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