"Science, Faculty of"@en . "Mathematics, Department of"@en . "DSpace"@en . "UBCV"@en . "Shukla, Abhishek Kumar"@en . "2020-04-21T23:24:53Z"@en . "2020"@en . "Doctor of Philosophy - PhD"@en . "University of British Columbia"@en . "The overarching theme of this thesis is to assign, and sometimes find, numerical values which reflect complexity of algebraic objects. The main objects of interest are field extensions of finite degree, and more generally, etale algebras of finite degree over a ring.\r\nOf particular interest to us is the invariant known as essential dimension. The essential dimension of separable field extensions was introduced by J. Buhler and Z. Reichstein in their landmark paper. A major (still) open problem arising from that work is to determine the essential dimension of a general separable field extension of degree n (or equivalently, the essential dimension of the symmetric group). Loosening the separability assumption we arrive at the case of inseparable field extensions. In the first part of this thesis we study the problem of determining the essential dimension of inseparable field extensions. In the second part of this thesis, we study the essential dimension of the double covers of symmetric groups and alternating groups, respectively. These groups were first studied by I. Schur and their representations are closely related to projective representations of symmetric and alternating groups. In the third part, we study the problem of determining the minimum number of generators of an etale algebra over a ring. The minimum of number of generators of an etale algebra is a natural measure of its complexity."@en . "https://circle.library.ubc.ca/rest/handle/2429/74097?expand=metadata"@en . "Essential dimension and classifyingspaces of algebrasbyAbhishek Kumar ShuklaBS-MS, Indian Institute of Science Education and Research Pune, 2016A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinThe Faculty of Graduate and Postdoctoral Studies(Mathematics)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)April 2020\u00C2\u00A9 Abhishek Kumar Shukla 2020The following individuals certify that they have read, and recommend to the Faculty of Graduateand Postdoctoral Studies for acceptance, the dissertation entitled:Essential dimension and classifying spaces of algebrassubmitted by Abhishek Kumar Shukla in partial fulfillment of the requirements forthe degree ofDoctor of PhilosophyinMathematicsExamining Committee:Zinovy Reichstein, Professor, Department of Mathematics, UBCSupervisorKai Behrend, Professor, Department of Mathematics, UBCSupervisory Committee MemberRachel Ollivier, Professor, Department of Mathematics, UBCSupervisory Committee MemberBen Williams, Professor, Department of Mathematics, UBCUniversity ExaminerIan Blake, Honorary Professor, Department of Electrical and Computer Engineering, UBCUniversity ExamineriiAbstractThe overarching theme of this thesis is to assign, and sometimes find, numerical values whichreflect complexity of algebraic objects. The main objects of interest are field extensions of finitedegree, and more generally, e\u00C2\u00B4tale algebras of finite degree over a ring.Of particular interest to us is the invariant known as essential dimension. The essentialdimension of separable field extensions was introduced by J. Buhler and Z. Reichstein in theirlandmark paper [BR97]. A major (still) open problem arising from that work is to determinethe essential dimension of a general separable field extension of degree n (or equivalently, theessential dimension of the symmetric group). Loosening the separability assumption we arrive atthe case of inseparable field extensions. In the first part of this thesis we study the problem ofdetermining the essential dimension of inseparable field extensions. In the second part of thisthesis, we study the essential dimension of the double covers of symmetric groups and alternatinggroups, respectively. These groups were first studied by I. Schur and their representations areclosely related to projective representations of symmetric and alternating groups. In the thirdpart, we study the problem of determining the minimum number of generators of an e\u00C2\u00B4tale algebraover a ring. The minimum of number of generators of an e\u00C2\u00B4tale algebra is a natural measure of itscomplexity.iiiLay SummaryIn the second half of the 19th century the German mathematician Felix Klein proposed a way tostudy geometric shapes by looking at their symmetries, i.e., by considering transformations of theobjects that leave them invariant. For example, if we look at a square in the plane, we note thatit is invariant under a 90 degree rotation; moreover, we can consider all possible transformationswith that property and study this collection of transformations instead of the square. Objectswith highly complex symmetries frequently arise in mathematics and the natural sciences.Klein\u00E2\u0080\u0099s ideas turned out to be fruitful in algebra as well as geometry. For example, whilepolynomial equations in one variable are difficult to solve explicitly, one gains a great deal ofinformation about their solutions from studying their symmetries (the so-called Galois group).This thesis addressed the following question: how complicated is it to define a given algebraicstructure? We focus on two measures of complexity: the minimal number of generators and theessential dimension. Symmetry groups play a key role in both cases.ivPrefaceThis dissertation is a compilation of three related works.A version of Chapter 2 has been published: Zinovy Reichstein and Abhishek Kumar Shukla,Essential dimension of inseparable field extensions, Algebra Number Theory 13 (2019), no 2,513-530.The problem, which the paper answers, was originally posed by Zinovy Reichstein. Theresearch and manuscript preparation was done in equal parts by myself and Zinovy Reichstein.A version of Chapter 3 is submitted for publication:Zinovy Reichstein and Abhishek Kumar Shukla, Essential dimension of double covers ofsymmetric and alternating groups, arXiv e-prints (2019), arXiv:1906.03698.The research and manuscript preparation was done in equal parts by myself and ZinovyReichstein.A version of Chapter 4 has been published: Shukla, A., Williams, B. (2020). Clas-sifying spaces for e\u00C2\u00B4tale algebras with generators. Canadian Journal of Mathematics, 1-21.doi:10.4153/S0008414X20000206.The problem, which this paper solves, was originally posed by Zinovy Reichstein. The researchand manuscript preparation was done in equal parts by myself and Ben Williams.vTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Essential dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Double covers of alternating groups . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Generators of an e\u00C2\u00B4tale algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 Essential dimension of inseparable field extensions . . . . . . . . . . . . . . . . 52.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 Finite-dimensional algebras and their automorphisms . . . . . . . . . . . . . . . 72.3 Essential dimension of a functor . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.4 Field extensions of type (n, e) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.5 Proof of the upper bound of Theorem 2.1.2 . . . . . . . . . . . . . . . . . . . . . 122.6 Versal algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.7 Conclusion of the proof of Theorem 2.1.2 . . . . . . . . . . . . . . . . . . . . . . 162.8 Alternative proofs of Theorem 2.1.2 . . . . . . . . . . . . . . . . . . . . . . . . . 192.9 The case where e1 = \u00C2\u00B7 \u00C2\u00B7 \u00C2\u00B7 = er . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 Essential dimension of double covers of symmetric and alternating group . 213.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.2.1 Essential dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.2.2 The index of a central extension . . . . . . . . . . . . . . . . . . . . . . . 263.2.3 Sylow 2-subgroups of A\u00CB\u009Cn . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.3 Proof of Theorem 3.1.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283.4 Proof of Theorem 3.1.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.5 Proof of Theorem 3.1.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32viTable of Contents3.6 Comparison of essential dimensions of S\u00CB\u009C+n and S\u00CB\u009C\u00E2\u0088\u0092n . . . . . . . . . . . . . . . . . 353.7 Explanation of the entries in Table 3.1 . . . . . . . . . . . . . . . . . . . . . . . 364 Classifying spaces for e\u00C2\u00B4tale algebra with generators . . . . . . . . . . . . . . . 384.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384.1.1 Notation and other preliminaries . . . . . . . . . . . . . . . . . . . . . . . 394.2 E\u00C2\u00B4tale algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404.2.1 Generation of trivial algebras . . . . . . . . . . . . . . . . . . . . . . . . . 414.3 Classifying spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434.3.1 Construction of B(r;An) . . . . . . . . . . . . . . . . . . . . . . . . . . . 434.3.2 The functor represented by B(r,An) . . . . . . . . . . . . . . . . . . . . . 444.4 Stabilization in cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464.5 The motivic cohomology of the spaces B(r;A2) . . . . . . . . . . . . . . . . . . . 474.5.1 Change of coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474.5.2 The deleted quadric presentation . . . . . . . . . . . . . . . . . . . . . . 474.6 Relation to line bundles in the quadratic case . . . . . . . . . . . . . . . . . . . . 494.7 The example of Chase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504.7.1 The singular cohomology of the real points of B(r;A2) . . . . . . . . . . 514.7.2 Algebras over fields containing a square root of \u00E2\u0088\u00921 . . . . . . . . . . . . 535 Conclusions and Future Research Directions . . . . . . . . . . . . . . . . . . . 555.1 Essential dimension of inseparable field extensions . . . . . . . . . . . . . . . . . 555.2 Essential dimension of double covers of symmetric and alternating groups . . . . 565.3 Generators of e\u00C2\u00B4tale algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58viiList of Tables3.1 Essential dimension of A\u00CB\u009Cn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24viiiList of Figures2.1 Descent diagram for field extension . . . . . . . . . . . . . . . . . . . . . . . . . . 53.1 Projective representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.2 Schematic of spin groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22ixAcknowledgementsI would like to express my sincere gratitude towards my advisor Zinovy Reichstein for hismentorship and direction throughout the course of my Ph.D. I am also grateful to Ben Williamsfor his mentorship and encouragement.The content of Chapter 2, coauthored with Zinovy Reichstein, benefitted from discussionswith Madhav Nori, Julia Pevtsova, Federico Scavia, and Angelo Vistoli, to whom I am grateful.The content of Chapter 3, coauthored with Zinovy Reichstein, benefitted from discussionswith Eva Bayer-Flu\u00C2\u00A8ckiger, Vladimir Chernousov, Alexander Merkurjev, Jean-Pierre Serre, BurtTotaro, Alexander Vishik, and Angelo Vistoli, to whom I am grateful.The content of Chapter 4, coauthored with Ben Williams, benefitted from discussions withZinovy Reichstein, Uriya First and Manuel Ojanguren, to whom I am grateful.I was partially supported by SERB-UBC fellowship during the period of my Ph.D.I would like to thank Niny Arcila Maya for her warm and affable friendship. I would like tothank my partner, Daisy Mengxi Zhang, for her company and enormous support.Finally, I would like to acknowledge that I wrote this thesis while residing on the unceded,ancestral territory of the Musqueam people.xChapter 1IntroductionA primary objective of this thesis is to study the essential dimension of some naturally arisingfunctors in mathematics. Another objective is to define and study the classifying space for e\u00C2\u00B4talealgebras with generators.1.1 Essential dimensionEssential dimension was initially defined and studied by J. Buhler and Z. Reichstein in [BR97].Chapter 2 concerns the problem of determining the essential dimension of inseparable fieldextensions. Roughly speaking, the problem is to determine the minimal number of parametersneeded to define a general inseparable extension of fixed degree. To make this problem preciselet k be a base field and assume all other fields contain k. A field extension L/K of finite degreeis said to descend to a subfield K0 \u00E2\u008A\u0082 K if there exists a subfield K0 \u00E2\u008A\u0082 L0 \u00E2\u008A\u0082 L such that L0 andK generate L and [L0 : K0] = [L : K]. Equivalently, L is isomorphic to L0 \u00E2\u008A\u0097K0 K over K, as isshown in the following diagram.LL0 KK0We define edk(L/K) := min{trdegkK0 | L/K descends to L0/K0}.How many parameters are needed to define a general separable field extension L/K of degreen? To formalize this notion we define\u00CF\u0084(n) = max{ed(L/K) | L/K is a separable extension of degree n and k \u00E2\u008A\u0082 K}. (1.1.1)For example, if n = 2 a general quadratic separable extension L/K can be defined by asingle parameter c \u00E2\u0088\u0088 K (since L = K(\u00E2\u0088\u009Ac) for some c \u00E2\u0088\u0088 K). Then L/K descends to k(\u00E2\u0088\u009Ac)/k(c).Consequently \u00CF\u0084(2) \u00E2\u0089\u00A4 1. Similarly it can be shown that every degree 3 separable extension L/Karises by solving a polynomial of the form X3 \u00E2\u0088\u0092 aX + a and hence \u00CF\u0084(3) \u00E2\u0089\u00A4 1 too.It is shown in [BR97] that if char(k) = 0, then\u00E2\u008C\u008An2\u00E2\u008C\u008B6 \u00CF\u0084(n) 6 n\u00E2\u0088\u0092 3 for every n > 5.Now suppose L/K is an arbitrary inseparable (but not necessarily purely inseparable) fieldextension L/K of finite degree. Denote the separable closure of K in L by S. We will say thatL/K is of type (n, e) if [S : K] = n and the purely inseparable extension L/S is of type e.The type e of a purely inseparable extension L/S is finite sequence of positive integerse = (e1, . . . , er) associated to it in a natural fashion with the property that e1 \u00E2\u0089\u00A5 e2 \u00E2\u0089\u00A5 . . . \u00E2\u0089\u00A5 er.Every element in L (over S) satisfies a polynomial equation Zpm \u00E2\u0088\u0092 s \u00E2\u0088\u0088 S[Z]. Then e1 is the11.2. Double covers of alternating groupslargest exponent m of p occuring among all elements of L. Call such an element l1. Then e2is defined in a similar fashion for the extension L/S[l1]. We stop when L = S[l1, l2, . . . , lr] andS[l1, . . . , lr\u00E2\u0088\u00921] ( L. Thus arriving at a sequence (e1, . . . , er). By a theorem of Pickert [Pic49]this sequence is independent of the choice of elements l1, l2, . . . , lr. As an example, the type ofFp(x) over Fp(xpn) is (n), Fp(x, y) over Fp(xp2, yp) is (2, 1) and type of Fp(xp, yp, z, xz + y) overFp(xp2, yp2, zp2) is (2, 2, 1).By analogy with (1.1.1) it is natural to define\u00CF\u0084(n, e) = max{ed(L/K) | L/K is a field extension of type (n, e) and k \u00E2\u008A\u0082 K}.We are interested in determining \u00CF\u0084(n, e). That is how many parameters are needed to defineda general inseparable extension degree n and type e. We answer this question as follows:Theorem 1.1.1 ([RS19a]). Let k be a base field of characteristic p > 0, n > 1 and e1 > e2 >\u00C2\u00B7 \u00C2\u00B7 \u00C2\u00B7 > er > 1 be integers, e = (e1, . . . , er) and si = e1 + \u00C2\u00B7 \u00C2\u00B7 \u00C2\u00B7+ ei for i = 1, . . . , r. Then\u00CF\u0084(n, e) = nr\u00E2\u0088\u0091i=1psi\u00E2\u0088\u0092iei .The proof combines methods from field theory along with techniques from the theory ofessential dimension. We interpret inseparable extensions L/K of type (n, e) as forms of atruncated polynomial algebra \u00CE\u009Bn,e, which are then classified by torsors over Spec(K) for a certaingroup scheme Gn,e. Then \u00CF\u0084(n, e) is the essential dimension of Gn,e, much as \u00CF\u0084(n) is the essentialdimension of Sn in the separable case (see next paragraph for definition of essential dimension ofa group). The group scheme Gn,e is neither finite nor smooth; however, much to our surprise,computing its essential dimension turns out to be easier than computing the essential dimensionof Sn.1.2 Double covers of alternating groupsChapter 3 concerns the essential dimension of double cover of symmetric and alternating groups.To define essential dimension of groups we consider a more general definition of essential dimension,due to A. Merkurjev.Let F : Fieldsk \u00E2\u0086\u0092 Sets be a covariant functor from the category of field extensions K/kto the category of sets. Here K ranges over all fields containing k. We say that an objecta \u00E2\u0088\u0088 F(K) descends to a subfield K0 \u00E2\u008A\u0082 K if a lies in the image of the natural restriction mapF(K0)\u00E2\u0086\u0092 F(K). The essential dimension ed(a) of a is defined as minimal value of trdeg(K0/k),where k \u00E2\u008A\u0082 K0 and a descends to K0.The essential dimension of the functor F , denoted by ed(F), is the supremum of ed(a) for alla \u00E2\u0088\u0088 F (K), and all fields K in Fieldsk. There is a related notion of essential dimension at l wherel is a prime, denoted by ed( ; l).The example of foremost interest is when G is a group scheme over a base field k andFG : K \u00E2\u0086\u0092 H1(K,G) be the functor defined byFG(K) = {isomorphism classes of G-torsors T \u00E2\u0086\u0092 Spec(K)}.Here by a torsor we mean a torsor in the flat topology. If G is smooth, then H1(K,G) is the firstGalois cohomology set. The essential dimension ed(G) is, by definition, ed(FG). These numericalinvariants of G have been extensively studied.21.3. Generators of an e\u00C2\u00B4tale algebraSpecializing to the case G = Sn, the set FSn(K) can be identified with e\u00C2\u00B4tale algebras A/K ofdegree n (which are products of separable field extensions whose degrees add up to n). Moreover,it can be shown that ed(Sn) = \u00CF\u0084(n) and ed(Gn,e) = \u00CF\u0084(n, e).In [BRV10] P. Brosnan, A. Vistoli and Z. Reichstein showed that the essential dimension ofSpin groups, which are double covers of SOn, grows exponentially with n. I was curious to seeif this phenomenon has an analog over finite groups. In joint work with Zinovy Reichstein wefound finite group analog of the Spin behaviour. The group is A\u00CB\u009Cn which is the double cover ofthe alternating groups An. These groups fit in a schematic as below where the top sequence isobtained as a pullback via the permutation representation An \u00E2\u0086\u0092 SOn.1 // \u00C2\u00B52 // A\u00CB\u009Cn //\u000F\u000FAn //\u000F\u000F11 // \u00C2\u00B52 // Spinn // SOn // 1Theorem 1.2.1 ([RS19b]). Assume that the base field k is of characteristic 6= 2 and containsa primitive 8th root of unity, and let n > 4 be an integer. Write n = 2a1 + . . . + 2as, wherea1 > a2 > . . . > as > 0 . Then(a) ed(A\u00CB\u009Cn; 2) = 2b(n\u00E2\u0088\u0092s\u00E2\u0088\u00921)/2c.(b) 2b(n\u00E2\u0088\u0092s\u00E2\u0088\u00921)/2c 6 ed(A\u00CB\u009Cn) 6 ed(An) + 2b(n\u00E2\u0088\u0092s\u00E2\u0088\u00921)/2c.In the case char k = 2 the Spin groups continue to show exponential growth (see [Tot19]) butwe found that ed(A\u00CB\u009Cn) grows sub-linearly.1.3 Generators of an e\u00C2\u00B4tale algebraIn Chapter 4 we consider the problem of minimum number of generators of an e\u00C2\u00B4tale algebra.O. Forster [For64] proved that over a noetherian ring R, a finite module M of rank at mostn can be generated by n + dimR elements. Generalizing this in great measure U. First andZ. Reichstein [FR17] showed that any finite R-algebra A can be generated by n+ dimR elementsif each A\u00E2\u008A\u0097R k(m), for m \u00E2\u0088\u0088 MaxSpec(R), is generated by n elements. In particular, any finitee\u00C2\u00B4tale algebra A over R can be generated by 1 + dimR elements as an R-algebra.One may ask whether the upper bounds can be universally improved. Is it possible to geta better upper bound on number of generators possibly after considering rings R of dimension\u00E2\u0089\u00A5 n0, for some positive integer n0?For modules over rings R. Swan [Swa67] produced examples of rings and finite modules overthese rings which cannot be generated by less than n+ dimR elements. Moreover, the dimensionof the rings goes to infinity. His techniques are topological in nature and the dictionary betweentopological category and the algebraic one is called the Serre-Swan theorem.Ben Williams (UBC) and I showed that the First-Reichstein upper bound is indeed sharp inthe case of e\u00C2\u00B4tale algebras as well.Theorem 1.3.1 ([SW19]). Let k = R and d any positive integer. There exist examples of finitelygenerated k-algebras Rn with dim(Rn) = n\u00E2\u0086\u0092\u00E2\u0088\u009E and e\u00C2\u00B4tale algebras An over Rn of degree d suchthat An cannot be generated by fewer than n+ 1 elements as an Rn-algebra.31.3. Generators of an e\u00C2\u00B4tale algebraThe proof has two main ideas. First is to construct classifying spaces for e\u00C2\u00B4tale algebras withgenerators. More precisely we construct varieties B(r;An) such that \u00E2\u0080\u009Can e\u00C2\u00B4tale algebra A ofdegree n equipped wtih r generating sections\u00E2\u0080\u009D on X is obtained as pullback of tautological e\u00C2\u00B4talealgebra on B(r;An) via a unique map X \u00E2\u0086\u0092 B(r;An). We refer to such a map as a classifyingmorphism. An important example of classifying spaces with generators is the projective n-spacePn which classifies projective modules of rank 1 with generators. A projective module P of rank1 on a ring R can be described as a line bundle L = P\u00CB\u009C on X = SpecR and elements of P asglobal sections \u00CE\u0093(X,L) of the line bundle L. Then Pn is the classifying space (in the above sense)as there is a natural 1\u00E2\u0088\u0092 1 correspondence between maps X \u00E2\u0086\u0092 Pn and line bundles L on X withn+ 1 generating sections (s0, s1, . . . , sn) \u00E2\u0088\u0088 \u00CE\u0093(X,L).The second main idea is to produce cohomological invariants obstructing the existence of aclassifying map. R. Swan, in his proof in the case of modules, used topological obstructions suchas Stiefel-Whitney classes of line bundles on real manifolds. In our case the obstruction is givenby Chow ring of varieties. (An alternative proof may be given by using singular cohomology asobstruction in the spirit of Swan.)Chapters 2, 3 and 4 are self-contained; each may be read independently.4Chapter 2Essential dimension of inseparablefield extensions2.1 IntroductionThroughout this chapter k will denote a base field. All other fields will be assumed to contain k.A field extension L/K of finite degree is said to descend to a subfield K0 \u00E2\u008A\u0082 K if there exists asubfield K0 \u00E2\u008A\u0082 L0 \u00E2\u008A\u0082 L such that L0 and K generate L and [L0 : K0] = [L : K]. Equivalently, Lis isomorphic to L0 \u00E2\u008A\u0097K0 K over K, as is shown in the following diagram.LL0 KK0Figure 2.1: Descent diagram for field extensionThe essential dimension of L/K (over k) is defined ased(L/K) = min{trdeg(K0/k) | L/K descends to K0 and k \u00E2\u008A\u0082 K0}.Essential dimension of separable field extensions was studied in [BR97]. Of particular interestis\u00CF\u0084(n) = max{ed(L/K) | L/K is a separable extension of degree n and k \u00E2\u008A\u0082 K}, (2.1.1)known as the essential dimension of the symmetric group Sn. It is shown in [BR97] that ifchar(k) = 0, then bn2c 6 \u00CF\u0084(n) 6 n\u00E2\u0088\u0092 3 for every n > 5. 1 A. Duncan [Dun10] later strengthenedthe lower bound as follows.Theorem 2.1.1. If char(k) = 0, then bn+ 12c 6 \u00CF\u0084(n) 6 n\u00E2\u0088\u0092 3 for every n > 6.This chapter is a sequel to [BR97]. Here we will assume that char(k) = p > 0 and studyinseparable field extensions L/K. The role of the degree, n = [L : K] in the separable case will beplayed by a pair (n, e). The first component of this pair is the separable degree, n = [S : K], where1These inequalities hold for any base field k of characteristic 6= 2. On the other hand, the stronger lower boundof Theorem 2.1.1, due to Duncan, is known only in characteristic 0.52.1. IntroductionS is the separable closure of K in L. The second component is the so-called type e = (e1, . . . , er)of the purely inseparable extension [L : S], where e1 > e2 > \u00C2\u00B7 \u00C2\u00B7 \u00C2\u00B7 > er > 1 are integers; seeSection 2.4 for the definition. Note that the type e = (e1, . . . , er) uniquely determines theinseparable degree [L : S] = pe1+\u00C2\u00B7\u00C2\u00B7\u00C2\u00B7+er of L/K but not conversely. By analogy with (2.1.1) it isnatural to define\u00CF\u0084(n, e) = max{ed(L/K) | L/K is a field extension of type (n, e) and k \u00E2\u008A\u0082 K}. (2.1.2)Our main result is the following.Theorem 2.1.2. Let k be a base field of characteristic p > 0, n > 1 and e1 > e2 > \u00C2\u00B7 \u00C2\u00B7 \u00C2\u00B7 > er > 1be integers, e = (e1, . . . , er) and si = e1 + \u00C2\u00B7 \u00C2\u00B7 \u00C2\u00B7+ ei for i = 1, . . . , r. Then\u00CF\u0084(n, e) = nr\u00E2\u0088\u0091i=1psi\u00E2\u0088\u0092iei .Some remarks are in order.(1) Theorem 2.1.2 gives the exact value for \u00CF\u0084(n, e). This is in contrast to the separable case,where Theorem 2.1.1 gives only estimates and the exact value of \u00CF\u0084(n) is unknown for any n > 8.(2) A priori, the integers ed(L/K), \u00CF\u0084(n) and \u00CF\u0084(n, e) all depend on the base field k. However,Theorem 2.1.2 shows that for a fixed p = char(k), \u00CF\u0084(n, e) is independent of the choice of k.(3) Theorem 2.1.2 implies that for any inseparable extension L/K of finite degree,ed(L/K) 6 1p[L : K] ;see Remark 2.5.3. This is again in contrast to the separable case, where Theorem 2.1.1 tells usthat there exists an extension L/K of degree n such that ed(L/K) >12[L : K] for every oddn > 7 (assuming char(k) = 0).(4) We will also show that the formula for \u00CF\u0084(n, e) remains valid if we replace essential dimensioned(L/K) in the definition (2.1.2) by essential dimension at p, edp(L/K); see Theorem 2.7.1. Forthe definition of essential dimension at a prime, see Section 5 in [Rei10] or Section 2.3 below.The number \u00CF\u0084(n) has two natural interpretations. On the one hand, \u00CF\u0084(n) is the essentialdimension of the functor Etn which associates to a field K the set of isomorphism classes of e\u00C2\u00B4talealgebras of degree n over K. On the other hand, \u00CF\u0084(n) is the essential dimension of the symmetricgroup Sn. Recall that an e\u00C2\u00B4tale algebra L/K is a direct product L = L1 \u00C3\u0097 \u00C2\u00B7 \u00C2\u00B7 \u00C2\u00B7 \u00C3\u0097 Lm of separablefield extensions Li/K. Equivalently, an e\u00C2\u00B4tale algebra of degree n over K can be thought of as atwisted K-form of the split algebra kn = k \u00C3\u0097 \u00C2\u00B7 \u00C2\u00B7 \u00C2\u00B7 \u00C3\u0097 k (n times). The symmetric group Sn arisesas the automorphism group of this split algebra, so that Etn = H1(K,Sn); see Example 2.3.5.Our proof of Theorem 2.1.2 relies on interpreting \u00CF\u0084(n, e) in a similar manner. Here the roleof the split e\u00C2\u00B4tale algebra kn will be played by the algebra \u00CE\u009Bn,e, which is the direct product of ncopies of the truncated polynomial algebra\u00CE\u009Be = k[x1, . . . , xr]/(xpe11 , . . . , xperr ).Note that the k-algebra \u00CE\u009Bn,e is finite-dimensional, associative and commutative, but not semisim-ple. E\u00C2\u00B4tale algebras over K will get replaced by K-forms of \u00CE\u009Bn,e. The role of the symmetric62.2. Finite-dimensional algebras and their automorphismsgroup Sn will be played by the algebraic group scheme Gn,e = Autk(\u00CE\u009Bn,e) over k. We will showthat \u00CF\u0084(n, e) is the essential dimension of Gn,e, just like \u00CF\u0084(n) is the essential dimension of Sn inthe separable case. The group scheme Gn,e is neither finite nor smooth; however, much to oursurprise, computing its essential dimension turns out to be easier than computing the essentialdimension of Sn.The remainder of this chapter is structured as follows. Sections 2.2 and 2.3 contain preliminaryresults on finite-dimensional algebras, their automorphism groups and essential dimension. InSection 2.4 we recall the structure theory of inseparable field extensions. Section 2.6 is devoted toversal algebras. The upper bound of Theorem 2.1.2 is proved in Section 2.5; alternative proofs areoutlined in Section 2.8. The lower bound of Theorem 2.1.2 is established in Section 2.7; our proofrelies on the inequality (2.7.2) due to D. Tossici and A. Vistoli [TV13]. Finally, in Section 2.9we prove a stronger version of Theorem 2.1.2 in the special case, where n = 1, e1 = \u00C2\u00B7 \u00C2\u00B7 \u00C2\u00B7 = er, andk is perfect.2.2 Finite-dimensional algebras and their automorphismsRecall that in the introduction we defined the essential dimension of a field extension L/K offinite degree, where K contains k. The same definition is valid for any finite-dimensional algebraA/K. That is, we say that A descends to a subfield K0 if there exists a K0-algebra A0 suchthat A0 \u00E2\u008A\u0097K0 K is isomorphic to A (as a K-algebra). The essential dimension ed(A) is thenthe minimal value of trdeg(K0/k), where the minimum is taken over the intermediate fieldsk \u00E2\u008A\u0082 K0 \u00E2\u008A\u0082 K such that A descends to K0.Here by a K-algebra A we mean a K-vector space with a bilinear multiplication mapm : A\u00C3\u0097A\u00E2\u0086\u0092 A. Later on we will primarily be interested in commutative associative algebras with1, but at this stage m can be arbitrary: we will not assume that A is commutative, associativeor has an identity element. (For example, one can talk of the essential dimension of a finite-dimensional Lie algebra A/K.) Recall that to each basis x1, . . . , xn of A one can associate a setof n3 structure constants chij \u00E2\u0088\u0088 K, wherexi \u00C2\u00B7 xj =n\u00E2\u0088\u0091h=1chijxh . (2.2.1)Lemma 2.2.1. Let A be an n-dimensional K-algebra with structure constants chij (relative tosome K-basis of A). Suppose a subfield K0 \u00E2\u008A\u0082 K contains chij for every i, j, h = 1, . . . , n. Then Adescends to K0. In particular, ed(A) 6 trdeg(K0/k).Proof. Let A0 be the K0-vector space with basis b1, . . . , bn. Define the K0-algebra structure onA0 by (2.2.1). Clearly A0 \u00E2\u008A\u0097K0 K = A, and the lemma follows.The following lemma will be helpful to us in the sequel.Lemma 2.2.2. Suppose k \u00E2\u008A\u0082 K \u00E2\u008A\u0082 S are field extensions, such that S/K is a separable extensionof degree n. Let A be a finite-dimensional algebra over S. If A descends to a subfield S0 of Ssuch that K(S0) = S, thened(A/K) 6 n trdeg(S0/k) .Here ed(A/K) is the essential dimension of A, viewed as a K-algebra.72.2. Finite-dimensional algebras and their automorphismsProof. By our assumption there exists an S0-algebra A0 such that A = A0 \u00E2\u008A\u0097S0 S.Denote the normal closure of S over K by Snorm, and the associated Galois groups byG = Gal(Snorm/K), H = Gal(Snorm/S) \u00E2\u008A\u0082 G. Now define S1 = k(g(s) | s \u00E2\u0088\u0088 S0, g \u00E2\u0088\u0088 G). Choosea transcendence basis t1, . . . , td for S0 over k, where d = trdeg(S0/k). Clearly S1 is algebraic overk(g(ti) | g \u00E2\u0088\u0088 G, i = 1, . . . , d). Since H fixes every element of S, each ti has at most [G : H] = ndistinct translates of the form g(ti), g \u00E2\u0088\u0088 G. This shows that trdeg(S1/k) 6 nd.Now let K2 = SG1 \u00E2\u008A\u0082 K,S2 = S0(K2) and A2 = A0 \u00E2\u008A\u0097S0 S2. Since S2 is algebraic over K2, wehavetrdeg(K2/k) = trdeg(S2/k) 6 nd.Examining the diagramA0 A2 AS0 S2 SK2 K,we see that A/K descends to K2, and the lemma follows.Now let \u00CE\u009B be a finite-dimensional k-algebra with multiplication map m : \u00CE\u009B\u00C3\u0097 \u00CE\u009B\u00E2\u0086\u0092 \u00CE\u009B. Thegeneral linear group GLk(\u00CE\u009B) acts on the vector space \u00CE\u009B\u00E2\u0088\u0097 \u00E2\u008A\u0097k \u00CE\u009B\u00E2\u0088\u0097 \u00E2\u008A\u0097k \u00CE\u009B of bilinear maps \u00CE\u009B\u00C3\u0097\u00CE\u009B\u00E2\u0086\u0092 \u00CE\u009B.The automorphsim group scheme G = Autk(\u00CE\u009B) of \u00CE\u009B is defined as the stabilizer of m under thisaction. It is a closed subgroup scheme of GLk(\u00CE\u009B) defined over k. The reason we use the term\u00E2\u0080\u009Cgroup scheme\u00E2\u0080\u009D here, rather than \u00E2\u0080\u009Calgebraic group\u00E2\u0080\u009D, is that G may not be smooth; see the Remarkafter Lemma III.1.1 in [Ser02a].Proposition 2.2.3. Let \u00CE\u009B be a commutative finite-dimensional local k-algebra with residuefield k, and G = Autk(\u00CE\u009B) be its automorphism group scheme. Then the natural mapf : Gn o Sn \u00E2\u0086\u0092 Autk(\u00CE\u009Bn)is an isomorphism. Here Gn = G \u00C3\u0097 \u00C2\u00B7 \u00C2\u00B7 \u00C2\u00B7 \u00C3\u0097 G (n times) acts on \u00CE\u009Bn = \u00CE\u009B \u00C3\u0097 \u00C2\u00B7 \u00C2\u00B7 \u00C2\u00B7 \u00C3\u0097 \u00CE\u009B (n times)componentwise and Sn acts by permuting the factors.Before proceeding with the proof of the proposition, recall that an element \u00CE\u00B1 of a ring R iscalled an idempotent if \u00CE\u00B12 = \u00CE\u00B1.Lemma 2.2.4. Let \u00CE\u009B be a commutative finite-dimensional local k-algebra with residue field kand R be an arbitrary commutative k-algebra with 1. Then the only idempotents of \u00CE\u009BR = \u00CE\u009B\u00E2\u008A\u0097k Rare those in R (more precisely in 1\u00E2\u008A\u0097R).Proof. By Lemma 6.2 in [Wat79], the maximal ideal M of \u00CE\u009B consists of nilpotent elements.Tensoring the natural projection \u00CE\u009B\u00E2\u0086\u0092 \u00CE\u009B/M ' k with R, we obtain a surjective homomorphism\u00CE\u009BR \u00E2\u0086\u0092 R whose kernel again consists of nilpotent elements. By Proposition 7.14 in [Jac89], everyidempotent in R lifts to a unique idempotent in \u00CE\u009BR, and the lemma follows.82.3. Essential dimension of a functorProof of Proposition 2.2.3. Let \u00CE\u00B1i = (0, . . . , 1, . . . , 0) where 1 appears in the ith position. Then\u00E2\u008A\u0095ni=1R\u00CE\u00B1i is an R-subalgebra of \u00CE\u009BnR.Let f \u00E2\u0088\u0088 AutR(\u00CE\u009BnR). Since each \u00CE\u00B1i is an idempotent in \u00CE\u009BnR, so is each f(\u00CE\u00B1i). The componentsof each f(\u00CE\u00B1i) are idempotents in \u00CE\u009BR. By Lemma 2.2.4, they lie in R. Thus, f(\u00CE\u00B1i) \u00E2\u0088\u0088 \u00E2\u008A\u0095ni=1R\u00CE\u00B1ifor every i = 1, . . . , n. As a result, we obtain a morphismAutR(\u00CE\u009BnR)\u00CF\u0084R\u00E2\u0088\u0092\u00E2\u0088\u0092\u00E2\u0086\u0092 AutR(\u00E2\u008A\u0095ni=1R\u00CE\u00B1i) = Sn(R).(For the second equality, see, e.g., p. 59 in [Wat79].) These maps are functorial in R and thusgive rise to a morphism \u00CF\u0084 : Aut(\u00CE\u009Bn)\u00E2\u0086\u0092 Sn of group schemes over k. The kernel of \u00CF\u0084 is Aut(\u00CE\u009B)n,and \u00CF\u0084 clearly has a section. The lemma follows.Remark 2.2.5. The assumption that \u00CE\u009B is commutative in Proposition 2.2.3 can be dropped, aslong as we assume that the center of \u00CE\u009B is a finite-dimensional local k-algebra with residue fieldk. The proof proceeds along similar lines, except that we restrict f to an automorphism of thecenter Z(\u00CE\u009Bn) = Z(\u00CE\u009B)n and apply Lemma 2.2.4 to Z(\u00CE\u009B), rather than \u00CE\u009B itself. This more generalvariant of Proposition 2.2.3 will not be needed in the sequel.Remark 2.2.6. On the other hand, the assumption that the residue field of \u00CE\u009B is k cannot bedropped. For example, if \u00CE\u009B is a separable field extension of k of degree d, then Autk(\u00CE\u009Bn) is atwisted form ofAutk(\u00CE\u009Bn \u00E2\u008A\u0097k k) = Autk(kdn) = Snd .Here k denotes the separable closure of k. Similarly, Autk(\u00CE\u009B)noSd is a twisted form of (Sd)noSn.For d, n > 1, these groups have different orders, so they cannot be isomorphic.2.3 Essential dimension of a functorIn the sequel we will need the following general notion of essential dimension, due to A. Merkur-jev [BF03]. Let F : Fieldsk \u00E2\u0086\u0092 Sets be a covariant functor from the category of field extensionsK/k to the category of sets. Here k is assumed to be fixed throughout, and K ranges over allfields containing k. We say that an object a \u00E2\u0088\u0088 F(K) descends to a subfield K0 \u00E2\u008A\u0082 K if a lies inthe image of the natural restriction map F(K0)\u00E2\u0086\u0092 F(K). The essential dimension ed(a) of a isdefined as minimal value of trdeg(K0/k), where k \u00E2\u008A\u0082 K0 and a descends to K0. The essentialdimension of the functor F , denoted by ed(F), is the supremum of ed(a) for all a \u00E2\u0088\u0088 F (K), andall fields K in Fieldsk.If l is a prime, there is also a related notion of essential dimension at l, which we denote byedl. For an object a \u00E2\u0088\u0088 F , we define edl(a) as the minimal value of ed(a\u00E2\u0080\u00B2), where a\u00E2\u0080\u00B2 is the imageof a in F(K \u00E2\u0080\u00B2), and the minimum is taken over all field extensions K \u00E2\u0080\u00B2/K such that the degree[K \u00E2\u0080\u00B2 : K] is finite and prime to l. The essential dimension edl(F) of the functor F at l is definedas the supremum of edl(a) for all a \u00E2\u0088\u0088 F (K) and all fields K in Fieldsk. Note that the prime l inthis definition is unrelated to p = char(k); we allow both l = p and l 6= p.Example 2.3.1. Let G be a group scheme over a base field k and FG : K \u00E2\u0086\u0092 H1(K,G) be thefunctor defined byFG(K) = {isomorphism classes of G-torsors T \u00E2\u0086\u0092 Spec(K)}.92.4. Field extensions of type (n, e)Here by a torsor we mean a torsor in the flat (fppf) topology. If G is smooth, then H1(K,G) isthe first Galois cohomology set, as in [Ser02a]; see Section II.1. The essential dimension ed(G) is,by definition, ed(FG), and similarly for the essential dimension edl(G) of G at at prime l. Thesenumerical invariants of G have been extensively studied; see, e.g.,[Mer09] or [Rei10] for a survey.Example 2.3.2. Define the functor Algn : Fieldsk \u00E2\u0086\u0092 Sets byAlgn(K) = {isomorphism classes of n-dimensional K-algebras}.If A is an n-dimensional dimensional algebra, and [A] is its class in Algn(K), then ed([A])coincides with ed(A) defined at the beginning of Section 2.2. By Lemma 2.2.1, ed(Algn) 6 n3;the exact value is unknown (except for very small n).We will now restrict our attention to certain subfunctors of Algn which are better understood.Definition 2.3.3. Let \u00CE\u009B/k be a finite-dimensional algebra and K/k be a field extension (notnecessarily finite or separable). We say that an algebra A/K is a K-form of \u00CE\u009B if there exists afield L containing K such that \u00CE\u009B\u00E2\u008A\u0097k L is isomorphic to A\u00E2\u008A\u0097K L as an L-algebra. We will writeAlg\u00CE\u009B : Fieldsk \u00E2\u0086\u0092 Setsfor the functor which sends a field K/k to the set of K-isomorphism classes of K-forms of \u00CE\u009B.Proposition 2.3.4. Let \u00CE\u009B be a finite-dimensional k-algebra and G = Autk(\u00CE\u009B) \u00E2\u008A\u0082 GL(\u00CE\u009B) be itsautomorphism group scheme. Then the functors Alg\u00CE\u009B and FG = H1(\u00E2\u0088\u0097, G) are isomorphic. Inparticular, ed(Alg\u00CE\u009B) = ed(G) and edl(Alg\u00CE\u009B) = edl(G) for every prime l.Proof. For the proof of the first assertion, see Proposition X.2.4 in [Ser79] or Proposition III.2.2.2in [Knu91]. The second assertion is an immediate consequence of the first, since isomorphicfunctors have the same essential dimension.Example 2.3.5. The K-forms of \u00CE\u009Bn = k \u00C3\u0097 \u00C2\u00B7 \u00C2\u00B7 \u00C2\u00B7 \u00C3\u0097 k (n times) are called e\u00C2\u00B4tale algebras of degreen. An e\u00C2\u00B4tale algebra L/K of degree n is a direct products of separable field extensions,L = L1 \u00C3\u0097 \u00C2\u00B7 \u00C2\u00B7 \u00C2\u00B7 \u00C3\u0097 Lr, wherer\u00E2\u0088\u0091i=1[Li : K] = n.The functor Alg\u00CE\u009Bn is usually denoted by Etn. The automorphism group Autk(\u00CE\u009Bn) is thesymmetric group Sn, acting on \u00CE\u009Bn by permuting the n factors of k; see Proposition 2.2.3. ThusEtn = H1(K,Sn); see, e.g., Examples 2.1 and 3.2 in [Ser03].2.4 Field extensions of type (n, e)Let L/S be a purely inseparable extension of finite degree. For x \u00E2\u0088\u0088 L we define the exponentof x over S as the smallest integer e such that xpe \u00E2\u0088\u0088 S. We will denote this number by e(x, S).We will say that x \u00E2\u0088\u0088 L is normal in L/S if e(x, S) = max{e(y, S) | y \u00E2\u0088\u0088 L}. When the base fieldS is clear from context we will omit S in notation e(x, S). A sequence x1, . . . , xr in L is callednormal if each xi is normal in L/Li\u00E2\u0088\u00921 and xi /\u00E2\u0088\u0088 Li\u00E2\u0088\u00921. Here Li = S(x1, . . . , xi\u00E2\u0088\u00921) and L0 = S.If L = S(x1, . . . , xr), where x1, . . . , xr is a normal sequence in L/S, then we call x1, . . . , xr anormal generating sequence of L/S. We will say that this sequence is of type e = (e1, . . . , er) ifei := e(xi, Li\u00E2\u0088\u00921) for each i. Here Li = S(x1, . . . , xi), as above. It is clear that e1 > e2 > . . . > er.102.4. Field extensions of type (n, e)Proposition 2.4.1. (G. Pickert [Pic49]) Let L/S be a purely inseparable field extension of finitedegree.(a) For any generating set \u00CE\u00BB of L/S there exists a normal generating sequence x1, . . . , xr witheach xi \u00E2\u0088\u0088 \u00CE\u00BB.(b) If x1, . . . , xr and y1, . . . , ys are two normal generating sequences for L/S, of types(e1, . . . , er) and (f1, . . . , fs) respectively, then r = s and ei = fi for each i = 1, . . . , r.Proof. For modern proofs of both parts, see Propositions 6 and 8 in [Ras71] or Lemma 1.2 andCorollary 1.5 in [Kar89].Proposition 2.4.1 allows us to talk about the type of a purely inseparable extension L/S. Wesay that L/S is of type e = (e1, . . . , er) if it admits a normal generating sequence x1, . . . , xr oftype e.Now suppose L/K is an arbitrary inseparable (but not necessarily purely inseparable) fieldextension L/K of finite degree. Denote the separable closure of K in L by S. We will say thatL/K is of type (n, e) if [S : K] = n and the purely inseparable extension L/S is of type e.Remark 2.4.2. Note that we will assume throughout that r > 1, i.e., that L/K is not separable.In particular, a finite field K does not admit an extension of type (n, e) for any n and e.Remark 2.4.3. It is easy to see that any proper subset of a normal generating sequence{x1, . . . , xr} of purely inseparable extension L/K generates a proper subfield of L. In otherwords, a normal generating sequence is a minimal generating set of L/K. By Theorem 6 in[BM40] we have [L : K(Lp)] = pr. Here K(Lp) denotes the subfield of L generated by Lp and K.Lemma 2.4.4. Let n > 1 and e1 > e2 > \u00C2\u00B7 \u00C2\u00B7 \u00C2\u00B7 > er > 1 be integers. Then there exist(a) a separable field extension E/F of degree n with k \u00E2\u008A\u0082 F ,(b) a field extension L/K of type (n, e) with k \u00E2\u008A\u0082 K and e = (e1, . . . , er).In particular, this lemma shows that the maxima in definitions (2.1.1) and (2.1.2) are takenover a non-empty set of integers.Proof. (a) Let x1, . . . , xn be independent variables over k. Set E = k(x1, . . . , xn) and F = EC ,where C is the cyclic group of order n acting on E by permuting the variables. Clearly E/F is aGalois (and hence, separable) extension of degree n.(b) Let E/F be as in part (a) and y1, . . . , yr be independent variables over F . Set L =E(y1, . . . , yr) and K = F (z1, . . . , zr), where zi = ypeii . One readily checks that S = E(z1, . . . , zn)is the separable closure of K in L and L/S is a purely inseparable extension of type e.Now suppose n > 1 and e = (e1, . . . , er) are as above, with e1 > e2 > \u00C2\u00B7 \u00C2\u00B7 \u00C2\u00B7 > er > 1. Thefollowing finite-dimensional commutative k-algebras will play an important role in the sequel:\u00CE\u009Bn,e = \u00CE\u009Be \u00C3\u0097 \u00C2\u00B7 \u00C2\u00B7 \u00C2\u00B7 \u00C3\u0097 \u00CE\u009Be (n times), where \u00CE\u009Be = k[x1, . . . , xr]/(xpe11 , . . . , xperr ) (2.4.1)is a truncated polynomial algebra.Lemma 2.4.5. \u00CE\u009Bn,e is isomorphic to \u00CE\u009Bm,f if and only if m = n and e = f .112.5. Proof of the upper bound of Theorem 2.1.2Proof. One direction is obvious: if m = n and e = f , then \u00CE\u009Bn,e is isomorphic to \u00CE\u009Bm,fTo prove the converse, note that \u00CE\u009Be is a finite-dimensional local k-algebra with residue fieldk. By Lemma 2.2.4, the only idempotents in \u00CE\u009Be are 0 and 1. This readily implies that the onlyidempotents in \u00CE\u009Bn,e are of the form (\u000F1, . . . , \u000Fn), where each \u000Fi is 0 or 1, and the only minimalidempotents are\u00CE\u00B11 = (1, 0, . . . , 0), . . . , \u00CE\u00B1n = (0, . . . , 0, 1).(Recall that idempotents \u00CE\u00B1 and \u00CE\u00B2 are called orthogonal if \u00CE\u00B1\u00CE\u00B2 = \u00CE\u00B2\u00CE\u00B1 = 0. If \u00CE\u00B1 and \u00CE\u00B2 are orthogonal,then one readily checks that \u00CE\u00B1+ \u00CE\u00B2 is also an idempotent. An idempotent is minimal if it cannotbe written as a sum of two orthogonal idempotents.)If \u00CE\u009Bn,e and \u00CE\u009Bm,f are isomorphic, then they have the same number of minimal idempotents;hence, m = n. Denote the minimal idempotents of \u00CE\u009Bm,f by\u00CE\u00B21 = (1, 0, . . . , 0), . . . , \u00CE\u00B2m = (0, . . . , 0, 1).A k-algebra isomorphism \u00CE\u009Bn,e \u00E2\u0086\u0092 \u00CE\u009Bm,f takes \u00CE\u00B11 to \u00CE\u00B2j for some j = 1, . . . , n and, hence, induces ak-algebra isomorphism between \u00CE\u00B11\u00CE\u009Bn,e ' \u00CE\u009Be and \u00CE\u00B2j\u00CE\u009Bm,f ' \u00CE\u009Bf . To complete the proof, we appealto Proposition 8 in [Ras71], which asserts that \u00CE\u009Be and \u00CE\u009Bf are isomorphic if and only if e = f .Lemma 2.4.6. Let L/K be a field extension of finite degree. Then the following are equivalent.(a) L/K is of type (n, e).(b) L is a K-form of \u00CE\u009Bn,e. In other words, L \u00E2\u008A\u0097K K \u00E2\u0080\u00B2 is isomorphic to \u00CE\u009Bn,e \u00E2\u008A\u0097k K \u00E2\u0080\u00B2 as anK \u00E2\u0080\u00B2-algebra for some field extension K \u00E2\u0080\u00B2/K.Proof. (a) =\u00E2\u0087\u0092 (b): Assume L/K is a field extension of type (n, e). Let S be the separableclosure of K in L and K \u00E2\u0080\u00B2 be an algebraic closure of S (which is also an algebraic closure of K).ThenL\u00E2\u008A\u0097K K \u00E2\u0080\u00B2 = L\u00E2\u008A\u0097S (S \u00E2\u008A\u0097K K \u00E2\u0080\u00B2) = (L\u00E2\u008A\u0097S K \u00E2\u0080\u00B2)\u00C3\u0097 \u00C2\u00B7 \u00C2\u00B7 \u00C2\u00B7 \u00C3\u0097 (L\u00E2\u008A\u0097S K \u00E2\u0080\u00B2) (n times).On the other hand, by [Ras71], Theorem 3, L\u00E2\u008A\u0097S K \u00E2\u0080\u00B2 is isomorphic to \u00CE\u009Be as a K \u00E2\u0080\u00B2-algebra, andpart (b) follows.(b) =\u00E2\u0087\u0092 (a): Assume L\u00E2\u008A\u0097K K \u00E2\u0080\u00B2 is isomorphic to \u00CE\u009Bn,e \u00E2\u008A\u0097k K \u00E2\u0080\u00B2 as an K \u00E2\u0080\u00B2-algebra for some fieldextension K \u00E2\u0080\u00B2/K. After replacing K \u00E2\u0080\u00B2 by a larger field, we may assume that K \u00E2\u0080\u00B2 contains the normalclosure of S over K. Since \u00CE\u009Bn,e \u00E2\u008A\u0097k K \u00E2\u0080\u00B2 is not separable over K \u00E2\u0080\u00B2, L is not separable over K. ThusL/K is of type (m, f) for some m > 1 and f = (f1, . . . , fs) with f1 > f2 > \u00C2\u00B7 \u00C2\u00B7 \u00C2\u00B7 > fs > 1. By part(a), L\u00E2\u008A\u0097K K \u00E2\u0080\u00B2\u00E2\u0080\u00B2 is isomorphic to \u00CE\u009Bm,f \u00E2\u008A\u0097k K \u00E2\u0080\u00B2\u00E2\u0080\u00B2 for a suitable field extension K \u00E2\u0080\u00B2\u00E2\u0080\u00B2/K. After enlargingK \u00E2\u0080\u00B2\u00E2\u0080\u00B2, we may assume without loss of generality that K \u00E2\u0080\u00B2 \u00E2\u008A\u0082 K \u00E2\u0080\u00B2\u00E2\u0080\u00B2. We conclude that \u00CE\u009Bn,e \u00E2\u008A\u0097k K \u00E2\u0080\u00B2\u00E2\u0080\u00B2 isisomorphic to \u00CE\u009Bm,f \u00E2\u008A\u0097k K \u00E2\u0080\u00B2\u00E2\u0080\u00B2 as a K \u00E2\u0080\u00B2\u00E2\u0080\u00B2-algebra. By Lemma 2.4.5, with k replaced by K \u00E2\u0080\u00B2\u00E2\u0080\u00B2, this is onlypossible if (n, e) = (m, f).2.5 Proof of the upper bound of Theorem 2.1.2In this section we will prove the following proposition.Proposition 2.5.1. Let n > 1 and e = (e1, . . . , er), where e1 > \u00C2\u00B7 \u00C2\u00B7 \u00C2\u00B7 > er > 1. Furthermore letsi =\u00E2\u0088\u0091j\u00E2\u0089\u00A4i ei. Then\u00CF\u0084(n, e) 6 nr\u00E2\u0088\u0091i=1psi\u00E2\u0088\u0092iei .122.5. Proof of the upper bound of Theorem 2.1.2Our proof of Proposition 2.5.1 will be facilitated by the following lemma.Lemma 2.5.2. Let K be an infinite field of characteristic p, let q be a power of p, S/K be aseparable field extension of finite degree, and 0 6= a \u00E2\u0088\u0088 S. Then there exists an s \u00E2\u0088\u0088 S such thatasq is a primitive element for S/K.Proof. Assume the contrary. It is well known that there are only finitely many intermediate fieldsbetween K and S; see e.g., [Lan02], Theorem V.4.6. Denote the intermediate fields properlycontained in S by S1, . . . , Sn ( S and let AK(S) be the affine space associated to S. (Here weview S as a K-vector space.) The non-generators of S/K may now be viewed as K-points of thefinite unionZ = \u00E2\u0088\u00AAni=1AK(Si) .Since we are assuming that every element of S of the form asq is a non-generator, and Kis an infinite field, the image of the K-morphism f : A(S) \u00E2\u0086\u0092 A(S) given by s \u00E2\u0086\u0092 asq lies inZ = \u00E2\u0088\u00AAni=1AK(Si). Since AK(S) is irreducible, we conclude that the image of f lies in one of theaffine subspaces AK(Si), say in AK(S1). Equivalently, asq \u00E2\u0088\u0088 S1 for every s \u00E2\u0088\u0088 S. Setting s = 1,we see that a \u00E2\u0088\u0088 S1. Dividing asq \u00E2\u0088\u0088 S1 by 0 6= a \u00E2\u0088\u0088 S1, we conclude that sq \u00E2\u0088\u0088 S1 for every s \u00E2\u0088\u0088 S.Thus S is purely inseparable over S1, contradicting our assumption that S/K is separable.Proof of Proposition 2.5.1. Let L/K be a field extension of type (n, e). Our goal is to show thated(L/K) 6 n\u00E2\u0088\u0091rj=1 psj\u00E2\u0088\u0092jej . By Remark 2.4.2, K is infinite.Let S be the separable closure of K in L and x1, . . . , xr be a normal generating sequencefor the purely inseparable extension L/S of type e. Set qi = pei . Recall that by the definitionof normal sequence, xq11 \u00E2\u0088\u0088 S. We are free to replace x1 by x1s for any 0 6= s \u00E2\u0088\u0088 S; clearlyx1s, x2, . . . , xr is another normal generating sequence. By Lemma 2.5.2, we may choose s \u00E2\u0088\u0088 Sso that (x1s)q1 is a primitive element for S/K. In other words, we may assume without loss ofgenerality that xq11 is a primitive element for S/K.By the structure theorem of Pickert, each xqii lies in S[xqi1 , . . . , xqii\u00E2\u0088\u00921], where qi = pei ; seeTheorem 1 in [Ras71]. In other words, for each i = 1, . . . , r,xqii =\u00E2\u0088\u0091ad1,...,di\u00E2\u0088\u00921xqid11 . . . xqidi\u00E2\u0088\u00921i\u00E2\u0088\u00921 (2.5.1)for some for some ad1,...,di\u00E2\u0088\u00921 \u00E2\u0088\u0088 S. Here the sum is taken over all integers d1, . . . , di\u00E2\u0088\u00921, where each0 6 dj 6 pej\u00E2\u0088\u0092ei \u00E2\u0088\u0092 1. Note that for i = 1 formula (2.5.1) reduces toxq11 = a\u00E2\u0088\u0085,for some a\u00E2\u0088\u0085 \u00E2\u0088\u0088 S. By Lemma 2.2.1, L (viewed as an S-algebra), descends toS0 = k(ad1,...,di\u00E2\u0088\u00921 | i = 1, . . . , r and 0 6 dj 6 pej\u00E2\u0088\u0092ei \u00E2\u0088\u0092 1) .Note that for each i = 1, . . . , r, there are exactlype1\u00E2\u0088\u0092ei \u00C2\u00B7 pe2\u00E2\u0088\u0092ei \u00C2\u00B7 . . . \u00C2\u00B7 pei\u00E2\u0088\u00921\u00E2\u0088\u0092ei = psi\u00E2\u0088\u0092ieichoices of the subscripts d1, . . . , di\u00E2\u0088\u00921. Hence, S0 is generated over k by\u00E2\u0088\u0091ri=1 psi\u00E2\u0088\u0092iei elementsand consequently,trdeg(S0/k) 6r\u00E2\u0088\u0091i=1psi\u00E2\u0088\u0092iei .132.6. Versal algebrasMoreover, since S0 contains a\u00E2\u0088\u0085 = xq1, which is a primitive element for S/K, we conclude thatK(S0) = S. Thus Lemma 2.2.2 can be applied to A = L; it yields ed(L/K) 6 n trdeg(S0/k),and the proposition follows.Remark 2.5.3. Suppose L/K is an extension of type (n, e), where e = (e1, . . . , er). Here, asusual, K is assumed to contain the base field k of characteristic p > 0. Dividing both sides ofthe inequality in Proposition 2.5.1 by [L : K] = npe1+\u00C2\u00B7\u00C2\u00B7\u00C2\u00B7+er , we readily deduce thated(L/K)[L : K]6 \u00CF\u0084(n, e)[L : K]6r\u00E2\u0088\u0091i=1p\u00E2\u0088\u0092iei\u00E2\u0088\u0092ei+1\u00E2\u0088\u0092\u00C2\u00B7\u00C2\u00B7\u00C2\u00B7\u00E2\u0088\u0092rr 6 rpr6 1p.In particular, ed(L/K) 6 12[L : K] for any inseparable extension [L : K] of finite degree, inany (positive) characteristic. As we pointed out in the introduction, this inequality fails incharacteristic 0 (even for k = C).2.6 Versal algebrasLet K be a field and A be a finite-dimensional associative K-algebra with 1. Every a \u00E2\u0088\u0088 A givesrise to the K-linear map la : A \u00E2\u0086\u0092 A given by la(x) = ax (left multiplication by a). Note thatlab = la \u00C2\u00B7 lb. It readily follows from this that a has a multiplicative inverse in A if and only if la isnon-singular.Proposition 2.6.1. Let l be a prime integer and \u00CE\u009B be a finite-dimensional associative k-algebrawith 1. Assume that there exists a field extension K/k and a K-form A of \u00CE\u009B such that A is adivision algebra. Then(a) there exists a field Kver containing k and a form Aver/Kver of \u00CE\u009B such thated(Aver) = ed(Alg\u00CE\u009B), edl(Aver) = edl(Alg\u00CE\u009B) for every prime integer l, andAver is a division algebra.(b) If G is the automorphism group scheme of \u00CE\u009B, thened(G) = ed(Alg\u00CE\u009B) = max{ed(A/K) |A is a K-form of \u00CE\u009B and a division algebra}andedl(G) = edl(Alg\u00CE\u009B) = max{edl(A/K) |A is a K-form of \u00CE\u009B and a division algebra}.Here the subscript \u00E2\u0080\u009Cver\u00E2\u0080\u009D is meant to indicate that Aver/Kver is a versal object for Alg\u00CE\u009B =H1(\u00E2\u0088\u0097, G). For a discussion of versal torsors, see Section I.5 in [Ser03], [BF03] or [DR15].Proof. (a) We begin by constructing of a versal G-torsor Tver \u00E2\u0086\u0092 Spec(Kver). Recall thatG = Autk(\u00CE\u009B) is defined as a closed subgroup of the general linear group GLk(\u00CE\u009B). This generallinear group admits a generically free linear action on some vector space V (e.g., we can takeV = Endk(\u00CE\u009B), with the natural left G-action). Restricting to G we obtain a generically freerepresentation G \u00E2\u0086\u0092 GL(V ). We can now choose a dense open G-invariant subscheme U \u00E2\u008A\u0082 Vover k which is the total space of a G-torsor pi : U \u00E2\u0086\u0092 B; see, e.g., Section 4 in [BF03]. Passing to142.6. Versal algebrasthe generic point of B, we obtain a G-torsor Tver \u00E2\u0086\u0092 Spec(Kver), where Kver is the function fieldof B over k. Then ed(Tver/Kver) = ed(G) and edl(Tver/Kver) = edl(G) (see [BF03, Corollary6.16]).Let T \u00E2\u0086\u0092 Spec(K) be the torsor associated to the K-algebra A and Aver be the Kver-algebra associated to Tver \u00E2\u0086\u0092 Spec(Kver) under the isomorphism between the functors Alg\u00CE\u009Band H1(\u00E2\u0088\u0097, G) of Proposition 2.3.4. By the characteristic-free version of the no-name Lemma,proved in [RV06], Section 2, T \u00C3\u0097 VK is G-equivariantly birationally isomorphic to T \u00C3\u0097AdK , whered = dim(V ) = dim(VK) and G acts trivially on AdK . In other words, we have a Cartesian diagramof rational maps defined over kT \u00C3\u0097 AdK ' //\u000F\u000FT \u00C3\u0097 VK pr2 // UK\u000F\u000FAdK // BK .Here all direct products are over Spec(K), and pr2 denotes the rational G-equivariant projectionmap taking (t, v) \u00E2\u0088\u0088 T \u00C3\u0097 V to v \u00E2\u0088\u0088 V for v \u00E2\u0088\u0088 U . The map AdK = Spec(K) \u00C3\u0097 Ad 99K B in thebottom row is induced from the dominant G-equivariant map T \u00C3\u0097 AdK 99K UK on top. Passingto generic points, we obtain an inclusion of field Kver \u00E2\u0086\u00AA\u00E2\u0086\u0092 K.Kver \u00E2\u0086\u00AA\u00E2\u0086\u0092 K(x1, . . . , xd) such that theinduced map H1(Kver, G)\u00E2\u0086\u0092 H1(K(x1, . . . , xd), G) sends the class of Tver \u00E2\u0086\u0092 Spec(Kver) to theclass associated to T \u00C3\u0097 AdK \u00E2\u0086\u0092 AdK . Under the isomorphism of Proposition 2.3.4 between thefunctors Alg\u00CE\u009B and FG = H1(\u00E2\u0088\u0097, G), this translates toAver \u00E2\u008A\u0097Kver K(x1, . . . , xd) ' A\u00E2\u008A\u0097K K(x1, . . . , xd)as K(x1, . . . , xd)-algebras.For simplicity we will write A(x1, . . . , xd) in place of A \u00E2\u008A\u0097K K(x1, . . . , xd). Since A is adivision algebra, so is A(x1, . . . , xd). Thus the linear map la : A(x1, . . . , xd)\u00E2\u0086\u0092 A(x1, . . . , xd) isnon-singular (i.e., has trivial kernel) for every a \u00E2\u0088\u0088 Aver. Hence, the same is true for the restrictionof la to Aver. We conclude that Aver is a division algebra. Remembering that Aver correspondsto Tver under the isomorphism of functors between Alg\u00CE\u009B and FG, we see thated(Aver) = ed(Tver/Kver) = ed(G) = ed(Alg\u00CE\u009B)andedl(Aver) = edl(Tver/Kver) = edl(G) = edl(Alg\u00CE\u009B) ,as desired.(b) The first equality in both formulas follows from Proposition 2.3.4, and the second frompart (a).We will now revisit the finite-dimensional k-algebras \u00CE\u009Be and \u00CE\u009Bn,e = \u00CE\u009Be \u00C3\u0097 \u00C2\u00B7 \u00C2\u00B7 \u00C2\u00B7 \u00C3\u0097 \u00CE\u009Be (ntimes) defined in Section 2.4; see (2.4.1). We will write Gn,e = Aut(\u00CE\u009Bn,e) \u00E2\u008A\u0082 GLk(\u00CE\u009Bn,e) for theautomorphism group scheme of \u00CE\u009Bn,e and Algn,e for the functor Alg\u00CE\u009Bn,e : Fieldsk \u00E2\u0086\u0092 Sets. Recallthat this functor associates to a field K/k the set of isomorphism classes of K-forms of \u00CE\u009Bn,e.Replacing essential dimension by essential dimension at a prime l in the definitions (2.1.1)and (2.1.1), we set\u00CF\u0084l(n) = max{edl(L/K) | L/K is a separable field extension of degree n and k \u00E2\u008A\u0082 K}.152.7. Conclusion of the proof of Theorem 2.1.2and\u00CF\u0084l(n, e) = max{edl(L/K) | L/K is a field extension of type (n, e) and k \u00E2\u008A\u0082 K}.Corollary 2.6.2. Let l be a prime integer. Then(a) ed(Sn) = ed(Etn) = \u00CF\u0084(n) and edl(Sn) = edl(Etn) = \u00CF\u0084l(n). Here Etn is the functor ofn-dimensional e\u00C2\u00B4tale algebras, as in Example 2.3.5.(b) ed(Gn,e) = ed(Algn,e) = \u00CF\u0084(n, e) and edl(Gn,e) = edl(Algn,e) = \u00CF\u0084l(n, e).Proof. (a) Recall that e\u00C2\u00B4tale algebra are, by definition, commutative and associative with identity.For such algebras \u00E2\u0080\u009Cdivision algebra\u00E2\u0080\u009D is the same as \u00E2\u0080\u009Cfield\u00E2\u0080\u009D. By Lemma 2.4.4(a) there existsa separable field extension E/F of degree n with k \u00E2\u008A\u0082 F . The desired equality follows fromProposition 2.6.1(b).(b) The same argument as in part (a) goes through, with part (a) of Lemma 2.4.4 replacedby part (b).Remark 2.6.3. The value of edl(Sn) is known for every integer n > and every prime l > 2:edl(Sn) =\u00EF\u00A3\u00B1\u00EF\u00A3\u00B4\u00EF\u00A3\u00B4\u00EF\u00A3\u00B2\u00EF\u00A3\u00B4\u00EF\u00A3\u00B4\u00EF\u00A3\u00B3bnlc, if char(k) 6= l, see Corollary 4.2 in [MR09],1, if char(k) = l 6 n, see Theorem 1 in [RV18], and0, if char(k) = l > n, see Lemma 4.1 in [MR09] or Theorem 1 in [RV18].2.7 Conclusion of the proof of Theorem 2.1.2In this section we will prove Theorem 2.1.2 in the following strengthened form.Theorem 2.7.1. Let k be a base field of characteristic p > 0, n > 1 and e1 > e2 > \u00C2\u00B7 \u00C2\u00B7 \u00C2\u00B7 > er > 1be integers, e = (e1, . . . , er) and si = e1 + \u00C2\u00B7 \u00C2\u00B7 \u00C2\u00B7+ ei for i = 1, . . . , r. Then\u00CF\u0084p(n, e) = \u00CF\u0084(n, e) = nr\u00E2\u0088\u0091i=1psi\u00E2\u0088\u0092iei .By definition \u00CF\u0084p(n, e) 6 \u00CF\u0084(n, e) and by Proposition 2.5.1, \u00CF\u0084(n, e) 6 n\u00E2\u0088\u0091ri=1 psi\u00E2\u0088\u0092iei . Moreover,by Corollary 2.6.2(b), \u00CF\u0084p(n, e) = edp(Gn,e). It thus remains to show thatedp(Gn,e) > nr\u00E2\u0088\u0091i=1psi\u00E2\u0088\u0092iei . (2.7.1)Our proof of (2.7.1) will be based on the following general inequality, due to Tossici andVistoli [TV13]:edp(G) > dim(Lie(G))\u00E2\u0088\u0092 dim(G) (2.7.2)for any group scheme G of finite type over a field k of characteristic p. Now recall thatGe = Autk(\u00CE\u009Be), and Gn,e = Autk(\u00CE\u009Bn,e), where \u00CE\u009Bn,e = \u00CE\u009Bne . Since \u00CE\u009Be is is a commutative localk-algebra with residue field k, Proposition 2.2.3 tells us that Gn,e = Gne oSn (see also Proposition5.1 in [SdS00]). We conclude thatdim(Gn,e) = n dim(Ge) and dim(Lie(Gn,e)) = n dim(Lie(Ge)).Substituting these formulas into (2.7.2), we see that the proof of the inequality (2.7.1) (and thusof Theorem 2.7.1) reduces to the following.162.7. Conclusion of the proof of Theorem 2.1.2Proposition 2.7.2. Let e = (e1, . . . , er), where e1 > \u00C2\u00B7 \u00C2\u00B7 \u00C2\u00B7 > er > 1 are integers. Then(a) dim(Lie(Ge)) = rpe1+\u00C2\u00B7\u00C2\u00B7\u00C2\u00B7+er , and(b) dim(Ge) = rpe1+\u00C2\u00B7\u00C2\u00B7\u00C2\u00B7+er \u00E2\u0088\u0092\u00E2\u0088\u0091ri=1 psi\u00E2\u0088\u0092iei .The remainder of this section will be devoted to proving Proposition 2.7.2. We will use thefollowing notations.1. We fix the type e = (e1, . . . , er) and set qi = pei .2. The infinitesimal group scheme \u00CE\u00B1pj over a commutative ring S of characteristic p is definedas the kernel of the j-th power of the Frobenius map, Ga \u00E2\u0086\u0092 Ga, x 7\u00E2\u0086\u0092 xpj , viewed as ahomomorphism of group schemes over S. We will be particularly interested in the case,where S = \u00CE\u009Be.3. Suppose X is a scheme over \u00CE\u009B, where \u00CE\u009B is a finite-dimensional commutative k-algebra. Wewill denote the Weil restriction of the \u00CE\u009B-scheme X to k by R\u00CE\u009B/k(X). For generalities onWeil restriction, see Chapter 2 and the Appendix in [Mil17].4. We will denote by End(\u00CE\u009Be) the functorCommk \u00E2\u0088\u0092\u00E2\u0086\u0092 SetsR \u00E2\u0088\u0092\u00E2\u0086\u0092 EndR\u00E2\u0088\u0092alg(\u00CE\u009Be \u00E2\u008A\u0097k R)of algebra endomorphisms of \u00CE\u009Be. Here Commk denotes the category of of commutativeassociative k-algebras with 1 and Sets denotes the category of sets.Lemma 2.7.3. (a) The functor End(\u00CE\u009Be) is represented by an irreducible (but non-reduced) affinek-scheme Xe.(b) dim(Xe) = rpe1+\u00C2\u00B7\u00C2\u00B7\u00C2\u00B7+er \u00E2\u0088\u0092\u00E2\u0088\u0091ri=1 psi\u00E2\u0088\u0092iei.(c) dim(T\u00CE\u00B3(Xe)) = rpe1+\u00C2\u00B7\u00C2\u00B7\u00C2\u00B7+er for any k-point \u00CE\u00B3 of Xe. Here T\u00CE\u00B3(Xe) denotes the tangentspace to Xe at \u00CE\u00B3.Proof. An endomorphism F in End(\u00CE\u009Be)(R) is uniquely determined by the imagesF (x1), F (x2), . . . , F (xr) \u00E2\u0088\u0088 \u00CE\u009Be(R)of the generators x1, . . . , xr of \u00CE\u009Be. These elements of \u00CE\u009Be satisfy F (xi)qi = 0. Conversely, anyr elements F1, . . . , Fr in \u00CE\u009Be \u00E2\u008A\u0097R satisfying F qii = 0, give rise to an algebra endomorphism F inEnd(\u00CE\u009Be)(R). We thus haveEnd(\u00CE\u009Be)(R) = HomR\u00E2\u0088\u0092alg(\u00CE\u009Be \u00E2\u008A\u0097k R,\u00CE\u009Be \u00E2\u008A\u0097R)\u00E2\u0088\u00BC= \u00CE\u00B1q1(\u00CE\u009Be \u00E2\u008A\u0097R)\u00C3\u0097 . . .\u00C3\u0097 \u00CE\u00B1qr(\u00CE\u009Be \u00E2\u008A\u0097R)\u00E2\u0088\u00BC= R\u00CE\u009Be/k(\u00CE\u00B1q1)(R)\u00C3\u0097 . . .\u00C3\u0097R\u00CE\u009Be/k(\u00CE\u00B1qr)(R)\u00E2\u0088\u00BC=r\u00E2\u0088\u008Fi=1R\u00CE\u009Be/k(\u00CE\u00B1qi)(R)172.7. Conclusion of the proof of Theorem 2.1.2We conclude that End(\u00CE\u009Be) is represented by an affine k-scheme Xe =\u00E2\u0088\u008Fri=1R\u00CE\u009Be/k(\u00CE\u00B1qi). (Notethat Xe is isomorphic to\u00E2\u0088\u008Fri=1R\u00CE\u009Be/k(\u00CE\u00B1qi) as a k-scheme only, not as a group scheme.) Tocomplete the proof of the lemma it remains to establish the following assertions:For any qj \u00E2\u0088\u0088 {q1, . . . , qr} we have that(a\u00E2\u0080\u00B2) R\u00CE\u009Be/k(\u00CE\u00B1qj ) is irreducible,(b\u00E2\u0080\u00B2) dim(R\u00CE\u009Be/k(\u00CE\u00B1qj ))= pe1+\u00C2\u00B7\u00C2\u00B7\u00C2\u00B7+er \u00E2\u0088\u0092 psj\u00E2\u0088\u0092jej and(c\u00E2\u0080\u00B2) dim(T\u00CE\u00B3(R\u00CE\u009Be/k(\u00CE\u00B1qj ))) = pe1+\u00C2\u00B7\u00C2\u00B7\u00C2\u00B7+er for any k-point \u00CE\u00B3 of R\u00CE\u009Be/k(\u00CE\u00B1qj ).To prove (a\u00E2\u0080\u00B2), (b\u00E2\u0080\u00B2) and (c\u00E2\u0080\u00B2), we will write out explicit equations for R\u00CE\u009Be/k(\u00CE\u00B1qj ) in R\u00CE\u009Be/k(A1) 'Ak(\u00CE\u009Be). We will work in the basis {xi11 xi22 . . . xirr } of monomials in \u00CE\u009Be, where 0 6 i1 < q1,0 6 i2 < q2, . . ., 0 6 ir < qr. Over \u00CE\u009Be, \u00CE\u00B1qj is cut out (scheme-theoretically) in A1 by the singleequation tqj = 0, where t is a coordinate function on A1. Since xqii = 0 for every i, writingt =\u00E2\u0088\u0091yi1,...,irxi11 xi22 . . . xirrand expandingtqj =\u00E2\u0088\u0091yqji1,...,irxqji11 xqji22 . . . xqjirrwe see that the only monomials appearing in the above sum are those for whichqji1 < q1, qji2 < q2, . . . , qjir < qr.Thus R\u00CE\u009Be/k(\u00CE\u00B1qj ) is cut out (again, scheme-theoretically) in R\u00CE\u009Be/k(A1) ' A(\u00CE\u009Be) byyqji1,...,ij\u00E2\u0088\u00921,0,...,0 = 0 for 0 6 i1 r. Assume the contrary: L/K descends toL0/K0 with trdeg(K0/k) < r. By Lemma 2.2.1, L0/K0 further descends to L1/K1, where K1 isfinitely generated over k. By Lemma 2.4.6, L1/K1 is a purely inseparable extension of type e.After replacing L/K by L1/K1, it remains to prove the following:Lemma 2.9.2. Let k be a perfect field and K/k be a finitely generated field extension oftranscendence degree < r. There there does not exist a purely inseparable field extension L/K oftype e = (e1, . . . , er), where e1 > \u00C2\u00B7 \u00C2\u00B7 \u00C2\u00B7 > er > 1.Proof. Assume the contrary. Let a1, . . . , as be a transcendence basis for K/k. That is, a1, . . . , asare algebraically independent over k, K is algebraic and finitely generated (hence, finite) overk(a1, . . . , as) and s 6 r \u00E2\u0088\u0092 1. By Remark 2.4.3,[L : Lp] > [L : (Lp \u00C2\u00B7K)] = pr. (2.9.3)On the other hand, since [L : k(a1, . . . , as)] <\u00E2\u0088\u009E, Theorem 3 in [BM40] tells us that[L : Lp] = [k(a1, . . . , as) : k(a1, . . . , as)p] = [k(a1, . . . , as) : k(ap1, . . . , aps)] = ps < pr. (2.9.4)Note that the second equality relies on our assumption that k is perfect. The contradictionbetween (2.9.3) and (2.9.4) completes the proof of Lemma 2.9.2 and thus of Proposition 2.9.1.20Chapter 3Essential dimension of double coversof symmetric and alternating group3.1 IntroductionI. Schur [Sch04] studied central extensions1 // Z/2Z // S\u00CB\u009C\u00C2\u00B1n\u00CF\u0086\u00C2\u00B1 // Sn // 1 (3.1.1)of the symmetric group Sn. Representations of these groups are closely related to projectiverepresentations of Sn: over an algebraically closed field of characteristic zero, every projectiverepresentation \u00CF\u0081 : Sn \u00E2\u0086\u0092 PGL(V ) lifts to linear representations \u00CF\u0081+ : S\u00CB\u009C+n \u00E2\u0086\u0092 GL(V ) and \u00CF\u0081\u00E2\u0088\u0092 : S\u00CB\u009C\u00E2\u0088\u0092n \u00E2\u0086\u0092GL(V ); see [HH92, Theorem 1.3]. That is, the following diagram commutes.S\u00CB\u009C\u00C2\u00B1n\u00CF\u0086\u00C2\u00B1\u000F\u000F\u00CF\u0081\u00C2\u00B1 // GL(V )\u000F\u000FSn\u00CF\u0081 // PGL(V )Figure 3.1: Projective representationMoreover, the groups S\u00CB\u009C\u00C2\u00B1n are minimal central extensions of Sn with this property. They arecalled representation groups of Sn. In terms of generators and relations,S\u00CB\u009C+n =\u00E2\u008C\u00A9z, s1, s2, . . . , sn\u00E2\u0088\u00921 | z2 = s2i = 1, [z, si] = 1, (sisj)2 = z if |i\u00E2\u0088\u0092 j| > 1, (sisi+1)3 = 1\u00E2\u008C\u00AAandS\u00CB\u009C\u00E2\u0088\u0092n =\u00E2\u008C\u00A9z, t1, t2, . . . , tn\u00E2\u0088\u00921 | z2 = 1, t2i = z, (titj)2 = z if |i\u00E2\u0088\u0092 j| > 1, (titi+1)3 = z\u00E2\u008C\u00AA.Here z is a central element of order 2 in S\u00CB\u009C+n (respectively, S\u00CB\u009C\u00E2\u0088\u0092n ) generating Ker(\u00CF\u0086+) (respectively,Ker(\u00CF\u0086\u00E2\u0088\u0092)), and \u00CF\u0086+(si) = \u00CF\u0086\u00E2\u0088\u0092(ti) is the transposition (i, i+ 1) in Sn. The preimage of An under\u00CF\u0086+ in S\u00CB\u009C+n is isomorphic to the preimage of An under \u00CF\u0086\u00E2\u0088\u0092 in S\u00CB\u009C\u00E2\u0088\u0092n ; see [Ser08, Section 9.1.3]. We willdenote this group by A\u00CB\u009Cn; it is a representation group of An. For modern expositions of Schur\u00E2\u0080\u0099stheory, see [HH92] or [Ste89].The purpose of this chapter is to study the essential dimension of the covering groups S\u00CB\u009C\u00C2\u00B1n andA\u00CB\u009Cn. We will assume that n > 4 throughout. As usual, we will denote the essential dimension ofa linear algebraic group G by ed(G) and the essential dimension of G at a prime p by ed(G; p).These numbers depend on the base field k; we will sometimes write edk(G) and edk(G; p) in213.1. Introductionplace of ed(G) and ed(G; p) to emphasize this dependence. We refer the reader to Section 3.2 forthe definition of essential dimension, some of its properties and further references.Our interest in the covering groups S\u00CB\u009C\u00C2\u00B1n , A\u00CB\u009Cn was motivated by their close connection to twofamilies of groups whose essential dimension was previously found to behave in interesting ways,namely permutation groups and spinor groups. The connection with spinor groups is summarizedin the following diagram. Here the base field k is assumed to be of characteristic 6= 2 and tocontain a primitive 8th root of unity, On is the orthogonal group associated to the quadratic formx21 + \u00C2\u00B7 \u00C2\u00B7 \u00C2\u00B7+ x2n, Sn \u00E2\u0086\u0092 On is the natural n-dimensional representation, \u00CE\u0093n is the Clifford group, andS\u00CB\u009C\u00C2\u00B1n , A\u00CB\u009Cn are the preimages of Sn and An under the double covers Pin\u00C2\u00B1n \u00E2\u0086\u0092 On. The groups Pin\u00C2\u00B1nare defined as the kernels of the homomorphisms N\u00C2\u00B1 : \u00CE\u0093n \u00E2\u0086\u0092 Gm given by N+(x) = x.xT andN\u00E2\u0088\u0092(x) = x.\u00CE\u00B3(xT ), where (x1 \u00E2\u008A\u0097 x2 \u00E2\u008A\u0097 . . .\u00E2\u008A\u0097 xn)T = xn \u00E2\u008A\u0097 . . .\u00E2\u008A\u0097 x2 \u00E2\u008A\u0097 x1 and \u00CE\u00B3 is the automorphismof the Clifford algebra which acts on degree 1 component by \u00E2\u0088\u00921. Both appear in literature asPin groups (Pin+ in [Ser84] and Pin\u00E2\u0088\u0092 in [ABS64], see also [GG86]).On\u00CE\u0093nGmPin+n OnZ/2Z Pin\u00E2\u0088\u0092nS\u00CB\u009C+n SnZ/2Z S\u00CB\u009C\u00E2\u0088\u0092nAnA\u00CB\u009CnZ/2ZFigure 3.2: Schematic of spin groupsThe essential dimension of Sn and An is known to be sublinear in n: in particular, n > 5, wehaveed(An) 6 ed(Sn) 6 n\u00E2\u0088\u0092 3;see [BR97]*Theorem 6.5(c). On the other hand, the essential dimension of Spinn increasesexponentially with n. If we write n = 2am, where m is odd, thened(Spinn) = ed(Spinn; 2) =\u00EF\u00A3\u00B1\u00EF\u00A3\u00B4\u00EF\u00A3\u00B4\u00EF\u00A3\u00B4\u00EF\u00A3\u00B4\u00EF\u00A3\u00B2\u00EF\u00A3\u00B4\u00EF\u00A3\u00B4\u00EF\u00A3\u00B4\u00EF\u00A3\u00B4\u00EF\u00A3\u00B32(n\u00E2\u0088\u00921)/2 \u00E2\u0088\u0092 n(n\u00E2\u0088\u0092 1)2, if a = 0,2(n\u00E2\u0088\u00922)/2 \u00E2\u0088\u0092 n(n\u00E2\u0088\u0092 1)2, if a = 1,2(n\u00E2\u0088\u00922)/2 + 2a \u00E2\u0088\u0092 n(n\u00E2\u0088\u0092 1)2, if a > 2;223.1. Introductionsee [BRV10], [CM14] and [Tot19].Question 3.1.1. What is the asymptotic behavior of ed(S\u00CB\u009C\u00C2\u00B1n ), ed(A\u00CB\u009Cn), ed(S\u00CB\u009C\u00C2\u00B1n ; 2) and ed(A\u00CB\u009Cn; 2)as n \u00E2\u0088\u0092\u00E2\u0086\u0092\u00E2\u0088\u009E? Do these numbers grow sublinearly, like ed(Sn), or exponentially, like ed(Spinn)?Note that for odd primes p, S\u00CB\u009C+n , S\u00CB\u009C\u00E2\u0088\u0092n and Sn have isomorphic Sylow p-subgroups, andthus ed(S\u00CB\u009C\u00C2\u00B1n ; p) = ed(Sn; p); see Lemma 3.2.2. Moreover, these numbers are known (see, e.g.,[RS19a]*Remark 6.3 and the references there), and similarly for An. Thus it remains to under-stand ed(S\u00CB\u009C\u00C2\u00B1n ; p) and ed(A\u00CB\u009Cn; p) when p = 2.In this chapter we answer Question 3.1.1 as follows: ed(S\u00CB\u009C\u00C2\u00B1n ), ed(A\u00CB\u009Cn), ed(S\u00CB\u009C\u00C2\u00B1n ; 2) grow exponen-tially if char(k) 6= 2 and sublinearly if char(k) = 2. This follows from Theorems 3.1.2 and 3.1.4below.Theorem 3.1.2. Assume that the base field k is of characteristic 6= 2 and contains a primitive 8throot of unity, and let n > 4 be an integer. Write n = 2a1 + . . .+2as , where a1 > a2 > . . . > as > 0and let S\u00CB\u009Cn be either S\u00CB\u009C+n or S\u00CB\u009C\u00E2\u0088\u0092n . Then(a) ed(A\u00CB\u009Cn) 6 ed(S\u00CB\u009Cn) 6 2b(n\u00E2\u0088\u00921)/2c.(b) ed(S\u00CB\u009Cn; 2) = 2b(n\u00E2\u0088\u0092s)/2c,(c) ed(A\u00CB\u009Cn; 2) = 2b(n\u00E2\u0088\u0092s\u00E2\u0088\u00921)/2c.(d) 2b(n\u00E2\u0088\u0092s)/2c 6 ed(S\u00CB\u009Cn) 6 ed(Sn) + 2b(n\u00E2\u0088\u0092s)/2c(e) 2b(n\u00E2\u0088\u0092s\u00E2\u0088\u00921)/2c 6 ed(A\u00CB\u009Cn) 6 ed(An) + 2b(n\u00E2\u0088\u0092s\u00E2\u0088\u00921)/2c.For s = 1 and 2, upper and lower bounds of Theorem 3.1.2 meet, and we obtain the followingexact values.Corollary 3.1.3. Assume that the base field k contains a primitive 8th root of unity.(a) If n = 2a, where a > 2, then ed(S\u00CB\u009Cn) = ed(S\u00CB\u009Cn; 2) = ed(A\u00CB\u009Cn) = ed(A\u00CB\u009Cn; 2) = 2n\u00E2\u0088\u009222 .(b) If n = 2a1 + 2a2 , where a1 > a2 > 1, then ed(S\u00CB\u009Cn) = ed(S\u00CB\u009Cn; 2) = 2n\u00E2\u0088\u009222 .Note that exact values of ed(Sn) or ed(An) are known only for n 6 7; see [Mer13, Section3i] for a summary. One may thus say that we know more about ed(S\u00CB\u009Cn) and ed(A\u00CB\u009Cn) than we doabout ed(Sn) and ed(An).Theorem 3.1.4. Let S\u00CB\u009Cn be either S\u00CB\u009C+n or S\u00CB\u009C\u00E2\u0088\u0092n . Assume char k = 2. Then(a) ed(Sn) 6 ed(S\u00CB\u009Cn) 6 ed(Sn) + 1.(b) ed(An) 6 ed(A\u00CB\u009Cn) 6 ed(An) + 1.(c) ed(S\u00CB\u009Cn; 2) = ed(A\u00CB\u009Cn; 2) = 1.Our proof shows that, more generally, central extensions by Z/pZ make little difference tothe essential dimension of a group over a field of characteristic p; see Lemma 3.4.1.One possible explanation for the slow growth of ed(S\u00CB\u009Cn) and ed(A\u00CB\u009Cn) in characteristic 2 isthat the connection between S\u00CB\u009C\u00C2\u00B1n (respectively, A\u00CB\u009Cn) and Pin\u00C2\u00B1n (respectively, Spinn) outlined abovebreaks down in this setting; see Remark 3.4.2.233.1. Introductionn 4 5 6 7 8 9 10 11 12 13 14 15 16edC(An) 2 2 3 4 4-5 4-6 5-7 6-8 6-9 6-10 7-11 8-12 8-13edC(A\u00CB\u009Cn; 2) 2 2 2 2 8 8 8 8 16 16 32 32 128edC(A\u00CB\u009Cn) 2 2 4 4 8 8-14 8-15 8-16 16-25 16-26 32-43 32-44 128Table 3.1: Essential dimension of A\u00CB\u009CnSome values of ed(An), ed(A\u00CB\u009Cn) and ed(A\u00CB\u009Cn; 2) over the field C of complex numbers are shownin Table 3.1. Here an entry of the form x-y means that the integer in question lies in the interval[x, y], and the exact value is unknown.As an application of Theorem 3.1.2, we will prove the following result in quadratic form theory.As usual, we will denote the non-degenerate diagonal form q(x1, . . . , xn) = a1x21 + \u00C2\u00B7 \u00C2\u00B7 \u00C2\u00B7 + anx2ndefined over a field F of characteristic 6= 2 by \u00E3\u0080\u0088a1, . . . , an\u00E3\u0080\u0089. Here a1, . . . , an \u00E2\u0088\u0088 F \u00E2\u0088\u0097. We willabbreviate \u00E3\u0080\u0088a, . . . , a\u00E3\u0080\u0089 (m times) as m\u00E3\u0080\u0088a\u00E3\u0080\u0089. Recall that the Hasse invariant w2(q) of q = \u00E3\u0080\u0088a1, . . . , an\u00E3\u0080\u0089(otherwise known as the second Stiefel-Whitney class of q) is given byw2(q) = \u00CE\u00A316i \u00C2\u00B7 \u00C2\u00B7 \u00C2\u00B7 > as > 0 (as in Theorem 3.1.2). Then every n-dimensional traceform q contains s\u00E3\u0080\u00881\u00E3\u0080\u0089 as a subform; see, e.g., [Ser84, Proposition 4]. This necessary condition for ann-dimensional quadratic form to be a trace form is not sufficient; see Remark 3.5.4. Nevertheless,Theorem 3.1.5 below, tells us that in some ways a general n-dimensional trace forms behaves likea general n-dimensional quadratic form that contain s\u00E3\u0080\u00881\u00E3\u0080\u0089 as a subform.Theorem 3.1.5. Let k be a field containing a primitive 8th root of unity, n > 4 be an integer,and n = 2a1 + \u00C2\u00B7 \u00C2\u00B7 \u00C2\u00B7+ 2as be the dyadic expansion of n, where a1 > \u00C2\u00B7 \u00C2\u00B7 \u00C2\u00B7 > as > 0. Then(a) maxF, qind(w2(q)) = maxF, tind(w2(t)) = 2b(n\u00E2\u0088\u0092s)/2c,(b) maxF, q1ind (w2(q1)) = maxF, t1ind(w2(t1)) = 2b(n\u00E2\u0088\u0092s\u00E2\u0088\u00921)/2c.Here the maxima are taken as\u00E2\u0080\u00A2 F ranges over all fields containing k,243.2. Preliminaries\u00E2\u0080\u00A2 q ranges over n-dimensional non-degenerate quadratic forms over F containing s \u00E3\u0080\u00881\u00E3\u0080\u0089,\u00E2\u0080\u00A2 q1 ranges over n-dimensional quadratic forms of discriminant 1 over F containing s \u00E3\u0080\u00881\u00E3\u0080\u0089,\u00E2\u0080\u00A2 t ranges over n-dimensional trace forms over F , and\u00E2\u0080\u00A2 t1 ranges over n-dimensional trace forms of discriminant 1 over F ,Note that if q = r \u00E2\u008A\u0095 s\u00E3\u0080\u00881\u00E3\u0080\u0089, then q and r have the same discriminant and the same Hasseinvariant. Thus in the statement of Theorem (a) we could replace ind(w2(q)) by ind(w2(r)),where r ranges over the (n\u00E2\u0088\u0092s)-dimensional non-degenerate quadratic forms over F , and similarlyin part (b).The remainder of this chapter is structured as follows. Section 3.2 gives a summary of knownresults which will be needed later on. Theorem 3.1.2 is proved in Section 3.3, Theorem 3.1.4 inSection 3.4 and Theorem 3.1.5 in Section 3.5. In Section 3.6 we compare the essential dimensionsof S\u00CB\u009C+n and S\u00CB\u009C\u00E2\u0088\u0092n , and in Section 3.7 we explain the entries in Table 3.1.3.2 Preliminaries3.2.1 Essential dimensionRecall that the essential dimension of a linear algebraic group G is defined as follows. Let V be agenerically free linear representation of G and let X be a G-variety, i.e., an algebraic variety withan action of G. Here G, V , X and the G-actions on V and X are assumed to be defined over thebase field k. We will say that X is generically free if the G-action on X is generically free. Theessential dimension ed(G) of G is the minimal value of dim(X)\u00E2\u0088\u0092 dim(G), where X ranges overall generically free G-varieties admitting a G-equivariant dominant rational map V 99K X. Thisnumber depends only on G and k and not on the choice of the generically free representation V .We will sometimes write edk(G) instead of ed(G) to emphasize the dependence on k.We will also be interested in the related notion of essential dimension ed(G; p) of G at aprime integer p. The essential dimension of G at p is defined in the same way as ed(G), as theminimal value of dim(X)\u00E2\u0088\u0092 dim(G), where X is a generically free G-variety, except that insteadof requiring that X admits a G-equivariant dominant rational map V 99K X, we only require thatit admits a G-equivariant dominant correspondence V X whose degree is prime to p. Here bya dominant correspondence V X of degree d we mean a diagram of dominant G-equivariantrational maps,V \u00E2\u0080\u00B2d:1\u000F\u000F ''V X.In Chapter 2 Example 2.3.1, the given definition of essential dimension of G agrees with thedefinition above. See, for example, [BF03, Remark 6.4 & Corollary 6.16] for a proof.We will now recall the properties of essential dimension that will be needed in the sequel.For a detailed discussion of essential dimension and its variants, we refer the reader to thesurveys [Mer13] and [Rei10].253.2. PreliminariesLemma 3.2.1. Let G \u00E2\u0086\u00AA\u00E2\u0086\u0092 GL(V ) be a generically free representation. Thened(G) 6 dim(V )\u00E2\u0088\u0092 dim(G).The proof is immediate from the definition of essential dimension.Lemma 3.2.2. Let H be a closed subgroup of an algebraic group G. If the index [G : H] is finiteand prime to p, then ed(G; p) = ed(H; p).Proof. See [MR09, Lemma 4.1].Lemma 3.2.3. Let G1 \u00E2\u0086\u0092 G2 be a homomorphism of algebraic groups. If the induced mapH1(K,G1)\u00E2\u0086\u0092 H1(K,G2)is surjective for all field extensions K of k, then ed(G1) > ed(G2) and ed(G1; p) > ed(G2; p) forevery prime p.Proof. See [Rei10, (1.1)] or [Mer13, Proposition 2.3].Lemma 3.2.4. Suppose H is a subgroup of G. Then(a) ed(G) > ed(H)\u00E2\u0088\u0092 dim(G) + dim(H),(b) ed(G; p) > ed(H; p)\u00E2\u0088\u0092 dim(G) + dim(H).Proof. See [BRV10, Lemma 2.2].3.2.2 The index of a central extensionAssume char(k) 6= p. Let G be a finite group and1 // Z/pZ // G // G // 1be a central exact sequence. This exact sequence gives rise to a connecting morphism\u00CE\u00B4K : H1(K,G)\u00E2\u0086\u0092 H2(K,Z/pZ)for every field K containing k. If K contains a primitive pth root of unity, then H2(K,Z/pZ)can be identified with the p-torsion subgroup Brp(K) of the Brauer group Br(K). In particular,we can talk about the index ind(\u00CE\u00B4K(\u00CE\u00B1)) for any \u00CE\u00B1 \u00E2\u0088\u0088 H1(K,G). Let ind(G,Z/pZ) denote themaximal value of ind(\u00CE\u00B4K(t)), as K ranges over all field extensions of k and t ranges over theelements of H1(K,G).Lemma 3.2.5. Assume that the base field k contains a primitive pth root of unity.(a) If Gp is a Sylow p-subgroup of G, then ind(G,Z/pZ) = ind(Gp,Z/pZ).(b) Suppose the center Z(Gp) is cyclic. Then ind(Gp,Z/pZ) = ed(Gp) = ed(Gp; p).(c) ed(G) 6 ed(G) + ind(G,Z/pZ).263.2. PreliminariesProof. (a) The diagram1 // Z/pZ // G // G // 11 // Z/pZ //'OOGp //OOGp //OO1,where the rows are central exact sequences and the vertical maps are natural inclusions gives riseto a commutative diagramH1(K,G)\u00CE\u00B4K // H2(K,Z/pZ)H1(K,Gp)\u00CE\u00BDK //i\u00E2\u0088\u0097OOH2(K,Z/pZ)'OO(3.2.1)of Galois cohomology sets for any field K/k. Here \u00CE\u00B4 and \u00CE\u00BD denote connecting morphisms. It isclear from (3.2.1) that ind(Gp,Z/pZ) 6 ind(G,Z/pZ). To prove the opposite inequality, choosea field extension K/k and an element t \u00E2\u0088\u0088 H1(K,G) such that \u00CE\u00B4K(t) has the maximal possibleindex in H2(K,Z/pZ); that is,ind(\u00CE\u00B4K(t)) = ind(G,Z/pZ).Since [G : Gp] = [G : Gp] is prime to p, after passing to a suitable finite extension L/K of degreeprime to p, we may assume that tL \u00E2\u0088\u0088 H1(L,G) is the image of some s \u00E2\u0088\u0088 H1(L,Gp). Here asusual, tL denotes the image of t \u00E2\u0088\u0088 H1(K,G) under the restriction map H1(K,G)\u00E2\u0086\u0092 H1(L,G).Since [L : K] is prime to p, we haveind(G,Z/pZ) = ind(\u00CE\u00B4K(t)) = ind(\u00CE\u00B4L(tL)) = ind(\u00CE\u00B4K(t)L) = ind(\u00CE\u00BDL(s)) 6 ind(Gp,Z/pZ) ,as desired.(b) is a variant of a theorem of N. Karpenko and A. Merkurjev: the equalityind(Gp,Z/pZ) = ed(Gp)is a special case of [KM08, Theorem 4.4], and the equality ed(Gp) = ed(Gp; p) is a part of thestatement of [KM06, Theorem 4.1].(c) See [Mer13, Corollaries 5.8 and 5.12]; cf. also [CR15, Proposition 2.1].3.2.3 Sylow 2-subgroups of A\u00CB\u009CnLemma 3.2.6. Let H\u00CB\u009Cn be a Sylow 2-subgroup of A\u00CB\u009Cn.(a) If n = 4 or 5, then H\u00CB\u009Cn is isomorphic to the quaternion groupQ8 =\u00E2\u008C\u00A9x, y, c | x2 = y2 = c, c2 = 1, cx = xc, cy = yc, xy = cyx \u00E2\u008C\u00AA .(b) If n = 6 or 7, then H\u00CB\u009Cn is isomorphic to the generalized quaternion groupQ16 =\u00E2\u008C\u00A9x, y | x8 = y4 = 1, y2 = x4, yxy\u00E2\u0088\u00921 = x\u00E2\u0088\u00921 \u00E2\u008C\u00AA .273.3. Proof of Theorem 3.1.2Proof. We will view A\u00CB\u009Cn (and thus H\u00CB\u009Cn) as a subgroup of S\u00CB\u009C+n and use the generators and relationsfor S\u00CB\u009C+n given in the introduction.(a) For n = 4 or 5, we can take H\u00CB\u009Cn to be the group of order 8 generated by \u00CF\u0083 = (s1s2s3)2 and\u00CF\u0084 = s1s3. These elements project to (1 3)(2 4) and (1 2)(3 4) in An, respectively. One readilychecks that \u00CF\u00832 = \u00CF\u00842 = z and \u00CF\u0083\u00CF\u0084 = z\u00CF\u0084\u00CF\u0083. An isomorphism Q8 \u00E2\u0086\u0092 H\u00CB\u009Cn can now be defined byx 7\u00E2\u0086\u0092 \u00CF\u0083, y 7\u00E2\u0086\u0092 \u00CF\u0084, c 7\u00E2\u0086\u0092 z.(b) For n = 6 or 7, we can take H\u00CB\u009Cn to be the group of order 16 generated by \u00CF\u0083 = s1s2s3s5and \u00CF\u0084 = s1s3. These elements project to (1 2 3 4)(5 6) and (1 2)(3 4) in An, respectively. Anisomorphism Q16 \u00E2\u0086\u0092 H\u00CB\u009Cn can now be given by x 7\u00E2\u0086\u0092 \u00CF\u0083 and y 7\u00E2\u0086\u0092 \u00CF\u0084 .Proposition 3.2.7. Let n > 4 be an integer, S\u00CB\u009Cn be either S\u00CB\u009C+n or S\u00CB\u009C\u00E2\u0088\u0092n , and P\u00CB\u009Cn, H\u00CB\u009Cn be Sylow2-subgroups of S\u00CB\u009Cn, A\u00CB\u009Cn, respectively. Denote the centers of P\u00CB\u009Cn and H\u00CB\u009Cn by Z(P\u00CB\u009Cn) and Z(H\u00CB\u009Cn),respectively. Then Z(P\u00CB\u009Cn) = Z(H\u00CB\u009Cn) = \u00E3\u0080\u0088z\u00E3\u0080\u0089 is a cyclic group of order 2.Proof. By [Wag77, Lemma 3.2] that Z(P\u00CB\u009Cn) = \u00E3\u0080\u0088z\u00E3\u0080\u0089 for every n > 4 and Z(H\u00CB\u009Cn) = \u00E3\u0080\u0088z\u00E3\u0080\u0089 for everyn > 8.It remains to show that Z(H\u00CB\u009Cn) = \u00E3\u0080\u0088z\u00E3\u0080\u0089 for 4 6 n 6 7. Clearly z \u00E2\u0088\u0088 Z(H\u00CB\u009Cn), so we only need toshow that Z(H\u00CB\u009Cn) is of order 2. We will use the description of the groups H\u00CB\u009Cn from Lemma 3.2.6.If n = 4 and 5, then H\u00CB\u009Cn is isomorphic to the quaternion group Q8, and the center of Q8 is clearlyof order 2. If n = 6 or 7, then H\u00CB\u009Cn ' Q16, and the center of Q16 is readily seen to be the cyclicgroup\u00E2\u008C\u00A9x4\u00E2\u008C\u00AAof order 2.3.3 Proof of Theorem 3.1.2We recall the theorem from the introduction.Theorem 3.3.1. Assume that the base field k is of characteristic 6= 2 and contains a primitive 8throot of unity, and let n > 4 be an integer. Write n = 2a1 + . . .+2as , where a1 > a2 > . . . > as > 0and let S\u00CB\u009Cn be either S\u00CB\u009C+n or S\u00CB\u009C\u00E2\u0088\u0092n . Then(a) ed(A\u00CB\u009Cn) 6 ed(S\u00CB\u009Cn) 6 2b(n\u00E2\u0088\u00921)/2c.(b) ed(S\u00CB\u009Cn; 2) = 2b(n\u00E2\u0088\u0092s)/2c,(c) ed(A\u00CB\u009Cn; 2) = 2b(n\u00E2\u0088\u0092s\u00E2\u0088\u00921)/2c.(d) 2b(n\u00E2\u0088\u0092s)/2c 6 ed(S\u00CB\u009Cn) 6 ed(Sn) + 2b(n\u00E2\u0088\u0092s)/2c(e) 2b(n\u00E2\u0088\u0092s\u00E2\u0088\u00921)/2c 6 ed(A\u00CB\u009Cn) 6 ed(An) + 2b(n\u00E2\u0088\u0092s\u00E2\u0088\u00921)/2c.(a) The first inequality follows from the fact that A\u00CB\u009Cn is contained in S\u00CB\u009Cn; see Lemma 3.2.4.For the second inequality, apply Lemma 3.2.1 to the so-called basic spin representation of S\u00CB\u009Cn.This representation is obtained by restricting a representation of the Clifford algebra Cn\u00E2\u0088\u00921 intoMat2bn\u00E2\u0088\u009212 c(k); see [Ste89, Section 3] for details. (Note that [Ste89] assumes k = C but the samemorphism works over any field containing square root of \u00E2\u0088\u00921).283.3. Proof of Theorem 3.1.2(b) Let Pn be a Sylow 2-subgroups of Sn. The preimage P\u00CB\u009Cn of Pn is a Sylow 2-subgroup of S\u00CB\u009Cn.By Lemma 3.2.2, ed(S\u00CB\u009Cn; 2) = ed(P\u00CB\u009Cn; 2). Moreover, by the Karpenko-Merkurjev theorem [KM08,Theorem 4.1],ed(P\u00CB\u009Cn) = ed(P\u00CB\u009Cn; 2) = dim(V ),where V is a faithful linear representation of P\u00CB\u009Cn of minimal dimension.By Proposition 3.2.7, the center of Z(P\u00CB\u009Cn) = \u00E3\u0080\u0088z\u00E3\u0080\u0089 is of order 2. Consequently, a faithfulrepresentation V of minimal dimension is automatically irreducible; see [MR10, Theorem 1.2].On the other hand, an irreducible representation \u00CF\u0081 of P\u00CB\u009Cn is faithful if and only if \u00CF\u0081(z) 6= 1; see,e.g., [Wag77, Lemma 4.1]. We will now consider several cases.Case 1: Suppose k = C is the field of complex numbers. By [Wag77, Lemma 4.2] everyirreducible representation \u00CF\u0081 : P\u00CB\u009Cn \u00E2\u0086\u0092 GL(V ) with \u00CF\u0081(z) 6= 1 is of dimension 2b(n\u00E2\u0088\u0092s)/2c. This provespart (b) for k = C.Case 2: Assume k \u00E2\u008A\u0082 C is a field containing a primitive root of unity \u00CE\u00B62d of degree 2d, where2d is the exponent of P\u00CB\u009Cn. By a theorem of R. Brauer [Ser77, 12.3.24], every irreducible complexrepresentation of P\u00CB\u009Cn is, in fact, defined over k. Thus the dimension of the minimal faithfulirreducible representation over k is the same as over C, i.e., 2b(n\u00E2\u0088\u0092s)/2c, and part (b) holds over k.Case 3: Now suppose that k is a field of characteristic 0 containing \u00CE\u00B68 (but possibly not \u00CE\u00B62d).Set l = k(\u00CE\u00B62d). Thenedk(P\u00CB\u009Cn) > edl(P\u00CB\u009Cn) = 2b(n\u00E2\u0088\u0092s)/2c.To prove the opposite inequality, let V be a faithful irreducible representation of P\u00CB\u009Cn dimension2b(n\u00E2\u0088\u0092s)/2c defined over Q(\u00CE\u00B62d). Such a representation exists by Case 2. We claim that V is, infact, defined over Q(\u00CE\u00B68). In particular, V is defined over k and thusedk(P\u00CB\u009Cn) 6 2b(n\u00E2\u0088\u0092s)/2c,as desired. We will prove the claim in two steps.First we will show that the character \u00CF\u0087 : P\u00CB\u009Cn \u00E2\u0086\u0092 Q(\u00CE\u00B62d) of V takes all of its values in Q(\u00CE\u00B68).By [Wag77, Lemma 4.2], there are either one or two faithful irreducible characters of P\u00CB\u009Cn ofdimension 2b(n\u00E2\u0088\u0092s)/2c. The Galois group G = Gal(Q(\u00CE\u00B62d)/Q) ' Z/2Z\u00C3\u0097Z/2d\u00E2\u0088\u00922Z acts on this set ofcharacters. Thus for any \u00CF\u0083 \u00E2\u0088\u0088 P\u00CB\u009Cn, the G-orbit of \u00CF\u0087(\u00CF\u0083) has either one or two elements. Consequently,[Q(\u00CF\u0087(\u00CF\u0083)) : Q] = 1 or 2. Note that G has exactly three subgroups of index 2. Under the Galoiscorrespondence these subgroups correspond to the subfields Q(\u00E2\u0088\u009A\u00E2\u0088\u00921), Q(\u00E2\u0088\u009A2) and Q(\u00E2\u0088\u009A\u00E2\u0088\u00922) ofQ(\u00CE\u00B62d). Thus \u00CF\u0087(\u00CF\u0083) lies in one of these three fields; in particular, \u00CF\u0087(\u00CF\u0083) \u00E2\u0088\u0088 Q(\u00E2\u0088\u009A\u00E2\u0088\u00921,\u00E2\u0088\u009A2) = Q(\u00CE\u00B68) forevery \u00CF\u0083 \u00E2\u0088\u0088 G. In other words, \u00CF\u0087 takes all of its values in Q(\u00CE\u00B68), as desired.Now observe that since P\u00CB\u009Cn is a 2-group, the Schur index of \u00CF\u0087 over Q(\u00CE\u00B68) is 1; see [Yam74,Corollary 9.6]. Since the character \u00CF\u0087 of V is defined over Q(\u00CE\u00B68) and the Schur index of \u00CF\u0087 is 1, weconclude that V itself is defined over k. This completes the proof of part (b) in Case 3 (i.e., incharacteristic 0).Case 4: Now assume that k is a perfect field of characteristic p > 2 containing \u00CE\u00B68. LetA = W (k) be the ring of Witt vectors of k. Recall that A is a complete discrete valuation ringof characteristic zero, whose residue field is k. By Hensel\u00E2\u0080\u0099s Lemma, \u00CE\u00B68 lifts to a primitive 8throot of unity in A. Denote the fraction field of A by K and the maximal ideal by M . SinceP\u00CB\u009Cn is a 2-group and char(k) = p is an odd prime, every d-dimensional k[P\u00CB\u009Cn]-module W liftsto a unique A[P\u00CB\u009Cn]-module WA, which is free of rank d over A. Moreover, the lifting operationV 7\u00E2\u0086\u0092 VK := VA\u00E2\u008A\u0097K and the \u00E2\u0080\u009Creduction mod M\u00E2\u0080\u009D operation give rise to mutually inverse bijections293.3. Proof of Theorem 3.1.2between k[P\u00CB\u009Cn]-modules and K[P\u00CB\u009Cn]-modules; see [Ser77, Section 15.5]. These bijections preservedimension and faithfulness of modules. Since K is a field of characteristic 0 containing a primitive8th root of unity, Case 3 tells us that the minimal dimension of a faithful K[P\u00CB\u009Cn]-module is2b(n\u00E2\u0088\u0092s)/2c. Hence, the minimal dimension of a faithful k[P\u00CB\u009Cn]-module is also 2b(n\u00E2\u0088\u0092s)/2c. This provespart (b) in Case 4.Case 5: Now assume that k is an arbitrary field of characteristic p > 2 containing \u00CE\u00B68. Denotethe prime field of k by Fp. Then k can be sandwiched between two perfect fields, k1 \u00E2\u008A\u0082 k \u00E2\u008A\u0082 k2,where k1 = Fp(\u00CE\u00B68) is a finite field, and k2 is the algebraic closure of k. Thenedk1(P\u00CB\u009Cn) > edk(P\u00CB\u009Cn) > edk2(P\u00CB\u009Cn).By Case 4, edk1(P\u00CB\u009Cn) = edk2(P\u00CB\u009Cn) = 2b(n\u00E2\u0088\u0092s)/2c. We conclude that edk(P\u00CB\u009Cn) = 2b(n\u00E2\u0088\u0092s)/2c. The proofof part (b) is now complete.(c) Let Hn be the Sylow 2-subgroups of An. Its preimage H\u00CB\u009Cn is a Sylow 2-subgroup of A\u00CB\u009Cn. Bythe Karpenko-Merkurjev theorem [KM08, Theorem 4.1], ed(H\u00CB\u009Cn) = ed(H\u00CB\u009Cn; 2) = dim(W ), whereW is a faithful linear representation of H\u00CB\u009Cn of minimal dimension.The rest of the argument in part (b) goes through with only minor changes. Once again, byProposition 3.2.7, the center of Z(H\u00CB\u009Cn) = \u00E3\u0080\u0088z\u00E3\u0080\u0089 is of order 2. Thus W is irreducible. Moreover, anirreducible representation \u00CF\u0081 of H\u00CB\u009Cn is faithful if and only if \u00CF\u0081(z) 6= 1.If k = C is the field of complex numbers, it is shown in [Wag77, Lemma 4.3] that everyirreducible representation \u00CF\u0081 of H\u00CB\u009Cn with \u00CF\u0081(z) 6= 1 is of dimension 2b(n\u00E2\u0088\u0092s\u00E2\u0088\u00921)/2c. This proves part(c) for k = C. Moreover, depending on the parity of n \u00E2\u0088\u0092 s, there are either one or two suchrepresentations. For other base fields k containing a primitive 8th root of unity (in Cases 2-5)the arguments of part (b) go through unchanged.(d) Since ed(S\u00CB\u009Cn) > ed(S\u00CB\u009Cn; 2), the lower bound of part (d) follows immediately from part (b).To prove the upper bound, we apply Lemma 3.2.5 to the exact sequence (3.1.1). Here p = 2, andZ/2Z = \u00E3\u0080\u0088z\u00E3\u0080\u0089 is the center of Gp = P\u00CB\u009Cn by Proposition 3.2.7.(e) The lower bound follows from part (c) and the inequality ed(A\u00CB\u009Cn) > ed(A\u00CB\u009Cn; 2). The upperbound is obtained by applying Lemma 3.2.5 to the exact sequence1 // \u00E3\u0080\u0088z\u00E3\u0080\u0089 // A\u00CB\u009Cn // An // 1,in the same way as in part (d).Remark 3.3.2. If n\u00E2\u0088\u0092 s is even, then S\u00CB\u009Cn has only one faithful irreducible complex representationof dimension 2b(n\u00E2\u0088\u0092s)/2c; see [Wag77, Lemma 4.2]. In this case we can relax the assumption onk in part (b) a little bit: our proof goes through for any base field k containing \u00CE\u00B64 =\u00E2\u0088\u009A\u00E2\u0088\u00921.(Similarly for part (c) in the case where n\u00E2\u0088\u0092 s is odd; see [Wag77, Lemma 4.3].)However, in general part (b) fails if we do not assume that \u00CE\u00B68 \u00E2\u0088\u0088 k. For example, in thecase, where s = 1 (i.e., n > 4 is a power of 2), the inequality ed(S\u00CB\u009Cn; 2) 6 2b(n\u00E2\u0088\u0092s)/2c = 2(n\u00E2\u0088\u00922)/2is equivalent to the existence of a faithful irreducible representation V of P\u00CB\u009Cn of degree 2(n\u00E2\u0088\u00922)/2defined over k. There are two such representations, and [HH92, Theorem 8.7] shows that someof their character values are not contained in Q(\u00CE\u00B64).303.4. Proof of Theorem 3.1.43.4 Proof of Theorem 3.1.4Part (c) follows directly from [RV18, Theorem 1], which says that if G is a finite group and k isa field of characteristic p > 0, thenedk(G; p) ={1, if the order of G is divisible by p, andedk(G; p) = 0, otherwise.In particular, edk(S\u00CB\u009Cn; 2) = edk(A\u00CB\u009Cn; 2) = 1 for any field k of characteristic 2.Parts (a) and (b) are consequences of the following lemma. In the case, where G is a finitep-group, this lemma is due to A. Ledet; see [Led04, Theorem 1].Lemma 3.4.1. Let k be a field of characteristic p, G be a linear algebraic group defined over kand1 // Z/pZ // G // G // 1be a central exact sequence. Then edk(G) 6 edk(G) 6 edk(G) + 1.Proof. (a) Consider the induced exact sequenceH1(K,G) // H1(K,G)\u00CE\u00B4K // H2(K,Z/pZ)in Galois cohomology (or flat cohomology, if G is not smooth), where \u00CE\u00B4K denotes the boundarymap. Here K/k is an arbitrary field extension K/k. Since K is a field of characteristic p, itscohomological p-dimension is 6 1 and thus H2(K,Z/pZ) = 1; see [Ser02b, Proposition II.2.2.3].In other words, the map H1(K,G)\u00E2\u0086\u0092 H1(K,G) is surjective for any field K containing k. ByLemma 3.2.3, this impliesedk(G) > edk(G).On the other hand, since Z/pZ is unipotent in characteristic p, [TV13, Lemma 3.4] tells us thatedk(G) 6 edk(G) + edk(Z/pZ) = edk(G) + 1;see also [L1\u00C2\u00A83, Corollary 3.5].Remark 3.4.2. Note that in characteristic 2 the group A\u00CB\u009Cn is no longer isomorphic to thepreimage of An \u00E2\u008A\u0082 SOn in Spinn. The scheme-theoretic preimage of An in Spinn is an extensionof a constant group scheme An by an infinitesimal group scheme \u00C2\u00B52. Any such extension is splitover a perfect base field; see, e.g., [Mil17, Proposition 15.22]. Thus, over a perfect field k, thepreimage of An \u00E2\u008A\u0082 SOn in Spinn is the direct product An\u00C3\u0097\u00C2\u00B52.Remark 3.4.3. As we mentioned in the introduction, the exact values of ed(Sn) and ed(An) incharacteristic 0 are not known for any n > 8 . In characteristic 2, even less is known. The upperbound,edk(An) 6 edk(Sn) 6 n\u00E2\u0088\u0092 3 for any n > 5,is valid over an arbitrary field k. It is also known that if G is a finite group and G does nothave a non-trivial normal 2-subgroup, then edk(G) 6 edC(G) for any field k of characteristic 2containing the algebraic closure of F2; see [BRV18, Corollary 3.4(b)]. In particular, this appliesto G = Sn or An for any n > 5.313.5. Proof of Theorem 3.1.5In characteristic 0, ed(Sn) > bn/2c for any n > 1 and ed(Sn) > b(n+ 1)/2c for any n > 7. Itis not known if these inequalities remain true in characteristic 2. On the other hand, since Ancontains (Z/3Z)r, where r = bn/3c, it is easy to see that the weaker inequalityed(Sn) > ed(An) > bn/3cremains valid in characteristic 2. For general n, this is the best lower bound we know.Example 3.4.4. Assume that the base field k is infinite of characteristic 2. We claim thatedk(S4) = 2.Let P4 be a Sylow 2-subgroup of S4. Recall that P4 is isomorphic to the dihedral group of order 8.By [Led07, Proposition 7], edk(P4) > 2 and thus edk(S4) > 2. To prove the opposite inequality,consider the faithful 3-dimensional representation of S4 given byV = {(x1, x2, x3, x4) |x1 + x2 + x3 + x4 = 0}.Here S4 acts on V by permuting x1, . . . , x4. The natural compression V 99K P(V ) shows thatedk(S4) 6 2. This proves the claim. By Theorem 3.1.4 we conclude that2 6 edk(S\u00CB\u009C4) 6 3.We do not know whether edk(S\u00CB\u009C4) = 2 or 3. Note however, that by a conjecture of Ledet [Led04,p. 4], edk(Z/2nZ) = n for every integer n > 1. Since S\u00CB\u009C4 contains an element of order 8 (thepreimage of a 4-cycle in S4), Ledet\u00E2\u0080\u0099s conjecture implies that edk(S\u00CB\u009C4) = 3. Note also that byCorollary 3.1.3, edl(S\u00CB\u009C4) = 2 for any base field l of characteristic 6= 2 containing a primitive 8throot of unity.3.5 Proof of Theorem 3.1.5Let q = \u00E3\u0080\u0088a1, . . . , an\u00E3\u0080\u0089 be a non-degenerate n-dimensional quadratic form over a field F of charac-terstic 6= 2. Recall from the Introduction that the Hasse invariant w2(q) is given byw2(q) = \u00CE\u00A316i 2b(n\u00E2\u0088\u0092s)/2c and ed(A\u00CB\u009Cn) > 2b(n\u00E2\u0088\u0092s\u00E2\u0088\u00921)/2c entirely within theframework of quadratic form theory, without the representation-theoretic input from [Wag77].Remark 3.5.3. Let F be a field of characteristic 6= 2 containing a primitive 8th root of unity,and q be an n-dimensional non-degenerate quadratic form over F . As we pointed out in theIntroduction, a necessary condition for q to be a trace form is that it should contain s\u00E3\u0080\u00881\u00E3\u0080\u0089 as asubform. Theorem 3.1.5 suggests that this condition might be sufficient. The following exampleshows that, in fact, this condition is not sufficient. In this example, n = 4 = 22 and thuss = 1. Let k be an arbitrary base field of characteristic 6= 2, a, b, c be independent variables,F = k(a, b, c), and q = \u00E3\u0080\u00881, a, b, c\u00E3\u0080\u0089 be a 4-dimensional non-singular quadratic form over F . Clearlyq contains s\u00E3\u0080\u00881\u00E3\u0080\u0089 = \u00E3\u0080\u00881\u00E3\u0080\u0089 as a subform. On the other hand, q is not isomorphic to the trace form t343.6. Comparison of essential dimensions of S\u00CB\u009C+n and S\u00CB\u009C\u00E2\u0088\u0092nof any 4-dimensional etale algebra E/F . Indeed, edk(t) 6 edk(E/F ) 6 edk(S4) = 2 (see [BR97,Theorem 6.5(a)]), whereas edk(q) = 3 (see [CS06, Proposition 6]).Remark 3.5.4. Recall that by a theorem of Merkurjev [Mer81], w2 gives rise to an isomorphismbetween I2(K)/I3(K) and H2(K,Z/2Z); cf. [Lam05, p. 115]. This is a special case of Milnor\u00E2\u0080\u0099sconjecture, which asserts the existence of an isomorphismer : Ir(F )/Ir+1(F )\u00E2\u0086\u0092 Hr(F,Z/2Z)for any r > 0, with the property that er takes the r-fold Pfister form \u00E3\u0080\u00881, a1\u00E3\u0080\u0089 \u00E2\u008A\u0097 \u00C2\u00B7 \u00C2\u00B7 \u00C2\u00B7 \u00E2\u008A\u0097 \u00E3\u0080\u00881, ar\u00E3\u0080\u0089 tothe cup product (a1) \u00E2\u0088\u00AA (a2) \u00E2\u0088\u00AA \u00C2\u00B7 \u00C2\u00B7 \u00C2\u00B7 \u00E2\u0088\u00AA (ar); see [Pfi95, p. 33]. Milnor\u00E2\u0080\u0099s conjecture has been provedby V. Voevodsky; see [Kah97] for an overview. It is natural to ask if the following variant ofTheorem 3.1.5 remains valid for every r > 1.Let k be a base field containing a primitive 8th root of unity, and n = 2a1 + \u00C2\u00B7 \u00C2\u00B7 \u00C2\u00B7+ 2as be aneven positive integer, where a1 > \u00C2\u00B7 \u00C2\u00B7 \u00C2\u00B7 > as > 1. Is it true thatmaxF, qind(er(q)) = maxF, tind(er(t)) ? (3.5.6)Here the maximum is taken as F ranges over all fields containing k, q ranges over all n-dimensionalforms in Ir(F ) containing s \u00E3\u0080\u00881\u00E3\u0080\u00892 and t ranges over all n-dimensional trace forms in Ir(F ). Theindex of a class \u00CE\u00B1 \u00E2\u0088\u0088 Hr(F,Z/2Z) is the greatest common divisor of the degrees [E : F ], whereE/F ranges over splitting fields for \u00CE\u00B1 with [E : F ] <\u00E2\u0088\u009E.For r = 1, it is easy to see that (3.5.6) holds. In this case the Milnor map e1 : I1(F )/I2(F )\u00E2\u0086\u0092H1(F,Z/2Z) is the discriminant, the index of an element of \u00CE\u00B1 = H1(F,Z/2Z) = F \u00E2\u0088\u0097/(F \u00E2\u0088\u0097)2 is 1or 2, depending on whether \u00CE\u00B1 = 0 or not, and the question boils down to the existence of ann-dimensional e\u00C2\u00B4tale algebra E/F with non-trivial discriminant. In the case where r = 2, theequality (3.5.6) is given by Theorem 3.1.5(b) (where n is taken to be even).3.6 Comparison of essential dimensions of S\u00CB\u009C+n and S\u00CB\u009C\u00E2\u0088\u0092nWe believe that S\u00CB\u009C+n and S\u00CB\u009C\u00E2\u0088\u0092n should have the same essential dimension, but are only able toestablish the following slightly weaker assertion.Proposition 3.6.1. Let k be a field of characteristic 6= 2 containing \u00E2\u0088\u009A\u00E2\u0088\u00921. Then| edk(S\u00CB\u009C+n )\u00E2\u0088\u0092 edk(S\u00CB\u009C\u00E2\u0088\u0092n )| 6 1.Proof. Let V be the spin representation of S\u00CB\u009C+n , Sn \u00E2\u0086\u0092 PGL(V ) be the associated projectiverepresentation of Sn, and \u00CE\u0093 \u00E2\u008A\u0082 GL(V ) be the preimage of Sn under the natural projectionpi : GL(V )\u00E2\u0086\u0092 PGL(V ). Note that \u00CE\u0093 is a 1-dimensional group, and S\u00CB\u009C\u00C2\u00B1n are finite subgroups of \u00CE\u0093.By Lemma 3.2.4(a),edk(\u00CE\u0093) > edk(S\u00CB\u009Cn)\u00E2\u0088\u0092 1, (3.6.1)2If s is even, and q = r \u00E2\u008A\u0095 s\u00E3\u0080\u00881\u00E3\u0080\u0089, then q and r are Witt equivalent over F . Thus maxF, q ind(er(q)) can bereplaced by maxF, r ind(er(r)), as r ranges over all (n\u00E2\u0088\u0092 s)-dimensional forms in Ir(F ). The same is true if s isodd: here r ranges over the (n\u00E2\u0088\u0092 s)-dimensional forms in Ir(F ) such that r \u00E2\u008A\u0095 \u00E3\u0080\u00881\u00E3\u0080\u0089 is in Ir(F ).353.7. Explanation of the entries in Table 3.1where S\u00CB\u009Cn denotes S\u00CB\u009C+n or S\u00CB\u009C\u00E2\u0088\u0092n . On the other hand, since \u00CE\u0093 is generated by S\u00CB\u009Cn and Gm, and Gm iscentral in \u00CE\u0093, we see that S\u00CB\u009Cn is normal in \u00CE\u0093. The exact sequence1 // S\u00CB\u009Cn // \u00CE\u0093pi // Gm // 1induces an exact sequenceH1(K, S\u00CB\u009Cn) // H1(K,\u00CE\u0093)pi // H1(K,Gm) = 1of Galois cohomology sets. Here K is an arbitrary field containing k, and H1(K,Gm) = 1 byHilbert\u00E2\u0080\u0099s Theorem 90. Thus the map H1(K, S\u00CB\u009Cn) \u00E2\u0086\u0092 H1(K,\u00CE\u0093) is surjective for every K. ByLemma 3.2.3,edk(S\u00CB\u009Cn) > edk(\u00CE\u0093). (3.6.2)Combining the inequalities (3.6.1) and (3.6.2), we see that each of the integers edk(S\u00CB\u009C+n ) andedk(S\u00CB\u009C\u00E2\u0088\u0092n ) equals either edk(\u00CE\u0093) or edk(\u00CE\u0093) + 1. Hence, edk(S\u00CB\u009C+n ) and edk(S\u00CB\u009C\u00E2\u0088\u0092n ) differ by at most 1, asclaimed.Remark 3.6.2. The inequality | edk(S\u00CB\u009C+n ) \u00E2\u0088\u0092 edk(S\u00CB\u009C\u00E2\u0088\u0092n )| 6 1 of Lemma 3.6.1 remains valid ifchar(k) = 2; see Theorem 3.1.4(a).3.7 Explanation of the entries in Table 3.1Throughout this section we will assume that the base field k = C is the field of complex numbers.For the first row of the table, we used the following results:\u00E2\u0080\u00A2 ed(A4) = ed(A5) = 2, see [BR97, Theorem 6.7(b)],\u00E2\u0080\u00A2 ed(A6) = 3, see [Ser10, Proposition 3.6],\u00E2\u0080\u00A2 ed(A7) = 4, see [Dun10, Theorem 1],\u00E2\u0080\u00A2 ed(An+4) > ed(An) + 2 for any n > 4, see [BR97, Theorem 6.7(a)],\u00E2\u0080\u00A2 ed(An) 6 ed(Sn) 6 n\u00E2\u0088\u0092 3 for any n > 5; see Lemma 3.2.4 and [BR97, Theorem 6.5(c)].The values of ed(A\u00CB\u009Cn; 2) in the second row of Table 3.1 are given by Theorem 3.1.2(c).In the third row,\u00E2\u0080\u00A2 ed(A\u00CB\u009C4) = 2 by Corollary 3.1.3(a).\u00E2\u0080\u00A2 To show that ed(A\u00CB\u009C5) = 2, combine the inequalities ed(A\u00CB\u009C5) > ed(A\u00CB\u009C5; 2) > 2 of Theo-rem 3.1.2(c) and ed(A\u00CB\u009C5) 6 2 of Theorem 3.1.2(a). Alternatively, see [Pro17, Lemma2.5].\u00E2\u0080\u00A2 ed(A\u00CB\u009C6) = 4 by [Pro17, Proposition 2.7].\u00E2\u0080\u00A2 To show that ed(A\u00CB\u009C7) = 4, note that ed(A\u00CB\u009C7) > 4 because A\u00CB\u009C7 contains A\u00CB\u009C6 and ed(A\u00CB\u009C7) 6 4because A\u00CB\u009C7 has a faithful 4-dimensional representation; see [Pro17, Corollary 2.1.4].363.7. Explanation of the entries in Table 3.1\u00E2\u0080\u00A2 The values of ed(A\u00CB\u009C8) = 8 and ed(A\u00CB\u009C16) = 128 are taken from Corollary 3.1.3(a).\u00E2\u0080\u00A2 When 9 6 n 6 15 the range of values for ed(A\u00CB\u009Cn) is given by the inequalityed(A\u00CB\u009Cn) 6 ed(A\u00CB\u009Cn) + ed(An) 6 ed(A\u00CB\u009Cn) + n\u00E2\u0088\u0092 3;see Theorem 3.1.2(e).37Chapter 4Classifying spaces for e\u00C2\u00B4tale algebrawith generators4.1 IntroductionGiven a topological group G, one may form the classifying space, well-defined up to homotopyequivalence, as the base space of any principal G-bundle EG \u00E2\u0086\u0092 BG where the total space iscontractible. If G is a finite nontrivial group, then BG is necessarily infinite dimensional as atopological space, [Swa60], and so there is no hope of producing BG as a variety even over C.Nonetheless, as in [Tot99], one can approximate BG by taking a large representation V of Gon which G acts freely outside of a high-codimension closed set Z, and such that (V \u00E2\u0088\u0092 Z)/Gis defined as a quasiprojective variety. The higher the codimension of Z in V , the better anapproximation (V \u00E2\u0088\u0092 Z)/G is to the notional BG.In this chapter, we consider the case of G = Sn, the symmetric group on n letters. Therepresentations we consider as our V s are the most obvious ones, r-copies of the permutationrepresentation of Sn on An. The closed loci we consider minimal: the loci where the action is notfree. We use the language of e\u00C2\u00B4tale algebras to give an interpretation of the resulting spaces. Fora fixed Sn and field k, and for a given multiple r of the permutation representation, the k-varietyB(r;An) := (V \u00E2\u0088\u0092 Z)/Sn produced by this machine represents \u00E2\u0080\u009Ce\u00C2\u00B4tale algebras equipped with rgenerating global sections\u00E2\u0080\u009D up to isomorphism of these data. The varieties B(r;An) are thereforein the same relation to the group Sn as the projective spaces Pr are to the group scheme Gm.Section 4.2 is concerned with preliminary results on generation of e\u00C2\u00B4tale algebras. The mainconstruction of the chapter, that of B(r;An), is made in Section 4.3, and the functor it representsis described.A choice of r global sections generating an e\u00C2\u00B4tale algebra A of degree n on a k-variety Xcorresponds to a map \u00CF\u0086 : X \u00E2\u0086\u0092 B(r;An). While the map \u00CF\u0086 is dependent on the chosen generatingsections, we show in Section 4.4 that if one is prepared to pass to a limit, in a sense made precisethere, that the A1-homotopy class of a composite \u00CF\u0086\u00CB\u009C : X \u00E2\u0086\u0092 B(r;An)\u00E2\u0086\u0092 B(\u00E2\u0088\u009E;An) depends onlyon the isomorphism class of A and not the generators. As a practical matter, this means thatfor a wide range of cohomology theories, E\u00E2\u0088\u0097, the map E\u00E2\u0088\u0097(\u00CF\u0086\u00CB\u009C) depends only on A and not on thegenerators used to define it.In Section 4.5, we observe that the motivic cohomology, and therefore the Chow groups, ofthe varieties B(r;A2) has already been calculated in [DI07].A degree-2 or quadratic e\u00C2\u00B4tale algebra A over a ring R carries an involution \u00CF\u0083 and a tracemap Tr : A \u00E2\u0086\u0092 R. There is a close connection between A and the rank-1 projective moduleL = ker(Tr). In Section 4.6, we show that the algebra A can be generated by r elements if andonly if the projective module L can be generated by r elements.A famous counterexample of S. Chase, appearing in [Swa67], shows that there is a smoothaffine r \u00E2\u0088\u0092 1-dimensional R-variety SpecR and a line bundle L on SpecR requiring r global384.1. Introductionsections to generate. This shows a that a bound of O. Forster [For64] on the minimal number ofsections required to generate a line bundle on SpecR, namely dimR + 1, is sharp. In light ofSection 4.6, the same smooth affine R-variety of dimension r \u00E2\u0088\u0092 1 can be used to produce e\u00C2\u00B4talealgebras A, of arbitrary degree n, requiring r global sections to generate. This fact was observedindependently by M. Ojanguren. It shows that a bound established by U. First and Z. Reichsteinin [FR17] is sharp in the case of e\u00C2\u00B4tale algebras: they can always be generated by dimR + 1global sections and this cannot be improved in general. The details are worked out in Section4.7, and we incidentally show that the example of S. Chase follows easily from our constructionof B(r;A2) and some elementary calculations in the singular cohomology of B(r;A2)(R).Finally, we offer some thoughts about determining whether the bound of First and Reichsteinis sharp if one restricts to varieties over algebraically closed fields.4.1.1 Notation and other preliminaries1. All rings in this chapter are assumed to be unital, associative, and commutative.2. k denotes a base ring, which will later be assumed to be a field of characteristic differentfrom 2.3. k \u00E2\u0088\u0092Alg denotes the category of commutative, unital, associative k-algebras and k-algebramorphisms.4. A variety X is a geometrically reduced, separated scheme of finite type over a field. Wedo not require the base field to be algebraically closed, nor do we require varieties to beirreducible.5. k \u00E2\u0088\u0092Var denotes the category of varieties over k and k-scheme morphisms.6. C2 denotes the cyclic group of order 2.We use the functor-of-points formalism heavily throughout, which is to say we view a schemeX as the presheaf of sets it represents on a category of schemesX(U) = MorSch(U,X).In fact, the presheaf X(\u00C2\u00B7) is a sheaf on the big Zariski site of all schemes, which is to say thatif W =\u00E2\u008B\u0083i\u00E2\u0088\u0088I Ui is a cover of a scheme by Zariski open subschemes, thenX(W )\u00E2\u0086\u0092\u00E2\u0088\u008Fi\u00E2\u0088\u0088IX(Ui)\u00E2\u0087\u0092\u00E2\u0088\u008F(i,j)\u00E2\u0088\u0088I2X(Ui \u00E2\u0088\u00A9 Uj)is a coequalizer diagram.Remark 4.1.1. The scheme A1 = SpecZ[t] represents the functorX 7\u00E2\u0086\u0092 A1(X) = \u00CE\u0093(X,OX) = OX(X).Similarly, An represents the functorX 7\u00E2\u0086\u0092 (OX(X))n .This can be deduced from the case of affine X, where one hasHomZ\u00E2\u0088\u0092Alg(Z[t1, . . . , tn], R) = Rnas sets.394.2. E\u00C2\u00B4tale algebras4.2 E\u00C2\u00B4tale algebrasLet R be a ring and S an R-algebra. Then there is a morphism of rings \u00C2\u00B5 : S \u00E2\u008A\u0097R S \u00E2\u0086\u0092 S sendinga\u00E2\u008A\u0097 b to ab. We obtain an exact sequence0\u00E2\u0086\u0092 ker(\u00C2\u00B5)\u00E2\u0086\u0092 S \u00E2\u008A\u0097R S \u00C2\u00B5\u00E2\u0088\u0092\u00E2\u0086\u0092 S \u00E2\u0086\u0092 0 (4.2.1)Definition 4.2.1. Let R be a ring. An R-algebra S is called finite e\u00C2\u00B4tale if S is finitely presented,flat as an R-module and S is projective S \u00E2\u008A\u0097R S-module, where S \u00E2\u008A\u0097R S acts through \u00C2\u00B5.As S is finitely generated and flat over R it is also a projective R-module. We say that thee\u00C2\u00B4tale algebra is of degree n if the rank of S as a projective R-module is n.It is clear that S is projective S \u00E2\u008A\u0097R S-module if and only if the sequence (4.2.1) splits.Remark 4.2.2. Over a ring R, and for any integer n > 0, there exists a \u00E2\u0080\u009Ctrivial\u00E2\u0080\u009D etale algebraRn with componentwise addition and multiplication.The following lemma states that all e\u00C2\u00B4tale algebras are fppf -locally isomorphic to the trivialone.Lemma 4.2.3. Let R be a ring and S an R-algebra. Let S be a faithful R-module. Then thefollowing are equivalent:1. S is an e\u00C2\u00B4tale algebra of degree n.2. There is a faithfully flat R-algebra T such that S \u00E2\u008A\u0097R T \u00E2\u0088\u00BC= Tn as T -algebras.A proof may be found in [For17].We may extend this definition to schemes.Definition 4.2.4. Let X be a k-scheme. Let A be a locally free sheaf of OX -algebras. Forsimplicity, we assume A has a constant degree n. We say that A is an e\u00C2\u00B4tale X-algebra or e\u00C2\u00B4talealgebra over X if for every open affine subset U \u00E2\u008A\u0082 X the A(U) is an e\u00C2\u00B4tale algebra, and we call nthe rank of A.Remark 4.2.5. For X a k-scheme and n a positive integer there exists a trivial e\u00C2\u00B4tale algebraOnX with componentwise addition and multiplication.Lemma 4.2.6. Let X be a k-scheme and A be a coherent OX-algebra. Then the following areequivalent:1. A is an e\u00C2\u00B4tale X-algebra of degree n.2. There is an affine flat cover {Ui fi\u00E2\u0088\u0092\u00E2\u0086\u0092 X} such that f\u00E2\u0088\u0097i A \u00E2\u0088\u00BC= OnUi as OUi-algebras.Proof. This is immediate from Lemma 4.2.3.Definition 4.2.7. If A is an e\u00C2\u00B4tale algebra over a ring R, then a subset \u00CE\u009B \u00E2\u008A\u0082 A is said to generateA over R if no strict R-subalgebra of A contains \u00CE\u009B.If \u00CE\u009B = {a1, . . . , ar} \u00E2\u008A\u0082 A is a finite subset, then the smallest subalgebra of A containing \u00CE\u009Bagrees with the image of the evaluation map R[x1, . . . , xr](a1,...,ar)\u00E2\u0086\u0092 A. Therefore, saying that \u00CE\u009Bgenerates A is equivalent to saying this map is surjective.404.2. E\u00C2\u00B4tale algebrasProposition 4.2.8. Let \u00CE\u009B = {a1, . . . , ar} be a finite set of elements of A, an e\u00C2\u00B4tale algebra overa ring R. The following are equivalent:1. \u00CE\u009B generates A as an R-algebra.2. There exists a set of elements {f1, . . . , fn} \u00E2\u008A\u0082 R that generate the unit ideal and such that,for each i \u00E2\u0088\u0088 {1, . . . , n}, the image of \u00CE\u009B in Afi generates Afi as an Rfi-algebra.3. For each m \u00E2\u0088\u0088 MaxSpecR, the image of \u00CE\u009B in Am generates Am as an Rm-algebra.4. Let k(m) denote the residue field of the local ring Rm. For each m \u00E2\u0088\u0088 MaxSpecR, the imageof \u00CE\u009B in A\u00E2\u008A\u0097R k(m) generates A\u00E2\u008A\u0097R k(m) as a k(m)-algebra.Proof. In the case of a finite subset, \u00CE\u009B = {a1, . . . , ar}, the condition that \u00CE\u009B generates A isequivalent to the surjectivity of the evaluation map R[x1, . . . , xr]\u00E2\u0086\u0092 A.The question of generation is therefore a question of whether a certain map is an epimorphismin the category of R-modules, and conditions (2)-(4) are well-known equivalent conditions sayingthat this map is an epimorphism.Using Proposition 4.2.8, we extend the definition of \u00E2\u0080\u009Cgeneration of an algebra\u00E2\u0080\u009D from the casewhere the base is affine to the case of a general scheme.Definition 4.2.9. Let A be an e\u00C2\u00B4tale algebra over a scheme X. For \u00CE\u009B \u00E2\u008A\u0082 \u00CE\u0093(X,A) we say that \u00CE\u009Bgenerates A if, for each open affine U \u00E2\u008A\u0082 X the OX(U)-algebra A(U) is generated by restrictionof sections in \u00CE\u009B to U .4.2.1 Generation of trivial algebrasLet n \u00E2\u0089\u00A5 2 and r \u00E2\u0089\u00A5 1. Consider the trivial e\u00C2\u00B4tale algebra OnX over a scheme X. A global section ofthis algebra is equivalent to a morphism X \u00E2\u0086\u0092 An, and an r-tuple \u00CE\u009B of sections is a morphismX \u00E2\u0086\u0092 (An)r. One might hope that the subfunctor F \u00E2\u008A\u0086 (An)r of r-tuples of sections generatingOnX as an e\u00C2\u00B4tale algebra is representable, and this turns out to be the case.In order to define subschemes of (An)r, it will be necessary to name coordinates:(x11, x12, . . . , x1n, x21, . . . , x2n, . . . , xr1, . . . , xrn)It will also be useful to retain the grouping into n-tuples, so we define ~xl = (xl1, xl2, . . . , xln).Notation 4.2.10. Fix n and r as above. For (i, j) \u00E2\u0088\u0088 {1, . . . , n}2 with i < j, let Zij \u00E2\u008A\u0082 (An)rdenote the closed subscheme given by the intersection of the vanishing loci\u00E2\u008B\u0082rk=1 V (xki \u00E2\u0088\u0092 xkj).Write U(r;An), or U(r) when n is clear from the context, for the open subscheme of (An)rgiven byU(r;An) = (An)r \u00E2\u0088\u0092\u00E2\u008B\u0083i 0 there exists Frobenius self map. The kernels of these maps, called Frobenius kernelsGp, are infinitesimal group schemes whose essential dimension is poorly understood. By work of[TV13] we obtain a lower bound on their essential dimension i.eed(Gp) \u00E2\u0089\u00A5 dimG. (5.1.1)There is also the general upper bound,ed(Gp) \u00E2\u0089\u00A4 ed(G) + dim(G). (5.1.2)Recall that a group G is special if all G-torsors over a field are trivial. For special groups (likeGLn, SLn,Spn) we have ed(G) = 0. Combining this with (5.1.1) and (5.1.2), we obtain thefollowing.Lemma 5.1.3. For a special group scheme G the essential dimension of its first Frobenius kernelis equal to dim(G).The problem is non-trivial for non-special groups. Orthogonal and projective linear groupsare of particular interest. Here only partial results are know; in particular, the lemma below.Lemma 5.1.4 (Najmmuddin Fakhruddin). Let k be a field of characteristc 2. Let PGL(1)2 denotethe first Frobenius kernel of PGL2. Then edk(PGL(1)2 ) = 3.Fakhruddin\u00E2\u0080\u0099s proof relies on demonstrating that the group PGL(1)2 is a semi-direct product of\u00CE\u00B122 and \u00C2\u00B52. The essential dimension of latter can be inferred from [TV13].5.2 Essential dimension of double covers of symmetric andalternating groupsIn Chapter 3 we determined the essential dimension of double covers of symmetric and alternatinggroups. Note that for most integers n > 2, there is a gap between the upper and lower boundsin Theorem 3.1.2(d) and (e), and the exact value of ed(S\u00CB\u009Cn) and ed(A\u00CB\u009Cn) remains open. It wouldbe interesting to close this gap. Note that ed(S\u00CB\u009Cn)\u00E2\u0088\u0092 ed2(S\u00CB\u009Cn) \u00E2\u0089\u00A4 ed(Sn). Determination of ed(S\u00CB\u009Cn)minght thus lead to interesting lower bounds for ed(Sn).The results of Chapter 3 also give rise to interesting applications and questions in quadraticform theory.We work over field k of characteristic 6= 2. All other fields contain k. Let W (F ) denotethe Witt group over F and I(F ) the fundamental ideal in W (F ) consisting of classes of evendimensional forms. Quadratic forms of rank n over a field F are classified by the first Galoiscohomology set H1(F,On).To an e\u00C2\u00B4tale algebra A of rank n over F we can attach a trace form x \u00E2\u0086\u0092 Tr(x2) which isa quadratic form. This association may also be obtained as the induced map between Galoiscohomology sets by the permutation representation Sn \u00E2\u0086\u0092 On. An important question in quadraticform theory to determine which quadratic forms are trace forms (see [BF94], [GMS03, ChapterIX]).565.3. Generators of e\u00C2\u00B4tale algebrasBy a theorem of Merkurjev, the Hasse-Witt invariant w2 gives rise to an isomorphismbetween I2(F )/I3(F ) and H2(F,Z/2Z). This is a special case of Milnor\u00E2\u0080\u0099s conjecture (proven byVoevodsky), which asserts the map er : Ir(F )/Ir+1(F )\u00E2\u0086\u0092 Hr(F,Z/2Z), sending the r-fold Pfisterform \u00E3\u0080\u00881, a1\u00E3\u0080\u0089\u00E2\u008A\u0097\u00C2\u00B7 \u00C2\u00B7 \u00C2\u00B7\u00E2\u008A\u0097\u00E3\u0080\u00881, ar\u00E3\u0080\u0089 to the cup product (a1)\u00E2\u0088\u00AA(a2)\u00E2\u0088\u00AA\u00C2\u00B7 \u00C2\u00B7 \u00C2\u00B7\u00E2\u0088\u00AA(ar), is an isomorphism. The Hasse-Witt invariant of a quadratic form is its image via the connecting map H1(F,On)\u00E2\u0086\u0092 H2(F,Z/2Z)through the \u00E2\u0080\u009C Pin \u00E2\u0080\u009D exact sequence. So in the program of distinguishing trace forms from arbitraryquadratic forms we may first try to distinguish them via their corresponding cohomology classes.To this end we define the index of a class \u00CE\u00B1 \u00E2\u0088\u0088 Hr(F,Z/2Z) as the greatest common divisorof the degrees [E : F ], where E/F ranges over splitting fields for \u00CE\u00B1 with [E : F ] <\u00E2\u0088\u009E.The project is to understand the difference between trace form and a general quadratic formbased on their respective maximal indices in Hr(F,Z/2Z) as F varies over all field extensions ofk. Note that there is a natural isomorphism er : Ir/Ir+1 \u00E2\u0086\u0092 Hr(F,Z/2Z) (due to V. Voedvodksy).In the case r = 1 we see that quadratic forms and trace forms cannot be distinguished (in theabove sense) using their discriminants. Our work [RS19b, Theorem 1.5] shows that it is also thecase if we consider r = 2. The proof relies on interpreting the maximal index of trace forms asessential dimension of double covers of symmetric group Sn and our computation of ed(S\u00CB\u009Cn; 2).Thus we may ask the same question for higher cohomology classes. That is, do rth cohomologyclassHr(F,Z/2Z) distinguish between trace forms and arbitrary quadratic forms in Ir(F )/Ir+1(F )in the above sense? For the case r = 3 the corresponding invariant of the quadratic form is theArason invariant. This problem also has connections to the torsion index of Spin groups. I wouldlike to tackle this problem for the case when r = 3.5.3 Generators of e\u00C2\u00B4tale algebrasIn Chapter 4 we solved the problem of determining the sharpness of First-Reichstein bound onthe minimum number of generators for an e\u00C2\u00B4tale algebra.However, we do not know if Theorem 1.3.1 continues to hold if we replace R by C. Moregenerally if we restrict attention to the category of affine k-varieties, where k is algebraicallyclosed of characteristic 6= 2, we may ask if the bound of First and Reichstein is sharp if we onlyconsider rings R coming from this category. We have been able to show that the bound is NOTsharp if we only consider degree 2 e\u00C2\u00B4tale algebras over rings lying in this category (see Proposition4.7.14). The next natural step is to study the invariants of classifying spaces for degree 3 e\u00C2\u00B4talealgebras with generators i.e of the space B(r,A3).More generally, the sharpness of First-Reichstein bound in the context of other algebras ismostly open. In the context of Azumaya algebras, the First-Reichstein bound states that anyAzumaya algebra A over a noetherian ring R of dimension d can be generated by d+ 2 elements.In [Wil18], Ben Williams obtained lower bounds on the number of generators of Azumayaalgebras.Theorem 5.3.1 ([Wil18]). For each n \u00E2\u0089\u00A5 1, and for all d there exists ring Rd of dimension d andAzumaya algebra Ad over Rd of degree n which cannot be generated by less than b d2(n\u00E2\u0088\u0092 1)c+ 2elements.In a recent unpublished work U. First and Z. Reichstein have improved the upper bound onthe number of generators of Azumaya algebras from d+ 2 to b dn\u00E2\u0088\u0092 1c+ 2.57Bibliography[ABS64] M. F. Atiyah, R. Bott, and A. Shapiro, Clifford modules, Topology 3 (1964), no. suppl.1, 3\u00E2\u0080\u009338. MR 0167985[Bae17] Sanghoon Baek, A lower bound on the essential dimension of PGL4 in characteristic2, J. Algebra Appl. 16 (2017), no. 4, 1750063, 6. MR 3635112[BF94] E. Bayer-Fluckiger, Galois cohomology and the trace form, Jahresber. Deutsch. Math.-Verein. 96 (1994), no. 2, 35\u00E2\u0080\u009355. MR 1273460[BF03] Gre\u00C2\u00B4gory Berhuy and Giordano Favi, Essential dimension: a functorial point of view(after A. Merkurjev), Doc. Math. 8 (2003), 279\u00E2\u0080\u0093330. MR 2029168[BM40] M. F. Becker and S. MacLane, The minimum number of generators for inseparablealgebraic extensions, Bull. Amer. Math. Soc. 46 (1940), 182\u00E2\u0080\u0093186. MR 0001218[BR97] J. Buhler and Z. Reichstein, On the essential dimension of a finite group, CompositioMath. 106 (1997), no. 2, 159\u00E2\u0080\u0093179. MR 1457337[BRV10] Patrick Brosnan, Zinovy Reichstein, and Angelo Vistoli, Essential dimension, spinorgroups, and quadratic forms, Ann. of Math. (2) 171 (2010), no. 1, 533\u00E2\u0080\u0093544. MR 2630047[BRV18] , Essential dimension in mixed characteristic, Doc. Math. 23 (2018), 1587\u00E2\u0080\u00931600.MR 3890962[CM14] Vladimir Chernousov and Alexander Merkurjev, Essential dimension of spinor andClifford groups, Algebra Number Theory 8 (2014), no. 2, 457\u00E2\u0080\u0093472. MR 3212863[CR15] Shane Cernele and Zinovy Reichstein, Essential dimension and error-correcting codes,Pacific J. Math. 279 (2015), no. 1-2, 155\u00E2\u0080\u0093179, With an appendix by Athena Nguyen.MR 3437774[CS06] Vladimir Chernousov and Jean-Pierre Serre, Lower bounds for essential dimensionsvia orthogonal representations, J. Algebra 305 (2006), no. 2, 1055\u00E2\u0080\u00931070.[DI07] Daniel Dugger and Daniel C. Isaksen, The Hopf condition for bilinear forms overarbitrary fields, Ann. of Math. (2) 165 (2007), no. 3, 943\u00E2\u0080\u0093964. MR 2335798[DR15] Alexander Duncan and Zinovy Reichstein, Versality of algebraic group actions andrational points on twisted varieties, J. Algebraic Geom. 24 (2015), no. 3, 499\u00E2\u0080\u0093530, Withan appendix containing a letter from J.-P. Serre. MR 3344763[Dun10] Alexander Duncan, Essential dimensions of A7 and S7, Math. Res. Lett. 17 (2010),no. 2, 263\u00E2\u0080\u0093266. MR 264437358Bibliography[EK94] Martin Epkenhans and Martin Kru\u00C2\u00A8skemper, On trace forms of e\u00C2\u00B4tale algebras and fieldextensions, Math. Z. 217 (1994), no. 3, 421\u00E2\u0080\u0093434. MR 1306669[For64] Otto Forster, U\u00C2\u00A8ber die Anzahl der Erzeugenden eines Ideals in einem NoetherschenRing, Math. Z. 84 (1964), 80\u00E2\u0080\u009387. MR 0163932[For17] Timothy J. Ford, Separable algebras, Graduate Studies in Mathematics, vol. 183,American Mathematical Society, Providence, RI, 2017. MR 3618889[FR17] Uriya A. First and Zinovy Reichstein, On the number of generators of an algebra, C.R. Math. Acad. Sci. Paris 355 (2017), no. 1, 5\u00E2\u0080\u00939. MR 3590278[GG86] Stephen M. Gagola, Jr. and Sidney C. Garrison, III, Real characters, double covers,and the multiplier. II, J. Algebra 98 (1986), no. 1, 38\u00E2\u0080\u009375. MR 825134[GMS03] Skip Garibaldi, Alexander Merkurjev, and Jean-Pierre Serre, Cohomological invariantsin Galois cohomology, University Lecture Series, vol. 28, American MathematicalSociety, Providence, RI, 2003.[Hat02] Allen Hatcher, Algebraic topology, Cambridge University Press, Cambridge, 2002. MR1867354[HH92] P. N. Hoffman and J. F. Humphreys, Projective representations of the symmetricgroups, Oxford Mathematical Monographs, The Clarendon Press, Oxford UniversityPress, New York, 1992, Q-functions and shifted tableaux, Oxford Science Publications.MR 1205350[Jac89] Nathan Jacobson, Basic algebra. II, second ed., W. H. Freeman and Company, NewYork, 1989. MR 1009787[Kah97] Bruno Kahn, La conjecture de Milnor (d\u00E2\u0080\u0099apre`s V. Voevodsky), Aste\u00C2\u00B4risque (1997),no. 245, Exp. No. 834, 5, 379\u00E2\u0080\u0093418, Se\u00C2\u00B4minaire Bourbaki, Vol. 1996/97. MR 1627119[Kar89] Gregory Karpilovsky, Topics in field theory, North-Holland Mathematics Studies,vol. 155, North-Holland Publishing Co., Amsterdam, 1989, Notas de Matema\u00C2\u00B4tica[Mathematical Notes], 124. MR 982265[KM90] N. A. Karpenko and A. S. Merkurjev, Chow groups of projective quadrics, Algebra iAnaliz 2 (1990), no. 3, 218\u00E2\u0080\u0093235.[KM06] Nikita A. Karpenko and Alexander S. Merkurjev, Canonical p-dimension of algebraicgroups, Adv. Math. 205 (2006), no. 2, 410\u00E2\u0080\u0093433.[KM08] , Essential dimension of finite p-groups, Invent. Math. 172 (2008), no. 3,491\u00E2\u0080\u0093508. MR MR2393078 (2009b:12009)[Knu91] Max-Albert Knus, Quadratic and Hermitian forms over rings, Grundlehren der Math-ematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol.294, Springer-Verlag, Berlin, 1991, With a foreword by I. Bertuccioni. MR 1096299[L1\u00C2\u00A83] Roland Lo\u00C2\u00A8tscher, A fiber dimension theorem for essential and canonical dimension,Compos. Math. 149 (2013), no. 1, 148\u00E2\u0080\u0093174. MR 301188159Bibliography[Lam05] T. Y. Lam, Introduction to quadratic forms over fields, Graduate Studies in Mathe-matics, vol. 67, American Mathematical Society, Providence, RI, 2005. MR 2104929[Lan02] Serge Lang, Algebra, third ed., Graduate Texts in Mathematics, vol. 211, Springer-Verlag, New York, 2002. MR 1878556[Led04] Arne Ledet, On the essential dimension of p-groups, Galois theory and modular forms,Dev. Math., vol. 11, Kluwer Acad. Publ., Boston, MA, 2004, pp. 159\u00E2\u0080\u0093172. MR 2059762[Led07] , Finite groups of essential dimension one, J. Algebra 311 (2007), no. 1, 31\u00E2\u0080\u009337.MR 2309876[McK17] Kelly McKinnie, Essential dimension of generic symbols in characteristic p, ForumMath. Sigma 5 (2017), e14, 30. MR 3668468[Mer81] A. S. Merkurjev, On the norm residue symbol of degree 2, Dokl. Akad. Nauk SSSR261 (1981), no. 3, 542\u00E2\u0080\u0093547. MR 638926[Mer09] Alexander S. Merkurjev, Essential dimension, Quadratic forms\u00E2\u0080\u0094algebra, arithmetic,and geometry, Contemp. Math., vol. 493, Amer. Math. Soc., Providence, RI, 2009,pp. 299\u00E2\u0080\u0093325. MR 2537108[Mer13] , Essential dimension: a survey, Transform. Groups 18 (2013), no. 2, 415\u00E2\u0080\u0093481.MR 3055773[Mil80] James S. Milne, E\u00C2\u00B4tale cohomology, Princeton Mathematical Series, vol. 33, PrincetonUniversity Press, Princeton, N.J., 1980. MR 559531[Mil17] J. S. Milne, Algebraic groups, Cambridge Studies in Advanced Mathematics, vol. 170,Cambridge University Press, Cambridge, 2017, The theory of group schemes of finitetype over a field. MR 3729270[MR09] Aurel Meyer and Zinovy Reichstein, The essential dimension of the normalizer of amaximal torus in the projective linear group, Algebra Number Theory 3 (2009), no. 4,467\u00E2\u0080\u0093487. MR 2525560[MR10] , Some consequences of the Karpenko-Merkurjev theorem, Doc. Math. Extravol.: Andrei A. Suslin sixtieth birthday (2010), 445\u00E2\u0080\u0093457. MR 2804261[Mum08] David Mumford, Abelian varieties, Tata Institute of Fundamental Research Studiesin Mathematics, vol. 5, Published for the Tata Institute of Fundamental Research,Bombay; by Hindustan Book Agency, New Delhi, 2008, With appendices by C. P.Ramanujam and Yuri Manin, Corrected reprint of the second (1974) edition. MR2514037[Mur94] M. Pavaman Murthy, Zero cycles and projective modules, Ann. of Math. (2) 140 (1994),no. 2, 405\u00E2\u0080\u0093434. MR 1298718[MV99] Fabien Morel and Vladimir Voevodsky, A1-homotopy theory of schemes, Inst. HautesE\u00C2\u00B4tudes Sci. Publ. Math. (1999), no. 90, 45\u00E2\u0080\u0093143 (2001). MR 181322460Bibliography[Pfi95] Albrecht Pfister, Quadratic forms with applications to algebraic geometry and topology,London Mathematical Society Lecture Note Series, vol. 217, Cambridge UniversityPress, Cambridge, 1995. MR 1366652[Pic49] G. Pickert, Inseparable Ko\u00C2\u00A8rpererweiterungen, Math. Z. 52 (1949), 81\u00E2\u0080\u0093136. MR 0032596[Pro17] Yuri Prokhorov, Quasi-simple finite groups of essential dimension 3,https://arxiv.org/abs/1703.10780, 2017.[Ras71] Richard Rasala, Inseparable splitting theory, Trans. Amer. Math. Soc. 162 (1971),411\u00E2\u0080\u0093448. MR 0284421[Rei10] Zinovy Reichstein, Essential dimension, Proceedings of the International Congress ofMathematicians. Volume II, Hindustan Book Agency, New Delhi, 2010, pp. 162\u00E2\u0080\u0093188.MR 2827790[RS19a] Zinovy Reichstein and Abhishek Shukla, Essential dimension of inseparable fieldextensions, Algebra Number Theory 13 (2019), no. 2, 513\u00E2\u0080\u0093530. MR 3927055[RS19b] Zinovy Reichstein and Abhishek Kumar Shukla, Essential dimension of double coversof symmetric and alternating groups, arXiv e-prints (2019), arXiv:1906.03698.[RV06] Zinovy Reichstein and Angelo Vistoli, Birational isomorphisms between twisted groupactions, J. Lie Theory 16 (2006), no. 4, 791\u00E2\u0080\u0093802. MR 2270660[RV18] , Essential dimension of finite groups in prime characteristic, C. R. Math. Acad.Sci. Paris 356 (2018), no. 5, 463\u00E2\u0080\u0093467. MR 3790415[Sch04] J. Schur, U\u00C2\u00A8ber die Darstellung der endlichen Gruppen durch gebrochen lineare Substi-tutionen, J. Reine Angew. Math. 127 (1904), 20\u00E2\u0080\u009350. MR 1580631[SdS00] Pedro J. Sancho de Salas, Automorphism scheme of a finite field extension, Trans.Amer. Math. Soc. 352 (2000), no. 2, 595\u00E2\u0080\u0093608. MR 1615958[Ser77] Jean-Pierre Serre, Linear representations of finite groups, Springer-Verlag, New York-Heidelberg, 1977, Translated from the second French edition by Leonard L. Scott,Graduate Texts in Mathematics, Vol. 42. MR 0450380[Ser79] , Local fields, Graduate Texts in Mathematics, vol. 67, Springer-Verlag, NewYork-Berlin, 1979, Translated from the French by Marvin Jay Greenberg. MR 554237[Ser84] , L\u00E2\u0080\u0099invariant de Witt de la forme Tr(x2), Comment. Math. Helv. 59 (1984),no. 4, 651\u00E2\u0080\u0093676. MR 780081[Ser02a] , Galois cohomology, english ed., Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2002, Translated from the French by Patrick Ion and revised by theauthor. MR 1867431[Ser02b] , Galois cohomology, English ed., Springer Monographs in Mathematics,Springer-Verlag, Berlin, 2002, Translated from the French by Patrick Ion and re-vised by the author. MR MR1867431 (2002i:12004)61Bibliography[Ser03] , Cohomological invariants, Witt invariants, and trace forms, Cohomologi-cal invariants in Galois cohomology, Univ. Lecture Ser., vol. 28, Amer. Math. Soc.,Providence, RI, 2003, Notes by Skip Garibaldi, pp. 1\u00E2\u0080\u0093100. MR 1999384[Ser08] , Topics in Galois theory, second ed., Research Notes in Mathematics, vol. 1, AK Peters, Ltd., Wellesley, MA, 2008, With notes by Henri Darmon. MR 2363329[Ser10] , Le groupe de Cremona et ses sous-groupes finis, Aste\u00C2\u00B4risque (2010), no. 332,Exp. No. 1000, vii, 75\u00E2\u0080\u0093100, Se\u00C2\u00B4minaire Bourbaki. Volume 2008/2009. Expose\u00C2\u00B4s 997\u00E2\u0080\u00931011.MR 2648675[Ste89] John R. Stembridge, Shifted tableaux and the projective representations of symmetricgroups, Adv. Math. 74 (1989), no. 1, 87\u00E2\u0080\u0093134. MR 991411[SW19] Abhishek Kumar Shukla and Ben Williams, Classifying spaces for e\u00C2\u00B4tale algebras withgenerators, arXiv e-prints (2019), arXiv:1902.07745.[Swa60] Richard G. Swan, The nontriviality of the restriction map in the cohomology of groups,Proceedings of the American Mathematical Society 11 (1960), 885\u00E2\u0080\u0093887.[Swa67] , The number of generators of a module, Math. Z. 102 (1967), 318\u00E2\u0080\u0093322. MR0218347[Swe68] Moss Eisenberg Sweedler, Structure of inseparable extensions, Ann. of Math. (2) 87(1968), 401\u00E2\u0080\u0093410. MR 0223343[Tot99] Burt Totaro, The Chow ring of a classifying space, Algebraic K-Theory (Seattle, WA,1997), Proc. Sympos. Pure Math., vol. 67, Amer. Math. Soc., Providence, RI, 1999,pp. 249\u00E2\u0080\u0093281.[Tot19] , Essential dimension of the spin groups in characteristic 2, Comment. Math.Helv. 94 (2019), 1\u00E2\u0080\u009320.[TV13] Dajano Tossici and Angelo Vistoli, On the essential dimension of infinitesimal groupschemes, Amer. J. Math. 135 (2013), no. 1, 103\u00E2\u0080\u0093114. MR 3022958[Voe03] Vladimir Voevodsky, Reduced power operations in motivic cohomology, Publ. Math.Inst. Hautes E\u00C2\u00B4tudes Sci. (2003), no. 98, 1\u00E2\u0080\u009357. MR 2031198[Wag77] Ascher Wagner, An observation on the degrees of projective representations of thesymmetric and alternating group over an arbitrary field, Arch. Math. (Basel) 29 (1977),no. 6, 583\u00E2\u0080\u0093589. MR 0460451[Wat79] William C. Waterhouse, Introduction to affine group schemes, Graduate Texts inMathematics, vol. 66, Springer-Verlag, New York-Berlin, 1979. MR 547117[Wil18] Ben Williams, Bounding the minimal number of generators of an azumaya algebra,2018.[Yam74] Toshihiko Yamada, The Schur subgroup of the Brauer group, Lecture Notes in Mathe-matics, Vol. 397, Springer-Verlag, Berlin-New York, 1974. MR 034795762"@en . "Thesis/Dissertation"@en . "2020-05"@en . "10.14288/1.0389897"@en . "eng"@en . "Mathematics"@en . "Vancouver : University of British Columbia Library"@en . "University of British Columbia"@en . "Attribution-NonCommercial-NoDerivatives 4.0 International"@* . "http://creativecommons.org/licenses/by-nc-nd/4.0/"@* . "Graduate"@en . "Essential dimension and classifying spaces of algebras"@en . "Text"@en . "http://hdl.handle.net/2429/74097"@en .