Essential dimension and classifying spaces of algebras

by

Abhishek Kumar Shukla

BS-MS, Indian Institute of Science Education and Research Pune, 2016

A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF

DOCTOR OF PHILOSOPHY

in

The Faculty of Graduate and Postdoctoral Studies

(Mathematics)

THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver) April 2020

© Abhishek Kumar Shukla 2020 The following individuals certify that they have read, and recommend to the Faculty of Graduate and Postdoctoral Studies for acceptance, the dissertation entitled:

Essential dimension and classifying spaces of algebras submitted by Abhishek Kumar Shukla in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Mathematics

Examining Committee:

Zinovy Reichstein, Professor, Department of Mathematics, UBC Supervisor

Kai Behrend, Professor, Department of Mathematics, UBC Supervisory Committee Member

Rachel Ollivier, Professor, Department of Mathematics, UBC Supervisory Committee Member

Ben Williams, Professor, Department of Mathematics, UBC University Examiner

Ian Blake, Honorary Professor, Department of Electrical and Computer Engineering, UBC University Examiner

ii Abstract

The overarching theme of this thesis is to assign, and sometimes find, numerical values which reflect complexity of algebraic objects. The main objects of interest are field extensions of finite degree, and more generally, ´etalealgebras of finite degree over a ring. Of particular interest to us is the invariant known as essential dimension. The essential dimension of separable field extensions was introduced by J. Buhler and Z. Reichstein in their landmark paper [BR97]. A major (still) open problem arising from that work is to determine the essential dimension of a general separable field extension of degree n (or equivalently, the essential dimension of the symmetric group). Loosening the separability assumption we arrive at the case of inseparable field extensions. In the first part of this thesis we study the problem of determining the essential dimension of inseparable field extensions. In the second part of this thesis, we study the essential dimension of the double covers of symmetric groups and alternating groups, respectively. These groups were first studied by I. Schur and their representations are closely related to projective representations of symmetric and alternating groups. In the third part, we study the problem of determining the minimum number of generators of an ´etalealgebra over a ring. The minimum of number of generators of an ´etalealgebra is a natural measure of its complexity.

iii Lay Summary

In the second half of the 19th century the German mathematician Felix Klein proposed a way to study geometric shapes by looking at their symmetries, i.e., by considering transformations of the objects that leave them invariant. For example, if we look at a square in the plane, we note that it is invariant under a 90 degree rotation; moreover, we can consider all possible transformations with that property and study this collection of transformations instead of the square. Objects with 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, while polynomial equations in one variable are difficult to solve explicitly, one gains a great deal of information 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 algebraic structure? We focus on two measures of complexity: the minimal number of generators and the essential dimension. Symmetry groups play a key role in both cases.

iv Preface

This 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. The research 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 of symmetric and alternating groups, arXiv e-prints (2019), arXiv:1906.03698. The research and manuscript preparation was done in equal parts by myself and Zinovy Reichstein. A version of Chapter 4 has been published: Shukla, A., Williams, B. (2020). Clas- sifying spaces for ´etalealgebras 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 research and manuscript preparation was done in equal parts by myself and Ben Williams.

v Table of Contents

Abstract ...... iii

Lay Summary ...... iv

Preface ...... v

Table of Contents ...... vi

List of Tables ...... viii

List of Figures ...... ix

Acknowledgements ...... x

1 Introduction ...... 1 1.1 Essential dimension ...... 1 1.2 Double covers of alternating groups ...... 2 1.3 Generators of an ´etalealgebra ...... 3

2 Essential dimension of inseparable field extensions ...... 5 2.1 Introduction ...... 5 2.2 Finite-dimensional algebras and their automorphisms ...... 7 2.3 Essential dimension of a functor ...... 9 2.4 Field extensions of type (n, e)...... 10 2.5 Proof of the upper bound of Theorem 2.1.2 ...... 12 2.6 Versal algebras ...... 14 2.7 Conclusion of the proof of Theorem 2.1.2 ...... 16 2.8 Alternative proofs of Theorem 2.1.2 ...... 19 2.9 The case where e1 = ··· = er ...... 19

3 Essential dimension of double covers of symmetric and alternating group . 21 3.1 Introduction ...... 21 3.2 Preliminaries ...... 25 3.2.1 Essential dimension ...... 25 3.2.2 The index of a central extension ...... 26 3.2.3 Sylow 2-subgroups of Ae n ...... 27 3.3 Proof of Theorem 3.1.2 ...... 28 3.4 Proof of Theorem 3.1.4 ...... 31 3.5 Proof of Theorem 3.1.5 ...... 32

vi Table of Contents

+ − 3.6 Comparison of essential dimensions of eSn and eSn ...... 35 3.7 Explanation of the entries in Table 3.1 ...... 36

4 Classifying spaces for ´etalealgebra with generators ...... 38 4.1 Introduction ...... 38 4.1.1 Notation and other preliminaries ...... 39 4.2 Etale´ algebras ...... 40 4.2.1 Generation of trivial algebras ...... 41 4.3 Classifying spaces ...... 43 n 4.3.1 Construction of B(r; A )...... 43 n 4.3.2 The functor represented by B(r, A )...... 44 4.4 Stabilization in cohomology ...... 46 2 4.5 The motivic cohomology of the spaces B(r; A )...... 47 4.5.1 Change of coordinates ...... 47 4.5.2 The deleted quadric presentation ...... 47 4.6 Relation to line bundles in the quadratic case ...... 49 4.7 The example of Chase ...... 50 2 4.7.1 The singular cohomology of the real points of B(r; A )...... 51 4.7.2 Algebras over fields containing a square root of −1 ...... 53

5 Conclusions and Future Research Directions ...... 55 5.1 Essential dimension of inseparable field extensions ...... 55 5.2 Essential dimension of double covers of symmetric and alternating groups . . . . 56 5.3 Generators of ´etalealgebras ...... 57

Bibliography ...... 58

vii List of Tables

3.1 Essential dimension of Ae n ...... 24

viii List of Figures

2.1 Descent diagram for field extension ...... 5

3.1 Projective representation ...... 21 3.2 Schematic of spin groups ...... 22

ix Acknowledgements

I would like to express my sincere gratitude towards my advisor Zinovy Reichstein for his mentorship and direction throughout the course of my Ph.D. I am also grateful to Ben Williams for his mentorship and encouragement. The content of Chapter 2, coauthored with Zinovy Reichstein, benefitted from discussions with 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 discussions with Eva Bayer-Fl¨uckiger, Vladimir Chernousov, Alexander Merkurjev, Jean-Pierre Serre, Burt Totaro, Alexander Vishik, and Angelo Vistoli, to whom I am grateful. The content of Chapter 4, coauthored with Ben Williams, benefitted from discussions with Zinovy 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 to thank 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.

x Chapter 1

Introduction

A primary objective of this thesis is to study the essential dimension of some naturally arising functors in mathematics. Another objective is to define and study the classifying space for ´etale algebras with generators.

1.1 Essential dimension

Essential 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 field extensions. Roughly speaking, the problem is to determine the minimal number of parameters needed to define a general inseparable extension of fixed degree. To make this problem precise let k be a base field and assume all other fields contain k. A field extension L/K of finite degree is said to descend to a subfield K0 ⊂ K if there exists a subfield K0 ⊂ L0 ⊂ L such that L0 and

K generate L and [L0 : K0] = [L : K]. Equivalently, L is isomorphic to L0 ⊗K0 K over K, as is shown in the following diagram. L

L0 K

K0

We define edk(L/K) := min{trdegk K0 | L/K descends to L0/K0}. How many parameters are needed to define a general separable field extension L/K of degree n? To formalize this notion we define

τ(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 a √ √ single 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/K arises by solving a polynomial of the form X3 − aX + a and hence τ(3) ≤ 1 too. jnk It is shown in [BR97] that if char(k) = 0, then τ(n) n − 3 for every n 5. 2 6 6 > Now suppose L/K is an arbitrary inseparable (but not necessarily purely inseparable) field extension L/K of finite degree. Denote the separable closure of K in L by S. We will say that L/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 integers e = (e1, . . . , er) associated to it in a natural fashion with the property that e1 ≥ e2 ≥ ... ≥ er. pm Every element in L (over S) satisfies a polynomial equation Z − s ∈ S[Z]. Then e1 is the

1 1.2. Double covers of alternating groups

largest exponent m of p occuring among all elements of L. Call such an element l1. Then e2 is defined in a similar fashion for the extension L/S[l1]. We stop when L = S[l1, l2, . . . , lr] and S[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 of pn p2 p p p Fp(x) over Fp(x ) is (n), Fp(x, y) over Fp(x , y ) is (2, 1) and type of Fp(x , y , z, xz + y) over p2 p2 p2 Fp(x , y , z ) is (2, 2, 1). By analogy with (1.1.1) it is natural to define τ(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 defined a 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. Then r X τ(n, e) = n psi−iei . i=1 The proof combines methods from field theory along with techniques from the theory of essential dimension. We interpret inseparable extensions L/K of type (n, e) as forms of a truncated polynomial algebra Λn,e, which are then classified by torsors over Spec(K) for a certain group scheme Gn,e. Then τ(n, e) is the essential dimension of Gn,e, much as τ(n) is the essential dimension of Sn in the separable case (see next paragraph for definition of essential dimension of a 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 dimension of Sn.

1.2 Double covers of alternating groups

Chapter 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/k to the category of sets. Here K ranges over all fields containing k. We say that an object a ∈ F(K) descends to a subfield K0 ⊂ K if a lies in the image of the natural restriction map F(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 all a ∈ F (K), and all fields K in Fieldsk. There is a related notion of essential dimension at l where l is a prime, denoted by ed( ; l). The example of foremost interest is when G is a group scheme over a base field k and 1 FG : K → H (K,G) be the functor defined by

FG(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 first Galois cohomology set. The essential dimension ed(G) is, by definition, ed(FG). These numerical invariants of G have been extensively studied.

2 1.3. Generators of an ´etalealgebra

Specializing to the case G = Sn, the set FSn (K) can be identified with ´etalealgebras A/K of degree 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 of Spin groups, which are double covers of SOn, grows exponentially with n. I was curious to see if this phenomenon has an analog over finite groups. In joint work with Zinovy Reichstein we found finite group analog of the Spin behaviour. The group is Ae n which is the double cover of the alternating groups An. These groups fit in a schematic as below where the top sequence is obtained as a pullback via the permutation representation An → SOn.

1 / µ2 / Ae n / An / 1

  1 / µ2 / Spinn / SOn / 1 Theorem 1.2.1 ([RS19b]). Assume that the base field k is of characteristic 6= 2 and contains a a a primitive 8th root of unity, and let n > 4 be an integer. Write n = 2 1 + ... + 2 s , where a1 > a2 > . . . > as > 0 . Then

b(n−s−1)/2c (a) ed(Ae n; 2) = 2 .

b(n−s−1)/2c b(n−s−1)/2c (b)2 6 ed(Ae n) 6 ed(An) + 2 . In the case char k = 2 the Spin groups continue to show exponential growth (see [Tot19]) but we found that ed(Ae n) grows sub-linearly.

1.3 Generators of an ´etalealgebra

In Chapter 4 we consider the problem of minimum number of generators of an ´etalealgebra. O. Forster [For64] proved that over a noetherian ring R, a finite module M of rank at most n can be generated by n + dim R elements. Generalizing this in great measure U. First and Z. Reichstein [FR17] showed that any finite R-algebra A can be generated by n + dim R elements if each A ⊗R k(m), for m ∈ MaxSpec(R), is generated by n elements. In particular, any finite ´etalealgebra A over R can be generated by 1 + dim R elements as an R-algebra. One may ask whether the upper bounds can be universally improved. Is it possible to get a 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 over these rings which cannot be generated by less than n + dim R elements. Moreover, the dimension of the rings goes to infinity. His techniques are topological in nature and the dictionary between topological 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 in the case of ´etalealgebras as well.

Theorem 1.3.1 ([SW19]). Let k = R and d any positive integer. There exist examples of finitely generated k-algebras Rn with dim(Rn) = n → ∞ and ´etalealgebras An over Rn of degree d such that An cannot be generated by fewer than n + 1 elements as an Rn-algebra.

3 1.3. Generators of an ´etalealgebra

The proof has two main ideas. First is to construct classifying spaces for ´etalealgebras with n generators. More precisely we construct varieties B(r; A ) such that “an ´etalealgebra A of degree n equipped wtih r generating sections” on X is obtained as pullback of tautological ´etale n n algebra on B(r; A ) via a unique map X → B(r; A ). We refer to such a map as a classifying morphism. An important example of classifying spaces with generators is the projective n-space n P which classifies projective modules of rank 1 with generators. A projective module P of rank 1 on a ring R can be described as a line bundle L = P˜ on X = Spec R and elements of P as n global sections Γ(X, L) of the line bundle L. Then P is the classifying space (in the above sense) n as there is a natural 1 − 1 correspondence between maps X → P and line bundles L on X with n + 1 generating sections (s0, s1, . . . , sn) ∈ Γ(X, L). The second main idea is to produce cohomological invariants obstructing the existence of a classifying map. R. Swan, in his proof in the case of modules, used topological obstructions such as Stiefel-Whitney classes of line bundles on real manifolds. In our case the obstruction is given by Chow ring of varieties. (An alternative proof may be given by using singular cohomology as obstruction in the spirit of Swan.) Chapters 2, 3 and 4 are self-contained; each may be read independently.

4 Chapter 2

Essential dimension of inseparable field extensions

2.1 Introduction

Throughout 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 a subfield K0 ⊂ L0 ⊂ L such that L0 and K generate L and [L0 : K0] = [L : K]. Equivalently, L is isomorphic to L0 ⊗K0 K over K, as is shown in the following diagram.

L

L0 K

K0

Figure 2.1: Descent diagram for field extension

The essential dimension of L/K (over k) is defined as

ed(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 interest is

τ(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 if n char(k) = 0, then b c τ(n) n − 3 for every n 5. 1 A. Duncan [Dun10] later strengthened 2 6 6 > the lower bound as follows. n + 1 Theorem 2.1.1. If char(k) = 0, then b c τ(n) n − 3 for every n 6. 2 6 6 > This chapter is a sequel to [BR97]. Here we will assume that char(k) = p > 0 and study inseparable field extensions L/K. The role of the degree, n = [L : K] in the separable case will be played by a pair (n, e). The first component of this pair is the separable degree, n = [S : K], where

1These inequalities hold for any base field k of characteristic 6= 2. On the other hand, the stronger lower bound of Theorem 2.1.1, due to Duncan, is known only in characteristic 0.

5 2.1. Introduction

S 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; see Section 2.4 for the definition. Note that the type e = (e1, . . . , er) uniquely determines the inseparable degree [L : S] = pe1+···+er of L/K but not conversely. By analogy with (2.1.1) it is natural to define

τ(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 > 1 be integers, e = (e1, . . . , er) and si = e1 + ··· + ei for i = 1, . . . , r. Then

r X τ(n, e) = n psi−iei . i=1 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, 1 ed(L/K) [L : K]; 6 p see Remark 2.5.3. This is again in contrast to the separable case, where Theorem 2.1.1 tells us 1 that there exists an extension L/K of degree n such that ed(L/K) > [L : K] for every odd 2 n > 7 (assuming char(k) = 0). (4) We will also show that the formula for τ(n, e) remains valid if we replace essential dimension ed(L/K) in the definition (2.1.2) by essential dimension at p, edp(L/K); see Theorem 2.7.1. For the 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 essential dimension of the functor Etn which associates to a field K the set of isomorphism classes of ´etale algebras of degree n over K. On the other hand, τ(n) is the essential dimension of the symmetric group Sn. Recall that an ´etalealgebra L/K is a direct product L = L1 × · · · × Lm of separable field extensions Li/K. Equivalently, an ´etalealgebra of degree n over K can be thought of as a n twisted K-form of the split algebra k = k × · · · × k (n times). The symmetric group Sn arises 1 as the automorphism group of this split algebra, so that Etn = H (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 role n of the split ´etalealgebra k will be played by the algebra Λn,e, which is the direct product of n copies of the truncated polynomial algebra

e e p1 p r Λe = k[x1, . . . , xr]/(x1 , . . . , xr ).

Note that the k-algebra Λn,e is finite-dimensional, associative and commutative, but not semisim- ple. Etale´ algebras over K will get replaced by K-forms of Λn,e. The role of the symmetric

6 2.2. Finite-dimensional algebras and their automorphisms

group Sn will be played by the algebraic group scheme Gn,e = Autk(Λn,e) over k. We will show that τ(n, e) is the essential dimension of Gn,e, just like τ(n) is the essential dimension of Sn in the separable case. 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 dimension of Sn. The remainder of this chapter is structured as follows. Sections 2.2 and 2.3 contain preliminary results on finite-dimensional algebras, their automorphism groups and essential dimension. In Section 2.4 we recall the structure theory of inseparable field extensions. Section 2.6 is devoted to versal algebras. The upper bound of Theorem 2.1.2 is proved in Section 2.5; alternative proofs are outlined in Section 2.8. The lower bound of Theorem 2.1.2 is established in Section 2.7; our proof relies on the inequality (2.7.2) due to D. Tossici and A. Vistoli [TV13]. Finally, in Section 2.9 we prove a stronger version of Theorem 2.1.2 in the special case, where n = 1, e1 = ··· = er, and k is perfect.

2.2 Finite-dimensional algebras and their automorphisms

Recall that in the introduction we defined the essential dimension of a field extension L/K of finite degree, where K contains k. The same definition is valid for any finite-dimensional algebra A/K. That is, we say that A descends to a subfield K0 if there exists a K0-algebra A0 such that A0 ⊗K0 K is isomorphic to A (as a K-algebra). The essential dimension ed(A) is then the minimal value of trdeg(K0/k), where the minimum is taken over the intermediate fields k ⊂ K0 ⊂ K such that A descends to K0. Here by a K-algebra A we mean a K-vector space with a bilinear multiplication map m: A×A → A. Later on we will primarily be interested in commutative associative algebras with 1, but at this stage m can be arbitrary: we will not assume that A is commutative, associative or 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 set 3 h of n structure constants cij ∈ K, where

n X h xi · xj = cijxh . (2.2.1) h=1

h Lemma 2.2.1. Let A be an n-dimensional K-algebra with structure constants cij (relative to h some K-basis of A). Suppose a subfield K0 ⊂ K contains cij for every i, j, h = 1, . . . , n. Then A descends 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 on

A0 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 extension of degree n. Let A be a finite-dimensional algebra over S. If A descends to a subfield S0 of S such that K(S0) = S, then ed(A/K) 6 n trdeg(S0/k) . Here ed(A/K) is the essential dimension of A, viewed as a K-algebra.

7 2.2. Finite-dimensional algebras and their automorphisms

Proof. 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 by norm norm G = Gal(S /K), H = Gal(S /S) ⊂ G. Now define S1 = k(g(s) | s ∈ S0, g ∈ G). Choose a transcendence basis t1, . . . , td for S0 over k, where d = trdeg(S0/k). Clearly S1 is algebraic over k(g(ti) | g ∈ G, i = 1, . . . , d). Since H fixes every element of S, each ti has at most [G : H] = n distinct translates of the form g(ti), g ∈ G. This shows that trdeg(S1/k) 6 nd. G Now let K2 = S1 ⊂ K,S2 = S0(K2) and A2 = A0 ⊗S0 S2. Since S2 is algebraic over K2, we have trdeg(K2/k) = trdeg(S2/k) 6 nd. Examining the diagram A0 A2 A

S0 S2 S

K2 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. The ∗ ∗ general 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 this action. 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 Remark after Lemma III.1.1 in [Ser02a].

Proposition 2.2.3. Let Λ be a commutative finite-dimensional local k-algebra with residue field k, and G = Autk(Λ) be its automorphism group scheme. Then the natural map

n n f : G o Sn → Autk(Λ )

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 is called an idempotent if α2 = α.

Lemma 2.2.4. Let Λ be a commutative finite-dimensional local k-algebra with residue field k and R be an arbitrary commutative k-algebra with 1. Then the only idempotents of ΛR = Λ ⊗k R are 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 homomorphism ΛR → R whose kernel again consists of nilpotent elements. By Proposition 7.14 in [Jac89], every idempotent in R lifts to a unique idempotent in ΛR, and the lemma follows.

8 2.3. Essential dimension of a functor

th Proof of Proposition 2.2.3. Let αi = (0,..., 1,..., 0) where 1 appears in the i position. Then n n ⊕i=1 Rαi is an R-subalgebra of ΛR. n n Let f ∈ AutR(ΛR). Since each αi is an idempotent in ΛR, so is each f(αi). The components n of each f(αi) are idempotents in ΛR. By Lemma 2.2.4, they lie in R. Thus, f(αi) ∈ ⊕i=1 Rαi for every i = 1, . . . , n. As a result, we obtain a morphism

n τR n AutR(ΛR) −−→ AutR(⊕i=1 Rαi) = Sn(R).

(For the second equality, see, e.g., p. 59 in [Wat79].) These maps are functorial in R and thus n n give rise to a morphism τ : Aut(Λ ) → Sn of group schemes over k. The kernel of τ is Aut(Λ) , 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, as long as we assume that the center of Λ is a finite-dimensional local k-algebra with residue field k. The proof proceeds along similar lines, except that we restrict f to an automorphism of the center Z(Λn) = Z(Λ)n and apply Lemma 2.2.4 to Z(Λ), rather than Λ itself. This more general variant 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 be n dropped. For example, if Λ is a separable field extension of k of degree d, then Autk(Λ ) is a twisted form of n dn Autk(Λ ⊗k k) = Autk(k ) = Snd . n n Here k denotes the separable closure of k. Similarly, Autk(Λ) oSd is a twisted form of (Sd) oSn. For d, n > 1, these groups have different orders, so they cannot be isomorphic.

2.3 Essential dimension of a functor

In 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 extensions K/k to the category of sets. Here k is assumed to be fixed throughout, and K ranges over all fields containing k. We say that an object a ∈ F(K) descends to a subfield K0 ⊂ K if a lies in the image of the natural restriction map F(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 all a ∈ F (K), and all fields K in Fieldsk. If l is a prime, there is also a related notion of essential dimension at l, which we denote by 0 0 edl. For an object a ∈ F, we define edl(a) as the minimal value of ed(a ), where a is the image of a in F(K0), and the minimum is taken over all field extensions K0/K such that the degree 0 [K : K] is finite and prime to l. The essential dimension edl(F) of the functor F at l is defined as the supremum of edl(a) for all a ∈ F (K) and all fields K in Fieldsk. Note that the prime l in this definition is unrelated to p = char(k); we allow both l = p and l 6= p.

1 Example 2.3.1. Let G be a group scheme over a base field k and FG : K → H (K,G) be the functor defined by

FG(K) = {isomorphism classes of G-torsors T → Spec(K)}.

9 2.4. 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) is the 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. These numerical 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 by

Algn(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]) 3 coincides with ed(A) defined at the beginning of Section 2.2. By Lemma 2.2.1, ed(Algn) 6 n ; 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 (not necessarily finite or separable). We say that an algebra A/K is a K-form of Λ if there exists a field L containing K such that Λ ⊗k L is isomorphic to A ⊗K L as an L-algebra. We will write

AlgΛ : Fieldsk → Sets for 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 its 1 automorphism group scheme. Then the functors AlgΛ and FG = H (∗,G) are isomorphic. In particular, 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.2.2.2 in [Knu91]. The second assertion is an immediate consequence of the first, since isomorphic functors have the same essential dimension.

Example 2.3.5. The K-forms of Λn = k × · · · × k (n times) are called ´etale algebras of degree n. An ´etalealgebra L/K of degree n is a direct products of separable field extensions, r X L = L1 × · · · × Lr, where [Li : K] = n. i=1

The functor AlgΛn is usually denoted by Etn. The automorphism group Autk(Λn) is the symmetric group Sn, acting on Λn by permuting the n factors of k; see Proposition 2.2.3. Thus 1 Etn = H (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 exponent of 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 field S is clear from context we will omit S in notation e(x, S). A sequence x1, . . . , xr in L is called normal 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 a normal generating sequence of L/S. We will say that this sequence is of type e = (e1, . . . , er) if ei := e(xi,Li−1) for each i. Here Li = S(x1, . . . , xi), as above. It is clear that e1 > e2 > ... > er.

10 2.4. Field extensions of type (n, e)

Proposition 2.4.1. (G. Pickert [Pic49]) Let L/S be a purely inseparable field extension of finite degree.

(a) For any generating set λ of L/S there exists a normal generating sequence x1, . . . , xr with each 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 and Corollary 1.5 in [Kar89].

Proposition 2.4.1 allows us to talk about the type of a purely inseparable extension L/S. We say that L/S is of type e = (e1, . . . , er) if it admits a normal generating sequence x1, . . . , xr of type e. Now suppose L/K is an arbitrary inseparable (but not necessarily purely inseparable) field extension L/K of finite degree. Denote the separable closure of K in L by S. We will say that L/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 other words, 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 taken over a non-empty set of integers.

C Proof. (a) Let x1, . . . , xn be independent variables over k. Set E = k(x1, . . . , xn) and F = E , where C is the cyclic group of order n acting on E by permuting the variables. Clearly E/F is a Galois (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 = pei E(y1, . . . , yr) and K = F (z1, . . . , zr), where zi = yi . 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. The following finite-dimensional commutative k-algebras will play an important role in the sequel:

pe1 per Λn,e = Λe × · · · × Λe (n times), where Λe = k[x1, . . . , xr]/(x1 , . . . , xr ) (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.

11 2.5. Proof of the upper bound of Theorem 2.1.2

Proof. One direction is obvious: if m = n and e = f, then Λn,e is isomorphic to Λm,f To prove the converse, note that Λe is a finite-dimensional local k-algebra with residue field k. By Lemma 2.2.4, the only idempotents in Λe are 0 and 1. This readily implies that the only idempotents in Λn,e are of the form (1, . . . , n), where each i is 0 or 1, and the only minimal idempotents are α1 = (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 cannot be 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 by

β1 = (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 a k-algebra isomorphism between α1Λn,e ' Λe and βjΛm,f ' Λf . To complete the proof, we appeal to 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). 0 0 (b) L is a K-form of Λn,e. In other words, L ⊗K K is isomorphic to Λn,e ⊗k K as an K0-algebra for some field extension K0/K. Proof. (a) =⇒ (b): Assume L/K is a field extension of type (n, e). Let S be the separable closure of K in L and K0 be an algebraic closure of S (which is also an algebraic closure of K). Then 0 0 0 0 L ⊗K K = L ⊗S (S ⊗K K ) = (L ⊗S K ) × · · · × (L ⊗S K )(n times). 0 0 On the other hand, by [Ras71], Theorem 3, L ⊗S K is isomorphic to Λe as a K -algebra, and part (b) follows. 0 0 0 (b) =⇒ (a): Assume L ⊗K K is isomorphic to Λn,e ⊗k K as an K -algebra for some field extension K0/K. After replacing K0 by a larger field, we may assume that K0 contains the normal 0 0 closure of S over K. Since Λn,e ⊗k K is not separable over K , L is not separable over K. Thus L/K is of type (m, f) for some m > 1 and f = (f1, . . . , fs) with f1 > f2 > ··· > fs > 1. By part 00 00 00 (a), L ⊗K K is isomorphic to Λm,f ⊗k K for a suitable field extension K /K. After enlarging 00 0 00 00 K , we may assume without loss of generality that K ⊂ K . We conclude that Λn,e ⊗k K is 00 00 00 isomorphic to Λm,f ⊗k K as a K -algebra. By Lemma 2.4.5, with k replaced by K , this is only possible if (n, e) = (m, f).

2.5 Proof of the upper bound of Theorem 2.1.2

In this section we will prove the following proposition.

Proposition 2.5.1. Let n > 1 and e = (e1, . . . , er), where e1 > ··· > er > 1. Furthermore let P si = j≤i ei. Then r X s −ie τ(n, e) 6 n p i i . i=1

12 2.5. Proof of the upper bound of Theorem 2.1.2

Our 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 a separable field extension of finite degree, and 0 6= a ∈ S. Then there exists an s ∈ S such that asq is a primitive element for S/K. Proof. Assume the contrary. It is well known that there are only finitely many intermediate fields between K and S; see e.g., [Lan02], Theorem V.4.6. Denote the intermediate fields properly contained in S by S1,...,Sn ( S and let AK (S) be the affine space associated to S. (Here we view S as a K-vector space.) The non-generators of S/K may now be viewed as K-points of the finite union n Z = ∪i=1 AK (Si) . Since we are assuming that every element of S of the form asq is a non-generator, and K q is an infinite field, the image of the K-morphism f : A(S) → A(S) given by s → as lies in n Z = ∪i=1 AK (Si). Since AK (S) is irreducible, we conclude that the image of f lies in one of the q affine subspaces AK (Si), say in AK (S1). Equivalently, as ∈ S1 for every s ∈ S. Setting s = 1, q q we see that a ∈ S1. Dividing as ∈ S1 by 0 6= a ∈ S1, we conclude that s ∈ 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 that Pr sj −jej ed(L/K) 6 n j=1 p . 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 sequence e for the purely inseparable extension L/S of type e. Set qi = p i . Recall that by the definition q1 of normal sequence, x1 ∈ S. We are free to replace x1 by x1s for any 0 6= s ∈ S; clearly x1s, x2, . . . , xr is another normal generating sequence. By Lemma 2.5.2, we may choose s ∈ S q so that (x1s) 1 is a primitive element for S/K. In other words, we may assume without loss of q1 generality that x1 is a primitive element for S/K. qi qi qi ei By the structure theorem of Pickert, each xi lies in S[x1 , . . . , xi−1], where qi = p ; see Theorem 1 in [Ras71]. In other words, for each i = 1, . . . , r,

qi X qid1 qidi−1 xi = ad1,...,di−1 x1 . . . xi−1 (2.5.1) for some for some ad1,...,di−1 ∈ S. Here the sum is taken over all integers d1, . . . , di−1, where each e −e 0 6 dj 6 p j i − 1. Note that for i = 1 formula (2.5.1) reduces to

q1 x1 = a∅, for some a∅ ∈ S. By Lemma 2.2.1, L (viewed as an S-algebra), descends to

ej −ei S0 = k(ad1,...,di−1 | i = 1, . . . , r and 0 6 dj 6 p − 1) . Note that for each i = 1, . . . , r, there are exactly

pe1−ei · pe2−ei · ... · pei−1−ei = psi−iei

Pr si−iei choices of the subscripts d1, . . . , di−1. Hence, S0 is generated over k by i=1 p elements and consequently, r X si−iei trdeg(S0/k) 6 p . i=1

13 2.6. Versal algebras

q Moreover, since S0 contains a∅ = x1, which is a primitive element for S/K, we conclude that K(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, as usual, K is assumed to contain the base field k of characteristic p > 0. Dividing both sides of the inequality in Proposition 2.5.1 by [L : K] = npe1+···+er , we readily deduce that

r ed(L/K) τ(n, e) X r 1 p−iei−ei+1−···−rr . [L : K] 6 [L : K] 6 6 pr 6 p i=1 1 In particular, ed(L/K) [L : K] for any inseparable extension [L : K] of finite degree, in 6 2 any (positive) characteristic. As we pointed out in the introduction, this inequality fails in characteristic 0 (even for k = C).

2.6 Versal algebras

Let K be a field and A be a finite-dimensional associative K-algebra with 1. Every a ∈ A gives rise to the K-linear map la : A → A given by la(x) = ax (left multiplication by a). Note that lab = la · lb. It readily follows from this that a has a multiplicative inverse in A if and only if la is non-singular.

Proposition 2.6.1. Let l be a prime integer and Λ be a finite-dimensional associative k-algebra with 1. Assume that there exists a field extension K/k and a K-form A of Λ such that A is a division algebra. Then

(a) there exists a field Kver containing k and a form Aver/Kver of Λ such that

ed(Aver) = ed(AlgΛ), edl(Aver) = edl(AlgΛ) for every prime integer l, and

Aver is a division algebra. (b) If G is the automorphism group scheme of Λ, then

ed(G) = ed(AlgΛ) = max{ed(A/K) | A is a K-form of Λ and a division algebra} and

edl(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 that G = Autk(Λ) is defined as a closed subgroup of the general linear group GLk(Λ). This general linear group admits a generically free linear action on some vector space V (e.g., we can take V = Endk(Λ), with the natural left G-action). Restricting to G we obtain a generically free representation G → GL(V ). We can now choose a dense open G-invariant subscheme U ⊂ V over k which is the total space of a G-torsor π : U → B; see, e.g., Section 4 in [BF03]. Passing to

14 2.6. Versal algebras

the generic point of B, we obtain a G-torsor Tver → Spec(Kver), where Kver is the function field of B over k. Then ed(Tver/Kver) = ed(G) and edl(Tver/Kver) = edl(G) (see [BF03, Corollary 6.16]). 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 AlgΛ and H1(∗,G) of Proposition 2.3.4. By the characteristic-free version of the no-name Lemma, d proved in [RV06], Section 2, T × VK is G-equivariantly birationally isomorphic to T × AK , where d d = dim(V ) = dim(VK ) and G acts trivially on AK . In other words, we have a Cartesian diagram of rational maps defined over k

d ' pr2 T × AK / T × VK / UK

 d  AK / BK .

Here all direct products are over Spec(K), and pr2 denotes the rational G-equivariant projection d d map taking (t, v) ∈ T × V to v ∈ V for v ∈ U. The map AK = Spec(K) × A 99K B in the d bottom row is induced from the dominant G-equivariant map T × AK 99K UK on top. Passing to generic points, we obtain an inclusion of field Kver ,→ K.Kver ,→ K(x1, . . . , xd) such that the 1 1 induced map H (Kver,G) → H (K(x1, . . . , xd),G) sends the class of Tver → Spec(Kver) to the d d class associated to T × AK → AK . Under the isomorphism of Proposition 2.3.4 between the 1 functors AlgΛ and FG = H (∗,G), this translates to

Aver ⊗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 a division algebra, so is A(x1, . . . , xd). Thus the linear map la : A(x1, . . . , xd) → A(x1, . . . , xd) is non-singular (i.e., has trivial kernel) for every a ∈ Aver. Hence, the same is true for the restriction of la to Aver. We conclude that Aver is a division algebra. Remembering that Aver corresponds to Tver under the isomorphism of functors between AlgΛ and FG, we see that

ed(Aver) = ed(Tver/Kver) = ed(G) = ed(AlgΛ) and edl(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 from part (a).

We will now revisit the finite-dimensional k-algebras Λe and Λn,e = Λe × · · · × Λe (n times) defined in Section 2.4; see (2.4.1). We will write Gn,e = Aut(Λn,e) ⊂ GLk(Λn,e) for the automorphism group scheme of Λn,e and Algn,e for the functor AlgΛn,e : Fieldsk → Sets. Recall that 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 set

τl(n) = max{edl(L/K) | L/K is a separable field extension of degree n and k ⊂ K}.

15 2.7. Conclusion of the proof of Theorem 2.1.2 and

τl(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 of n-dimensional ´etalealgebras, 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 ´etalealgebra 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 exists a separable field extension E/F of degree n with k ⊂ F . The desired equality follows from Proposition 2.6.1(b). (b) The same argument as in part (a) goes through, with part (a) of Lemma 2.4.4 replaced by part (b).

Remark 2.6.3. The value of edl(Sn) is known for every integer n > and every prime l > 2:  n b c, if char(k) 6= l, see Corollary 4.2 in [MR09],  l edl(Sn) = 1, if char(k) = l 6 n, see Theorem 1 in [RV18], and  0, if char(k) = l > n, see Lemma 4.1 in [MR09] or Theorem 1 in [RV18].

2.7 Conclusion of the proof of Theorem 2.1.2

In 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 > 1 be integers, e = (e1, . . . , er) and si = e1 + ··· + ei for i = 1, . . . , r. Then r X si−iei τp(n, e) = τ(n, e) = n p . i=1

Pr si−iei By definition τp(n, e) 6 τ(n, e) and by Proposition 2.5.1, τ(n, e) 6 n i=1 p . Moreover, by Corollary 2.6.2(b), τp(n, e) = edp(Gn,e). It thus remains to show that r X si−iei edp(Gn,e) > n p . (2.7.1) i=1 Our proof of (2.7.1) will be based on the following general inequality, due to Tossici and Vistoli [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 that n Ge = Autk(Λe), and Gn,e = Autk(Λn,e), where Λn,e = Λe . Since Λe is is a commutative local n k-algebra with residue field k, Proposition 2.2.3 tells us that Gn,e = Ge oSn (see also Proposition 5.1 in [SdS00]). We conclude that

dim(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 thus of Theorem 2.7.1) reduces to the following.

16 2.7. Conclusion of the proof of Theorem 2.1.2

Proposition 2.7.2. Let e = (e1, . . . , er), where e1 > ··· > er > 1 are integers. Then e +···+er (a) dim(Lie(Ge)) = rp 1 , and e1+···+er Pr si−iei (b) dim(Ge) = rp − i=1 p . The remainder of this section will be devoted to proving Proposition 2.7.2. We will use the following notations.

e 1. We fix the type e = (e1, . . . , er) and set qi = p i .

2. The infinitesimal group scheme αpj over a commutative ring S of characteristic p is defined pj as the kernel of the j-th power of the Frobenius map, Ga → Ga, x 7→ x , viewed as a homomorphism 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. We will denote the Weil restriction of the Λ-scheme X to k by RΛ/k(X). For generalities on Weil restriction, see Chapter 2 and the Appendix in [Mil17].

4. We will denote by End(Λe) the functor

Commk −→ Sets

R −→ EndR−alg(Λe ⊗k R)

of algebra endomorphisms of Λe. Here Commk denotes the category of of commutative associative 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) affine k-scheme Xe. e1+···+er Pr si−iei (b) dim(Xe) = rp − i=1 p . e +···+er (c) dim(Tγ(Xe)) = rp 1 for any k-point γ of Xe. Here Tγ(Xe) denotes the tangent space to Xe at γ.

Proof. An endomorphism F in End(Λe)(R) is uniquely determined by the images

F (x1),F (x2),...,F (xr) ∈ Λe(R)

q of the generators x1, . . . , xr of Λe. These elements of Λe satisfy F (xi) i = 0. Conversely, any qi r elements F1,...,Fr in Λe ⊗ R satisfying Fi = 0, give rise to an algebra endomorphism F in End(Λe)(R). We thus have

End(Λe)(R) = HomR−alg(Λe ⊗k R, Λe ⊗ R) ∼ = αq1 (Λe ⊗ R) × ... × αqr (Λe ⊗ R) ∼ = RΛe/k(αq1 )(R) × ... × RΛe/k(αqr )(R) r ∼ Y = RΛe/k(αqi )(R) i=1

17 2.7. Conclusion of the proof of Theorem 2.1.2

Qr We conclude that End(Λe) is represented by an affine k-scheme Xe = i=1 RΛe/k(αqi ). (Note Qr that Xe is isomorphic to i=1 RΛe/k(αqi ) as a k-scheme only, not as a group scheme.) To complete the proof of the lemma it remains to establish the following assertions:

For any qj ∈ {q1, . . . , qr} we have that 0 (a ) RΛe/k(αqj ) is irreducible, 0
e1+···+er sj −jej (b ) dim RΛe/k(αqj ) = p − p and 0 e1+···+er (c ) dim(Tγ(RΛe/k(αqj ))) = p for any k-point γ of RΛe/k(αqj ). 0 0 0 1 To prove (a ), (b ) and (c ), we will write out explicit equations for RΛe/k(αqj ) in RΛe/k(A ) ' i1 i2 ir Ak(Λe). We will work in the basis {x1 x2 . . . xr } of monomials in Λe, where 0 6 i1 < q1, 1 0 6 i2 < q2, ..., 0 6 ir < qr. Over Λe, αqj is cut out (scheme-theoretically) in A by the single qj 1 qi equation t = 0, where t is a coordinate function on A . Since xi = 0 for every i, writing

X i1 i2 ir t = yi1,...,ir x1 x2 . . . xr and expanding

X q q i1 q i2 q ir tqj = y j x j x j . . . x j i1,...,ir 1 2 r we see that the only monomials appearing in the above sum are those for which

qji1 < q1, qji2 < q2, . . . , qjir < qr. 1 Thus RΛe/k(αqj ) is cut out (again, scheme-theoretically) in RΛe/k(A ) ' A(Λe) by q q yqj = 0 for 0 i < 1 , ..., 0 i < j−1 , i1,...,ij−1,0,...,0 6 1 6 j−1 qj qj where yi1,...,ir are the coordinates in A(Λe). In other words, RΛe/k(αqj ) is the subscheme of 1 pe1+···+er RΛe/k(A ) ' Ak(Λe) ' Ak cut out (again, scheme-theoretically) by qjth powers of

q1 q2 qj−1 ... = psj −jej qj qj qj distinct coordinate functions. The reduced scheme RΛe/k(αqj )red is thus isomorphic to an affine e1+···+er Pr sj −jej space of dimension p − j=1 p . On the other hand, since qj is a power of p, the

Jacobian criterion tells us that the tangent space to RΛe/k(αql ) at any k-point is the same as the pe1+···+er 0 0 0 tangent space to A(Λe) = A , and (a ), (b ), (c ) follow.

Conclusion of the proof of Proposition 2.7.2. The automorphism group scheme Ge is the group of invertible elements in End(Λe). In other words, the natural diagram

Ge / GLN

  End(Λe) / MatN×N

e +...+er where N = dim(Λe) = p 1 , is Cartesian. Hence, Ge is an open subscheme of Xe. Since Xe is irreducible, Proposition 2.7.2 follows from Lemma 2.7.3. This completes the proof of Proposition 2.7.2 and thus of Theorem 2.7.1.

18 2.8. Alternative proofs of Theorem 2.1.2

2.8 Alternative proofs of Theorem 2.1.2

The proof of the lower bound of Theorem 2.1.2 given in Section 2.7 section is the only one we know. However, we have two other proofs for the upper bound (Proposition 2.5.1), in addition to the one given in Section 2.5. In this section we will briefly outline these arguments for the interested reader. Our first alternative proof of Proposition 2.5.1 is based on an explicit construction of the versal algebra Aver of type (n, e) whose existence is asserted by Proposition 2.6.1. This construction is via generators and relations, by taking “the most general” structure constants in (2.5.1). Versality of Aver constructed this way takes some work to prove; however, once versality is established, it is easy to see directly that Aver is a field and thus

r X si−iei τ(n, e) = ed(Aver) 6 trdeg(Kver/k) = n p . i=1 Our second alternative proof of Proposition 2.5.1 is based on showing that the natural r representation of Gn,e on V = Λn,e is generically free. Intuitively speaking, this is clear: Λn,e is generated by r elements as a k-algebra, so r-tuples of generators of Λn,e are dense in V and have trivial stabilizer in Gn,e. The actual proof involves checking that the stabilizer in general position is trivial scheme-theoretically and not just on the level of points. Once generic freeness of this linear action is established, the upper bound of Proposition 2.5.1 follows from the inequality

ed(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 this inequality, recall that

τ(n, e) = ed(Gn,e) (see Corollary 2.6.2(b)), e +···+er dim(V ) = r dim(Λn,e) = nr dim(Λe) = nrp 1 (clear from the definition), and e1+···+er Pr si−iei dim(Gn,e) = n dim(Ge) = nrp − n i=1 p (see Proposition 2.7.2(b)).

2.9 The case where e1 = ··· = er

In the special case, where n = 1 and e1 = ··· = er, Theorem 2.1.2 tells us that τ(n, e) = r. In this section, we will give a short proof of the following stronger assertion (under the assumption that k is perfect).

Proposition 2.9.1. Let e = (e, . . . , e)(r times) and L/K be purely inseparable extension of type 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 a field extension L/K of type e. Setting k = K, we see that ed(L/K) = 0, and the proposition fails. The remainder of this section will be devoted to proving Proposition 2.9.1. We begin with two reductions. (1) It suffices to show that

ed(L/K) = r for every field extension L/K of type e; (2.9.1)

19 2.9. The case where e1 = ··· = er

the identity edp(L/K) will then follow. Indeed, edp(L/K) is defined as the minimal value of 0 0 0 0 0 ed(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, L0 is a field, so (2.9.1) tells us that ed(L0/K0) = 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. For the sake of completeness we reproduce it here. Let x1, . . . , xr be a normal generating sequence q q e for L/K. By a theorem of Pickert (Theorem 1 in [Ras71]), x1, . . . , xr ∈ K, where q = p . Set q d1 dr ai = xi and K0 = k(a1, . . . , ar). The structure constants of L relative to the K-basis x1 . . . xr of 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 to L0/K0 with trdeg(K0/k) < r. By Lemma 2.2.1, L0/K0 further descends to L1/K1, where K1 is finitely 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 of transcendence degree < r. There there does not exist a purely inseparable field extension L/K of type e = (e1, . . . , er), where e1 > ··· > er > 1.

Proof. Assume the contrary. Let a1, . . . , as be a transcendence basis for K/k. That is, a1, . . . , as are algebraically independent over k, K is algebraic and finitely generated (hence, finite) over k(a1, . . . , as) and s 6 r − 1. By Remark 2.4.3,

p p r [L : L ] > [L :(L · K)] = p . (2.9.3)

On the other hand, since [L : k(a1, . . . , as)] < ∞, Theorem 3 in [BM40] tells us that

p p p p s r [L : L ] = [k(a1, . . . , as): k(a1, . . . , as) ] = [k(a1, . . . , as): k(a1, . . . , as)] = p < p . (2.9.4)

Note that the second equality relies on our assumption that k is perfect. The contradiction between (2.9.3) and (2.9.4) completes the proof of Lemma 2.9.2 and thus of Proposition 2.9.1.

20 Chapter 3

Essential dimension of double covers of symmetric and alternating group

3.1 Introduction

I. Schur [Sch04] studied central extensions

± ± φ 1 / Z/2Z / Sen / Sn / 1 (3.1.1) of the symmetric group Sn. Representations of these groups are closely related to projective representations of Sn: over an algebraically closed field of characteristic zero, every projective + + − − representation ρ:Sn → PGL(V ) lifts to linear representations ρ : eSn → GL(V ) and ρ : eSn → GL(V ); see [HH92, Theorem 1.3]. That is, the following diagram commutes.

± ρ± eSn / GL(V )

φ±  ρ  Sn / PGL(V )

Figure 3.1: Projective representation

± Moreover, the groups eSn are minimal central extensions of Sn with this property. They are called representation groups of Sn. In terms of generators and relations,

+ 2 2 2 3 eSn = z, s1, s2, . . . , sn−1 | z = si = 1, [z, si] = 1, (sisj) = z if |i − j| > 1, (sisi+1) = 1 and − 2 2 2 3 eSn = z, t1, t2, . . . , tn−1 | z = 1, ti = z, (titj) = z if |i − j| > 1, (titi+1) = z .

+ − + Here z is a central element of order 2 in eSn (respectively, eSn ) generating Ker(φ ) (respectively, − + − Ker(φ )), and φ (si) = φ (ti) is the transposition (i, i + 1) in Sn. The preimage of An under + + − − φ in eSn is isomorphic to the preimage of An under φ in eSn ; see [Ser08, Section 9.1.3]. We will denote this group by Ae n; it is a representation group of An. For modern expositions of Schur’s theory, see [HH92] or [Ste89]. ± The purpose of this chapter is to study the essential dimension of the covering groups eSn and Ae n. We will assume that n > 4 throughout. As usual, we will denote the essential dimension of a 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) in

21 3.1. Introduction place of ed(G) and ed(G; p) to emphasize this dependence. We refer the reader to Section 3.2 for the definition of essential dimension, some of its properties and further references. ± Our interest in the covering groups eSn , Ae n was motivated by their close connection to two families 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 summarized in the following diagram. Here the base field k is assumed to be of characteristic 6= 2 and to contain a primitive 8th root of unity, On is the orthogonal group associated to the quadratic form 2 2 x1 + ··· + xn, Sn → On is the natural n-dimensional representation, Γn is the Clifford group, and ± ± ± eSn , Ae n are the preimages of Sn and An under the double covers Pinn → On. The groups Pinn ± + T are defined as the kernels of the homomorphisms N :Γn → Gm given by N (x) = x.x and − T T N (x) = x.γ(x ), where (x1 ⊗ x2 ⊗ ... ⊗ xn) = xn ⊗ ... ⊗ x2 ⊗ x1 and γ is the automorphism of the Clifford algebra which acts on degree 1 component by −1. Both appear in literature as Pin groups (Pin+ in [Ser84] and Pin− in [ABS64], see also [GG86]).

On

Γn

Gm

+ Pinn On

− Z/2Z Pinn

+ eSn Sn

− Z/2Z eSn

An

Ae n

Z/2Z

Figure 3.2: Schematic of spin groups

The essential dimension of Sn and An is known to be sublinear in n: in particular, n > 5, we have ed(An) 6 ed(Sn) 6 n − 3; see [BR97]*Theorem 6.5(c). On the other hand, the essential dimension of Spinn increases exponentially with n. If we write n = 2am, where m is odd, then  n(n − 1) 2(n−1)/2 − , if a = 0,  2  (n−2)/2 n(n − 1) ed(Spinn) = ed(Spinn; 2) = 2 − , if a = 1,  2  n(n − 1) 2(n−2)/2 + 2a − , if a 2; 2 >

22 3.1. Introduction see [BRV10], [CM14] and [Tot19]. ± ± Question 3.1.1. What is the asymptotic behavior of ed(eSn ), ed(Ae n), ed(eSn ; 2) and ed(Ae n; 2) as n −→ ∞? Do these numbers grow sublinearly, like ed(Sn), or exponentially, like ed(Spinn)? + − Note that for odd primes p, eSn , eSn and Sn have isomorphic Sylow p-subgroups, and ± thus ed(eSn ; 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(eSn ; p) and ed(Ae n; p) when p = 2. ± ± In this chapter we answer Question 3.1.1 as follows: ed(eSn ), ed(Ae n), ed(eSn ; 2) grow exponen- tially if char(k) 6= 2 and sublinearly if char(k) = 2. This follows from Theorems 3.1.2 and 3.1.4 below. Theorem 3.1.2. Assume that the base field k is of characteristic 6= 2 and contains a primitive 8th a as root of unity, and let n > 4 be an integer. Write n = 2 1 +...+2 , where a1 > a2 > . . . > as > 0 + − and let eSn be either eSn or eSn . Then

b(n−1)/2c (a) ed(Ae n) 6 ed(eSn) 6 2 .

b(n−s)/2c (b) ed(eSn; 2) = 2 ,

b(n−s−1)/2c (c) ed(Ae n; 2) = 2 .

b(n−s)/2c b(n−s)/2c (d)2 6 ed(eSn) 6 ed(Sn) + 2

b(n−s−1)/2c b(n−s−1)/2c (e)2 6 ed(Ae n) 6 ed(An) + 2 . For s = 1 and 2, upper and lower bounds of Theorem 3.1.2 meet, and we obtain the following exact values. Corollary 3.1.3. Assume that the base field k contains a primitive 8th root of unity. a n−2 (a) If n = 2 , where a > 2, then ed(eSn) = ed(eSn; 2) = ed(Ae n) = ed(Ae n; 2) = 2 2 . a a n−2 (b) If n = 2 1 + 2 2 , where a1 > a2 > 1, then ed(eSn) = ed(eSn; 2) = 2 2 .

Note that exact values of ed(Sn) or ed(An) are known only for n 6 7; see [Mer13, Section 3i] for a summary. One may thus say that we know more about ed(eSn) and ed(Ae n) than we do about ed(Sn) and ed(An). + − Theorem 3.1.4. Let eSn be either eSn or eSn . Assume char k = 2. Then (a) ed(Sn) 6 ed(eSn) 6 ed(Sn) + 1. (b) ed(An) 6 ed(Ae n) 6 ed(An) + 1. (c) ed(eSn; 2) = ed(Ae n; 2) = 1.

Our proof shows that, more generally, central extensions by Z/pZ make little difference to the 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(eSn) and ed(Ae n) in characteristic 2 is ± ± that the connection between eSn (respectively, Ae n) and Pinn (respectively, Spinn) outlined above breaks down in this setting; see Remark 3.4.2.

23 3.1. Introduction

n 4 5 6 7 8 9 10 11 12 13 14 15 16

edC(An) 2 2 3 4 4-5 4-6 5-7 6-8 6-9 6-10 7-11 8-12 8-13

edC(Ae n; 2) 2 2 2 2 8 8 8 8 16 16 32 32 128

edC(Ae n) 2 2 4 4 8 8-14 8-15 8-16 16-25 16-26 32-43 32-44 128

Table 3.1: Essential dimension of Ae n

Some values of ed(An), ed(Ae n) and ed(Ae n; 2) over the field C of complex numbers are shown in 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. 2 2 As usual, we will denote the non-degenerate diagonal form q(x1, . . . , xn) = a1x1 + ··· + anxn ∗ defined over a field F of characteristic 6= 2 by ha1, . . . , ani. Here a1, . . . , an ∈ F . We will abbreviate ha, . . . , ai (m times) as mhai. Recall that the Hasse invariant w2(q) of q = ha1, . . . , ani (otherwise known as the second Stiefel-Whitney class of q) is given by

2 w2(q) = Σ16i ··· > as > 0 (as in Theorem 3.1.2). Then every n-dimensional trace form q contains sh1i as a subform; see, e.g., [Ser84, Proposition 4]. This necessary condition for an n-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 like a general n-dimensional quadratic form that contain sh1i as a subform.

Theorem 3.1.5. Let k be a field containing a primitive 8th root of unity, n > 4 be an integer, a as and n = 2 1 + ··· + 2 be the dyadic expansion of n, where a1 > ··· > as > 0. Then b(n−s)/2c (a) max ind(w2(q)) = max ind(w2(t)) = 2 , F, q F, t b(n−s−1)/2c (b) max ind (w2(q1)) = max ind(w2(t1)) = 2 . F, q1 F, t1 Here the maxima are taken as

• F ranges over all fields containing k,

24 3.2. Preliminaries

• q ranges over n-dimensional non-degenerate quadratic forms over F containing s h1i,

• q1 ranges over n-dimensional quadratic forms of discriminant 1 over F containing s h1i, • 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 ⊕ sh1i, then q and r have the same discriminant and the same Hasse invariant. 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 similarly in part (b). The remainder of this chapter is structured as follows. Section 3.2 gives a summary of known results which will be needed later on. Theorem 3.1.2 is proved in Section 3.3, Theorem 3.1.4 in Section 3.4 and Theorem 3.1.5 in Section 3.5. In Section 3.6 we compare the essential dimensions + − of eSn and eSn , and in Section 3.7 we explain the entries in Table 3.1.

3.2 Preliminaries

3.2.1 Essential dimension Recall that the essential dimension of a linear algebraic group G is defined as follows. Let V be a generically free linear representation of G and let X be a G-variety, i.e., an algebraic variety with an action of G. Here G, V , X and the G-actions on V and X are assumed to be defined over the base field k. We will say that X is generically free if the G-action on X is generically free. The essential dimension ed(G) of G is the minimal value of dim(X) − dim(G), where X ranges over all generically free G-varieties admitting a G-equivariant dominant rational map V 99K X. This number 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 a prime integer p. The essential dimension of G at p is defined in the same way as ed(G), as the minimal value of dim(X) − dim(G), where X is a generically free G-variety, except that instead of requiring that X admits a G-equivariant dominant rational map V 99K X, we only require that it admits a G-equivariant dominant correspondence V X whose degree is prime to p. Here by a dominant correspondence V X of degree d we mean a diagram of dominant G-equivariant rational maps,

V 0

d:1  VX.'

In Chapter 2 Example 2.3.1, the given definition of essential dimension of G agrees with the definition 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 the surveys [Mer13] and [Rei10].

25 3.2. Preliminaries

Lemma 3.2.1. Let G,→ GL(V ) be a generically free representation. Then

ed(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 finite and 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 map

1 1 H (K,G1) → H (K,G2) is surjective for all field extensions K of k, then ed(G1) > ed(G2) and ed(G1; p) > ed(G2; p) for every 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 extension Assume char(k) 6= p. Let G be a finite group and

1 / Z/pZ / G / G / 1 be a central exact sequence. This exact sequence gives rise to a connecting morphism

1 2 δK : H (K, G) → H (K, Z/pZ)

2 for every field K containing k. If K contains a primitive pth root of unity, then H (K, Z/pZ) can be identified with the p-torsion subgroup Brp(K) of the Brauer group Br(K). In particular, 1 we can talk about the index ind(δK (α)) for any α ∈ H (K, G). Let ind(G, Z/pZ) denote the maximal value of ind(δK (t)), as K ranges over all field extensions of k and t ranges over the elements 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).

26 3.2. Preliminaries

Proof. (a) The diagram

1 / Z/pZ / G / G / 1 O O O '

1 / Z/pZ / Gp / Gp / 1, where the rows are central exact sequences and the vertical maps are natural inclusions gives rise to a commutative diagram 1 δK 2 H (K, G) / H (K, Z/pZ) (3.2.1) O O i∗ '

1 νK 2 H (K, Gp) / H (K, Z/pZ) of Galois cohomology sets for any field K/k. Here δ and ν denote connecting morphisms. It is clear from (3.2.1) that ind(Gp, Z/pZ) 6 ind(G, Z/pZ). To prove the opposite inequality, choose 1 a field extension K/k and an element t ∈ H (K, G) such that δK (t) has the maximal possible 2 index in H (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 degree 1 1 prime to p, we may assume that tL ∈ H (L, G) is the image of some s ∈ H (L, Gp). Here as 1 1 1 usual, tL denotes the image of t ∈ H (K, G) under the restriction map H (K, G) → H (L, G). Since [L : K] is prime to p, we have

ind(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 equality

ind(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 the statement 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 Ae n

Lemma 3.2.6. Let Hen be a Sylow 2-subgroup of Ae n. (a) If n = 4 or 5, then Hen is isomorphic to the quaternion group

2 2 2 Q8 = x, y, c | x = y = c, c = 1, cx = xc, cy = yc, xy = cyx .

(b) If n = 6 or 7, then Hen is isomorphic to the generalized quaternion group

8 4 2 4 −1 −1 Q16 = x, y | x = y = 1, y = x , yxy = x .

27 3.3. Proof of Theorem 3.1.2

+ Proof. We will view Ae n (and thus Hen) as a subgroup of eSn and use the generators and relations + for eSn given in the introduction. 2 (a) For n = 4 or 5, we can take Hen to be the group of order 8 generated by σ = (s1s2s3) and τ = s1s3. These elements project to (1 3)(2 4) and (1 2)(3 4) in An, respectively. One readily 2 2 checks that σ = τ = z and στ = zτσ. An isomorphism Q8 → Hen can now be defined by

x 7→ σ, y 7→ τ, c 7→ z.

(b) For n = 6 or 7, we can take Hen to be the group of order 16 generated by σ = s1s2s3s5 and τ = s1s3. These elements project to (1 2 3 4)(5 6) and (1 2)(3 4) in An, respectively. An isomorphism Q16 → Hen can now be given by x 7→ σ and y 7→ τ.

+ − Proposition 3.2.7. Let n > 4 be an integer, Sen be either Sen or Sen , and Pen, Hen be Sylow 2-subgroups of Sen, Ae n, respectively. Denote the centers of Pen and Hen by Z(Pen) and Z(Hen), respectively. Then Z(Pen) = Z(Hen) = hzi is a cyclic group of order 2.

Proof. By [Wag77, Lemma 3.2] that Z(Pen) = hzi for every n > 4 and Z(Hen) = hzi for every n > 8. It remains to show that Z(Hen) = hzi for 4 6 n 6 7. Clearly z ∈ Z(Hen), so we only need to show that Z(Hen) is of order 2. We will use the description of the groups Hen from Lemma 3.2.6. If n = 4 and 5, then Hen is isomorphic to the quaternion group Q8, and the center of Q8 is clearly of order 2. If n = 6 or 7, then Hen ' Q16, and the center of Q16 is readily seen to be the cyclic group x4 of order 2.

3.3 Proof of Theorem 3.1.2

We 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 8th a as root of unity, and let n > 4 be an integer. Write n = 2 1 +...+2 , where a1 > a2 > . . . > as > 0 + − and let eSn be either eSn or eSn . Then

b(n−1)/2c (a) ed(Ae n) 6 ed(eSn) 6 2 .

b(n−s)/2c (b) ed(eSn; 2) = 2 ,

b(n−s−1)/2c (c) ed(Ae n; 2) = 2 .

b(n−s)/2c b(n−s)/2c (d)2 6 ed(eSn) 6 ed(Sn) + 2

b(n−s−1)/2c b(n−s−1)/2c (e)2 6 ed(Ae n) 6 ed(An) + 2 .

(a) The first inequality follows from the fact that Ae n is contained in eSn; see Lemma 3.2.4. For the second inequality, apply Lemma 3.2.1 to the so-called basic spin representation of eSn. This representation is obtained by restricting a representation of the Clifford algebra Cn−1 into Mat n−1 (k); see [Ste89, Section 3] for details. (Note that [Ste89] assumes k = but the same 2b 2 c C morphism works over any field containing square root of −1).

28 3.3. Proof of Theorem 3.1.2

(b) Let Pn be a Sylow 2-subgroups of Sn. The preimage Pen of Pn is a Sylow 2-subgroup of eSn. By Lemma 3.2.2, ed(eSn; 2) = ed(Pen; 2). Moreover, by the Karpenko-Merkurjev theorem [KM08, Theorem 4.1], ed(Pen) = ed(Pen; 2) = dim(V ), where V is a faithful linear representation of Pen of minimal dimension. By Proposition 3.2.7, the center of Z(Pen) = hzi is of order 2. Consequently, a faithful representation V of minimal dimension is automatically irreducible; see [MR10, Theorem 1.2]. On the other hand, an irreducible representation ρ of Pen 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] every b(n−s)/2c irreducible representation ρ: Pen → GL(V ) with ρ(z) 6= 1 is of dimension 2 . This proves part (b) for k = C. d Case 2: Assume k ⊂ C is a field containing a primitive root of unity ζ2d of degree 2 , where d 2 is the exponent of Pen. By a theorem of R. Brauer [Ser77, 12.3.24], every irreducible complex representation of Pen is, in fact, defined over k. Thus the dimension of the minimal faithful b(n−s)/2c irreducible representation over k is the same as over C, i.e., 2 , 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 ). Then b(n−s)/2c edk(Pen) > edl(Pen) = 2 .

To prove the opposite inequality, let V be a faithful irreducible representation of Pen dimension b(n−s)/2c 2 defined over Q(ζ2d ). Such a representation exists by Case 2. We claim that V is, in fact, defined over Q(ζ8). In particular, V is defined over k and thus

b(n−s)/2c edk(Pen) 6 2 , as desired. We will prove the claim in two steps. First we will show that the character χ: Pen → 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 Pen of b(n−s)/2c d−2 dimension 2 . The Galois group G = Gal(Q(ζ2d )/Q) ' Z/2Z×Z/2 Z acts on this set of characters. Thus for any σ ∈ Pen, 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√ Galois correspondence these subgroups correspond to the subfields Q( −1), Q√( 2)√ and Q( −2) of Q(ζ2d ). Thus χ(σ) lies in one of these three fields; in particular, χ(σ) ∈ Q( −1, 2) = Q(ζ8) for every σ ∈ G. In other words, χ takes all of its values in Q(ζ8), as desired. Now observe that since Pen 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, we conclude that V itself is defined over k. This completes the proof of part (b) in Case 3 (i.e., in characteristic 0).

Case 4: Now assume that k is a perfect field of characteristic p > 2 containing ζ8. Let A = W (k) be the ring of Witt vectors of k. Recall that A is a complete discrete valuation ring of characteristic zero, whose residue field is k. By Hensel’s Lemma, ζ8 lifts to a primitive 8th root of unity in A. Denote the fraction field of A by K and the maximal ideal by M. Since Pen is a 2-group and char(k) = p is an odd prime, every d-dimensional k[Pen]-module W lifts to a unique A[Pen]-module WA, which is free of rank d over A. Moreover, the lifting operation V 7→ VK := VA ⊗K and the “reduction mod M” operation give rise to mutually inverse bijections

29 3.3. Proof of Theorem 3.1.2

between k[Pen]-modules and K[Pen]-modules; see [Ser77, Section 15.5]. These bijections preserve dimension and faithfulness of modules. Since K is a field of characteristic 0 containing a primitive 8th root of unity, Case 3 tells us that the minimal dimension of a faithful K[Pen]-module is b(n−s)/2c b(n−s)/2c 2 . Hence, the minimal dimension of a faithful k[Pen]-module is also 2 . This proves part (b) in Case 4.

Case 5: Now assume that k is an arbitrary field of characteristic p > 2 containing ζ8. Denote the 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. Then

edk1 (Pen) > edk(Pen) > edk2 (Pen).

b(n−s)/2c b(n−s)/2c By Case 4, edk1 (Pen) = edk2 (Pen) = 2 . We conclude that edk(Pen) = 2 . The proof of part (b) is now complete.

(c) Let Hn be the Sylow 2-subgroups of An. Its preimage Hen is a Sylow 2-subgroup of Ae n. By the Karpenko-Merkurjev theorem [KM08, Theorem 4.1], ed(Hen) = ed(Hen; 2) = dim(W ), where W is a faithful linear representation of Hen of minimal dimension. The rest of the argument in part (b) goes through with only minor changes. Once again, by Proposition 3.2.7, the center of Z(Hen) = hzi is of order 2. Thus W is irreducible. Moreover, an irreducible representation ρ of Hen 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 every b(n−s−1)/2c irreducible representation ρ of Hen with ρ(z) 6= 1 is of dimension 2 . This proves part (c) for k = C. Moreover, depending on the parity of n − s, there are either one or two such representations. 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(eSn) > ed(eSn; 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, and Z/2Z = hzi is the center of Gp = Pen by Proposition 3.2.7. (e) The lower bound follows from part (c) and the inequality ed(Ae n) > ed(Ae n; 2). The upper bound is obtained by applying Lemma 3.2.5 to the exact sequence

1 / hzi / Ae n / An / 1, in the same way as in part (d).

Remark 3.3.2. If n − s is even, then eSn has only one faithful irreducible complex representation b(n−s)/2c of dimension 2 ; see [Wag77, Lemma 4.2]. In this case we can relax the assumption√ on k 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 the b(n−s)/2c (n−2)/2 case, where s = 1 (i.e., n > 4 is a power of 2), the inequality ed(Sfn; 2) 6 2 = 2 (n−2)/2 is equivalent to the existence of a faithful irreducible representation V of Pen of degree 2 defined over k. There are two such representations, and [HH92, Theorem 8.7] shows that some of their character values are not contained in Q(ζ4).

30 3.4. Proof of Theorem 3.1.4

3.4 Proof of Theorem 3.1.4

Part (c) follows directly from [RV18, Theorem 1], which says that if G is a finite group and k is a field of characteristic p > 0, then ( 1, if the order of G is divisible by p, and edk(G; p) = edk(G; p) = 0, otherwise.

In particular, edk(eSn; 2) = edk(Ae 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 finite p-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 k and 1 / Z/pZ / G / G / 1 be a central exact sequence. Then edk(G) 6 edk(G) 6 edk(G) + 1. Proof. (a) Consider the induced exact sequence

1 1 δK 2 H (K,G) / H (K, G) / H (K, Z/pZ) in Galois cohomology (or flat cohomology, if G is not smooth), where δK denotes the boundary map. Here K/k is an arbitrary field extension K/k. Since K is a field of characteristic p, its 2 cohomological p-dimension is 6 1 and thus H (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. By Lemma 3.2.3, this implies edk(G) > edk(G). On the other hand, since Z/pZ is unipotent in characteristic p,[TV13, Lemma 3.4] tells us that

edk(G) 6 edk(G) + edk(Z/pZ) = edk(G) + 1; see also [L13,¨ Corollary 3.5].

Remark 3.4.2. Note that in characteristic 2 the group Ae n is no longer isomorphic to the preimage of An ⊂ SOn in Spinn. The scheme-theoretic preimage of An in Spinn is an extension of a constant group scheme An by an infinitesimal group scheme µ2. Any such extension is split over a perfect base field; see, e.g., [Mil17, Proposition 15.22]. Thus, over a perfect field k, the preimage 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) in characteristic 0 are not known for any n > 8 . In characteristic 2, even less is known. The upper bound, 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 not have a non-trivial normal 2-subgroup, then edk(G) 6 edC(G) for any field k of characteristic 2 containing the algebraic closure of F2; see [BRV18, Corollary 3.4(b)]. In particular, this applies to G = Sn or An for any n > 5.

31 3.5. Proof of Theorem 3.1.5

In characteristic 0, ed(Sn) > bn/2c for any n > 1 and ed(Sn) > b(n + 1)/2c for any n > 7. It is not known if these inequalities remain true in characteristic 2. On the other hand, since An r contains (Z/3Z) , where r = bn/3c, it is easy to see that the weaker inequality

ed(Sn) > ed(An) > bn/3c remains 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 that

edk(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 by

V = {(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 that edk(S4) 6 2. This proves the claim. By Theorem 3.1.4 we conclude that

2 6 edk(eS4) 6 3.

We do not know whether edk(eS4) = 2 or 3. Note however, that by a conjecture of Ledet [Led04, n p. 4], edk(Z/2 Z) = n for every integer n > 1. Since eS4 contains an element of order 8 (the preimage of a 4-cycle in S4), Ledet’s conjecture implies that edk(eS4) = 3. Note also that by Corollary 3.1.3, edl(eS4) = 2 for any base field l of characteristic 6= 2 containing a primitive 8th root of unity.

3.5 Proof of Theorem 3.1.5

Let q = ha1, . . . , ani 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 by

2 w2(q) = Σ16i

F {x, y}/(x2 = a, y2 = b, xy + yx = 0).

It is immediate from this definition that

w2(h1i ⊕ 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 bn/2c (a) ind(w2(q)) 6 2 . b(n−1)/2c (b) If q is of discriminant 1 over F , then ind(w2(q)) 6 2 .

32 3.5. Proof of Theorem 3.1.5

∗ Proof. Let q = ha1, a2, . . . , an−1, ani for some a1, . . . , an ∈ F . (a) We will consider the cases where n is odd and even separately. If n = 2m even, then q q  S = F − a1 ,..., − a2m−1 splits q. That is q ' nh1i. Hence, S also splits w (q). Since the a2 a2m S 2 2 index of an element α ∈ H (F, Z/2Z) is the minimal degree [K : F ] of a splitting field K/F of α, m we conclude that ind(w2(q)) 6 [S : F ] = 2 , as desired. q q  Now suppose that n = 2m + 1 is odd. Set S = F − a1 ,..., − a2m−1 , as before. Over S, a2 a2m

qS ' 2mh1i ⊕ ha2m+1i.

2 It now follows from (3.5.1) that w2(qS) = 0 in H (S, Z/2Z). Hence, w2(q) splits over S and m n−1 consequently, ind(w2(q)) 6 [S : F ] = 2 = 2 2 , as desired. (b) Since q = ha1, . . . , ani has discriminant 1, we may assume√ without loss of generality that a1 · ... · an = 1 in F . Let r = ha2, . . . , ani. Since K contains −1, the quaternion algebra (a1, a1) is split. Thus

w2(q) = (a1, a1) + w2(q) = (a1, a1 · ... · an) + w2(r)

= (a1, 1) + w2(r) = w2(r).

b(n−1)/2c By part (a), ind(w2(r)) 6 2 , 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 sh1i as a subform; see [Ser84, Proposition 4]. Thus

max ind(w2(t)) max ind(w2(q)). (3.5.2) F, t 6 F, q

On the other hand, by our assumption, q = s h1i ⊕ r, where r is a form of dimension n − s. b(n−s)/2c By (3.5.1), w2(q) = w2(r) and by Lemma 3.5.1(a), ind(w2(r)) 6 2 . Thus b(n−s)/2c max ind(w2(q)) 2 . (3.5.3) F, q 6

In view of the inequalities (3.5.2) and (3.5.3), it suffices to show that

b(n−s)/2c max ind(w2(t)) = 2 . (3.5.4) F, t

1 Recall that elements of H (F, Sn) are in a natural bijective correspondence with isomorphism 1 classes of n-dimensional ´etalealgebras E/F . Denote the class of E/F by [E/F ] ∈ H (F, Sn) and the trace form of E/F by t. By [Ser84, Th´eor`eme1],

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 field extensions of k and E/F ranges over all n-dimensional ´etale F -algebras, is precisely the integer ind(eSn, Z/2Z) defined in Section 3.2.2. Let Pen be a Sylow subgroup of eSn. By Proposition 3.2.7, the center Z(Pen) is cyclic. Thus by Lemma 3.2.5,

ind(eSn; Z/2Z) = ed(Pen) = ed(Pen; 2).

33 3.5. Proof of Theorem 3.1.5

b(n−s)/2c On the other hand, ed(Pen; 2) = ed(eSn; 2) by Lemma 3.2.2 and ed(eSn; 2) = 2 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 sh1i as a subform, max ind(w2(t1)) 6 max ind(w2(q1)). F, t1 F, q1 b(n−s−1)/2c On the other hand, max ind(w2(q1)) 6 2 by Lemma 3.5.1(b). It thus remains to show F, q1 that b(n−s−1)/2c max{ind(w2(t1))} 6 2 . (3.5.5) Consider the diagram

1 / Z/2Z / eSn / Sn / 1 O O i ? ? 1 / Z/2Z / Ae n / An / 1 + − where eSn can be either eSn or eSn . Since the rows are exact, the connecting morphisms fit into a commutative diagram 1 w2 2 H (F, Sn) / H (F, Z/2Z) O i∗

1 ∂F 2 H (F, An) / H (F, Z/2Z) 1 Once again, elements of H (F, Sn) are in a natural bijective correspondence with n-dimensional 1 1 etale algebras E/F . The image of the vertical map i∗ : H (F, An) → H (F, Sn) is readily seen to consist of etale algebras E/F of discriminant 1. Consequently,

max{ind(w2(t1))} = ind(Ae n, Z/2Z).

Let Hen be a Sylow subgroup of Ae n. By Proposition 3.2.7, the center Z(Hen) is cyclic. Thus by Lemma 3.2.5, ind(Ae n; Z/2Z) = ed(Hen) = ed(Hen; 2). b(n−s−1)/2c On the other hand, ed(Hen; 2) = ed(Ae n; 2) by Lemma 3.2.2 and ed(Ae n; 2) = 2 by Theorem 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 proof b(n−s)/2c of the upper bound ed(eSn) 6 2 of Theorem 3.1.2(b). Similarly for the upper bound b(n−s−1)/2c ed(Ae n) 6 2 in the proof of Theorem 3.1.2(c). On the other hand, we do not know b(n−s)/2c b(n−s−1)/2c how to prove the lower bounds ed(eSn) > 2 and ed(Ae n) > 2 entirely within the framework 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 the Introduction, a necessary condition for q to be a trace form is that it should contain sh1i as a subform. Theorem 3.1.5 suggests that this condition might be sufficient. The following example shows that, in fact, this condition is not sufficient. In this example, n = 4 = 22 and thus s = 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 = h1, a, b, ci be a 4-dimensional non-singular quadratic form over F . Clearly q contains sh1i = h1i as a subform. On the other hand, q is not isomorphic to the trace form t

34 + − 3.6. Comparison of essential dimensions of eSn and eSn of 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 isomorphism 2 3 2 between I (K)/I (K) and H (K, Z/2Z); cf. [Lam05, p. 115]. This is a special case of Milnor’s conjecture, which asserts the existence of an isomorphism

r r+1 r er : I (F )/I (F ) → H (F, Z/2Z) for any r > 0, with the property that er takes the r-fold Pfister form h1, a1i ⊗ · · · ⊗ h1, ari to the cup product (a1) ∪ (a2) ∪ · · · ∪ (ar); see [Pfi95, p. 33]. Milnor’s conjecture has been proved by V. Voevodsky; see [Kah97] for an overview. It is natural to ask if the following variant of Theorem 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 an even positive integer, where a1 > ··· > as > 1. Is it true that

max ind(er(q)) = max ind(er(t)) ? (3.5.6) F, q F, t

Here the maximum is taken as F ranges over all fields containing k, q ranges over all n-dimensional forms in Ir(F ) containing s h1i2 and t ranges over all n-dimensional trace forms in Ir(F ). The r index of a class α ∈ H (F, Z/2Z) is the greatest common divisor of the degrees [E : F ], where E/F ranges over splitting fields for α with [E : F ] < ∞. 1 2 For r = 1, it is easy to see that (3.5.6) holds. In this case the Milnor map e1 : I (F )/I (F ) → 1 1 ∗ ∗ 2 H (F, Z/2Z) is the discriminant, the index of an element of α = H (F, Z/2Z) = F /(F ) is 1 or 2, depending on whether α = 0 or not, and the question boils down to the existence of an n-dimensional ´etalealgebra E/F with non-trivial discriminant. In the case where r = 2, the equality (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 eSn and eSn

+ − We believe that eSn and eSn should have the same essential dimension, but are only able to establish the following slightly weaker assertion. √ Proposition 3.6.1. Let k be a field of characteristic 6= 2 containing −1. Then

+ − | edk(eSn ) − edk(eSn )| 6 1.

+ Proof. Let V be the spin representation of eSn , Sn → PGL(V ) be the associated projective representation of Sn, and Γ ⊂ GL(V ) be the preimage of Sn under the natural projection ± π : GL(V ) → PGL(V ). Note that Γ is a 1-dimensional group, and eSn are finite subgroups of Γ. By Lemma 3.2.4(a), edk(Γ) > edk(eSn) − 1, (3.6.1)

2 If s is even, and q = r ⊕ sh1i, then q and r are Witt equivalent over F . Thus maxF, q ind(er(q)) can be r replaced by maxF, r ind(er(r)), as r ranges over all (n − s)-dimensional forms in I (F ). The same is true if s is odd: here r ranges over the (n − s)-dimensional forms in Ir(F ) such that r ⊕ h1i is in Ir(F ).

35 3.7. Explanation of the entries in Table 3.1

+ − where eSn denotes eSn or eSn . On the other hand, since Γ is generated by eSn and Gm, and Gm is central in Γ, we see that Sfn is normal in Γ. The exact sequence

π 1 / eSn / Γ / Gm / 1 induces an exact sequence

1 1 π 1 H (K, eSn) / H (K, Γ) / H (K, Gm) = 1

1 of Galois cohomology sets. Here K is an arbitrary field containing k, and H (K, Gm) = 1 by 1 1 Hilbert’s Theorem 90. Thus the map H (K, eSn) → H (K, Γ) is surjective for every K. By Lemma 3.2.3, edk(eSn) > edk(Γ). (3.6.2) + Combining the inequalities (3.6.1) and (3.6.2), we see that each of the integers edk(eSn ) and − + − edk(eSn ) equals either edk(Γ) or edk(Γ) + 1. Hence, edk(eSn ) and edk(eSn ) differ by at most 1, as claimed. + − Remark 3.6.2. The inequality | edk(eSn ) − edk(eSn )| 6 1 of Lemma 3.6.1 remains valid if char(k) = 2; see Theorem 3.1.4(a).

3.7 Explanation of the entries in Table 3.1

Throughout 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(Ae n; 2) in the second row of Table 3.1 are given by Theorem 3.1.2(c). In the third row,

• ed(Ae 4) = 2 by Corollary 3.1.3(a).

• To show that ed(Ae 5) = 2, combine the inequalities ed(Ae 5) > ed(Ae 5; 2) > 2 of Theo- rem 3.1.2(c) and ed(Ae 5) 6 2 of Theorem 3.1.2(a). Alternatively, see [Pro17, Lemma 2.5].

• ed(Ae 6) = 4 by [Pro17, Proposition 2.7].

• To show that ed(Ae 7) = 4, note that ed(Ae 7) > 4 because Ae 7 contains Ae 6 and ed(Ae 7) 6 4 because Ae 7 has a faithful 4-dimensional representation; see [Pro17, Corollary 2.1.4].

36 3.7. Explanation of the entries in Table 3.1

• The values of ed(Ae 8) = 8 and ed(Ae 16) = 128 are taken from Corollary 3.1.3(a).

• When 9 6 n 6 15 the range of values for ed(Ae n) is given by the inequality

ed(Ae n) 6 ed(Ae n) + ed(An) 6 ed(Ae n) + n − 3; see Theorem 3.1.2(e).

37 Chapter 4

Classifying spaces for ´etalealgebra with generators

4.1 Introduction

Given a topological group G, one may form the classifying space, well-defined up to homotopy equivalence, as the base space of any principal G-bundle EG → BG where the total space is contractible. If G is a finite nontrivial group, then BG is necessarily infinite dimensional as a topological 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 G on which G acts freely outside of a high-codimension closed set Z, and such that (V − Z)/G is defined as a quasiprojective variety. The higher the codimension of Z in V , the better an approximation (V − Z)/G is to the notional BG. In this chapter, we consider the case of G = Sn, the symmetric group on n letters. The representations we consider as our V s are the most obvious ones, r-copies of the permutation n representation of Sn on A . The closed loci we consider minimal: the loci where the action is not free. We use the language of ´etalealgebras to give an interpretation of the resulting spaces. For a fixed Sn and field k, and for a given multiple r of the permutation representation, the k-variety n B(r; A ) := (V − Z)/Sn produced by this machine represents “´etalealgebras equipped with r n generating global sections” up to isomorphism of these data. The varieties B(r; A ) are therefore r in the same relation to the group Sn as the projective spaces P are to the group scheme Gm. Section 4.2 is concerned with preliminary results on generation of ´etalealgebras. The main n construction of the chapter, that of B(r; A ), is made in Section 4.3, and the functor it represents is described. A choice of r global sections generating an ´etalealgebra A of degree n on a k-variety X n corresponds to a map φ : X → B(r; A ). While the map φ is dependent on the chosen generating sections, we show in Section 4.4 that if one is prepared to pass to a limit, in a sense made precise 1 n n there, that the A -homotopy class of a composite φ˜ : X → B(r; A ) → B(∞; A ) depends only on the isomorphism class of A and not the generators. As a practical matter, this means that for a wide range of cohomology theories, E∗, the map E∗(φ˜) depends only on A and not on the generators used to define it. In Section 4.5, we observe that the motivic cohomology, and therefore the Chow groups, of 2 the varieties B(r; A ) has already been calculated in [DI07]. A degree-2 or quadratic ´etalealgebra A over a ring R carries an involution σ and a trace map Tr : A → R. There is a close connection between A and the rank-1 projective module L = ker(Tr). In Section 4.6, we show that the algebra A can be generated by r elements if and only 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 smooth affine r − 1-dimensional R-variety Spec R and a line bundle L on Spec R requiring r global

38 4.1. Introduction sections to generate. This shows a that a bound of O. Forster [For64] on the minimal number of sections required to generate a line bundle on Spec R, namely dim R + 1, is sharp. In light of Section 4.6, the same smooth affine R-variety of dimension r − 1 can be used to produce ´etale algebras A, of arbitrary degree n, requiring r global sections to generate. This fact was observed independently by M. Ojanguren. It shows that a bound established by U. First and Z. Reichstein in [FR17] is sharp in the case of ´etalealgebras: they can always be generated by dim R + 1 global sections and this cannot be improved in general. The details are worked out in Section 4.7, and we incidentally show that the example of S. Chase follows easily from our construction 2 2 of B(r; A ) and some elementary calculations in the singular cohomology of B(r; A )(R). Finally, we offer some thoughts about determining whether the bound of First and Reichstein is sharp if one restricts to varieties over algebraically closed fields.

4.1.1 Notation and other preliminaries 1. 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 different from 2. 3. k − Alg denotes the category of commutative, unital, associative k-algebras and k-algebra morphisms. 4. A variety X is a geometrically reduced, separated scheme of finite type over a field. We do not require the base field to be algebraically closed, nor do we require varieties to be irreducible. 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 scheme X as the presheaf of sets it represents on a category of schemes

X(U) = MorSch(U, X). In fact, the presheaf X(·) is a sheaf on the big Zariski site of all schemes, which is to say that S if W = i∈I Ui is a cover of a scheme by Zariski open subschemes, then Y Y X(W ) → X(Ui) ⇒ X(Ui ∩ Uj) i∈I (i,j)∈I2 is a coequalizer diagram. 1 Remark 4.1.1. The scheme A = Spec Z[t] represents the functor 1 X 7→ A (X) = Γ(X, OX ) = OX (X). n Similarly, A represents the functor n X 7→ (OX (X)) . This can be deduced from the case of affine X, where one has n HomZ−Alg(Z[t1, . . . , tn],R) = R as sets.

39 4.2. Etale´ algebras

4.2 Etale´ algebras

Let R be a ring and S an R-algebra. Then there is a morphism of rings µ : S ⊗R S → S sending a ⊗ b to ab. We obtain an exact sequence µ 0 → ker(µ) → S ⊗R S −→ S → 0 (4.2.1) Definition 4.2.1. Let R be a ring. An R-algebra S is called finite ´etale 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 the ´etalealgebra 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 algebra Rn with componentwise addition and multiplication. The following lemma states that all ´etalealgebras are fppf -locally isomorphic to the trivial one. Lemma 4.2.3. Let R be a ring and S an R-algebra. Let S be a faithful R-module. Then the following are equivalent: 1. S is an ´etalealgebra of degree n. ∼ n 2. There is a faithfully flat R-algebra T such that S ⊗R T = T 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. For simplicity, we assume A has a constant degree n. We say that A is an ´etale X-algebra or ´etale algebra over X if for every open affine subset U ⊂ X the A(U) is an ´etalealgebra, and we call n the rank of A. Remark 4.2.5. For X a k-scheme and n a positive integer there exists a trivial ´etalealgebra n OX 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 are equivalent: 1. A is an ´etale X-algebra of degree n.

f 2. There is an affine flat cover {U −→i X} such that f ∗A =∼ On as O -algebras. i i Ui Ui Proof. This is immediate from Lemma 4.2.3.

Definition 4.2.7. If A is an ´etalealgebra over a ring R, then a subset Λ ⊂ A is said to generate A 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 Λ (a1,...,ar) agrees with the image of the evaluation map R[x1, . . . , xr] → A. Therefore, saying that Λ generates A is equivalent to saying this map is surjective.

40 4.2. Etale´ algebras

Proposition 4.2.8. Let Λ = {a1, . . . , ar} be a finite set of elements of A, an ´etalealgebra over a 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 ∈ MaxSpec R, 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 ∈ MaxSpec R, the image of 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 is equivalent 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 epimorphism in the category of R-modules, and conditions (2)-(4) are well-known equivalent conditions saying that this map is an epimorphism.

Using Proposition 4.2.8, we extend the definition of “generation of an algebra” from the case where the base is affine to the case of a general scheme.

Definition 4.2.9. Let A be an ´etalealgebra over a scheme X. For Λ ⊂ Γ(X, A) we say that Λ generates A if, for each open affine U ⊂ X the OX (U)-algebra A(U) is generated by restriction of sections in Λ to U.

4.2.1 Generation of trivial algebras n Let n ≥ 2 and r ≥ 1. Consider the trivial ´etalealgebra OX over a scheme X. A global section of n this algebra is equivalent to a morphism X → A , and an r-tuple Λ of sections is a morphism n r n r X → (A ) . One might hope that the subfunctor F ⊆ (A ) of r-tuples of sections generating n OX as an ´etalealgebra is representable, and this turns out to be the case. n r In order to define subschemes of (A ) , 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).

2 n r Notation 4.2.10. Fix n and r as above. For (i, j) ∈ {1, . . . , n} with i < j, let Zij ⊂ (A ) Tr denote the closed subscheme given by the intersection of the vanishing loci k=1 V (xki − xkj). n n r Write U(r; A ), or U(r) when n is clear from the context, for the open subscheme of (A ) given by n n r [ U(r; A ) = (A ) − Zij i

n n r Proposition 4.2.11. Let n ≥ 2 and r ≥ 1. The open subscheme U(r; A ) ⊂ (A ) represents n the functor sending a scheme X to r-tuples (a1, . . . , ar) of global sections of OX that generate it as an OX -algebra.

41 4.2. Etale´ algebras

n r Proof. Temporarily, let F denote the subfunctor of (A ) defined by

n r n F(X) = {Λ ⊆ (Γ(X,OX ) | Λ generates OX }.

It follows from Proposition 4.2.8 and Definition 4.2.9 that F is actually a sheaf on the big Zariski site. n n r Both U(r; A ) and F are subsheaves of the sheaf represented by (A ) , and therefore in order n to show they agree, it suffices to show U(r; A )(R) = F(R) when R is a local ring. n Let R be a local ring. The set U(r; A )(R) consists of certain r-tuples (~a1, . . . ,~ar) of elements n of R . Letting aki denote the i-th element of ~ak, then the r-tuples are those with the property × that for each i 6= j, there exists some k such that aki − akj ∈ R . The proposition now follows from 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-tuple n of elements in R , and let aki denote the i-th element of ~ak. The following are equivalent:

n • The set {~a1, . . . ,~ar} generates the (trivial) ´etale R-algebra R .

• For each pair (i, j) satisfying 1 ≤ i < j ≤ n, there is some k ∈ {1, . . . , r} such that the × element aki − akj is a unit in R .

n n Proof. Suppose {~a1, . . . ,~ar} generates R as an algebra. That is, any n-tuple (r1, . . . , rn) ∈ R may be expressed by evaluating a polynomial p ∈ R[X1,...,Xr] at (~a1, . . . ,~ar), i.e., for any i ∈ {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 a polynomial p ∈ R[X1,...,Xr] such that p(a1i, a2i, . . . , ari) = 1 and p(a1j, a2j, . . . , arj) = 0. We remark 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 obtain

1 = p(a1i, a2i, . . . , ari) − p(a1j, a2j, . . . , arj) = 0 mod m a 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 a unit. For any pair i 6= j, we can find a polynomial pi,j ∈ R[x1, . . . , xr] with the property that pi,j(a1i, . . . , ari) = 1 and pi,j(a1j, . . . , arj) = 0 by taking

−1 pi,j = (aki − akj) (xk − akj) for instance. Q Let pi = j6=i pi,j. The polynomial pi has the property that ( 1 if i = j pi(a1j, . . . , arj) = 0 otherwise and from here it is immediate that evaluation at (~a1, . . . ,~ar) yields a surjection R[x1, . . . , xn] → Rn.

42 4.3. Classifying spaces

4.3 Classifying spaces

Fix n ≥ 2 and r ≥ 1. In this subsection we work over a fixed field k. n We tacitly change base from Spec Z to Spec k, so that U(r; A ) denotes the k-variety that n should properly be written U(r; A ) ×Spec Z Spec k. The reason we make this change of base is to use standard results about quotient varieties.

Notation 4.3.1. For a given k-variety X, a degree-n ´etalealgebra A with r generating sections denotes the data of a degree-n ´etalealgebra 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 isomorphism 0 0 0 0 ψ :(A, a1, . . . , ar) → (A , a1, . . . , ar) of such data consists of an isomorphism ψ : A → A of 0 ´etalealgebras over X such that ψ(ai) = ai 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 isomorphism classes of degree-n ´etalealgebras over X, and so there is a set of isomorphism classes of degree-n ´etalealgebras with r generating sections. Since generation is a local condition by Proposition 4.2.8, it follows that there is a functor

n F(r; A ): k-Var → Set, n F(r; A )(X) = {[A, a1, . . . , ar] | (A, a1, . . . , ar) is a degree-n ´etalealgebra over X and r generating sections}

n n The purpose of this section is to produce a variety B(r; A ) representing the functor F(r; A ), 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 ´etale F -algebra F n may be calculated as follows: The elements

n ei = (0,..., 0, 1, 0,..., 0) ∈ F

2 are determined, up to reordering, by the conditions that ei = ei, ei 6= 0, eiej = 0 for i 6= j and Pn n n i=1 ei = 1 in the ´etalealgebra structure on F . Any automorphism of the ´etale F -algebra F permutes the ei and is determined by this permutation, and from there it is immediate that n AutF -alg(F ) is the symmetric group, Sn.

n There is an action of the symmetric group Sn on A , given by permuting the coordinates, and n r from there, there is a diagonal action of Sn on (A ) , and one verifies that the action restricts to n one on the open subscheme U(r; A ). n Proposition 4.3.4. The action of Sn on U(r; A ) is scheme-theoretically free. n n Proof. Since U(r; A ) is a variety, it suffices to verify that the action is free on the sets U(r; A )(K) where K/k is a field extension. Here one is considering the diagonal Sn action on r-tuples n (~a1, . . . ,~ar) where each ~al ∈ K 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.

43 4.3. Classifying spaces

n n Notation 4.3.5. There is a free diagonal action of Sn on U(r; A )×A , such that the projection n n n n n p : U(r; A ) × A → U(r; A ) is equivariant. Write q : E(r; A ) → B(r; A ) for the induced map of quotient varieties. These are again quasiprojective varieties (by, for instance [Mum08, Section 7]) and there is a commutative square

n n π0 n U(r; A ) × A E(r; A ) p q

n π n U(r; A ) B(r; A )

n n n Remark 4.3.6. The sheaf of sections of the map p : U(r; A ) × A → U(r; A ) is the trivial degree-n ´etalealgebra On on U(r; n). The action of S on these sections is by algebra U(r;An) A n n n automorphisms, and so the sheaf of sections of the quotient map q : E(r; A ) → B(r; A ) is n n endowed with the structure of a degree-n ´etalealgebra E(r; A ) on B(r; A ). We will often n n n confuse the scheme E(r; A ) over B(r; A ) with the ´etalealgebra of sections E(r; A ). r The map p has r canonical sections {sj}j=1 given as follows:

sj(~x1, ~x2, . . . , ~xr) = ((~x1, ~x2, . . . , ~xr, ), ~xj).

n n r These sections are Sn-equivariant, and so descend to sections {ti : B(r; A ) → E(r; A )}i=1 of the map q. n n Remark 4.3.7. The quotient variety B(r; A ) is smooth since U(r; A ) is smooth and π is n ´etale—see[Mil80, Ch. I, Remark 2. 24]. Since π is finite it is proper. The variety B(r; A ) is n n quasiprojective but not projective. Indeed, if B(r; A ) → Spec(k) were proper then U(r; A ) → n Spec(k) would be proper too, but U(r; A ) is an open subvariety of an affine variety.

4.3.2 The functor represented by B(r, An) n n We now establish the canonical isomorphism of the functors B(r, A )(X) = F(r; A )(X). n n n By Remark 4.3.6, F(r; A )(B(r; A )) has a canonical element [E(r; A ), t1, . . . , tr]. n Lemma 4.3.8. If [A, s1, . . . , sr] ∈ F(r; A )(L) where L is a separably closed field over k, then n there exists a unique morphism of schemes φ : Spec(L) → B(r; A ) such that ∗ n ∗ ∗ [A, s1, . . . , sr] = [φ (E(r; A )), φ t1, . . . , φ tr]

ψ n n Proof. Since L is separably closed, there exists an L-isomorphism A −→ L . Let {ψ(si)} ⊂ L denote the corresponding sections of Ln. ˜ n We thus obtain a map φ : Spec(L) → U(r; A ) defined by giving the L-point (ψ(s1), . . . , ψ(sr)). n n Post-composing this map with the projection U(r; A ) → B(r; A ), we obtain a morphism n ∗ n ∗ φ : Spec(L) → B(r; A ). It is a tautology that φ (E(r; A )) = A and φ (ti) = si. It now behooves us to show that φ does not depend on the choices made in the construction. 0 n Suppose φ : Spec L → B(r; A ) is another morphism satisfying the conditions of the lemma. n 0 n We may lift this L-point of B(r; A ) to an L-point φ˜ : Spec L → U(r; A ), since π is finite and L is separably closed field. By hypothesis we have

0∗ n 0∗ 0∗ [A, s1, . . . , sr] = [φ (E(r; A )), φ t1, . . . , φ tr]. ˜ ˜0 n 0 Thus φ and φ differ by a Sn = AutL(L ) automorphism, which is to say φ = φ as required.

44 4.3. Classifying spaces

n The universality of B(r; A ) extends to all k-varieties, as follows. n Proposition 4.3.9. If X is a variety over k and if [A, s1, . . . , sr] ∈ F(r; A )(X) then there exists n a unique morphism of k-schemes φ : X → B(r; A ) such that

∗ n ∗ ∗ [A, s1, . . . , sr] = [φ (E(r; A )), φ t1, . . . , φ tr]. (4.3.1)

n Proof. If [A, s1, . . . , sr] ∈ F(r; A )(X) then there exists an ´etalecover U = {Spec(Ri) → X} ∼ = n of reduced affine schemes such that there exist isomorphisms ψi : A|Spec Ri −→ Ri . Write Ui = Spec Ri. Then we obtain a morphism

φ =(s | ,...,s | ) i 1 Ui r Ui n Ui −−−−−−−−−−−−→ U(r; A ).

n We post-compose with π to get maps φi : Ui → B(r; A ). The maps φi|Ui×Uj and φj|Ui×Uj agree on all geometric points by Lemma 4.3.8 and thus are equal. By ´etaledescent the maps φi n define a map φ : X → B(r; A ) satisfying (4.3.1). 0 n If φ : X → B(r; A ) is a different map satisfying (4.3.1), then φ and φ must differ on some geometric point. This is not possible by Lemma 4.3.8.

n Corollary 4.3.10. The functor F(r; A ) of Definition 4.3.2 is represented by the scheme n B(r; A ). Example 4.3.11. Let us consider the toy example where X = Spec K where K is a field containing k, n ≥ 2, and where r = 1. That is, we are considering ´etale algebras A/K along with a chosen generating element a ∈ A. After base change to the separable closure, Ks, we obtain a s Sn-equivariant isomorphism of K -algebras:

=∼ s ×n ψ : AKs −→ (K ) .

For the sake of the exposition, use ψ to identify source and target. The element a ∈ A yields a chosen generating element a˜ ∈ (Ks)n. The element a˜ is a vector of n pairwise distinct elements s s n of K . The element a˜ is a K -point of U(1; A ). In general, this point is not defined over K, but n its image in B(1; A ) is. n n n n n s Since U(1; A ) ⊆ A , and B(1; A ) = U(1; A )/ Sn, the image of a˜ in B(1; A )(K ) may be presented as the elementary symmetric polynomials in the ai. To say that the image of n a˜ = (a1, . . . , an) in B(1; A ) is defined over K is to say that the coefficients of the polynomial Qn i=1(x − ai) are defined in K. n The variety B(1; A ) 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 a field of characteristic different from 2, and n = 2. 2 −1 The variety B(1; A ) may be presented as spectrum of the C2-fixed subring of k[x, y, (x−y) ] under the action interchanging x and y. This is k[(x + y), (x − y)2, (x − y)−2], although it is more elegant to present it after the change of coordinates c1 = x + y and c0 = xy:

2 2 −1 B(1; A ) = Spec k[c1, c0, (c1 − 4c0) ]

45 4.4. Stabilization in cohomology

A quadratic ´etale k-algebra equipped with the generating element a corresponds to the point 2 2 (c1, c0) ∈ B(1; A )(k) where a satisfies the minimal polynomial a − c1a + c0 = 0. For instance if k = R, the quadratic ´etalealgebra of complex numbers C with generator 2 2 2 s + ti over R (here t 6= 0), corresponds to the point (2s, s + t ) ∈ B(1; A )(R), whereas R × R, 2 2 generated by (s + t, s − t) over R (again t 6= 0) , corresponds to the point (2s, s − t ).

4.4 Stabilization in cohomology

n We might wish to use the varieties B(r; A ) to define cohomological invariants of ´etalealgebras. The idea is the following: suppose given such an algebra A on a k-scheme X, and suppose one n can find generators (a1, . . . ar) for A. Then one has a classifying map φ : X → B(r; A ), and one may apply a cohomology functor E∗, such as Chow groups or algebraic K-theory, to obtain ∗ ∗ n ∗ “characteristic classes” for A-along-with-(a1, . . . , ar), in the form of φ : E (B(r; A )) → E (X). The dependence on the specific generators chosen is a nuisance, and we see in this section that this dependence goes away provided we are prepared to pass to a limit “B(∞)” and assume that ∗ 1 ∗ ∗ 1 the theory E is A -invariant, in that E (X) → E (X × A ) is an isomorphism. n n Definition 4.4.1. There are stabilization maps U(r; A ) → U(r+1; A ) obtained by augmenting an r-tuple of n-tuples by the n-tuple (0, 0,..., 0). These stabilization maps are Sn-equivariant n n and therefore descend to maps B(r; A ) → B(r + 1; A ). The stabilization maps defined above may be composed with one another, to yield maps n 0 n 0 B(r; A ) → B(r ; A ) for all r < r . These maps will also be called stabilization maps. n Proposition 4.4.2. Let X be a regular k-scheme. Suppose [A, a1, . . . , ar] ∈ F(r; A )(X) and 0 0 0 n ∼ 0 [A , a1, . . . , ar0 ] ∈ F(l; A )(X) have the property that A = A as ´etalealgebras. Let φ : X → n 0 0 n 0 B(r; A ) and φ : X → B(r ; A ) be the corresponding classifying morphism. For R = r + r , the n n 0 0 n n composite maps φ˜ : X → B(r; A ) → B(R; A ) and φ˜ : X → B(r ; A ) → B(R; A ) given by 1 stabilization are na¨ıvely A -homotopic. 1 0 1 An “elementary A -homotopy” between maps φ, φ : X → B is a map Φ : X × A → B 0 0 1 specializing to φ at 0 and φ at 1. Two maps φ, φ : X → B are “naively A -homotopic” if they may be joined by a finite sequence of elementary homotopies. Two naively homotopic maps 1 are identified in the A -homotopy theory of schemes of [MV99], but they do not account for all identifications in that theory. (Weak) equivalences in the homotopy category HA1 (k) of smooth 1 r pr2 k-schemes are called A -equivalences. In this category, one posits that X × A −−→ X is an 1 A -equivalence. Proof. We may assume that A = A0. We may also assume, by padding, that r = r0. 1 Write t for the parameter of A . Let A[t] denote the pull-back of A along the projection 1 X × A → X. 0 0 Consider the sections ((1 − t)a1,..., (1 − t)ar, ta1, . . . , tar) of A[t]. Since either t or (1 − t) is 1 a unit at all local rings of points A , by appeal to Proposition 4.2.8 and consideration of the 1 1 0 0 restrictions to X ×(A −{0}) and X ×(A −{1}), we see that ((1−t)a1,..., (1−t)ar, ta1, . . . , tar) furnish a set of generators for A[t]. At t = 0, they specialize to (a1, . . . , ar, 0,..., 0), viz., the n n generators specified by the stabilized map φ : X → B(r; A ) → B(2r; A ). At t = 1, they 0 0 specialize to (0,..., 0, a1, . . . , ar), which is not precisely the list of generators specified by 0 n n 1 φ : X → B(r; A ) → B(2r; A ), but may be brought to this form by another elementary A homotopy.

46 2 4.5. The motivic cohomology of the spaces B(r; A )

0 ∗ 1 Corollary 4.4.3. Let φ and φ be as in the previous proposition. If E denotes any A -invariant cohomology theory, then E∗(φ˜) = E∗(φ˜0).

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 motivic 2 cohomology of the spaces B(r; A ) has already been calculated in [DI07].

4.5.1 Change of coordinates 2 ∼ r r Lemma 4.5.1. There is an equivariant isomorphism U(r; A ) = A \{0} × A , where C2 acts r r as multiplication by −1 on first factor A \{0} and trivially on the second factor A . Taking 2 ∼ r r quotient by C2-action yields B(r; A ) = (A \{0})/C2 × A . Proof. By means of the change of coordinates

xi − yi = zi, xi + yi = wi

2 ∼ r r 2 we see that U(r; A ) = (A \{0})×A . Moreover, the action of C2 on U(r; A ) is given by zi 7→ −zi 2 2 ∼ r r and wi 7→ wi. We therefore obtain an isomorphism B(r; A ) = U(r; A )/C2 = (A \{0})/C2 × A . 2 r 2 ∼ 2 r Write V (r; A ) for A \{0}/C2. It is immediate that B(r; A ) = V (r; A ) × A , and so there is a 2 2 1 split inclusion V (r; A ) → B(r; A ) which is moreover an A -equivalence.

4.5.2 The deleted quadric presentation 2r−1 Definition 4.5.2. Endow P with the projective coordinates a1, . . . , ar, b1, . . . , br. Let Q2r−2 Pr denote the closed subvariety given by the vanishing of i=1 aibi, and let DQ2r−1 denote the 2r−1 open complement P \ Q2r−2. The main computation of [DI07] is a calculation of the modulo-2 motivic cohomology of DQ2r−1, and of a family of related spaces DQ2r (which are complements of the quadrics P 2 2r i aibi + c in P ). Denote the modulo-2 motivic cohomology of Spec k by M2. This is a n,i bigraded ring whose graded components M2 are concentrated in degrees 0 ≤ n ≤ i. There are 1,1 M 1,1 two notable classes, ρ ∈ M2 , the reduction modulo 2 of −1 ∈ K1 (k) = H (Spec k, Z), and 0,1 2 τ ∈ M2 , corresponding to the identity (−1) = 1. If −1 is a square in k, then ρ = 0, but τ is always a nonzero class.

Proposition 4.5.3 (Dugger–Isaksen, [DI07] Theorem 4.9). There is an isomorphism of graded rings [a, b] H∗,∗(DQ ; ) =∼ M2 2r−1 F2 (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 map ∗,∗ ∗,∗ H (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, owing ∗ to the Beilinson–Lichtenbaum conjecture [Voe03], it subsumes the calculation of H´et(DQ2r−1, F2).

47 2 4.5. The motivic cohomology of the spaces B(r; A )

∗ ∼ For instance, if k is algebraically closed, then M2 = F2[τ], and one deduces that H´et(DQ2r−1, F2) = 2 r 2r F2[a, b]/(a − b, b ) = F2[a]/(a ). 2n,n n In the second, since H (·, F2) is identified with CH (·) ⊗Z F2, the calculation of the proposition subsumes that of the Chow groups modulo 2. In fact, the extension problems that ∗,∗ prevented Dugger and Isaksen from calculating H (DQ2r−1; Z) do not arise in this range, and by reference to the appendix of [DI07], which in turn refers to [KM90], one can calculate the integral Chow rings. This is done in the first two paragraphs of the proof of [DI07, Theorem 4.9]. Proposition 4.5.4. One may present [b] CH∗(DQ ) = Z , |b| = 1. 2r−1 (2b, br)

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) generated by b. The reason we have explained all this is that there is a composite map

r 2 DQ2r−1 → (A \{0})/C2 → B(r; A ) (4.5.1) 1 both of which are A -equivalences, and so Propositions 4.5.3 and 4.5.4 amount to a calculation 2 of the motivic and ´etalecohomologies and Chow rings of B(r; A ). 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 this 2 section to compute the stable invariants of B(r; A ) in the sense of Section 4.4. 1 2 r The A -equivalence B(r; A ) → (A \{0})/C2 was constructed above in Lemma 4.5.1, so it remains to prove the following.

Lemma 4.5.5. Let r ≥ 1. The variety DQ2r−1 is affine and has coordinate ring

" #C2 k[x1, . . . , xr, y1, . . . , yr] R = Pr
(4.5.2) 1 − i=1 xiyi where the C2 action on xi and yi is by xi 7→ −xi and yi 7→ −yi. 2r−1 Proof. The variety DQ2r−1 is a complement of a hypersurface in P , and is therefore affine. Let Q denote a1b1 + ··· + arbr. The coordinate ring of DQ2r−1 is the ring of degree-0 terms −1 −1 in the graded ring S = k[a1, . . . , ar, b1, . . . , br,Q ], where |ai| = |bi| = 1 and |Q | = −2. This −1 −1 −1 ring is the subring of S generated by the terms aiajQ , aibjQ and bibjQ . Consider the ring k[x1, . . . , xr, y1, . . . , yr] T = Pr . (4.5.3) (1 − i=1 xiyi) One may define a map of rings φ : S → T by sending ai 7→ xi and bi 7→ yi, since

Q 7→ 1 under this assignment. Restricting to Γ(DQ2r−1, ODQ2r−1 ) ⊂ S, one obtains a map

Γ(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 → T contains only one homogeneous element, 0, so that the restriction of this map to the subring of degree-0 terms in S is injective. The kernel of φ is the ideal (Q − 1). Since S is an integral domain, degree considerations imply that no nonzero multiple of (Q − 1) is homogeneous.

48 4.6. Relation to line bundles in the quadratic case

1 r Proposition 4.5.6. For all r, there is an A -equivalence DQ2r−1 → (A \{0})/C2. Proof. Let T be as in the proof of Lemma 4.5.5. It is well known that Spec T is an affine vector r ∼ 1 r−1 bundle torsor over A \{0}. In fact, for each j ∈ {1, . . . , r}, if we define Uj = A \{0} × A to r be the open subscheme of A \{0} where the j-th coordinate is invertible, then we arrive at a pull-back diagram r−1 ∼ A × Uj = Spec T ×Ar\{0} Uj / Spec T

 r  Uj / A \{0}

Since Uj inherits a free C2-action, it follows that in the quotient we obtain a vector bundle r−1 r 1 (A × Uj)/C2 → Uj/C2, and so the map (Spec T )/C2 → (A \{0})/C2 is an A -equivalence, as claimed.

As a consequence of Proposition 4.5.6 we observe that the affine variety DQ2r−1 is an affine 2 approximation of B(r; A ).

4.6 Relation to line bundles in the quadratic case

We 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 ´etalealgebra is C2, the cyclic 1 group of order 2, which happens to be a subgroup of Gm. More explicitly, H´et(Spec R,C2) is an abelian group which is isomorphic to the isomorphism classes of quadratic ´etalealgebras on Spec R. On the other hand due to the Kummer sequence and C2 ⊂ Gm we have

∗ ∗2 1 0 → R /R → H´et(Spec R,C2) → 2 Pic(R) → 0

1 which means that H´et(Spec R,C2) is also isomorphic to isomorphism class of 2-torsion line bundle =∼ L 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 A be a quadratic ´etalealgebra on X. There is a trace map [Knu91, Section I.1]:

Tr : A → O and 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 X 0 → L → A → O → 0 (4.6.1) 1 is split short exact, where the splitting O → A is given on sections by x 7→ 2 x. 1 The construction of L from A gives an explicit instantiation of the map H´et(X,C2) → 1 H´et(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.

49 4.7. The example of Chase

Construction 4.6.2. Let X be as above, and let L be a line-bundle on X such that there is an isomorphism L ⊗ L → O. Let φ : L ⊗ L → O be a specific choice of isomorphism. From the data (L, φ), we may produce an ´etalealgebra A = O ⊕ L on which the multiplication is given, on sections, by (r, x) · (r0, x0) = (rr0 + φ(x ⊗ x0), rx0 + r0x). Proposition 4.6.3. Let X be a scheme such that 2 is invertible in all residue fields of points of X. Let A a quadratic ´etalealgebra on X. Let L be the associated line bundle to A, as in Construction 4.6.1. Then A can be generated by r global sections as an ´etalealgebra if and only if 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 the 1 projection q(a) = a − 2 Tr(a). The questions of generation of A and of L may be reduced to stalks at points of X, by Proposition 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 a quadratic ´etalealgebra. Since 2 is invertible, we may write A = R[z]/(z2 − a) for some element a ∈ R,[Knu91, Lemma 4.1.1]. In this presentation, σ(z) = −z and Tr(az + b) = 2b. The kernel of 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 and only if q(~a) = (a1z, . . . arz) do. This tuple generates A as an algebra if and only if at least one of × the 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 ´etalealgebra at the closed point of Spec R 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 at least 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. An 2 ´etalealgebra of degree 2 generated by r global sections corresponds to a map X → B(r; A ). r−1 A line bundle generated by r global sections corresponds to a map X → P . In the light of 2 r−1 Proposition 4.6.3, there must be a map of varieties B(r; A ) → P . This map is given by

2 =∼ r r p1 r r =∼ r−1 B(r; A ) → (A \{0})/C2 × A → (A \{0})/C2 → (A \{0})/Gm −→ P where the morphisms are, left to right, the isomorphism of Lemma 4.5.1, projection onto the second factor, and the map induced by the inclusion C2 ⊂ Gm.

4.7 The example of Chase

The following will be referred to as “the example of Chase”. Construction 4.7.1. Let [z , . . . , z ] S = R 1 r  Pr 2  i=1 zi − 1

C which is equipped with a C2-action given by zi 7→ −zi. Let R = S 2 . The dimension of both R and S is r − 1. The ring R carries a projective module of rank 1, i.e., a line bundle, that requires r global sections in order to generate it. This example given in [Swa67, Theorem 4].

50 4.7. The example of Chase

Remark 4.7.2. In fact, the line bundle in question is of order 2 in the Picard group, so Proposition 4.6.3 applies and there is an associated quadratic ´etalealgebra on Spec R = Y (r) requiring r generators. The algebra is, of course, dependent on a choice of trivialization of the square of the line bundle, but one may choose the trivialization so the ´etalealgebra in question is S itself as an R-algebra.

Remark 4.7.3. This construction shows that the bound of First and Reichstein, [FR17], on the number of generators required by an ´etalealgebra 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 ´etalealgebra over R requiring r elements to generate, so the bound is tight in the case of ´etalealgebras of arbitrary degrees. We owe this observation to Zinovy Reichstein.

The original method of proof that the line bundle in the example of Chase cannot be generated by fewer than r global sections uses the Borsuk–Ulam theorem. Here we show that a variation on that proof follows naturally from our general theory of classifying objects. The Borsuk–Ulam r theorem is a theorem about the topology of RP , so it can be no surprise that it is replaced here r by facts about the singular cohomology of RP .

4.7.1 The singular cohomology of the real points of B(r; A2) 2 In addition to the general results about the motivic cohomology of B(r; A ), we can give a 2 complete description of the homotopy type of the real points B(r; A )(R). If X is a nonsingular R-variety, then it is possible to produce a complex manifold from X by first extending scalars to C and then employing the usual Betti realization functor to produce a manifold X(C). Since X is defined over R, however, the resulting manifold is equipped with an ∼ action 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 if 1 0 1 f, g : X → Y are two maps of varieties and H : X × A → X is an A -homotopy between them, then f(R), g(R) are homotopic maps of varieties, via the homotopy obtained by restricting 1 0 H(R): X(R) × A (R) = X(R) × R → X (R) to the subspace X(R) × [0, 1]. 2 Using Lemma 4.5.1, present U(r; A ) as the variety of 2r-tuples (z1, . . . , zr, w1, . . . , wr) such that (z1, . . . , zr) 6= (0,..., 0). This variety carries an action by C2 sending zi 7→ −zi and fixing the 2 2 r r wi. We know U(r; A ) and B(r; A ) are naively homotopy equivalent to A \{0} and A \{0}/C2 respectively.

Construction 4.7.5. We now consider an inclusion that is not, in general, an equivalence. Let r P (r) = Spec S denote the subvariety of A \{0} consisting of r-tuples (z1, . . . , zr) such that Pr 2 r i=1 zi = 1. This is an (r − 1)-dimensional closed affine subscheme of A \{0}, invariant under r the C2 action on A \{0}. The quotient of P (r) by C2 is Y (r) = Spec R, and is equipped with r 2 an evident map Y (r) → (A \{0})/C2 → B(r; A ). Here S and R take on the same meanings as in Construction 4.7.1.

2 Proposition 4.7.6. Let notation be as in Construction 4.7.5. The real manifold B(r; A )(R) has the homotopy type of 2 r−1 a r−1 B(r; A )(R) ' RP RP .

51 4.7. The example of Chase

2 2 The closed inclusion Y (r) → B(r; A ) includes Y (r)(R) → B(r; A )(R) as a deformation retract of one of the connected components. 2 Proof. By Lemma 4.5.1 and Remark 4.7.4, the manifold B(r, A )(R) is homotopy equivalent to r r A {0}/C2(R). The manifold A {0}/C2(C) consists of equivalence classes of r-tuples of complex numbers (z1, ..., zr), where the zi are not all 0, under the relation

(z1, ..., zr) ∼ (−z1, ..., −zr). r The real points of A {0}/C2 consist of Galois-invariant equivalence classes. There are two components of this manifold: either the terms in (z1, ..., zr) are all real or they are all imaginary. r−1 In either case, the connected component is homeomorphic to the manifold RP . 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 complex Pr 2 manifold of r-tuples (z1, . . . , zr) satisfying i=1 zi = 1. Again, in an R-points, the zi are either Pr 2 all real or all purely imaginary. The condition i=1 zi = 1 is incompatible with purely imaginary Pr 2 zi, so Y (r)(R) is the manifold of r-tuples of real numbers (z1, . . . , zr) satisfying i=1 zi = 1, r r−1 taken up to sign. In short, Y (R) = RP . 2 As for the inclusion Y (r)(R) → B(r; A )(R), it admits the following description, as can be seen by tracing through all the morphisms defined so far. Suppose given an equivalence class of Pr 2 real numbers (z1, . . . , zr), satisfying i=1 zi = 1, taken up to sign. Then embed (z1, . . . , zr) as 2 r−1 the point of B(r; A )(R) given by the class of (z1, z2, . . . , zr, 0,..., 0). That is, embed RP in r r−1
r−1 r R × R \{0} /C2 by embedding RP ⊂ (R \{0})/C2 as a deformation retract, and then embedding the latter space as the zero section of the trivial bundle. It is elementary that this composite is also a deformation retract.

Remark 4.7.7. We remark that the functor X X(R) does not commute with colimits. For 2 2 instance U(r; A )(R)/C2, which is connected, is not the same as B(r; A )(R). 2 In fact, the two components of B(r; A )(R) as calculated above correspond to two isomorphism classes of quadratic ´etale R-algebras: one component corresonds to the split algebra R × R, and the other to the nonsplit C. ∗ r We will need two properties of H (RP ; F2) here. Both are standard and may be found in [Hat02]. ∗ r ∼ r+1 1.H (RP ; F2) = F2[θ]/(θ ) where |θ| = 1. r r+1 2. The standard inclusion of RP ,→ RP given by augmenting by 0 induces the evident reduction map θ 7→ θ on cohomology. 2 2 Proposition 4.7.8. We continue to work over k = R. Let sr : B(r; A ) → B(r + 1; A ) be the stabilization map of Definition 4.4.1. The induced map on cohomology groups ∗ j 2 j 2 sr :H (B(r + 1; A )(R); F2) → H (B(r; A )(R); F2) is an isomorphism when j ≤ r and is 0 otherwise ∗ 2 2 Proof. The map sr is arrived at by considering the inclusion U(r; A ) → U(r + 1; A ), which is given by augmenting an r-tuple of pairs (a1, b1, . . . , ar, br) by (0, 0), and then taking the quotient 2 2 by C2. After R-realization, one is left with a map B(r; A )(R) → B(r + 1; A )(R) which on each r r+1 connected component is homotopy equivalent to the standard inclusion RP → RP . The result follows.

52 4.7. The example of Chase

Proposition 4.7.9 (Ojanguren). Let S and R be as in Construction 4.7.1. The quadratic ´etale algebra S/R cannot be generated by fewer than r elements.

2 Sketch of proof. Write Y (r) = Spec R as in Construction 4.7.5. The morphism Y (r) → B(r; A ) of Construction 4.7.5 classifies a quadratic ´etalealgebra over Y (r), and we can identify this algebra as S. 2 2 The map φ : Y (r) → B(r; A ) induces stable maps φ˜ : Y (r) → B(R; A ). Any such stable map induces a surjective map

˜∗ ∗ 2 ∗ φ :H (B(R; A )(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 φ0 : Y (r) → 2 B(r − 1; A ), from which one can produce a stable map ˜0 ∗ ∗ 2 ∗ 2 ∗ (φ ) :H (B(R; A )(R): F2) → H (B(r − 1; A ); F2) → H (Y (r)(R); F2). By reference to Corollary 4.4.3, for sufficiently large values of R, the maps φ˜∗ and (φ˜0)∗ agree. ˜0 ∗ ∗ 2 But (φ ) induces the 0-map when ∗ = r − 1, since H (B(r − 1; A )(R); F2) is a direct sum of r−1 ˜∗ two copies of F2[θ]/(θ ). This contradicts the surjectivity of φ in this degree.

4.7.2 Algebras over fields containing a square root of −1 Remark 4.7.10. When the field k contains a square root i of −1, the analogous construction to that of Chase exhibits markedly different behaviour. For simplicity, suppose r is an even integer. Consider the ring k[z , . . . , z ] S0 = 1 r Pr 2
i=1 zi − 1 0 0 C with the action of C2 given by zi 7→ −zi. Let R = (S ) 2 . After making the change of variables 0 xj = z2j−1 + iz2j and yj = z2j−1 − iz2j, we see that S is isomorphic to

k[x1, . . . , xr/2, y1, . . . , yr/2]  Pr/2  j=1 xjyj − 1 and R0 is isomorphic to the subring consisting of terms of even degree. The smallest R0-subalgebra 0 of S containing the r/2-terms x1, . . . , xr/2 contains each of the yj because of the relation

r/2 X yj = xl(ylyj) l=1

0 0 0 so S may be generated over R by r/2 elements. In fact, R is the coordinate ring of DQr−1, by Lemma 4.5.5. In Proposition 4.7.13 below, we show that S0 cannot be generated by fewer than r/2 elements over R0. 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 Spec R and an ´etale algebra A over Spec R such that A cannot be generated by fewer than d+1 elements?

53 4.7. The example of Chase

The 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 produced using any smooth affine curve Y for which 2 Pic(Y ) 6= 0. Specifically, one may take a smooth elliptic curve and discard a point to produce such a Y . A nontrivial 2-torsion line bundle L on Y cannot be generated by 1 section, since it is not trivial. One may also choose a trivialization φ : L ⊗ L → O, and therefore endow L ⊕ O with the structure of a quadratic ´etalealgebra, as in Construction 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 ring

k[x1, . . . , xr, y1, . . . , yr] T = Pr ( i=1 xiyi − 1)

C endowed with the C2 action given by xi 7→ −xi and yi 7→ −yi. Let R = T 2 . Then the quadratic ´etalealgebra T over R can be generated by the r elements x1, . . . , xr, but cannot be generated by fewer than r elements.

Proof. The ring R is the coordinate ring of the variety DQ2r−1 in Lemma 4.5.5. In particular, 1 2 there is an A -equivalence φ : DQ2r−1 → B(r; A ), as in equation (4.5.1). Tracing through this composite, one sees it classifies the quadratic ´etalealgebra generated by x1, . . . , xr, i.e., T itself—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. 0 2 Let φ : DQ2r−1 → B(r − 1; A ) be a classifying map for some such r − 1-tuple of generators. ˜ ˜0 2 Let φ and φ denote the composite maps DQ2r−1 → B(2r − 1; A ). By Corollary 4.4.3, these maps induce the same map on Chow groups. But in degree r − 1, the map φ˜∗ : CHr−1(B(2r − 2 r−1 2 1; A )) → CH (B(r; A )) → CH(DQ2r−1) is an isomorphism of cyclic groups of order 2, by 0 ∗ r−1 2 reference to Proposition 4.5.4, while by the same proposition, (φ˜ ) : CH (B(2r − 1; A )) → r−1 2 CH (B(r − 1; A )) → CH(DQ2r−1) is 0. The following shows that the bound of [FR17] is not quite sharp when applied to quadratic ´etalealgebras 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 Spec R an n- dimensional smooth affine k¯-variety. If A is a quadratic ´etalealgebra on Spec R, then A may be generated by n global sections.

Proof. Let L be a torsion line bundle on Spec R, or, equivalently, a rank-1 projective module on R. A result of Murthy’s, [Mur94, Corollary 3.16], implies that L may be generated by n n elements if and only if c1(L) = 0. By another result of Murthy’s, [Mur94, Theorem 2.14], the group CHn(R) is torsion free, so it follows that if L is a 2-torsion line bundle, then L can be generated by n elements. The proposition follows by Proposition 4.6.3.

54 Chapter 5

Conclusions and Future Research Directions

In this chapter we discuss the conclusions, lingering questions and future directions arising from each chapter. We also place our work in the context of contemporary research and discuss recent advances.

5.1 Essential dimension of inseparable field extensions

The 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 is bounded from above by the essential dimension of N/K where N is the normal closure of L/K. In particular N/K is a Galois extension with automorphism group G and we have ed(L/K) = ed(N/K) ≤ ed(G). So for the purposes of determing essential dimension of separable extensions we may only worry about the Galois ones. In that case, ed(G) is an upper bound and scope of techniques which can be applied increases. ∼ A purely inseparable extension L/K is called modular if L = K(x1) ⊗ ... ⊗ K(xn) where pei xi ∈ K. Modular extensions are the simplest kind of purely inseparable extensions. A purely inseparable extension L/K can never be Galois since automorphisms of L which fix K is only the trivial one. In particular there cannot be any Galois correspondence between subfields of L/K and subgroups of Aut(L/K). However, at least in some cases, there exists a Galois correspondence 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 inseparable extensions called the modular closure, reducing to normal closure in the case the extension is separable. For L/K a purely inseparable extension the modular closure M/K is the smallest field extension of L such that M/K is modular. The modular closure of a finite inseparable field extension was first defined by Sweedler in [Swe68], who also proved its existence. The definition and construction may be found in [Swe68] or [Ras71]. The modular closure M/K of a purely 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 closure M/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 it true 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).

55 5.2. Essential dimension of double covers of symmetric and alternating groups

Another direction this work leads to is the problem of determining essential dimension of certain infinitesimal group schemes. Recently there has been lot of interest in essential dimension of group schemes in positive characteristic (see [Tot19], [Bae17], [McK17]). For group schemes in char p > 0 there exists Frobenius self map. The kernels of these maps, called Frobenius kernels Gp, are infinitesimal group schemes whose essential dimension is poorly understood. By work of [TV13] we obtain a lower bound on their essential dimension i.e

ed(Gp) ≥ dim G. (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 (like GLn, SLn, Spn) we have ed(G) = 0. Combining this with (5.1.1) and (5.1.2), we obtain the following. Lemma 5.1.3. For a special group scheme G the essential dimension of its first Frobenius kernel is equal to dim(G). The problem is non-trivial for non-special groups. Orthogonal and projective linear groups are of particular interest. Here only partial results are know; in particular, the lemma below.

(1) Lemma 5.1.4 (Najmmuddin Fakhruddin). Let k be a field of characteristc 2. Let PGL2 denote (1) the first Frobenius kernel of P GL2. Then edk(PGL2 ) = 3. (1) Fakhruddin’s proof relies on demonstrating that the group PGL2 is a semi-direct product of 2 α2 and µ2. The essential dimension of latter can be inferred from [TV13].

5.2 Essential dimension of double covers of symmetric and alternating groups

In Chapter 3 we determined the essential dimension of double covers of symmetric and alternating groups. Note that for most integers n > 2, there is a gap between the upper and lower bounds in Theorem 3.1.2(d) and (e), and the exact value of ed(eSn) and ed(Ae n) remains open. It would be interesting to close this gap. Note that ed(eSn) − ed2(eSn) ≤ ed(Sn). Determination of ed(eSn) minght thus lead to interesting lower bounds for ed(Sn). The results of Chapter 3 also give rise to interesting applications and questions in quadratic form theory. We work over field k of characteristic 6= 2. All other fields contain k. Let W (F ) denote the Witt group over F and I(F ) the fundamental ideal in W (F ) consisting of classes of even dimensional forms. Quadratic forms of rank n over a field F are classified by the first Galois 1 cohomology set H (F, On). To an ´etalealgebra A of rank n over F we can attach a trace form x → Tr(x2) which is a quadratic form. This association may also be obtained as the induced map between Galois cohomology sets by the permutation representation Sn → On. An important question in quadratic form theory to determine which quadratic forms are trace forms (see [BF94], [GMS03, Chapter IX]).

56 5.3. Generators of ´etalealgebras

By a theorem of Merkurjev, the Hasse-Witt invariant w2 gives rise to an isomorphism 2 3 2 between I (F )/I (F ) and H (F, Z/2Z). This is a special case of Milnor’s conjecture (proven by r r+1 r Voevodsky), which asserts the map er : I (F )/I (F ) → H (F, Z/2Z), sending the r-fold Pfister form h1, a1i⊗· · ·⊗h1, ari to the cup product (a1)∪(a2)∪· · ·∪(ar), is an isomorphism. The Hasse- 1 2 Witt invariant of a quadratic form is its image via the connecting map H (F, On) → H (F, Z/2Z) through the “ Pin ” exact sequence. So in the program of distinguishing trace forms from arbitrary quadratic forms we may first try to distinguish them via their corresponding cohomology classes. r To this end we define the index of a class α ∈ H (F, Z/2Z) as the greatest common divisor of 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 form r based on their respective maximal indices in H (F, Z/2Z) as F varies over all field extensions of r r+1 r k. Note that there is a natural isomorphism er : I /I → H (F, Z/2Z) (due to V. Voedvodksy). In the case r = 1 we see that quadratic forms and trace forms cannot be distinguished (in the above sense) using their discriminants. Our work [RS19b, Theorem 1.5] shows that it is also the case if we consider r = 2. The proof relies on interpreting the maximal index of trace forms as essential dimension of double covers of symmetric group Sn and our computation of ed(eSn; 2). Thus we may ask the same question for higher cohomology classes. That is, do rth cohomology r r r+1 class H (F, Z/2Z) distinguish between trace forms and arbitrary quadratic forms in I (F )/I (F ) in the above sense? For the case r = 3 the corresponding invariant of the quadratic form is the Arason invariant. This problem also has connections to the torsion index of Spin groups. I would like to tackle this problem for the case when r = 3.

5.3 Generators of ´etalealgebras

In Chapter 4 we solved the problem of determining the sharpness of First-Reichstein bound on the minimum number of generators for an ´etalealgebra. However, we do not know if Theorem 1.3.1 continues to hold if we replace R by C. More generally if we restrict attention to the category of affine k-varieties, where k is algebraically closed of characteristic 6= 2, we may ask if the bound of First and Reichstein is sharp if we only consider rings R coming from this category. We have been able to show that the bound is NOT sharp if we only consider degree 2 ´etale algebras over rings lying in this category (see Proposition 4.7.14). The next natural step is to study the invariants of classifying spaces for degree 3 ´etale 3 algebras with generators i.e of the space B(r, A ). More generally, the sharpness of First-Reichstein bound in the context of other algebras is mostly open. In the context of Azumaya algebras, the First-Reichstein bound states that any Azumaya 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 Azumaya algebras.

Theorem 5.3.1 ([Wil18]). For each n ≥ 1, and for all d there exists ring Rd of dimension d and d Azumaya algebra A over R of degree n which cannot be generated by less than b c + 2 d d 2(n − 1) elements.

In a recent unpublished work U. First and Z. Reichstein have improved the upper bound on d the number of generators of Azumaya algebras from d + 2 to b c + 2. n − 1

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