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Essential dimension and classifying spaces of algebras Shukla, Abhishek Kumar 2020

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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© 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´tale 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´tale algebraover a ring. The minimum of number of generators of an e´tale 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’s 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´tale 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´tale 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 = · · · = 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˜n . . . . . . . . . . . . . . . . . . . . . . . . . . . 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˜+n and S˜−n . . . . . . . . . . . . . . . . . 353.7 Explanation of the entries in Table 3.1 . . . . . . . . . . . . . . . . . . . . . . . 364 Classifying spaces for e´tale algebra with generators . . . . . . . . . . . . . . . 384.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384.1.1 Notation and other preliminaries . . . . . . . . . . . . . . . . . . . . . . . 394.2 E´tale 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 −1 . . . . . . . . . . . . 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´tale algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58viiList of Tables3.1 Essential dimension of A˜n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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¨ckiger, 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´talealgebras 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 ⊂ K if there exists a subfield K0 ⊂ L0 ⊂ L such that L0 andK generate L and [L0 : K0] = [L : K]. Equivalently, L is isomorphic to L0 ⊗K0 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 defineτ(n) = max{ed(L/K) | L/K is a separable extension of degree n and k ⊂ K}. (1.1.1)For example, if n = 2 a general quadratic separable extension L/K can be defined by asingle parameter c ∈ K (since L = K(√c) for some c ∈ K). Then L/K descends to k(√c)/k(c).Consequently τ(2) ≤ 1. Similarly it can be shown that every degree 3 separable extension L/Karises by solving a polynomial of the form X3 − aX + a and hence τ(3) ≤ 1 too.It is shown in [BR97] that if char(k) = 0, then⌊n2⌋6 τ(n) 6 n− 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 ≥ e2 ≥ . . . ≥ er.Every element in L (over S) satisfies a polynomial equation Zpm − s ∈ 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−1] ( 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 defineτ(n, e) = max{ed(L/K) | L/K is a field extension of type (n, e) and k ⊂ K}.We are interested in determining τ(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 >· · · > er > 1 be integers, e = (e1, . . . , er) and si = e1 + · · ·+ ei for i = 1, . . . , r. Thenτ(n, e) = nr∑i=1psi−iei .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 Λn,e, which are then classified by torsors over Spec(K) for a certaingroup scheme Gn,e. Then τ(n, e) is the essential dimension of Gn,e, much as τ(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 → 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 ∈ F(K) descends to a subfield K0 ⊂ K if a lies in the image of the natural restriction mapF(K0)→ F(K). The essential dimension ed(a) of a is defined as minimal value of trdeg(K0/k),where k ⊂ 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 ∈ 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 → H1(K,G) be the functor defined byFG(K) = {isomorphism classes of G-torsors T → 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´tale algebraSpecializing to the case G = Sn, the set FSn(K) can be identified with e´tale 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) = τ(n) and ed(Gn,e) = τ(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˜n 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 → SOn.1 // µ2 // A˜n //An //11 // µ2 // 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˜n; 2) = 2b(n−s−1)/2c.(b) 2b(n−s−1)/2c 6 ed(A˜n) 6 ed(An) + 2b(n−s−1)/2c.In the case char k = 2 the Spin groups continue to show exponential growth (see [Tot19]) butwe found that ed(A˜n) grows sub-linearly.1.3 Generators of an e´tale algebraIn Chapter 4 we consider the problem of minimum number of generators of an e´tale 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⊗R k(m), for m ∈ MaxSpec(R), is generated by n elements. In particular, any finitee´tale 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≥ 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´tale 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→∞ and e´tale 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´tale algebraThe proof has two main ideas. First is to construct classifying spaces for e´tale algebras withgenerators. More precisely we construct varieties B(r;An) such that “an e´tale algebra A ofdegree n equipped wtih r generating sections” on X is obtained as pullback of tautological e´talealgebra on B(r;An) via a unique map X → 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˜ on X = SpecR and elements of P asglobal sections Γ(X,L) of the line bundle L. Then Pn is the classifying space (in the above sense)as there is a natural 1− 1 correspondence between maps X → Pn and line bundles L on X withn+ 1 generating sections (s0, s1, . . . , sn) ∈ Γ(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 ⊂ K if there exists asubfield K0 ⊂ L0 ⊂ L such that L0 and K generate L and [L0 : K0] = [L : K]. Equivalently, Lis isomorphic to L0 ⊗K0 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 ⊂ K0}.Essential dimension of separable field extensions was studied in [BR97]. Of particular interestisτ(n) = max{ed(L/K) | L/K is a separable extension of degree n and k ⊂ 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 τ(n) 6 n− 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 τ(n) 6 n− 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 > · · · > er > 1 are integers; seeSection 2.4 for the definition. Note that the type e = (e1, . . . , er) uniquely determines theinseparable degree [L : S] = pe1+···+er of L/K but not conversely. By analogy with (2.1.1) it isnatural to defineτ(n, e) = max{ed(L/K) | L/K is a field extension of type (n, e) and k ⊂ 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 > · · · > er > 1be integers, e = (e1, . . . , er) and si = e1 + · · ·+ ei for i = 1, . . . , r. Thenτ(n, e) = nr∑i=1psi−iei .Some remarks are in order.(1) Theorem 2.1.2 gives the exact value for τ(n, e). This is in contrast to the separable case,where Theorem 2.1.1 gives only estimates and the exact value of τ(n) is unknown for any n > 8.(2) A priori, the integers ed(L/K), τ(n) and τ(n, e) all depend on the base field k. However,Theorem 2.1.2 shows that for a fixed p = char(k), τ(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 τ(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 τ(n) has two natural interpretations. On the one hand, τ(n) is the essentialdimension of the functor Etn which associates to a field K the set of isomorphism classes of e´talealgebras of degree n over K. On the other hand, τ(n) is the essential dimension of the symmetricgroup Sn. Recall that an e´tale algebra L/K is a direct product L = L1 × · · · × Lm of separablefield extensions Li/K. Equivalently, an e´tale algebra of degree n over K can be thought of as atwisted K-form of the split algebra kn = k × · · · × 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 τ(n, e) in a similar manner. Here the roleof the split e´tale algebra kn will be played by the algebra Λn,e, which is the direct product of ncopies of the truncated polynomial algebraΛe = k[x1, . . . , xr]/(xpe11 , . . . , xperr ).Note that the k-algebra Λn,e is finite-dimensional, associative and commutative, but not semisim-ple. E´tale algebras over K will get replaced by K-forms of Λn,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(Λn,e) over k. We will showthat τ(n, e) is the essential dimension of Gn,e, just like τ(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 = · · · = 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 ⊗K0 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 ⊂ K0 ⊂ 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×A→ 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 ∈ K, wherexi · xj =n∑h=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 ⊂ 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 ⊗K0 K = A, and the lemma follows.The following lemma will be helpful to us in the sequel.Lemma 2.2.2. Suppose k ⊂ K ⊂ 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 ⊗S0 S.Denote the normal closure of S over K by Snorm, and the associated Galois groups byG = Gal(Snorm/K), H = Gal(Snorm/S) ⊂ G. Now define S1 = k(g(s) | s ∈ S0, g ∈ G). Choosea transcendence basis t1, . . . , td for S0 over k, where d = trdeg(S0/k). Clearly S1 is algebraic overk(g(ti) | g ∈ 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 ∈ G. This shows that trdeg(S1/k) 6 nd.Now let K2 = SG1 ⊂ K,S2 = S0(K2) and A2 = A0 ⊗S0 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 Λ be a finite-dimensional k-algebra with multiplication map m : Λ× Λ→ Λ. Thegeneral linear group GLk(Λ) acts on the vector space Λ∗ ⊗k Λ∗ ⊗k Λ of bilinear maps Λ×Λ→ Λ.The automorphsim group scheme G = Autk(Λ) of Λ is defined as the stabilizer of m under thisaction. It is a closed subgroup scheme of GLk(Λ) defined over k. The reason we use the term“group scheme” here, rather than “algebraic group”, is that G may not be smooth; see the Remarkafter Lemma III.1.1 in [Ser02a].Proposition 2.2.3. Let Λ be a commutative finite-dimensional local k-algebra with residuefield k, and G = Autk(Λ) be its automorphism group scheme. Then the natural mapf : Gn o Sn → Autk(Λn)is an isomorphism. Here Gn = G × · · · × G (n times) acts on Λn = Λ × · · · × Λ (n times)componentwise and Sn acts by permuting the factors.Before proceeding with the proof of the proposition, recall that an element α of a ring R iscalled an idempotent if α2 = α.Lemma 2.2.4. Let Λ 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 ΛR = Λ⊗k Rare those in R (more precisely in 1⊗R).Proof. By Lemma 6.2 in [Wat79], the maximal ideal M of Λ consists of nilpotent elements.Tensoring the natural projection Λ→ Λ/M ' k with R, we obtain a surjective homomorphismΛR → R whose kernel again consists of nilpotent elements. By Proposition 7.14 in [Jac89], everyidempotent in R lifts to a unique idempotent in ΛR, and the lemma follows.82.3. Essential dimension of a functorProof of Proposition 2.2.3. Let αi = (0, . . . , 1, . . . , 0) where 1 appears in the ith position. Then⊕ni=1Rαi is an R-subalgebra of ΛnR.Let f ∈ AutR(ΛnR). Since each αi is an idempotent in ΛnR, so is each f(αi). The componentsof each f(αi) are idempotents in ΛR. By Lemma 2.2.4, they lie in R. Thus, f(αi) ∈ ⊕ni=1Rαifor every i = 1, . . . , n. As a result, we obtain a morphismAutR(ΛnR)τR−−→ AutR(⊕ni=1Rαi) = 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 τ : Aut(Λn)→ Sn of group schemes over k. The kernel of τ is Aut(Λ)n,and τ clearly has a section. The lemma follows.Remark 2.2.5. The assumption that Λ is commutative in Proposition 2.2.3 can be dropped, aslong as we assume that the center of Λ 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(Λn) = Z(Λ)n and apply Lemma 2.2.4 to Z(Λ), rather than Λ 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 Λ is k cannot bedropped. For example, if Λ is a separable field extension of k of degree d, then Autk(Λn) is atwisted form ofAutk(Λn ⊗k k) = Autk(kdn) = Snd .Here k denotes the separable closure of k. Similarly, Autk(Λ)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 → 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 ∈ F(K) descends to a subfield K0 ⊂ K if a lies inthe image of the natural restriction map F(K0)→ F(K). The essential dimension ed(a) of a isdefined as minimal value of trdeg(K0/k), where k ⊂ 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 ∈ 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 ∈ F , we define edl(a) as the minimal value of ed(a′), where a′ is the imageof a in F(K ′), and the minimum is taken over all field extensions K ′/K such that the degree[K ′ : 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 ∈ 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 → H1(K,G) be thefunctor defined byFG(K) = {isomorphism classes of G-torsors T → 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 → 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 Λ/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 Λ if there exists afield L containing K such that Λ⊗k L is isomorphic to A⊗K L as an L-algebra. We will writeAlgΛ : Fieldsk → Setsfor the functor which sends a field K/k to the set of K-isomorphism classes of K-forms of Λ.Proposition 2.3.4. Let Λ be a finite-dimensional k-algebra and G = Autk(Λ) ⊂ GL(Λ) be itsautomorphism group scheme. Then the functors AlgΛ and FG = H1(∗, G) are isomorphic. Inparticular, ed(AlgΛ) = ed(G) and edl(AlgΛ) = 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 Λn = k × · · · × k (n times) are called e´tale algebras of degreen. An e´tale algebra L/K of degree n is a direct products of separable field extensions,L = L1 × · · · × Lr, wherer∑i=1[Li : K] = n.The functor AlgΛn is usually denoted by Etn. The automorphism group Autk(Λn) is thesymmetric group Sn, acting on Λn 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 ∈ L we define the exponentof x over S as the smallest integer e such that xpe ∈ S. We will denote this number by e(x, S).We will say that x ∈ L is normal in L/S if e(x, S) = max{e(y, S) | y ∈ 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−1 and xi /∈ Li−1. Here Li = S(x1, . . . , xi−1) 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−1) 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 λ of L/S there exists a normal generating sequence x1, . . . , xr witheach xi ∈ λ.(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 > · · · > er > 1 be integers. Then there exist(a) a separable field extension E/F of degree n with k ⊂ F ,(b) a field extension L/K of type (n, e) with k ⊂ 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 > · · · > er > 1. Thefollowing finite-dimensional commutative k-algebras will play an important role in the sequel:Λn,e = Λe × · · · × Λe (n times), where Λe = k[x1, . . . , xr]/(xpe11 , . . . , xperr ) (2.4.1)is a truncated polynomial algebra.Lemma 2.4.5. Λn,e is isomorphic to Λm,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 Λn,e is isomorphic to Λm,fTo prove the converse, note that Λe is a finite-dimensional local k-algebra with residue fieldk. By Lemma 2.2.4, the only idempotents in Λe are 0 and 1. This readily implies that the onlyidempotents in Λn,e are of the form (1, . . . , n), where each i is 0 or 1, and the only minimalidempotents areα1 = (1, 0, . . . , 0), . . . , αn = (0, . . . , 0, 1).(Recall that idempotents α and β are called orthogonal if αβ = βα = 0. If α and β are orthogonal,then one readily checks that α+ β is also an idempotent. An idempotent is minimal if it cannotbe written as a sum of two orthogonal idempotents.)If Λn,e and Λm,f are isomorphic, then they have the same number of minimal idempotents;hence, m = n. Denote the minimal idempotents of Λm,f byβ1 = (1, 0, . . . , 0), . . . , βm = (0, . . . , 0, 1).A k-algebra isomorphism Λn,e → Λm,f takes α1 to βj for some j = 1, . . . , n and, hence, induces ak-algebra isomorphism between α1Λn,e ' Λe and βjΛm,f ' Λf . To complete the proof, we appealto Proposition 8 in [Ras71], which asserts that Λe and Λf 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 Λn,e. In other words, L ⊗K K ′ is isomorphic to Λn,e ⊗k K ′ as anK ′-algebra for some field extension K ′/K.Proof. (a) =⇒ (b): Assume L/K is a field extension of type (n, e). Let S be the separableclosure of K in L and K ′ be an algebraic closure of S (which is also an algebraic closure of K).ThenL⊗K K ′ = L⊗S (S ⊗K K ′) = (L⊗S K ′)× · · · × (L⊗S K ′) (n times).On the other hand, by [Ras71], Theorem 3, L⊗S K ′ is isomorphic to Λe as a K ′-algebra, andpart (b) follows.(b) =⇒ (a): Assume L⊗K K ′ is isomorphic to Λn,e ⊗k K ′ as an K ′-algebra for some fieldextension K ′/K. After replacing K ′ by a larger field, we may assume that K ′ contains the normalclosure of S over K. Since Λn,e ⊗k K ′ is not separable over K ′, L is not separable over K. ThusL/K is of type (m, f) for some m > 1 and f = (f1, . . . , fs) with f1 > f2 > · · · > fs > 1. By part(a), L⊗K K ′′ is isomorphic to Λm,f ⊗k K ′′ for a suitable field extension K ′′/K. After enlargingK ′′, we may assume without loss of generality that K ′ ⊂ K ′′. We conclude that Λn,e ⊗k K ′′ isisomorphic to Λm,f ⊗k K ′′ as a K ′′-algebra. By Lemma 2.4.5, with k replaced by K ′′, 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 > · · · > er > 1. Furthermore letsi =∑j≤i ei. Thenτ(n, e) 6 nr∑i=1psi−iei .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 ∈ S. Then there exists an s ∈ 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 = ∪ni=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) → A(S) given by s → asq lies inZ = ∪ni=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 ∈ S1 for every s ∈ S. Setting s = 1,we see that a ∈ S1. Dividing asq ∈ S1 by 0 6= a ∈ S1, we conclude that sq ∈ S1 for every s ∈ 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∑rj=1 psj−jej . 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 ∈ S. We are free to replace x1 by x1s for any 0 6= s ∈ S; clearlyx1s, x2, . . . , xr is another normal generating sequence. By Lemma 2.5.2, we may choose s ∈ 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−1], where qi = pei ; seeTheorem 1 in [Ras71]. In other words, for each i = 1, . . . , r,xqii =∑ad1,...,di−1xqid11 . . . xqidi−1i−1 (2.5.1)for some for some ad1,...,di−1 ∈ S. Here the sum is taken over all integers d1, . . . , di−1, where each0 6 dj 6 pej−ei − 1. Note that for i = 1 formula (2.5.1) reduces toxq11 = a∅,for some a∅ ∈ S. By Lemma 2.2.1, L (viewed as an S-algebra), descends toS0 = k(ad1,...,di−1 | i = 1, . . . , r and 0 6 dj 6 pej−ei − 1) .Note that for each i = 1, . . . , r, there are exactlype1−ei · pe2−ei · . . . · pei−1−ei = psi−ieichoices of the subscripts d1, . . . , di−1. Hence, S0 is generated over k by∑ri=1 psi−iei elementsand consequently,trdeg(S0/k) 6r∑i=1psi−iei .132.6. Versal algebrasMoreover, since S0 contains a∅ = 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+···+er , we readily deduce thated(L/K)[L : K]6 τ(n, e)[L : K]6r∑i=1p−iei−ei+1−···−rr 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 ∈ A givesrise to the K-linear map la : A → A given by la(x) = ax (left multiplication by a). Note thatlab = la · 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 Λ be a finite-dimensional associative k-algebrawith 1. Assume that there exists a field extension K/k and a K-form A of Λ such that A is adivision algebra. Then(a) there exists a field Kver containing k and a form Aver/Kver of Λ such thated(Aver) = ed(AlgΛ), edl(Aver) = edl(AlgΛ) for every prime integer l, andAver is a division algebra.(b) If G is the automorphism group scheme of Λ, thened(G) = ed(AlgΛ) = max{ed(A/K) |A is a K-form of Λ and a division algebra}andedl(G) = edl(AlgΛ) = max{edl(A/K) |A is a K-form of Λ and a division algebra}.Here the subscript “ver” is meant to indicate that Aver/Kver is a versal object for AlgΛ =H1(∗, 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 → Spec(Kver). Recall thatG = Autk(Λ) is defined as a closed subgroup of the general linear group GLk(Λ). This generallinear group admits a generically free linear action on some vector space V (e.g., we can takeV = Endk(Λ), with the natural left G-action). Restricting to G we obtain a generically freerepresentation G → GL(V ). We can now choose a dense open G-invariant subscheme U ⊂ Vover k which is the total space of a G-torsor pi : U → B; see, e.g., Section 4 in [BF03]. Passing to142.6. Versal algebrasthe generic point of B, we obtain a G-torsor Tver → 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 → Spec(K) be the torsor associated to the K-algebra A and Aver be the Kver-algebra associated to Tver → Spec(Kver) under the isomorphism between the functors AlgΛand H1(∗, G) of Proposition 2.3.4. By the characteristic-free version of the no-name Lemma,proved in [RV06], Section 2, T × VK is G-equivariantly birationally isomorphic to T ×AdK , whered = dim(V ) = dim(VK) and G acts trivially on AdK . In other words, we have a Cartesian diagramof rational maps defined over kT × AdK ' //T × VK pr2 // UKAdK // BK .Here all direct products are over Spec(K), and pr2 denotes the rational G-equivariant projectionmap taking (t, v) ∈ T × V to v ∈ V for v ∈ U . The map AdK = Spec(K) × Ad 99K B in thebottom row is induced from the dominant G-equivariant map T × AdK 99K UK on top. Passingto generic points, we obtain an inclusion of field Kver ↪→ K.Kver ↪→ K(x1, . . . , xd) such that theinduced map H1(Kver, G)→ H1(K(x1, . . . , xd), G) sends the class of Tver → Spec(Kver) to theclass associated to T × AdK → AdK . Under the isomorphism of Proposition 2.3.4 between thefunctors AlgΛ and FG = H1(∗, G), this translates toAver ⊗Kver K(x1, . . . , xd) ' A⊗K K(x1, . . . , xd)as K(x1, . . . , xd)-algebras.For simplicity we will write A(x1, . . . , xd) in place of A ⊗K K(x1, . . . , xd). Since A is adivision algebra, so is A(x1, . . . , xd). Thus the linear map la : A(x1, . . . , xd)→ A(x1, . . . , xd) isnon-singular (i.e., has trivial kernel) for every a ∈ 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 AlgΛ and FG, we see thated(Aver) = ed(Tver/Kver) = ed(G) = ed(AlgΛ)andedl(Aver) = edl(Tver/Kver) = edl(G) = edl(AlgΛ) ,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 Λe and Λn,e = Λe × · · · × Λe (ntimes) defined in Section 2.4; see (2.4.1). We will write Gn,e = Aut(Λn,e) ⊂ GLk(Λn,e) for theautomorphism group scheme of Λn,e and Algn,e for the functor AlgΛn,e : Fieldsk → Sets. Recallthat this functor associates to a field K/k the set of isomorphism classes of K-forms of Λn,e.Replacing essential dimension by essential dimension at a prime l in the definitions (2.1.1)and (2.1.1), we setτl(n) = max{edl(L/K) | L/K is a separable field extension of degree n and k ⊂ K}.152.7. Conclusion of the proof of Theorem 2.1.2andτl(n, e) = max{edl(L/K) | L/K is a field extension of type (n, e) and k ⊂ K}.Corollary 2.6.2. Let l be a prime integer. Then(a) ed(Sn) = ed(Etn) = τ(n) and edl(Sn) = edl(Etn) = τl(n). Here Etn is the functor ofn-dimensional e´tale algebras, as in Example 2.3.5.(b) ed(Gn,e) = ed(Algn,e) = τ(n, e) and edl(Gn,e) = edl(Algn,e) = τl(n, e).Proof. (a) Recall that e´tale algebra are, by definition, commutative and associative with identity.For such algebras “division algebra” is the same as “field”. By Lemma 2.4.4(a) there existsa separable field extension E/F of degree n with k ⊂ 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) =bnlc, 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 > · · · > er > 1be integers, e = (e1, . . . , er) and si = e1 + · · ·+ ei for i = 1, . . . , r. Thenτp(n, e) = τ(n, e) = nr∑i=1psi−iei .By definition τp(n, e) 6 τ(n, e) and by Proposition 2.5.1, τ(n, e) 6 n∑ri=1 psi−iei . Moreover,by Corollary 2.6.2(b), τp(n, e) = edp(Gn,e). It thus remains to show thatedp(Gn,e) > nr∑i=1psi−iei . (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))− dim(G) (2.7.2)for any group scheme G of finite type over a field k of characteristic p. Now recall thatGe = Autk(Λe), and Gn,e = Autk(Λn,e), where Λn,e = Λne . Since Λe 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 > · · · > er > 1 are integers. Then(a) dim(Lie(Ge)) = rpe1+···+er , and(b) dim(Ge) = rpe1+···+er −∑ri=1 psi−iei .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 αpj over a commutative ring S of characteristic p is definedas the kernel of the j-th power of the Frobenius map, Ga → Ga, x 7→ xpj , viewed as ahomomorphism of group schemes over S. We will be particularly interested in the case,where S = Λe.3. Suppose X is a scheme over Λ, where Λ is a finite-dimensional commutative k-algebra. Wewill denote the Weil restriction of the Λ-scheme X to k by RΛ/k(X). For generalities onWeil restriction, see Chapter 2 and the Appendix in [Mil17].4. We will denote by End(Λe) the functorCommk −→ SetsR −→ EndR−alg(Λe ⊗k R)of algebra endomorphisms of Λe. 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(Λe) is represented by an irreducible (but non-reduced) affinek-scheme Xe.(b) dim(Xe) = rpe1+···+er −∑ri=1 psi−iei.(c) dim(Tγ(Xe)) = rpe1+···+er for any k-point γ of Xe. Here Tγ(Xe) denotes the tangentspace to Xe at γ.Proof. An endomorphism F in End(Λe)(R) is uniquely determined by the imagesF (x1), F (x2), . . . , F (xr) ∈ Λe(R)of the generators x1, . . . , xr of Λe. These elements of Λe satisfy F (xi)qi = 0. Conversely, anyr elements F1, . . . , Fr in Λe ⊗R satisfying F qii = 0, give rise to an algebra endomorphism F inEnd(Λe)(R). We thus haveEnd(Λe)(R) = HomR−alg(Λe ⊗k R,Λe ⊗R)∼= αq1(Λe ⊗R)× . . .× αqr(Λe ⊗R)∼= RΛe/k(αq1)(R)× . . .×RΛe/k(αqr)(R)∼=r∏i=1RΛe/k(αqi)(R)172.7. Conclusion of the proof of Theorem 2.1.2We conclude that End(Λe) is represented by an affine k-scheme Xe =∏ri=1RΛe/k(αqi). (Notethat Xe is isomorphic to∏ri=1RΛe/k(αqi) 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 ∈ {q1, . . . , qr} we have that(a′) RΛe/k(αqj ) is irreducible,(b′) dim(RΛe/k(αqj ))= pe1+···+er − psj−jej and(c′) dim(Tγ(RΛe/k(αqj ))) = pe1+···+er for any k-point γ of RΛe/k(αqj ).To prove (a′), (b′) and (c′), we will write out explicit equations for RΛe/k(αqj ) in RΛe/k(A1) 'Ak(Λe). We will work in the basis {xi11 xi22 . . . xirr } of monomials in Λe, where 0 6 i1 < q1,0 6 i2 < q2, . . ., 0 6 ir < qr. Over Λe, αqj 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 =∑yi1,...,irxi11 xi22 . . . xirrand expandingtqj =∑yqji1,...,irxqji11 xqji22 . . . xqjirrwe see that the only monomials appearing in the above sum are those for whichqji1 < q1, qji2 < q2, . . . , qjir < qr.Thus RΛe/k(αqj ) is cut out (again, scheme-theoretically) in RΛe/k(A1) ' A(Λe) byyqji1,...,ij−1,0,...,0 = 0 for 0 6 i1 <q1qj, . . ., 0 6 ij−1 <qj−1qj,where yi1,...,ir are the coordinates in A(Λe). In other words, RΛe/k(αqj ) is the subscheme ofRΛe/k(A1) ' Ak(Λe) ' Ape1+···+erk cut out (again, scheme-theoretically) by qjth powers ofq1qjq2qj. . .qj−1qj= psj−jejdistinct coordinate functions. The reduced scheme RΛe/k(αqj )red is thus isomorphic to an affinespace of dimension pe1+···+er −∑rj=1 psj−jej . On the other hand, since qj is a power of p, theJacobian criterion tells us that the tangent space to RΛe/k(αql) at any k-point is the same as thetangent space to A(Λe) = Ape1+···+er , and (a′), (b′), (c′) follow.Conclusion of the proof of Proposition 2.7.2. The automorphism group scheme Ge is the groupof invertible elements in End(Λe). In other words, the natural diagramGe //GLNEnd(Λe) //MatN×Nwhere N = dim(Λe) = pe1+...+er , is Cartesian. Hence, Ge is an open subscheme of Xe. SinceXe is irreducible, Proposition 2.7.2 follows from Lemma 2.7.3. This completes the proof ofProposition 2.7.2 and thus of Theorem Alternative proofs of Theorem Alternative proofs of Theorem 2.1.2The proof of the lower bound of Theorem 2.1.2 given in Section 2.7 section is the only one weknow. However, we have two other proofs for the upper bound (Proposition 2.5.1), in additionto the one given in Section 2.5. In this section we will briefly outline these arguments for theinterested reader.Our first alternative proof of Proposition 2.5.1 is based on an explicit construction of the versalalgebra Aver of type (n, e) whose existence is asserted by Proposition 2.6.1. This construction isvia generators and relations, by taking “the most general” structure constants in (2.5.1). Versalityof Aver constructed this way takes some work to prove; however, once versality is established, itis easy to see directly that Aver is a field and thusτ(n, e) = ed(Aver) 6 trdeg(Kver/k) = nr∑i=1psi−iei .Our second alternative proof of Proposition 2.5.1 is based on showing that the naturalrepresentation of Gn,e on V = Λrn,e is generically free. Intuitively speaking, this is clear: Λn,e isgenerated by r elements as a k-algebra, so r-tuples of generators of Λn,e are dense in V and havetrivial stabilizer in Gn,e. The actual proof involves checking that the stabilizer in general positionis trivial scheme-theoretically and not just on the level of points. Once generic freeness of thislinear action is established, the upper bound of Proposition 2.5.1 follows from the inequalityed(Gn,e) 6 dim(V )− dim(Gn,e)see, e.g., Proposition 4.11 in [BF03]. To deduce the upper bound of Proposition 2.5.1 from thisinequality, recall thatτ(n, e) = ed(Gn,e) (see Corollary 2.6.2(b)),dim(V ) = r dim(Λn,e) = nr dim(Λe) = nrpe1+···+er (clear from the definition), anddim(Gn,e) = n dim(Ge) = nrpe1+···+er − n∑ri=1 psi−iei (see Proposition 2.7.2(b)).2.9 The case where e1 = · · · = erIn the special case, where n = 1 and e1 = · · · = er, Theorem 2.1.2 tells us that τ(n, e) = r. Inthis section, we will give a short proof of the following stronger assertion (under the assumptionthat k is perfect).Proposition 2.9.1. Let e = (e, . . . , e) (r times) and L/K be purely inseparable extension oftype e, with k ⊂ K. Assume that the base field k is perfect. Then edp(L/K) = ed(L/K) = r.The assumption that k is perfect is crucial here. Indeed, by Lemma 2.4.4(b), there exists afield extension L/K of type e. Setting k = K, we see that ed(L/K) = 0, and the propositionfails.The remainder of this section will be devoted to proving Proposition 2.9.1. We begin withtwo reductions.(1) It suffices to show thated(L/K) = r for every field extension L/K of type e; (2.9.1)192.9. The case where e1 = · · · = erthe identity edp(L/K) will then follow. Indeed, edp(L/K) is defined as the minimal value ofed(L′/K ′) taken over all finite extensions K ′/K of degree prime to p. Here L′ = L⊗K K ′. Since[L : K] is a power of p, L′ is a field, so (2.9.1) tells us that ed(L′/K ′) = r.(2) The proof of the upper bound,ed(L/K) 6 r (2.9.2)is the same as in Section 2.5, but in this special case the argument is much simplified. Forthe sake of completeness we reproduce it here. Let x1, . . . , xr be a normal generating sequencefor L/K. By a theorem of Pickert (Theorem 1 in [Ras71]), xq1, . . . , xqr ∈ K, where q = pe. Setai = xqi and K0 = k(a1, . . . , ar). The structure constants of L relative to the K-basis xd11 . . . xdrrof L, with 0 6 d1, . . . , dr 6 q − 1 all lie in K0. Clearly trdeg(K0/k) 6 r; the inequality (2.9.2)now follows from Lemma 2.2.1.It remains to prove the lower bound, ed(L/K) > 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 > · · · > 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 − 1. By Remark 2.4.3,[L : Lp] > [L : (Lp ·K)] = pr. (2.9.3)On the other hand, since [L : k(a1, . . . , as)] <∞, 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 3Essential dimension of double coversof symmetric and alternating group3.1 IntroductionI. Schur [Sch04] studied central extensions1 // Z/2Z // S˜±nφ± // 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 ρ : Sn → PGL(V ) lifts to linear representations ρ+ : S˜+n → GL(V ) and ρ− : S˜−n →GL(V ); see [HH92, Theorem 1.3]. That is, the following diagram commutes.S˜±nφ±ρ± // GL(V )Snρ // PGL(V )Figure 3.1: Projective representationMoreover, the groups S˜±n are minimal central extensions of Sn with this property. They arecalled representation groups of Sn. In terms of generators and relations,S˜+n =〈z, s1, s2, . . . , sn−1 | z2 = s2i = 1, [z, si] = 1, (sisj)2 = z if |i− j| > 1, (sisi+1)3 = 1〉andS˜−n =〈z, t1, t2, . . . , tn−1 | z2 = 1, t2i = z, (titj)2 = z if |i− j| > 1, (titi+1)3 = z〉.Here z is a central element of order 2 in S˜+n (respectively, S˜−n ) generating Ker(φ+) (respectively,Ker(φ−)), and φ+(si) = φ−(ti) is the transposition (i, i+ 1) in Sn. The preimage of An underφ+ in S˜+n is isomorphic to the preimage of An under φ− in S˜−n ; see [Ser08, Section 9.1.3]. We willdenote this group by A˜n; it is a representation group of An. For modern expositions of Schur’stheory, see [HH92] or [Ste89].The purpose of this chapter is to study the essential dimension of the covering groups S˜±n andA˜n. 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˜±n , A˜n 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 + · · ·+ x2n, Sn → On is the natural n-dimensional representation, Γn is the Clifford group, andS˜±n , A˜n are the preimages of Sn and An under the double covers Pin±n → On. The groups Pin±nare defined as the kernels of the homomorphisms N± : Γn → Gm given by N+(x) = x.xT andN−(x) = x.γ(xT ), where (x1 ⊗ x2 ⊗ . . .⊗ xn)T = xn ⊗ . . .⊗ x2 ⊗ x1 and γ is the automorphismof the Clifford algebra which acts on degree 1 component by −1. Both appear in literature asPin groups (Pin+ in [Ser84] and Pin− in [ABS64], see also [GG86]).OnΓnGmPin+n OnZ/2Z Pin−nS˜+n SnZ/2Z S˜−nAnA˜nZ/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− 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) =2(n−1)/2 − n(n− 1)2, if a = 0,2(n−2)/2 − n(n− 1)2, if a = 1,2(n−2)/2 + 2a − n(n− 1)2, if a > 2;223.1. Introductionsee [BRV10], [CM14] and [Tot19].Question 3.1.1. What is the asymptotic behavior of ed(S˜±n ), ed(A˜n), ed(S˜±n ; 2) and ed(A˜n; 2)as n −→∞? Do these numbers grow sublinearly, like ed(Sn), or exponentially, like ed(Spinn)?Note that for odd primes p, S˜+n , S˜−n and Sn have isomorphic Sylow p-subgroups, andthus ed(S˜±n ; 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˜±n ; p) and ed(A˜n; p) when p = 2.In this chapter we answer Question 3.1.1 as follows: ed(S˜±n ), ed(A˜n), ed(S˜±n ; 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˜n be either S˜+n or S˜−n . Then(a) ed(A˜n) 6 ed(S˜n) 6 2b(n−1)/2c.(b) ed(S˜n; 2) = 2b(n−s)/2c,(c) ed(A˜n; 2) = 2b(n−s−1)/2c.(d) 2b(n−s)/2c 6 ed(S˜n) 6 ed(Sn) + 2b(n−s)/2c(e) 2b(n−s−1)/2c 6 ed(A˜n) 6 ed(An) + 2b(n−s−1)/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˜n) = ed(S˜n; 2) = ed(A˜n) = ed(A˜n; 2) = 2n−22 .(b) If n = 2a1 + 2a2 , where a1 > a2 > 1, then ed(S˜n) = ed(S˜n; 2) = 2n−22 .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˜n) and ed(A˜n) than we doabout ed(Sn) and ed(An).Theorem 3.1.4. Let S˜n be either S˜+n or S˜−n . Assume char k = 2. Then(a) ed(Sn) 6 ed(S˜n) 6 ed(Sn) + 1.(b) ed(An) 6 ed(A˜n) 6 ed(An) + 1.(c) ed(S˜n; 2) = ed(A˜n; 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˜n) and ed(A˜n) in characteristic 2 isthat the connection between S˜±n (respectively, A˜n) and Pin±n (respectively, Spinn) outlined abovebreaks down in this setting; see Remark 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˜n; 2) 2 2 2 2 8 8 8 8 16 16 32 32 128edC(A˜n) 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˜nSome values of ed(An), ed(A˜n) and ed(A˜n; 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 + · · · + anx2ndefined over a field F of characteristic 6= 2 by 〈a1, . . . , an〉. Here a1, . . . , an ∈ F ∗. We willabbreviate 〈a, . . . , a〉 (m times) as m〈a〉. Recall that the Hasse invariant w2(q) of q = 〈a1, . . . , an〉(otherwise known as the second Stiefel-Whitney class of q) is given byw2(q) = Σ16i<j6n(ai, aj) in H2(F,Z/2Z) = Br2(F ),where (a, b) is the class of the quaternion algebra F{x, y}/(x2 = a, y2 = b, xy + yx = 0) inH2(F,Z/2Z); see [Lam05, Section V.3].Let E/F be an n-dimensional e´tale algebra. The trace form qE/F is the n-dimensionalnon-degenerate quadratic form given by x 7→ TrE/F (x2). Trace forms have been much studied;for an overview of this research area, see [BF94]. An important basic (but still largely open)problem is: Which n-dimensional quadratic forms over F are trace forms?A classification of trace forms of dimension 6 7, due to J.-P. Serre, can be found in [Ser03,Chapter IX]. Note that by a theorem of M. Epkenhans and K. Kru¨skemper [EK94], for aHilbertian field F of characteristic 0 it suffices to consider only field extensions E/F . That is,the trace form of any n-dimensional e´tale algebra E/F will occur as the trace form of some fieldextension E′/F of degree n. We shall not use this result in the sequel.Now suppose F contains a primitive 8th root of unity, and n = 2a1 + · · ·+ 2as is a dyadicexpansion of n, where a1 > · · · > as > 0 (as in Theorem 3.1.2). Then every n-dimensional traceform q contains s〈1〉 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〈1〉 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 + · · ·+ 2as be the dyadic expansion of n, where a1 > · · · > as > 0. Then(a) maxF, qind(w2(q)) = maxF, tind(w2(t)) = 2b(n−s)/2c,(b) maxF, q1ind (w2(q1)) = maxF, t1ind(w2(t1)) = 2b(n−s−1)/2c.Here the maxima are taken as• F ranges over all fields containing k,243.2. Preliminaries• q ranges over n-dimensional non-degenerate quadratic forms over F containing s 〈1〉,• q1 ranges over n-dimensional quadratic forms of discriminant 1 over F containing s 〈1〉,• t ranges over n-dimensional trace forms over F , and• t1 ranges over n-dimensional trace forms of discriminant 1 over F ,Note that if q = r ⊕ s〈1〉, 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−s)-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˜+n and S˜−n , and in Section 3.7 we explain the entries in Table 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)− 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)− 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 ′d:1 ''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 ↪→ GL(V ) be a generically free representation. Thened(G) 6 dim(V )− 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 → G2 be a homomorphism of algebraic groups. If the induced mapH1(K,G1)→ 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)− dim(G) + dim(H),(b) ed(G; p) > ed(H; p)− 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 morphismδK : H1(K,G)→ 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(δK(α)) for any α ∈ H1(K,G). Let ind(G,Z/pZ) denote themaximal value of ind(δK(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)δK // H2(K,Z/pZ)H1(K,Gp)νK //i∗OOH2(K,Z/pZ)'OO(3.2.1)of Galois cohomology sets for any field K/k. Here δ and ν 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 ∈ H1(K,G) such that δK(t) has the maximal possibleindex in H2(K,Z/pZ); that is,ind(δK(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 ∈ H1(L,G) is the image of some s ∈ H1(L,Gp). Here asusual, tL denotes the image of t ∈ H1(K,G) under the restriction map H1(K,G)→ H1(L,G).Since [L : K] is prime to p, we haveind(G,Z/pZ) = ind(δK(t)) = ind(δL(tL)) = ind(δK(t)L) = ind(νL(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˜nLemma 3.2.6. Let H˜n be a Sylow 2-subgroup of A˜n.(a) If n = 4 or 5, then H˜n is isomorphic to the quaternion groupQ8 =〈x, y, c | x2 = y2 = c, c2 = 1, cx = xc, cy = yc, xy = cyx 〉 .(b) If n = 6 or 7, then H˜n is isomorphic to the generalized quaternion groupQ16 =〈x, y | x8 = y4 = 1, y2 = x4, yxy−1 = x−1 〉 .273.3. Proof of Theorem 3.1.2Proof. We will view A˜n (and thus H˜n) as a subgroup of S˜+n and use the generators and relationsfor S˜+n given in the introduction.(a) For n = 4 or 5, we can take H˜n to be the group of order 8 generated by σ = (s1s2s3)2 andτ = s1s3. These elements project to (1 3)(2 4) and (1 2)(3 4) in An, respectively. One readilychecks that σ2 = τ2 = z and στ = zτσ. An isomorphism Q8 → H˜n can now be defined byx 7→ σ, y 7→ τ, c 7→ z.(b) For n = 6 or 7, we can take H˜n to be the group of order 16 generated by σ = s1s2s3s5and τ = s1s3. These elements project to (1 2 3 4)(5 6) and (1 2)(3 4) in An, respectively. Anisomorphism Q16 → H˜n can now be given by x 7→ σ and y 7→ τ .Proposition 3.2.7. Let n > 4 be an integer, S˜n be either S˜+n or S˜−n , and P˜n, H˜n be Sylow2-subgroups of S˜n, A˜n, respectively. Denote the centers of P˜n and H˜n by Z(P˜n) and Z(H˜n),respectively. Then Z(P˜n) = Z(H˜n) = 〈z〉 is a cyclic group of order 2.Proof. By [Wag77, Lemma 3.2] that Z(P˜n) = 〈z〉 for every n > 4 and Z(H˜n) = 〈z〉 for everyn > 8.It remains to show that Z(H˜n) = 〈z〉 for 4 6 n 6 7. Clearly z ∈ Z(H˜n), so we only need toshow that Z(H˜n) is of order 2. We will use the description of the groups H˜n from Lemma 3.2.6.If n = 4 and 5, then H˜n is isomorphic to the quaternion group Q8, and the center of Q8 is clearlyof order 2. If n = 6 or 7, then H˜n ' Q16, and the center of Q16 is readily seen to be the cyclicgroup〈x4〉of 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˜n be either S˜+n or S˜−n . Then(a) ed(A˜n) 6 ed(S˜n) 6 2b(n−1)/2c.(b) ed(S˜n; 2) = 2b(n−s)/2c,(c) ed(A˜n; 2) = 2b(n−s−1)/2c.(d) 2b(n−s)/2c 6 ed(S˜n) 6 ed(Sn) + 2b(n−s)/2c(e) 2b(n−s−1)/2c 6 ed(A˜n) 6 ed(An) + 2b(n−s−1)/2c.(a) The first inequality follows from the fact that A˜n is contained in S˜n; see Lemma 3.2.4.For the second inequality, apply Lemma 3.2.1 to the so-called basic spin representation of S˜n.This representation is obtained by restricting a representation of the Clifford algebra Cn−1 intoMat2bn−12 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 −1).283.3. Proof of Theorem 3.1.2(b) Let Pn be a Sylow 2-subgroups of Sn. The preimage P˜n of Pn is a Sylow 2-subgroup of S˜n.By Lemma 3.2.2, ed(S˜n; 2) = ed(P˜n; 2). Moreover, by the Karpenko-Merkurjev theorem [KM08,Theorem 4.1],ed(P˜n) = ed(P˜n; 2) = dim(V ),where V is a faithful linear representation of P˜n of minimal dimension.By Proposition 3.2.7, the center of Z(P˜n) = 〈z〉 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 ρ of P˜n is faithful if and only if ρ(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 ρ : P˜n → GL(V ) with ρ(z) 6= 1 is of dimension 2b(n−s)/2c. This provespart (b) for k = C.Case 2: Assume k ⊂ C is a field containing a primitive root of unity ζ2d of degree 2d, where2d is the exponent of P˜n. By a theorem of R. Brauer [Ser77, 12.3.24], every irreducible complexrepresentation of P˜n 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−s)/2c, and part (b) holds over k.Case 3: Now suppose that k is a field of characteristic 0 containing ζ8 (but possibly not ζ2d).Set l = k(ζ2d). Thenedk(P˜n) > edl(P˜n) = 2b(n−s)/2c.To prove the opposite inequality, let V be a faithful irreducible representation of P˜n dimension2b(n−s)/2c defined over Q(ζ2d). Such a representation exists by Case 2. We claim that V is, infact, defined over Q(ζ8). In particular, V is defined over k and thusedk(P˜n) 6 2b(n−s)/2c,as desired. We will prove the claim in two steps.First we will show that the character χ : P˜n → Q(ζ2d) of V takes all of its values in Q(ζ8).By [Wag77, Lemma 4.2], there are either one or two faithful irreducible characters of P˜n ofdimension 2b(n−s)/2c. The Galois group G = Gal(Q(ζ2d)/Q) ' Z/2Z×Z/2d−2Z acts on this set ofcharacters. Thus for any σ ∈ P˜n, the G-orbit of χ(σ) has either one or two elements. Consequently,[Q(χ(σ)) : Q] = 1 or 2. Note that G has exactly three subgroups of index 2. Under the Galoiscorrespondence these subgroups correspond to the subfields Q(√−1), Q(√2) and Q(√−2) ofQ(ζ2d). Thus χ(σ) lies in one of these three fields; in particular, χ(σ) ∈ Q(√−1,√2) = Q(ζ8) forevery σ ∈ G. In other words, χ takes all of its values in Q(ζ8), as desired.Now observe that since P˜n is a 2-group, the Schur index of χ over Q(ζ8) is 1; see [Yam74,Corollary 9.6]. Since the character χ of V is defined over Q(ζ8) and the Schur index of χ 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 ζ8. 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’s Lemma, ζ8 lifts to a primitive 8throot of unity in A. Denote the fraction field of A by K and the maximal ideal by M . SinceP˜n is a 2-group and char(k) = p is an odd prime, every d-dimensional k[P˜n]-module W liftsto a unique A[P˜n]-module WA, which is free of rank d over A. Moreover, the lifting operationV 7→ VK := VA⊗K and the “reduction mod M” operation give rise to mutually inverse bijections293.3. Proof of Theorem 3.1.2between k[P˜n]-modules and K[P˜n]-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˜n]-module is2b(n−s)/2c. Hence, the minimal dimension of a faithful k[P˜n]-module is also 2b(n−s)/2c. This provespart (b) in Case 4.Case 5: Now assume that k is an arbitrary field of characteristic p > 2 containing ζ8. Denotethe prime field of k by Fp. Then k can be sandwiched between two perfect fields, k1 ⊂ k ⊂ k2,where k1 = Fp(ζ8) is a finite field, and k2 is the algebraic closure of k. Thenedk1(P˜n) > edk(P˜n) > edk2(P˜n).By Case 4, edk1(P˜n) = edk2(P˜n) = 2b(n−s)/2c. We conclude that edk(P˜n) = 2b(n−s)/2c. The proofof part (b) is now complete.(c) Let Hn be the Sylow 2-subgroups of An. Its preimage H˜n is a Sylow 2-subgroup of A˜n. Bythe Karpenko-Merkurjev theorem [KM08, Theorem 4.1], ed(H˜n) = ed(H˜n; 2) = dim(W ), whereW is a faithful linear representation of H˜n 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˜n) = 〈z〉 is of order 2. Thus W is irreducible. Moreover, anirreducible representation ρ of H˜n is faithful if and only if ρ(z) 6= 1.If k = C is the field of complex numbers, it is shown in [Wag77, Lemma 4.3] that everyirreducible representation ρ of H˜n with ρ(z) 6= 1 is of dimension 2b(n−s−1)/2c. This proves part(c) for k = C. Moreover, depending on the parity of n − 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˜n) > ed(S˜n; 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 = 〈z〉 is the center of Gp = P˜n by Proposition 3.2.7.(e) The lower bound follows from part (c) and the inequality ed(A˜n) > ed(A˜n; 2). The upperbound is obtained by applying Lemma 3.2.5 to the exact sequence1 // 〈z〉 // A˜n // An // 1,in the same way as in part (d).Remark 3.3.2. If n− s is even, then S˜n has only one faithful irreducible complex representationof dimension 2b(n−s)/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 ζ4 =√−1.(Similarly for part (c) in the case where n− s is odd; see [Wag77, Lemma 4.3].)However, in general part (b) fails if we do not assume that ζ8 ∈ k. For example, in thecase, where s = 1 (i.e., n > 4 is a power of 2), the inequality ed(S˜n; 2) 6 2b(n−s)/2c = 2(n−2)/2is equivalent to the existence of a faithful irreducible representation V of P˜n of degree 2(n−2)/2defined over k. There are two such representations, and [HH92, Theorem 8.7] shows that someof their character values are not contained in Q(ζ4).303.4. Proof of Theorem 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˜n; 2) = edk(A˜n; 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)δK // H2(K,Z/pZ)in Galois cohomology (or flat cohomology, if G is not smooth), where δK 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)→ 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¨3, Corollary 3.5].Remark 3.4.2. Note that in characteristic 2 the group A˜n is no longer isomorphic to thepreimage of An ⊂ SOn in Spinn. The scheme-theoretic preimage of An in Spinn is an extensionof a constant group scheme An by an infinitesimal group scheme µ2. 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 ⊂ SOn in Spinn is the direct product An×µ2.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− 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˜4) 6 3.We do not know whether edk(S˜4) = 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˜4 contains an element of order 8 (thepreimage of a 4-cycle in S4), Ledet’s conjecture implies that edk(S˜4) = 3. Note also that byCorollary 3.1.3, edl(S˜4) = 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 = 〈a1, . . . , an〉 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) = Σ16i<j6n(ai, aj) in H2(F,Z/2Z) = Br2(F ),where (a, b) = (a) ∪ (b) is the class of the quaternion algebraF{x, y}/(x2 = a, y2 = b, xy + yx = 0).It is immediate from this definition thatw2(〈1〉 ⊕ q) = w2(q). (3.5.1)Our proof of Theorem 3.1.5 will be based on the following elementary lemma.Lemma 3.5.1. Let F be a field of characteristic 6= 2 containing a primitive 4th root of unity.Let q be an n-dimensional non-degenerate quadratic form over F . Then(a) ind(w2(q)) 6 2bn/2c.(b) If q is of discriminant 1 over F , then ind(w2(q)) 6 2b(n−1)/2c.323.5. Proof of Theorem 3.1.5Proof. Let q = 〈a1, a2, . . . , an−1, an〉 for some a1, . . . , an ∈ F ∗.(a) We will consider the cases where n is odd and even separately. If n = 2m even, thenS = F(√−a1a2 , . . . ,√−a2m−1a2m)splits q. That is qS ' n〈1〉. Hence, S also splits w2(q). Since theindex of an element α ∈ H2(F,Z/2Z) is the minimal degree [K : F ] of a splitting field K/F of α,we conclude that ind(w2(q)) 6 [S : F ] = 2m, as desired.Now suppose that n = 2m+ 1 is odd. Set S = F(√−a1a2 , . . . ,√−a2m−1a2m), as before. Over S,qS ' 2m〈1〉 ⊕ 〈a2m+1〉.It now follows from (3.5.1) that w2(qS) = 0 in H2(S,Z/2Z). Hence, w2(q) splits over S andconsequently, ind(w2(q)) 6 [S : F ] = 2m = 2n−12 , as desired.(b) Since q = 〈a1, . . . , an〉 has discriminant 1, we may assume without loss of generality thata1 · . . . ·an = 1 in F . Let r = 〈a2, . . . , an〉. Since K contains√−1, the quaternion algebra (a1, a1)is split. Thusw2(q) = (a1, a1) + w2(q) = (a1, a1 · . . . · an) + w2(r)= (a1, 1) + w2(r) = w2(r).By part (a), ind(w2(r)) 6 2b(n−1)/2c, and part (b) follows.We are now ready to proceed with the proof of Theorem 3.1.5.(a) As we pointed out in the Introduction, every n-dimensional trace form contains s〈1〉 as asubform; see [Ser84, Proposition 4]. ThusmaxF, tind(w2(t)) 6 maxF, qind(w2(q)). (3.5.2)On the other hand, by our assumption, q = s 〈1〉 ⊕ r, where r is a form of dimension n − s.By (3.5.1), w2(q) = w2(r) and by Lemma 3.5.1(a), ind(w2(r)) 6 2b(n−s)/2c. ThusmaxF, qind(w2(q)) 6 2b(n−s)/2c. (3.5.3)In view of the inequalities (3.5.2) and (3.5.3), it suffices to show thatmaxF, tind(w2(t)) = 2b(n−s)/2c. (3.5.4)Recall that elements of H1(F,Sn) are in a natural bijective correspondence with isomorphismclasses of n-dimensional e´tale algebras E/F . Denote the class of E/F by [E/F ] ∈ H1(F,Sn) andthe trace form of E/F by t. By [Ser84, The´ore`me 1],δ([E/F ]) = w2(t);cf. also [Ser08, Section 9.2]. Thus the largest value of the index of w2(t), as F ranges over all fieldextensions of k and E/F ranges over all n-dimensional e´tale F -algebras, is precisely the integerind(S˜n,Z/2Z) defined in Section 3.2.2. Let P˜n be a Sylow subgroup of S˜n. By Proposition 3.2.7,the center Z(P˜n) is cyclic. Thus by Lemma 3.2.5,ind(S˜n;Z/2Z) = ed(P˜n) = ed(P˜n; 2).333.5. Proof of Theorem 3.1.5On the other hand, ed(P˜n; 2) = ed(S˜n; 2) by Lemma 3.2.2 and ed(S˜n; 2) = 2b(n−s)/2c by Theo-rem 3.1.2(b). This completes the proof of (3.5.4) and thus of part (a) of Theorem 3.1.5.The proof of part (b) is similar. Once again, since t1 contains s〈1〉 as a subform, maxF, t1ind(w2(t1)) 6 maxF, q1ind(w2(q1)).On the other hand, maxF, q1ind(w2(q1)) 6 2b(n−s−1)/2c by Lemma 3.5.1(b). It thus remains to showthatmax{ind(w2(t1))} 6 2b(n−s−1)/2c. (3.5.5)Consider the diagram1 // Z/2Z // S˜n // Sn // 11 // Z/2Z // A˜n //?OOAn //?iOO1where S˜n can be either S˜+n or S˜−n . Since the rows are exact, the connecting morphisms fit into acommutative diagramH1(F,Sn)w2 // H2(F,Z/2Z)H1(F,An)∂F //i∗OOH2(F,Z/2Z)Once again, elements of H1(F,Sn) are in a natural bijective correspondence with n-dimensionaletale algebras E/F . The image of the vertical map i∗ : H1(F,An)→ H1(F,Sn) is readily seen toconsist of etale algebras E/F of discriminant 1. Consequently,max{ind(w2(t1))} = ind(A˜n,Z/2Z).Let H˜n be a Sylow subgroup of A˜n. By Proposition 3.2.7, the center Z(H˜n) is cyclic. Thus byLemma 3.2.5,ind(A˜n;Z/2Z) = ed(H˜n) = ed(H˜n; 2).On the other hand, ed(H˜n; 2) = ed(A˜n; 2) by Lemma 3.2.2 and ed(A˜n; 2) = 2b(n−s−1)/2c byTheorem 3.1.2(c). This completes the proof of (3.5.5).Remark 3.5.2. One can use the inequalities (3.5.2) and (3.5.3) to give an alternative proofof the upper bound ed(S˜n) 6 2b(n−s)/2c of Theorem 3.1.2(b). Similarly for the upper bounded(A˜n) 6 2b(n−s−1)/2c in the proof of Theorem 3.1.2(c). On the other hand, we do not knowhow to prove the lower bounds ed(S˜n) > 2b(n−s)/2c and ed(A˜n) > 2b(n−s−1)/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〈1〉 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 = 〈1, a, b, c〉 be a 4-dimensional non-singular quadratic form over F . Clearlyq contains s〈1〉 = 〈1〉 as a subform. On the other hand, q is not isomorphic to the trace form t343.6. Comparison of essential dimensions of S˜+n and S˜−nof 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’sconjecture, which asserts the existence of an isomorphismer : Ir(F )/Ir+1(F )→ Hr(F,Z/2Z)for any r > 0, with the property that er takes the r-fold Pfister form 〈1, a1〉 ⊗ · · · ⊗ 〈1, ar〉 tothe cup product (a1) ∪ (a2) ∪ · · · ∪ (ar); see [Pfi95, p. 33]. Milnor’s 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 + · · ·+ 2as be aneven positive integer, where a1 > · · · > 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 〈1〉2 and t ranges over all n-dimensional trace forms in Ir(F ). Theindex of a class α ∈ Hr(F,Z/2Z) is the greatest common divisor of the degrees [E : F ], whereE/F ranges over splitting fields for α with [E : F ] <∞.For r = 1, it is easy to see that (3.5.6) holds. In this case the Milnor map e1 : I1(F )/I2(F )→H1(F,Z/2Z) is the discriminant, the index of an element of α = H1(F,Z/2Z) = F ∗/(F ∗)2 is 1or 2, depending on whether α = 0 or not, and the question boils down to the existence of ann-dimensional e´tale 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˜+n and S˜−nWe believe that S˜+n and S˜−n 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 √−1. Then| edk(S˜+n )− edk(S˜−n )| 6 1.Proof. Let V be the spin representation of S˜+n , Sn → PGL(V ) be the associated projectiverepresentation of Sn, and Γ ⊂ GL(V ) be the preimage of Sn under the natural projectionpi : GL(V )→ PGL(V ). Note that Γ is a 1-dimensional group, and S˜±n are finite subgroups of Γ.By Lemma 3.2.4(a),edk(Γ) > edk(S˜n)− 1, (3.6.1)2If s is even, and q = r ⊕ s〈1〉, 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− s)-dimensional forms in Ir(F ). The same is true if s isodd: here r ranges over the (n− s)-dimensional forms in Ir(F ) such that r ⊕ 〈1〉 is in Ir(F ).353.7. Explanation of the entries in Table 3.1where S˜n denotes S˜+n or S˜−n . On the other hand, since Γ is generated by S˜n and Gm, and Gm iscentral in Γ, we see that S˜n is normal in Γ. The exact sequence1 // S˜n // Γpi // Gm // 1induces an exact sequenceH1(K, S˜n) // H1(K,Γ)pi // H1(K,Gm) = 1of Galois cohomology sets. Here K is an arbitrary field containing k, and H1(K,Gm) = 1 byHilbert’s Theorem 90. Thus the map H1(K, S˜n) → H1(K,Γ) is surjective for every K. ByLemma 3.2.3,edk(S˜n) > edk(Γ). (3.6.2)Combining the inequalities (3.6.1) and (3.6.2), we see that each of the integers edk(S˜+n ) andedk(S˜−n ) equals either edk(Γ) or edk(Γ) + 1. Hence, edk(S˜+n ) and edk(S˜−n ) differ by at most 1, asclaimed.Remark 3.6.2. The inequality | edk(S˜+n ) − edk(S˜−n )| 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:• ed(A4) = ed(A5) = 2, see [BR97, Theorem 6.7(b)],• ed(A6) = 3, see [Ser10, Proposition 3.6],• ed(A7) = 4, see [Dun10, Theorem 1],• ed(An+4) > ed(An) + 2 for any n > 4, see [BR97, Theorem 6.7(a)],• ed(An) 6 ed(Sn) 6 n− 3 for any n > 5; see Lemma 3.2.4 and [BR97, Theorem 6.5(c)].The values of ed(A˜n; 2) in the second row of Table 3.1 are given by Theorem 3.1.2(c).In the third row,• ed(A˜4) = 2 by Corollary 3.1.3(a).• To show that ed(A˜5) = 2, combine the inequalities ed(A˜5) > ed(A˜5; 2) > 2 of Theo-rem 3.1.2(c) and ed(A˜5) 6 2 of Theorem 3.1.2(a). Alternatively, see [Pro17, Lemma2.5].• ed(A˜6) = 4 by [Pro17, Proposition 2.7].• To show that ed(A˜7) = 4, note that ed(A˜7) > 4 because A˜7 contains A˜6 and ed(A˜7) 6 4because A˜7 has a faithful 4-dimensional representation; see [Pro17, Corollary 2.1.4].363.7. Explanation of the entries in Table 3.1• The values of ed(A˜8) = 8 and ed(A˜16) = 128 are taken from Corollary 3.1.3(a).• When 9 6 n 6 15 the range of values for ed(A˜n) is given by the inequalityed(A˜n) 6 ed(A˜n) + ed(An) 6 ed(A˜n) + n− 3;see Theorem 3.1.2(e).37Chapter 4Classifying spaces for e´tale 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 → 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 − Z)/Gis defined as a quasiprojective variety. The higher the codimension of Z in V , the better anapproximation (V − 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´tale 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 − Z)/Sn produced by this machine represents “e´tale algebras equipped with rgenerating global sections” 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´tale 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´tale algebra A of degree n on a k-variety Xcorresponds to a map φ : X → B(r;An). While the map φ 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 φ˜ : X → B(r;An)→ B(∞;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∗, the map E∗(φ˜) 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´tale algebra A over a ring R carries an involution σ and a tracemap Tr : A → 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 − 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 − 1 can be used to produce e´talealgebras 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´tale 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 −Alg 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 −Var 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(·) is a sheaf on the big Zariski site of all schemes, which is to say thatif W =⋃i∈I Ui is a cover of a scheme by Zariski open subschemes, thenX(W )→∏i∈IX(Ui)⇒∏(i,j)∈I2X(Ui ∩ Uj)is a coequalizer diagram.Remark 4.1.1. The scheme A1 = SpecZ[t] represents the functorX 7→ A1(X) = Γ(X,OX) = OX(X).Similarly, An represents the functorX 7→ (OX(X))n .This can be deduced from the case of affine X, where one hasHomZ−Alg(Z[t1, . . . , tn], R) = Rnas sets.394.2. E´tale algebras4.2 E´tale algebrasLet R be a ring and S an R-algebra. Then there is a morphism of rings µ : S ⊗R S → S sendinga⊗ b to ab. We obtain an exact sequence0→ ker(µ)→ S ⊗R S µ−→ S → 0 (4.2.1)Definition 4.2.1. Let R be a ring. An R-algebra S is called finite e´tale if S is finitely presented,flat as an R-module and S is projective S ⊗R S-module, where S ⊗R S acts through µ.As S is finitely generated and flat over R it is also a projective R-module. We say that thee´tale 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 ⊗R 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 “trivial” etale algebraRn with componentwise addition and multiplication.The following lemma states that all e´tale 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´tale algebra of degree n.2. There is a faithfully flat R-algebra T such that S ⊗R T ∼= 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´tale X-algebra or e´talealgebra over X if for every open affine subset U ⊂ X the A(U) is an e´tale 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´tale 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´tale X-algebra of degree n.2. There is an affine flat cover {Ui fi−→ X} such that f∗i A ∼= OnUi as OUi-algebras.Proof. This is immediate from Lemma 4.2.3.Definition 4.2.7. If A is an e´tale algebra over a ring R, then a subset Λ ⊂ A is said to generateA over R if no strict R-subalgebra of A contains Λ.If Λ = {a1, . . . , ar} ⊂ A is a finite subset, then the smallest subalgebra of A containing Λagrees with the image of the evaluation map R[x1, . . . , xr](a1,...,ar)→ A. Therefore, saying that Λgenerates A is equivalent to saying this map is surjective.404.2. E´tale algebrasProposition 4.2.8. Let Λ = {a1, . . . , ar} be a finite set of elements of A, an e´tale algebra overa ring R. The following are equivalent:1. Λ generates A as an R-algebra.2. There exists a set of elements {f1, . . . , fn} ⊂ R that generate the unit ideal and such that,for each i ∈ {1, . . . , n}, the image of Λ in Afi generates Afi as an Rfi-algebra.3. For each m ∈ MaxSpecR, the image of Λ in Am generates Am as an Rm-algebra.4. Let k(m) denote the residue field of the local ring Rm. For each m ∈ MaxSpecR, the imageof Λ in A⊗R k(m) generates A⊗R k(m) as a k(m)-algebra.Proof. In the case of a finite subset, Λ = {a1, . . . , ar}, the condition that Λ generates A isequivalent to the surjectivity of the evaluation map R[x1, . . . , xr]→ 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 “generation of an algebra” from the casewhere the base is affine to the case of a general scheme.Definition 4.2.9. Let A be an e´tale algebra over a scheme X. For Λ ⊂ Γ(X,A) we say that Λgenerates A if, for each open affine U ⊂ X the OX(U)-algebra A(U) is generated by restrictionof sections in Λ to U .4.2.1 Generation of trivial algebrasLet n ≥ 2 and r ≥ 1. Consider the trivial e´tale algebra OnX over a scheme X. A global section ofthis algebra is equivalent to a morphism X → An, and an r-tuple Λ of sections is a morphismX → (An)r. One might hope that the subfunctor F ⊆ (An)r of r-tuples of sections generatingOnX as an e´tale 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) ∈ {1, . . . , n}2 with i < j, let Zij ⊂ (An)rdenote the closed subscheme given by the intersection of the vanishing loci⋂rk=1 V (xki − 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 −⋃i<jZijProposition 4.2.11. Let n ≥ 2 and r ≥ 1. The open subscheme U(r;An) ⊂ (An)r representsthe functor sending a scheme X to r-tuples (a1, . . . , ar) of global sections of OnX that generate itas an OX -algebra.414.2. E´tale algebrasProof. Temporarily, let F denote the subfunctor of (An)r defined byF(X) = {Λ ⊆ (Γ(X,OnX)r | Λ generates OnX}.It follows from Proposition 4.2.8 and Definition 4.2.9 that F is actually a sheaf on the big Zariskisite.Both U(r;An) and F are subsheaves of the sheaf represented by (An)r, and therefore in orderto show they agree, it suffices to show U(r;An)(R) = F(R) when R is a local ring.Let R be a local ring. The set U(r;An)(R) consists of certain r-tuples (~a1, . . . ,~ar) of elementsof Rn. Letting aki denote the i-th element of ~ak, then the r-tuples are those with the propertythat for each i 6= j, there exists some k such that aki − akj ∈ R×. The proposition now followsfrom Lemma 4.2.12 below.Lemma 4.2.12. Let R be a local ring, with maximal ideal m. Let (~a1, . . . ,~ar) denote an r-tupleof elements in Rn, and let aki denote the i-th element of ~ak. The following are equivalent:• The set {~a1, . . . ,~ar} generates the (trivial) e´tale R-algebra Rn.• For each pair (i, j) satisfying 1 ≤ i < j ≤ n, there is some k ∈ {1, . . . , r} such that theelement aki − akj is a unit in R×.Proof. Suppose {~a1, . . . ,~ar} generates Rn as an algebra. That is, any n-tuple (r1, . . . , rn) ∈ Rnmay be expressed by evaluating a polynomial p ∈ R[X1, . . . , Xr] at (~a1, . . . ,~ar), i.e., for anyi ∈ {1, . . . , n}, we have p(a1i, . . . ari) = ri.In particular, for any pair of indices (i, j) with 1 ≤ i < j ≤ n, it is possible to find apolynomial p ∈ R[X1, . . . , Xr] such that p(a1i, a2i, . . . , ari) = 1 and p(a1j , a2j , . . . , arj) = 0. Weremark that reduction modulo m is a homomorphism of rings, so that the class of p(c1, . . . , cr)modulo m depends only on the classes of c1, . . . , cr modulo m.If ali − alj ∈ m for all l, then ali = alj mod m and we obtain1 = p(a1i, a2i, . . . , ari)− p(a1j , a2j , . . . , arj) = 0 mod ma contradiction, so there exists some l such that ali − alj is a unit in R.Conversely, suppose that for each pair i < j, we can find some k such that aki − akj is aunit. For any pair i 6= j, we can find a polynomial pi,j ∈ R[x1, . . . , xr] with the property thatpi,j(a1i, . . . , ari) = 1 and pi,j(a1j , . . . , arj) = 0 by takingpi,j = (aki − akj)−1(xk − akj)for instance.Let pi =∏j 6=i pi,j . The polynomial pi has the property thatpi(a1j , . . . , arj) ={1 if i = j0 otherwiseand from here it is immediate that evaluation at (~a1, . . . ,~ar) yields a surjection R[x1, . . . , xn]→Rn.424.3. Classifying spaces4.3 Classifying spacesFix n ≥ 2 and r ≥ 1. In this subsection we work over a fixed field k.We tacitly change base from SpecZ to Spec k, so that U(r;An) denotes the k-variety thatshould properly be written U(r;An)×SpecZ Spec k. The reason we make this change of base is touse standard results about quotient varieties.Notation 4.3.1. For a given k-variety X, a degree-n e´tale algebra A with r generating sectionsdenotes the data of a degree-n e´tale algebra A over X, and an r-tuple of sections (a1, . . . , ar) ∈Γ(X,A) that generate A. These data will be briefly denoted (A, a1, . . . , ar). An isomorphismψ : (A, a1, . . . , ar) → (A′, a′1, . . . , a′r) of such data consists of an isomorphism ψ : A → A′ ofe´tale algebras over X such that ψ(ai) = a′i for all i ∈ {1, . . . , r}. The isomorphism class of(A, a1, . . . , ar) will be denoted [A, a1, . . . , ar].Definition 4.3.2. For a given X, there is a set, rather than a proper class, of isomorphismclasses of degree-n e´tale algebras over X, and so there is a set of isomorphism classes of degree-ne´tale algebras with r generating sections. Since generation is a local condition by Proposition4.2.8, it follows that there is a functorF(r;An) : k-Var→ Set,F(r;An)(X) = {[A, a1, . . . , ar] | (A, a1, . . . , ar) is a degree-n e´tale algebra over Xand r generating sections}The purpose of this section is to produce a variety B(r;An) representing the functor F(r;An),on the category of k-varieties.4.3.1 Construction of B(r;An)Remark 4.3.3. Fix a field F . The automorphism group of the trivial e´tale F -algebra Fn maybe calculated as follows: The elementsei = (0, . . . , 0, 1, 0, . . . , 0) ∈ Fnare determined, up to reordering, by the conditions that e2i = ei, ei 6= 0, eiej = 0 for i 6= j and∑ni=1 ei = 1 in the e´tale algebra structure on Fn. Any automorphism of the e´tale F -algebra Fnpermutes the ei and is determined by this permutation, and from there it is immediate thatAutF -alg(Fn) is the symmetric group, Sn.There is an action of the symmetric group Sn on An, given by permuting the coordinates, andfrom there, there is a diagonal action of Sn on (An)r, and one verifies that the action restricts toone on the open subscheme U(r;An).Proposition 4.3.4. The action of Sn on U(r;An) is scheme-theoretically free.Proof. Since U(r;An) is a variety, it suffices to verify that the action is free on the sets U(r;An)(K)where K/k is a field extension. Here one is considering the diagonal Sn action on r-tuples(~a1, . . . ,~ar) where each ~al ∈ Kn is a vector and such that for all indices i 6= j, there exists some~al such that the i-th and j-th entries of ~al are different. The result follows.434.3. Classifying spacesNotation 4.3.5. There is a free diagonal action of Sn on U(r;An)×An, such that the projectionp : U(r;An)×An → U(r;An) is equivariant. Write q : E(r;An)→ B(r;An) for the induced mapof quotient varieties. These are again quasiprojective varieties (by, for instance [Mum08, Section7]) and there is a commutative squareU(r;An)× An E(r;An)U(r;An) B(r;An)ppi′qpiRemark 4.3.6. The sheaf of sections of the map p : U(r;An) × An → U(r;An) is the trivialdegree-n e´tale algebra OnU(r;An) on U(r;An). The action of Sn on these sections is by algebraautomorphisms, and so the sheaf of sections of the quotient map q : E(r;An) → B(r;An) isendowed with the structure of a degree-n e´tale algebra E(r;An) on B(r;An). We will oftenconfuse the scheme E(r;An) over B(r;An) with the e´tale algebra of sections E(r;An).The map p has r canonical sections {sj}rj=1 given as follows:sj(~x1, ~x2, . . . , ~xr) = ((~x1, ~x2, . . . , ~xr, ), ~xj).These sections are Sn-equivariant, and so descend to sections {ti : B(r;An)→ E(r;An)}ri=1of the map q.Remark 4.3.7. The quotient variety B(r;An) is smooth since U(r;An) is smooth and pi ise´tale—see [Mil80, Ch. I, Remark 2. 24]. Since pi is finite it is proper. The variety B(r;An) isquasiprojective but not projective. Indeed, if B(r;An)→ Spec(k) were proper then U(r;An)→Spec(k) would be proper too, but U(r;An) is an open subvariety of an affine variety.4.3.2 The functor represented by B(r,An)We now establish the canonical isomorphism of the functors B(r,An)(X) = F(r;An)(X).By Remark 4.3.6, F(r;An)(B(r;An)) has a canonical element [E(r;An), t1, . . . , tr].Lemma 4.3.8. If [A, s1, . . . , sr] ∈ F(r;An)(L) where L is a separably closed field over k, thenthere exists a unique morphism of schemes φ : Spec(L)→ B(r;An) such that[A, s1, . . . , sr] = [φ∗(E(r;An)), φ∗t1, . . . , φ∗tr]Proof. Since L is separably closed, there exists an L-isomorphism Aψ−→ Ln. Let {ψ(si)} ⊂ Lndenote the corresponding sections of Ln.We thus obtain a map φ˜ : Spec(L)→ U(r;An) defined by giving the L-point (ψ(s1), . . . , ψ(sr)).Post-composing this map with the projection U(r;An) → B(r;An), we obtain a morphismφ : Spec(L)→ B(r;An). It is a tautology that φ∗(E(r;An)) = A and φ∗(ti) = si.It now behooves us to show that φ does not depend on the choices made in the construction.Suppose φ′ : SpecL→ B(r;An) is another morphism satisfying the conditions of the lemma.We may lift this L-point of B(r;An) to an L-point φ˜′ : SpecL→ U(r;An), since pi is finite andL is separably closed field. By hypothesis we have[A, s1, . . . , sr] = [φ′∗(E(r;An)), φ′∗t1, . . . , φ′∗tr].Thus φ˜ and φ˜′ differ by a Sn = AutL(Ln) automorphism, which is to say φ = φ′ as required.444.3. Classifying spacesThe universality of B(r;An) extends to all k-varieties, as follows.Proposition 4.3.9. If X is a variety over k and if [A, s1, . . . , sr] ∈ F(r;An)(X) then there existsa unique morphism of k-schemes φ : X → B(r;An) such that[A, s1, . . . , sr] = [φ∗(E(r;An)), φ∗t1, . . . , φ∗tr]. (4.3.1)Proof. If [A, s1, . . . , sr] ∈ F(r;An)(X) then there exists an e´tale cover U = {Spec(Ri) → X}of reduced affine schemes such that there exist isomorphisms ψi : A|SpecRi∼=−→ Rni . WriteUi = SpecRi.Then we obtain a morphismUiφi=(s1|Ui ,...,sr|Ui )−−−−−−−−−−−−→ U(r;An).We post-compose with pi to get maps φi : Ui → B(r;An). The maps φi|Ui×Uj and φj |Ui×Ujagree on all geometric points by Lemma 4.3.8 and thus are equal. By e´tale descent the maps φidefine a map φ : X → B(r;An) satisfying (4.3.1).If φ′ : X → B(r;An) is a different map satisfying (4.3.1), then φ and φ must differ on somegeometric point. This is not possible by Lemma 4.3.8.Corollary 4.3.10. The functor F(r;An) of Definition 4.3.2 is represented by the schemeB(r;An).Example 4.3.11. Let us consider the toy example where X = SpecK where K is a fieldcontaining k, n ≥ 2, and where r = 1. That is, we are considering e´tale algebras A/K along witha chosen generating element a ∈ A. After base change to the separable closure, Ks, we obtain aSn-equivariant isomorphism of Ks-algebras:ψ : AKs∼=−→ (Ks)×n.For the sake of the exposition, use ψ to identify source and target. The element a ∈ A yields achosen generating element a˜ ∈ (Ks)n. The element a˜ is a vector of n pairwise distinct elementsof Ks. The element a˜ is a Ks-point of U(1;An). In general, this point is not defined over K, butits image in B(1;An) is.Since U(1;An) ⊆ An, and B(1;An) = U(1;An)/ Sn, the image of a˜ in B(1;An)(Ks) maybe presented as the elementary symmetric polynomials in the ai. To say that the image ofa˜ = (a1, . . . , an) in B(1;An) is defined over K is to say that the coefficients of the polynomial∏ni=1(x− ai) are defined in K.The variety B(1;An) is the k-variety parametrizing degree-n polynomials with distinct roots,i.e., with invertible discriminant.Example 4.3.12. To reduce the toy example even further, let us consider the case of k = K afield of characteristic different from 2, and n = 2.The variety B(1;A2) may be presented as spectrum of the C2-fixed subring of k[x, y, (x−y)−1]under the action interchanging x and y. This is k[(x + y), (x − y)2, (x − y)−2], although it ismore elegant to present it after the change of coordinates c1 = x+ y and c0 = xy:B(1;A2) = Spec k[c1, c0, (c21 − 4c0)−1]454.4. Stabilization in cohomologyA quadratic e´tale k-algebra equipped with the generating element a corresponds to the point(c1, c0) ∈ B(1;A2)(k) where a satisfies the minimal polynomial a2 − c1a+ c0 = 0.For instance if k = R, the quadratic e´tale algebra of complex numbers C with generators+ ti over R (here t 6= 0), corresponds to the point (2s, s2 + t2) ∈ B(1;A2)(R), whereas R× R,generated by (s+ t, s− t) over R (again t 6= 0) , corresponds to the point (2s, s2 − t2).4.4 Stabilization in cohomologyWe might wish to use the varieties B(r;An) to define cohomological invariants of e´tale algebras.The idea is the following: suppose given such an algebra A on a k-scheme X, and suppose onecan find generators (a1, . . . ar) for A. Then one has a classifying map φ : X → B(r;An), andone may apply a cohomology functor E∗, such as Chow groups or algebraic K-theory, to obtain“characteristic classes” for A-along-with-(a1, . . . , ar), in the form of φ∗ : E∗(B(r;An))→ E∗(X).The dependence on the specific generators chosen is a nuisance, and we see in this section thatthis dependence goes away provided we are prepared to pass to a limit “B(∞)” and assume thatthe theory E∗ is A1-invariant, in that E∗(X)→ E∗(X × A1) is an isomorphism.Definition 4.4.1. There are stabilization maps U(r;An)→ U(r+1;An) obtained by augmentingan r-tuple of n-tuples by the n-tuple (0, 0, . . . , 0). These stabilization maps are Sn-equivariantand therefore descend to maps B(r;An)→ B(r + 1;An).The stabilization maps defined above may be composed with one another, to yield mapsB(r;An)→ B(r′;An) for all r < r′. These maps will also be called stabilization maps.Proposition 4.4.2. Let X be a regular k-scheme. Suppose [A, a1, . . . , ar] ∈ F(r;An)(X) and[A′, a′1, . . . , a′r′ ] ∈ F(l;An)(X) have the property that A ∼= A′ as e´tale algebras. Let φ : X →B(r;An) and φ′ : X → B(r′;An) be the corresponding classifying morphism. For R = r + r′, thecomposite maps φ˜ : X → B(r;An) → B(R;An) and φ˜′ : X → B(r′;An) → B(R;An) given bystabilization are na¨ıvely A1-homotopic.An “elementary A1-homotopy” between maps φ, φ′ : X → B is a map Φ : X × A1 → Bspecializing to φ at 0 and φ′ at 1. Two maps φ, φ′ : X → B are “naively A1-homotopic” if theymay be joined by a finite sequence of elementary homotopies. Two naively homotopic mapsare identified in the A1-homotopy theory of schemes of [MV99], but they do not account for allidentifications in that theory. (Weak) equivalences in the homotopy category HA1(k) of smoothk-schemes are called A1-equivalences. In this category, one posits that X × Ar pr2−−→ X is anA1-equivalence.Proof. We may assume that A = A′. We may also assume, by padding, that r = r′.Write t for the parameter of A1. Let A[t] denote the pull-back of A along the projectionX × A1 → X.Consider the sections ((1− t)a1, . . . , (1− t)ar, ta′1, . . . , ta′r) of A[t]. Since either t or (1− t) isa unit at all local rings of points A1, by appeal to Proposition 4.2.8 and consideration of therestrictions to X×(A1−{0}) and X×(A1−{1}), we see that ((1−t)a1, . . . , (1−t)ar, ta′1, . . . , ta′r)furnish a set of generators for A[t]. At t = 0, they specialize to (a1, . . . , ar, 0, . . . , 0), viz., thegenerators specified by the stabilized map φ : X → B(r;An) → B(2r;An). At t = 1, theyspecialize to (0, . . . , 0, a′1, . . . , a′r), which is not precisely the list of generators specified byφ′ : X → B(r;An) → B(2r;An), but may be brought to this form by another elementary A1homotopy.464.5. The motivic cohomology of the spaces B(r;A2)Corollary 4.4.3. Let φ and φ′ be as in the previous proposition. If E∗ denotes any A1-invariantcohomology theory, then E∗(φ˜) = E∗(φ˜′).4.5 The motivic cohomology of the spaces B(r;A2)For this section, let k denote a fixed field of characteristic different from 2. The motiviccohomology of the spaces B(r;A2) has already been calculated in [DI07].4.5.1 Change of coordinatesLemma 4.5.1. There is an equivariant isomorphism U(r;A2) ∼= Ar \ {0} × Ar, where C2 actsas multiplication by −1 on first factor Ar \ {0} and trivially on the second factor Ar. Takingquotient by C2-action yields B(r;A2) ∼= (Ar \ {0})/C2 × Ar.Proof. By means of the change of coordinatesxi − yi = zi, xi + yi = wiwe see that U(r;A2) ∼= (Ar\{0})×Ar. Moreover, the action of C2 on U(r;A2) is given by zi 7→ −ziand wi 7→ wi. We therefore obtain an isomorphism B(r;A2) = U(r;A2)/C2 ∼= (Ar \{0})/C2×Ar.Write V (r;A2) for Ar \ {0}/C2. It is immediate that B(r;A2) ∼= V (r;A2)×Ar, and so there is asplit inclusion V (r;A2)→ B(r;A2) which is moreover an A1-equivalence.4.5.2 The deleted quadric presentationDefinition 4.5.2. Endow P2r−1 with the projective coordinates a1, . . . , ar, b1, . . . , br. Let Q2r−2denote the closed subvariety given by the vanishing of∑ri=1 aibi, and let DQ2r−1 denote theopen complement P2r−1 \Q2r−2.The main computation of [DI07] is a calculation of the modulo-2 motivic cohomology ofDQ2r−1, and of a family of related spaces DQ2r (which are complements of the quadrics∑i aibi + c2 in P2r). Denote the modulo-2 motivic cohomology of Spec k by M2. This is abigraded ring whose graded components Mn,i2 are concentrated in degrees 0 ≤ n ≤ i. There aretwo notable classes, ρ ∈ M1,12 , the reduction modulo 2 of −1 ∈ KM1 (k) = H1,1(Spec k,Z), andτ ∈M0,12 , corresponding to the identity (−1)2 = 1. If −1 is a square in k, then ρ = 0, but τ isalways a nonzero class.Proposition 4.5.3 (Dugger–Isaksen, [DI07] Theorem 4.9). There is an isomorphism of gradedringsH∗,∗(DQ2r−1;F2) ∼= M2[a, b](a2 − ρa− τb, br)where |a| = (1, 1) and |b| = (2, 1).Moreover, the inclusion DQ2r−1 → DQ2r+1 given by ar+1 = br+1 = 0 induces the mapH∗,∗(DQ2r+1;F2)→ H∗,∗(DQ2r−1;F2) sending a to a and b to b.This proposition subsumes two other notable calculations of invariants. In the first place, owingto the Beilinson–Lichtenbaum conjecture [Voe03], it subsumes the calculation of H∗e´t(DQ2r−1,F2).474.5. The motivic cohomology of the spaces B(r;A2)For instance, if k is algebraically closed, thenM2 = F2[τ ], and one deduces that H∗e´t(DQ2r−1,F2) ∼=F2[a, b]/(a2 − b, br) = F2[a]/(a2r).In the second, since H2n,n(·,F2) is identified with CHn(·) ⊗Z F2, the calculation of theproposition subsumes that of the Chow groups modulo 2. In fact, the extension problems thatprevented Dugger and Isaksen from calculating H∗,∗(DQ2r−1;Z) do not arise in this range, andby reference to the appendix of [DI07], which in turn refers to [KM90], one can calculate theintegral Chow rings. This is done in the first two paragraphs of the proof of [DI07, Theorem 4.9].Proposition 4.5.4. One may presentCH∗(DQ2r−1) =Z[b](2b, br), |b| = 1.As before, the map DQ2r−1 → DQ2r+1 given by adding 0s induces the map b 7→ b on Chow rings.Moreover CH∗(DQ2r−1)⊗Z F2 can be identified with the subring of H∗,∗(DQ2r−1;F2) generatedby b.The reason we have explained all this is that there is a composite mapDQ2r−1 → (Ar \ {0})/C2 → B(r;A2) (4.5.1)both of which are A1-equivalences, and so Propositions 4.5.3 and 4.5.4 amount to a calculationof the motivic and e´tale cohomologies and Chow rings of B(r;A2). Both maps in diagram (4.5.1)are compatible in the evident way with an increase in r, so that we may use the material of thissection to compute the stable invariants of B(r;A2) in the sense of Section 4.4.The A1-equivalence B(r;A2)→ (Ar \ {0})/C2 was constructed above in Lemma 4.5.1, so itremains to prove the following.Lemma 4.5.5. Let r ≥ 1. The variety DQ2r−1 is affine and has coordinate ringR =[k[x1, . . . , xr, y1, . . . , yr](1−∑ri=1 xiyi)]C2(4.5.2)where the C2 action on xi and yi is by xi 7→ −xi and yi 7→ −yi.Proof. The variety DQ2r−1 is a complement of a hypersurface in P2r−1, and is therefore affine.Let Q denote a1b1 + · · ·+ arbr. The coordinate ring of DQ2r−1 is the ring of degree-0 termsin the graded ring S = k[a1, . . . , ar, b1, . . . , br, Q−1], where |ai| = |bi| = 1 and |Q−1| = −2. Thisring is the subring of S generated by the terms aiajQ−1, aibjQ−1 and bibjQ−1.Consider the ringT =k[x1, . . . , xr, y1, . . . , yr](1−∑ri=1 xiyi) . (4.5.3)One may define a map of rings φ : S → T by sending ai 7→ xi and bi 7→ yi, sinceQ 7→ 1 under this assignment. Restricting to Γ(DQ2r−1,ODQ2r−1) ⊂ S, one obtains a mapΓ(DQ2r−1,ODQ2r−1)→ T for which the image is precisely the subring generated by terms xixj ,xiyj and yiyj , i.e., the fixed subring under the C2 action given by xi 7→ −xi and yi 7→ −yi.It remains to establish this map is injective. We show that the kernel of the map φ : S → Tcontains only one homogeneous element, 0, so that the restriction of this map to the subringof degree-0 terms in S is injective. The kernel of φ is the ideal (Q− 1). Since S is an integraldomain, degree considerations imply that no nonzero multiple of (Q− 1) is homogeneous.484.6. Relation to line bundles in the quadratic caseProposition 4.5.6. For all r, there is an A1-equivalence DQ2r−1 → (Ar \ {0})/C2.Proof. Let T be as in the proof of Lemma 4.5.5. It is well known that SpecT is an affine vectorbundle torsor over Ar \ {0}. In fact, for each j ∈ {1, . . . , r}, if we define Uj ∼= A1 \ {0} ×Ar−1 tobe the open subscheme of Ar \ {0} where the j-th coordinate is invertible, then we arrive at apull-back diagramAr−1 × Uj ∼= SpecT ×Ar\{0} Uj //SpecTUj // Ar \ {0}Since Uj inherits a free C2-action, it follows that in the quotient we obtain a vector bundle(Ar−1 × Uj)/C2 → Uj/C2, and so the map (SpecT )/C2 → (Ar \ {0})/C2 is an A1-equivalence,as claimed.As a consequence of Proposition 4.5.6 we observe that the affine variety DQ2r−1 is an affineapproximation of B(r;A2).4.6 Relation to line bundles in the quadratic caseWe continue to work over a field k, and to require that the characteristic of k be different from 2.In the case where n = 2, the structure group of the degree-2 e´tale algebra is C2, the cyclicgroup of order 2, which happens to be a subgroup of Gm. More explicitly, H1e´t(SpecR,C2) isan abelian group which is isomorphic to the isomorphism classes of quadratic e´tale algebras onSpecR. On the other hand due to the Kummer sequence and C2 ⊂ Gm we have0→ R∗/R∗2 → H1e´t(SpecR,C2)→ 2 Pic(R)→ 0which means that H1e´t(SpecR,C2) is also isomorphic to isomorphism class of 2-torsion line bundleL with a choice of trivialization φ : L ⊗ L ∼=−→ OR.This is the basis of the following construction.Construction 4.6.1. Let X be a scheme such that 2 is invertible in all residue fields, and let Abe a quadratic e´tale algebra on X. There is a trace map [Knu91, Section I.1]:Tr : A → Oand an involution σ : A → A given by σ = Tr− id. Define L to be the kernel of Tr : A → O.The sequence of sheaves on X0→ L → A→ O → 0 (4.6.1)is split short exact, where the splitting O → A is given on sections by x 7→ 12x.The construction of L from A gives an explicit instantiation of the map H1e´t(X,C2) →H1e´t(X,Gm) on isomorphism classes. We note that L must necessarily be a 2-torsion line bundle,in that L ⊗ L is trivial.It is partly possible to reverse the construction of L from A.494.7. The example of ChaseConstruction 4.6.2. Let X be as above, and let L be a line-bundle on X such that there isan isomorphism L ⊗ L → O. Let φ : L ⊗ L → O be a specific choice of isomorphism. From thedata (L, φ), we may produce an e´tale algebra A = O ⊕ L on which the multiplication is given,on sections, by (r, x) · (r′, x′) = (rr′ + φ(x⊗ x′), rx′ + r′x).Proposition 4.6.3. Let X be a scheme such that 2 is invertible in all residue fields of pointsof X. Let A a quadratic e´tale algebra on X. Let L be the associated line bundle to A, as inConstruction 4.6.1. Then A can be generated by r global sections as an e´tale algebra if and onlyif L can be generated by r global sections as a line bundle.Proof. Using the split exact sequence (4.6.1), we may write A = O⊕L. Write q : A → L for theprojection q(a) = a− 12 Tr(a).The questions of generation of A and of L may be reduced to stalks at points of X, byProposition 4.2.8 for the algebra and a similar result for the line bundle.We may therefore suppose (R,m) is a local ring in which 2 is a unit, and that A/R is aquadratic e´tale algebra. Since 2 is invertible, we may write A = R[z]/(z2 − a) for some elementa ∈ R, [Knu91, Lemma 4.1.1]. In this presentation, σ(z) = −z and Tr(az + b) = 2b. The kernelof the trace map is therefore Rz. The map q : A→ Rz is given by q(az + b) = az.An r-tuple ~a = (a1z + b1, . . . , arz + br) of elements of A generate it as an R-algebra if andonly if q(~a) = (a1z, . . . arz) do. This tuple generates A as an algebra if and only if at least one ofthe ai is not in m—if one ai ∈ R×, then the algebra generated contains z and therefore all of A,whereas if ai ∈ m for all i, they correspond to the zero element in the e´tale algebra at the closedpoint of SpecR and hence, by Proposition 4.2.8, cannot generate A.On the other hand, q(~a) = (a1z, . . . , arz), and this generates Rz = ker(Tr) if and only if atleast one of the ai is a unit.Remark 4.6.4. Let k be a field of characteristic different from 2. Let X be a k-variety. Ane´tale algebra of degree 2 generated by r global sections corresponds to a map X → B(r;A2).A line bundle generated by r global sections corresponds to a map X → Pr−1. In the light ofProposition 4.6.3, there must be a map of varieties B(r;A2)→ Pr−1. This map is given byB(r;A2)∼=→ (Ar \ {0})/C2 × Ar p1→ (Ar \ {0})/C2 → (Ar \ {0})/Gm∼=−→ Pr−1where the morphisms are, left to right, the isomorphism of Lemma 4.5.1, projection onto thesecond factor, and the map induced by the inclusion C2 ⊂ Gm.4.7 The example of ChaseThe following will be referred to as “the example of Chase”.Construction 4.7.1. LetS =R[z1, . . . , zr](∑ri=1 z2i − 1)which is equipped with a C2-action given by zi 7→ −zi. Let R = SC2 . The dimension of both Rand S is r − 1.The ring R carries a projective module of rank 1, i.e., a line bundle, that requires r globalsections in order to generate it. This example given in [Swa67, Theorem 4].504.7. The example of ChaseRemark 4.7.2. In fact, the line bundle in question is of order 2 in the Picard group, soProposition 4.6.3 applies and there is an associated quadratic e´tale algebra on SpecR = Y (r)requiring r generators. The algebra is, of course, dependent on a choice of trivialization of thesquare of the line bundle, but one may choose the trivialization so the e´tale algebra in questionis S itself as an R-algebra.Remark 4.7.3. This construction shows that the bound of First and Reichstein, [FR17], on thenumber of generators required by an e´tale algebra of degree 2 is tight. This was first observed,to the best of our knowledge, by M. Ojanguren in private communication.Even better, replacing S by S ×Rn−2 over R, one produces a degree-n e´tale algebra over Rrequiring r elements to generate, so the bound is tight in the case of e´tale algebras of arbitrarydegrees. We owe this observation to Zinovy Reichstein.The original method of proof that the line bundle in the example of Chase cannot be generatedby fewer than r global sections uses the Borsuk–Ulam theorem. Here we show that a variationon that proof follows naturally from our general theory of classifying objects. The Borsuk–Ulamtheorem is a theorem about the topology of RP r, so it can be no surprise that it is replaced hereby facts about the singular cohomology of RP r.4.7.1 The singular cohomology of the real points of B(r;A2)In addition to the general results about the motivic cohomology of B(r;A2), we can give acomplete description of the homotopy type of the real points B(r;A2)(R).If X is a nonsingular R-variety, then it is possible to produce a complex manifold from X byfirst extending scalars to C and then employing the usual Betti realization functor to produce amanifold X(C). Since X is defined over R, however, the resulting manifold is equipped with anaction of the Galois group Gal(C/R) ∼= C2. We write X(R) for the Galois-fixed points of X(C).Remark 4.7.4. The real realization functor X  X(R) preserves finite products, so that iff, g : X → Y are two maps of varieties and H : X × A1 → X ′ is an A1-homotopy betweenthem, then f(R), g(R) are homotopic maps of varieties, via the homotopy obtained by restrictingH(R) : X(R)× A1(R) = X(R)× R→ X ′(R) to the subspace X(R)× [0, 1].Using Lemma 4.5.1, present U(r;A2) as the variety of 2r-tuples (z1, . . . , zr, w1, . . . , wr) suchthat (z1, . . . , zr) 6= (0, . . . , 0). This variety carries an action by C2 sending zi 7→ −zi and fixing thewi. We know U(r;A2) and B(r;A2) are naively homotopy equivalent to Ar \{0} and Ar \{0}/C2respectively.Construction 4.7.5. We now consider an inclusion that is not, in general, an equivalence. LetP (r) = SpecS denote the subvariety of Ar \ {0} consisting of r-tuples (z1, . . . , zr) such that∑ri=1 z2i = 1. This is an (r − 1)-dimensional closed affine subscheme of Ar \ {0}, invariant underthe C2 action on Ar \ {0}. The quotient of P (r) by C2 is Y (r) = SpecR, and is equipped withan evident map Y (r)→ (Ar \ {0})/C2 → B(r;A2). Here S and R take on the same meanings asin Construction 4.7.1.Proposition 4.7.6. Let notation be as in Construction 4.7.5. The real manifold B(r;A2)(R)has the homotopy type ofB(r;A2)(R) ' RP r−1∐RP r−1.514.7. The example of ChaseThe closed inclusion Y (r)→ B(r;A2) includes Y (r)(R)→ B(r;A2)(R) as a deformation retractof one of the connected components.Proof. By Lemma 4.5.1 and Remark 4.7.4, the manifold B(r,A2)(R) is homotopy equivalent toAr{0}/C2(R). The manifold Ar{0}/C2(C) consists of equivalence classes of r-tuples of complexnumbers (z1, ..., zr), where the zi are not all 0, under the relation(z1, ..., zr) ∼ (−z1, ...,−zr).The real points of Ar{0}/C2 consist of Galois-invariant equivalence classes. There are twocomponents of this manifold: either the terms in (z1, ..., zr) are all real or they are all imaginary.In either case, the connected component is homeomorphic to the manifold RP r−1.We now consider the manifold Y (r)(R). This arises as the Galois-fixed points of Y (r)(C),which in turn is the quotient of P (r)(C) by a sign action. That is, P (r)(C) is the complexmanifold of r-tuples (z1, . . . , zr) satisfying∑ri=1 z2i = 1. Again, in an R-points, the zi are eitherall real or all purely imaginary. The condition∑ri=1 z2i = 1 is incompatible with purely imaginaryzi, so Y (r)(R) is the manifold of r-tuples of real numbers (z1, . . . , zr) satisfying∑ri=1 z2i = 1,taken up to sign. In short, Y r(R) = RP r−1.As for the inclusion Y (r)(R)→ B(r;A2)(R), it admits the following description, as can beseen by tracing through all the morphisms defined so far. Suppose given an equivalence class ofreal numbers (z1, . . . , zr), satisfying∑ri=1 z2i = 1, taken up to sign. Then embed (z1, . . . , zr) asthe point of B(r;A2)(R) given by the class of (z1, z2, . . . , zr, 0, . . . , 0). That is, embed RP r−1 inRr × (Rr−1 \ {0}) /C2 by embedding RP r−1 ⊂ (Rr \ {0})/C2 as a deformation retract, and thenembedding the latter space as the zero section of the trivial bundle. It is elementary that thiscomposite is also a deformation retract.Remark 4.7.7. We remark that the functor X  X(R) does not commute with colimits. Forinstance U(r;A2)(R)/C2, which is connected, is not the same as B(r;A2)(R).In fact, the two components of B(r;A2)(R) as calculated above correspond to two isomorphismclasses of quadratic e´tale R-algebras: one component corresonds to the split algebra R× R, andthe other to the nonsplit C.We will need two properties of H∗(RP r;F2) here. Both are standard and may be found in[Hat02].1. H∗(RP r;F2) ∼= F2[θ]/(θr+1) where |θ| = 1.2. The standard inclusion of RP r ↪→ RP r+1 given by augmenting by 0 induces the evidentreduction map θ 7→ θ on cohomology.Proposition 4.7.8. We continue to work over k = R. Let sr : B(r;A2)→ B(r + 1;A2) be thestabilization map of Definition 4.4.1. The induced map on cohomology groupss∗r : Hj(B(r + 1;A2)(R);F2)→ Hj(B(r;A2)(R);F2)is an isomorphism when j ≤ r and is 0 otherwiseProof. The map s∗r is arrived at by considering the inclusion U(r;A2)→ U(r + 1;A2), which isgiven by augmenting an r-tuple of pairs (a1, b1, . . . , ar, br) by (0, 0), and then taking the quotientby C2. After R-realization, one is left with a map B(r;A2)(R)→ B(r + 1;A2)(R) which on eachconnected component is homotopy equivalent to the standard inclusion RP r → RP r+1. Theresult follows.524.7. The example of ChaseProposition 4.7.9 (Ojanguren). Let S and R be as in Construction 4.7.1. The quadratic e´talealgebra S/R cannot be generated by fewer than r elements.Sketch of proof. Write Y (r) = SpecR as in Construction 4.7.5. The morphism Y (r)→ B(r;A2)of Construction 4.7.5 classifies a quadratic e´tale algebra over Y (r), and we can identify thisalgebra as S.The map φ : Y (r) → B(r;A2) induces stable maps φ˜ : Y (r) → B(R;A2). Any such stablemap induces a surjective mapφ˜∗ : H∗(B(R;A2)(R);F2)→ H∗(Y (r)(R);F2)by Proposition 4.7.6 and 4.7.8. In particular, it is a surjection when ∗ = r − 1.Suppose S can be generated by r − 1 elements, then there is a classifying map φ′ : Y (r)→B(r − 1;A2), from which one can produce a stable map(φ˜′)∗ : H∗(B(R;A2)(R) : F2)→ H∗(B(r − 1;A2);F2)→ H∗(Y (r)(R);F2).By reference to Corollary 4.4.3, for sufficiently large values of R, the maps φ˜∗ and (φ˜′)∗ agree.But (φ˜′)∗ induces the 0-map when ∗ = r − 1, since H∗(B(r − 1;A2)(R);F2) is a direct sum oftwo copies of F2[θ]/(θr−1). This contradicts the surjectivity of φ˜∗ in this degree.4.7.2 Algebras over fields containing a square root of −1Remark 4.7.10. When the field k contains a square root i of −1, the analogous construction tothat of Chase exhibits markedly different behaviour. For simplicity, suppose r is an even integer.Consider the ringS′ =k[z1, . . . , zr](∑ri=1 z2i − 1)with the action of C2 given by zi 7→ −zi. Let R′ = (S′)C2 . After making the change of variablesxj = z2j−1 + iz2j and yj = z2j−1 − iz2j , we see that S′ is isomorphic tok[x1, . . . , xr/2, y1, . . . , yr/2](∑r/2j=1 xjyj − 1)and R′ is isomorphic to the subring consisting of terms of even degree. The smallest R′-subalgebraof S′ containing the r/2-terms x1, . . . , xr/2 contains each of the yj because of the relationyj =r/2∑l=1xl(ylyj)so S′ may be generated over R′ by r/2 elements. In fact, R′ is the coordinate ring of DQr−1, byLemma 4.5.5. In Proposition 4.7.13 below, we show that S′ cannot be generated by fewer thanr/2 elements over R′.One may reasonably ask therefore, over a field k containing a square root of −1:Question 4.7.11. For a given dimension d, is there a smooth d-dimensional affine variety SpecRand an e´tale algebra A over SpecR such that A cannot be generated by fewer than d+1 elements?534.7. The example of ChaseThe result of [FR17] implies that if d+ 1 is increased, then the answer is negative.Remark 4.7.12. If d = 1, the answer to the question is positive. An example can be producedusing any smooth affine curve Y for which 2 Pic(Y ) 6= 0. Specifically, one may take a smoothelliptic curve and discard a point to produce such a Y . A nontrivial 2-torsion line bundle L onY cannot be generated by 1 section, since it is not trivial. One may also choose a trivializationφ : L⊗L → O, and therefore endow L⊕O with the structure of a quadratic e´tale algebra, as inConstruction 4.6.2, and this algebra also cannot be generated by 1 element.Proposition 4.7.13. Let k be a field containing a square root i of −1. Let T denote the ringT =k[x1, . . . , xr, y1, . . . , yr](∑ri=1 xiyi − 1)endowed with the C2 action given by xi 7→ −xi and yi 7→ −yi. Let R = TC2 . Then the quadratice´tale algebra T over R can be generated by the r elements x1, . . . , xr, but cannot be generatedby fewer than r elements.Proof. The ring R is the coordinate ring of the variety DQ2r−1 in Lemma 4.5.5. In particular,there is an A1-equivalence φ : DQ2r−1 → B(r;A2), as in equation (4.5.1). Tracing throughthis composite, one sees it classifies the quadratic e´tale algebra generated by x1, . . . , xr, i.e., Titself—the argument being as given for DQr−1 in Remark 4.7.10.Suppose for the sake of contradiction that T can be generated by r − 1 elements over R.Let φ′ : DQ2r−1 → B(r − 1;A2) be a classifying map for some such r − 1-tuple of generators.Let φ˜ and φ˜′ denote the composite maps DQ2r−1 → B(2r − 1;A2). By Corollary 4.4.3, thesemaps induce the same map on Chow groups. But in degree r − 1, the map φ˜∗ : CHr−1(B(2r −1;A2)) → CHr−1(B(r;A2)) → CH(DQ2r−1) is an isomorphism of cyclic groups of order 2, byreference to Proposition 4.5.4, while by the same proposition, (φ˜′)∗ : CHr−1(B(2r − 1;A2))→CHr−1(B(r − 1;A2))→ CH(DQ2r−1) is 0.The following shows that the bound of [FR17] is not quite sharp when applied to quadratice´tale algebras over smooth k¯-algebras where k¯ is an algebraically closed field.Proposition 4.7.14. Let k¯ be an algebraically closed field. Let n ≥ 2, and SpecR an n-dimensional smooth affine k¯-variety. If A is a quadratic e´tale algebra on SpecR, then A may begenerated by n global sections.Proof. Let L be a torsion line bundle on SpecR, or, equivalently, a rank-1 projective moduleon R. A result of Murthy’s, [Mur94, Corollary 3.16], implies that L may be generated by nelements if and only if c1(L)n = 0. By another result of Murthy’s, [Mur94, Theorem 2.14], thegroup CHn(R) is torsion free, so it follows that if L is a 2-torsion line bundle, then L can begenerated by n elements. The proposition follows by Proposition 5Conclusions and Future ResearchDirectionsIn this chapter we discuss the conclusions, lingering questions and future directions arising fromeach chapter. We also place our work in the context of contemporary research and discuss recentadvances.5.1 Essential dimension of inseparable field extensionsThe results of Chapter 2 lead to new questions on the nature of inseparable field extensions.In [BR97] it was shown that the essential dimension of a finite separable extension L/K isbounded from above by the essential dimension of N/K where N is the normal closure ofL/K. In particular N/K is a Galois extension with automorphism group G and we haveed(L/K) = ed(N/K) ≤ ed(G). So for the purposes of determing essential dimension of separableextensions we may only worry about the Galois ones. In that case, ed(G) is an upper bound andscope of techniques which can be applied increases.A purely inseparable extension L/K is called modular if L ∼= K(x1) ⊗ . . . ⊗K(xn) wherexpeii ∈ K. Modular extensions are the simplest kind of purely inseparable extensions. A purelyinseparable extension L/K can never be Galois since automorphisms of L which fix K is onlythe trivial one. In particular there cannot be any Galois correspondence between subfieldsof L/K and subgroups of Aut(L/K). However, at least in some cases, there exists a Galoiscorrespondence between subfields of L/K and Lie p-algebras of EndK(L) the L-algebra of K-linear endomorphisms of L. In addition, there is an analog of Galois closure for inseparableextensions called the modular closure, reducing to normal closure in the case the extension isseparable. For L/K a purely inseparable extension the modular closure M/K is the smallestfield extension of L such that M/K is modular. The modular closure of a finite inseparablefield extension was first defined by Sweedler in [Swe68], who also proved its existence. Thedefinition and construction may be found in [Swe68] or [Ras71]. The modular closure M/K of apurely inseparable extension L/K is unique and has the property that it is a torsor for some,not neccesarily unique, infinitesimal group scheme G. We may then ask if the modular closureM/K sufficiently captures the complexity of the corresponding inseparable extension.Question 5.1.1. For L/K a purely inseparable extension and M/K its modular closure is ittrue that ed(L/K) = ed(M/K)?.We have not been able to settle this question, but we do have the following partial result.Lemma 5.1.2. If L/K is purely inseparable extension we have ed(L/K) ≥ ed(M/K).555.2. Essential dimension of double covers of symmetric and alternating groupsAnother direction this work leads to is the problem of determining essential dimension ofcertain infinitesimal group schemes. Recently there has been lot of interest in essential dimensionof group schemes in positive characteristic (see [Tot19], [Bae17], [McK17]). For group schemes inchar p > 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) ≥ dimG. (5.1.1)There is also the general upper bound,ed(Gp) ≤ 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’s proof relies on demonstrating that the group PGL(1)2 is a semi-direct product ofα22 and µ2. 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˜n) and ed(A˜n) remains open. It wouldbe interesting to close this gap. Note that ed(S˜n)− ed2(S˜n) ≤ ed(Sn). Determination of ed(S˜n)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´tale algebra A of rank n over F we can attach a trace form x → Tr(x2) which isa quadratic form. This association may also be obtained as the induced map between Galoiscohomology sets by the permutation representation Sn → On. An important question in quadraticform theory to determine which quadratic forms are trace forms (see [BF94], [GMS03, ChapterIX]).565.3. Generators of e´tale 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’s conjecture (proven byVoevodsky), which asserts the map er : Ir(F )/Ir+1(F )→ Hr(F,Z/2Z), sending the r-fold Pfisterform 〈1, a1〉⊗· · ·⊗〈1, ar〉 to the cup product (a1)∪(a2)∪· · ·∪(ar), is an isomorphism. The Hasse-Witt invariant of a quadratic form is its image via the connecting map H1(F,On)→ H2(F,Z/2Z)through the “ Pin ” 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 α ∈ Hr(F,Z/2Z) as the greatest common divisorof the degrees [E : F ], where E/F ranges over splitting fields for α with [E : F ] <∞.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 → 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˜n; 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´tale algebrasIn Chapter 4 we solved the problem of determining the sharpness of First-Reichstein bound onthe minimum number of generators for an e´tale 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´tale 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´talealgebras 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]). 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