◦ dK(E)). Thenχ1, . . . , χm generate X(D)/pX(D), since pX(D) = ker(X(D) → X(Dp)).Thus χ1∗ ◦ dK(E), . . . , χm∗ ◦ dK(E) generate TE,K/pTE,K which by Lemma4.1 means they generate TE,K . Hence ind(E,K) ≤ ind(E ′, K ′).To see the reverse direction, choose a generating set (not necessarily ofminimal size) χ1, . . . , χn ∈ X(D) such that χi∗ ◦ dk(E) = bi for i ≤ l andχi∗ ◦ dk(E) = 0 for i > l. Let χ′i = χi|Dp . Observe that if we replace χi by τfor any τ ∈ X(D) with τ |Dp = χi|Dp then the resulting set {χ1, . . . , χn} willstill generate X(D)/pX(D) and hence {χ1∗ ◦ dK(E), . . . , χn∗ ◦ dK(E)} will68still generate TE,K . Thus by Lemma B.2 and the minimality of∑bi, we haveind(χi∗ ◦dK(E)) = ind(χ′i∗ ◦d′K′(E′)) for all i. Since the restriction map fromX(D) to X(Dp) is surjective, we have that {χ′1∗ ◦ d′K′(E′), . . . , χ′n∗ ◦ d′K′(E′)}generates TE′,K′ , and the result follows.Let K/k and E ∈ H1(K,H/D) be as chosen above. In particular,cdimK([E/H]; p) = maxA,L(cdimL([A/H]; p)) over all L/k, A ∈ H1(L,H).Corollary B.4. We haveedk(H; p) ≥ cdimK([E/H]; p) + edk(D; p)− dim(H/D).Proof. Let K ′/k and E ′ ∈ H1(K ′, H/Dp) also be as chosen above. From[BRV11, Corollary 3.3] (see also [M13, Corollary 5.7]) and [M13, Theorem5.11], we have:edK(H; p) ≥ edK′([E′/H]; p)− dim(H/Dp)= cdimK′([E′/H]; p) + edk(Dp; p)− dim(H)Since edk(Dp; p) = edk(D; p) + dim(D) (D is diagonalizable), the result fol-lows from the previous theorem.We finish this section with a proof of the upper bound from Theorem 2.7.Proof of Theorem 2.7.2. Suppose we are given a finitely generated field ex-tension K of k, and E ∈ H1(K,H). It suffices to show thatedk(E) ≤ edk(H) + maxA,L(cdimL([A/H])) + edk(D)Ket E be the image of E under the map H1(K,H) → H1(K,H). Bydefinition of essential dimension, we can find a k-subfield K0 of K and E0 ∈H1(K0, H), with trdegk(K0) ≤ edk(H), such that (E0)K = E. Further, if we69view E as an H-scheme over K then by Remark 2.6 we have [E/H](K) 6= ∅since E is in the image of H1(K,H) → H1(K,H). Thus we can find anintermediate field K1 with K0 ⊂ K1 ⊂ K such that [E0/H](K1) 6= ∅ andtrdegK0 K1 ≤ cdimK0 [E0/H]. Setting E1 = (E0)K1 ∈ H1(K1, H), then againby Remark 2.6 this means that there exists a preimage E1 of E1 under themap H1(K1, H)→ H1(K1, H).We would like to conclude (E1)K = E, however what we know is:H1(K,H) → H1(K,H)E 7→ E(E1)K 7→ EFrom [S97, I.5.7, Proposition 42], it follows that there exists a ∈ H1(K,D)such that, via the action of H1(K,D) on H1(K,H), we have a · (E1)K =E. Again by definition of essential dimension, there exists a field extensionK2/K1 of transcendence degree at most edK1(D), and b ∈ H1(K2, D) suchthat bK = a. If we define E2 = b · (E1)K2 ∈ H1(K2, H), then we have(E2)K = E.Hence,ed(E) ≤ trdegk(K2) = trdegk(K0) + trdegK0(K1) + trdegK1(K2)≤ edk(H) + cdimK0 [E0/H] + edK1(D)≤ edk(H) + maxA,L(cdimL([A/H])) + edk(D)as required.70Appendix C Products of Groups with p 6= 2In this section we will consider the case when p 6= 2 andGi ∈ {GOni ,Oni , SOni}.The key ingredient in these cases is that, since 2 - ni, the boundary mapH1(K,Gi)→ H2(K,Z(Gi)) is trivial. The results in this section hold underthe assumption that each ni is odd and at least 3 (but not necessarily a primepower). We first study the case where Z(G) is finite.Theorem C.1. Let Gi ∈ {Oni , SOni} for 1 ≤ i ≤ r. Let µ be a centralsubgroup of G. Thened(G/µ) = ed(G/µ; 2) =(r∑i=1si)− rank(µ)where si =ni if Gi = Onini − 1 if Gi = SOniProof. Since ni is odd, we have Oni ∼= SOni ×µ2, and Z(SOni) is trivial. Thusif m = |{i | Gi = Oni}|, we may write G ∼= SOn1 × · · · × SOnr × ((µ2)m/µ).Recall that for any algebraic groups H1 and H2 we have edk(H1 × H2) ≤edkH1 + edkH2, and that edk(SOn) = n − 1 for n ≥ 3. Since edk(µm2 /µ) =rank(µm2 /µ) = m− rank(µ), the upper bound follows.For the lower bound we proceed as in [RY00, Theorem 7.8 & Theorem 8.1](see also [GR09, Theorem 1.2 & Example 9.1]). The subgroup (A1, . . . , Ar, λ)of G, where each Ai a diagonal matrix in SOni with entries±1, and λ ∈ µm2 /µ,is a finite 2-subgroup of G of rank(r∑i=1(ni − 1))+m− rank(µ)71and this subgroup has a finite centralizer. Thus by [RY00, Theorem 7.8],edK(G; 2) ≥(r∑i=1(ni − 1))+m− rank(µ) =(r∑i=1si)− rank(µ)as required.We now study the case where Z(G) is connected.Lemma C.2. Let K/k and Gi = GOni for 1 ≤ i ≤ r. Let µ be a centralsubgroup of G. ThenH1(K,G) ∼= H1(K,G/µ)Proof. In this case, Z(G) is a torus. Thus we have an exact sequence1→ Z(G)/µ→ G/µ→ G→ 1which yields the following in cohomology:0→ H1(K,G/µ)γ→ H1(K,G)Since the boundary map is zero, γ is surjective. By [S97, I.5, Proposition42], γ is injective. Thus H1(K,G/µ) = H1(K,G) for any µ ≤ Z(G). Inparticular, H1(K,G) ∼= H1(K,G/µ).Lemma C.3. We have edk(GOn) ≤ n− 1.Proof. H1(K,GOn) classifies orthogonal involutions onMn(K) (see [KMRT98,Section 29]). An orthogonal involution is determined by a non-degeneratesymmetric bilinear form on Kn up to scalar multiples, and we can diagonal-ize the form and multiply by scalars so that it is represented by the diagonalGram matrix:72a1a2. . .an−11Thus we conclude edk(GOn) ≤ n− 1.Theorem C.4. Let Gi = GOni for 1 ≤ i ≤ r. Let µ be a central subgroupof G. Thened(G/µ) = ed(G/µ; 2) =r∑i=1(ni − 1).Proof. By Lemma C.2, we may assume µ is the trivial subgroup. The upperbound is now obvious using edk(H1 ×H2) ≤ edk(H1) + edk(H2) and LemmaC.3. Recall also that for any algebraic group H1 with subgroup H2 we haveedk(H1; 2) ≥ edk(H2; 2) − dim(H1) + dim(H2). Now, consider the subgroupH = On1 × · · · ×Onr . Then we haveedk(G; 2) ≥ edk(H; 2)− rBy Theorem C.1, edk(H; 2) =r∑i=1ni, and the result follows.Remark C.5. For all groups G/µ studied in this section, edk(G/µ; q) = 0for primes q 6= 2. This is because we have the exact sequenceH1(K,Z(G)/µ)→ H1(K,G/µ)→ H1(K,G)and any elements a ∈ H1(K,Z(G)/µ) and b ∈ H1(K,G) can be split byadjoining sufficiently many square roots to K. It follows that any element ofH1(K,G/µ) can also be split by an extension whose degree is a power of 2.73

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