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Abelian Affine Group Schemes, Plethories, and Arithmetic Topology

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Abelian Affine Group Schemes, Plethories, and Arithmetic Topology Abelian affine group schemes, plethories, and arithmetic topology MAGNUS CARLSON Doctoral Thesis Stockholm, Sweden 2018 KTH TRITA-MAT-A 2018:28 Institutionen f¨orMatematik ISRN KTH/MAT/A–18/28-SE 100 44 Stockholm ISBN 978-91-7729-831-1 SWEDEN Akademisk avhandling som med tillst˚andav Kungl Tekniska h¨ogskolan framl¨agges till offentlig granskning f¨or avl¨aggande av teknologie doktors- examen i matematik tisdagen den 12 Juni 2018 kl 13.00 i sal E3, Kungl Tekniska h¨ogskolan, Lindstedtsv¨agen 3, Stockholm. c Magnus Carlson, 2018 Tryck: Universitetsservice US AB iii Till Farfar, Morfar och Marcus. iv Abstract This thesis consists of four papers. In Paper A we classify plethories over a field of characteristic zero. All plethories over characteristic zero fields are “linear”, in the sense that they are free plethories on a bialgebra. For the proof of this clas- sification we need some facts from the theory of ring schemes where we extend previously known results. We also give a classification of plethories with trivial Verschiebung over a perfect field k of character- istic p > 0. In Paper B we study tensor products of abelian affine group schemes over a perfect field k. We first prove that the tensor product G1 ⊗ G2 of two abelian affine group schemes G1,G2 over a perfect field k exists. We then describe the multiplicative and unipotent part of the group scheme G1 ⊗G2. The multiplicative part is described in terms of Galois modules over the absolute Galois group of k. In characteristic zero the unipotent part of G1 ⊗ G2 is the group scheme whose primitive ele- ments are P (G1) ⊗ P (G2). In positive characteristic, we give a formula for the tensor product in terms of Dieudonn´etheory. In Paper C we use ideas from homotopy theory to define new ob- structions to solutions of embedding problems and compute the ´etale cohomology ring of the ring of integers of a totally imaginary number field with coefficients in Z/2Z. As an application of the obstruction- theoretical machinery, we give an infinite family of totally imaginary quadratic number fields such that Aut(PSL(2, q2)), for q an odd prime power, cannot be realized as an unramified Galois group over K, but its maximal solvable quotient can. In Paper D we compute the ´etalecohomology ring of an arbitrary number field with coefficients in Z/nZ for n an arbitrary positive inte- ger. This generalizes the computation in Paper C. As an application, we give a formula for an invariant defined by Minhyong Kim. v Sammanfattning Denna avhandling best˚arav fyra artiklar. I Artikel A studerar vi plethories ¨over en kropp och klassificerar plethories i karakteristik 0 och ger en partiell klassifikation i posi- tiv karakteristik. F¨or v˚araklassifikationsresultat anv¨ander vi metoder fr˚anteorin om algebraiska grupper. Vi visar att varje plethory i ka- rakteristik 0 ¨ar “linj¨ar”, i bem¨arkelsen att de ¨ar fria plethories p˚aett bialgebra. Vi klassificerar ¨aven plethories med trivial Verschiebung i karakteristik p > 0. I Artikel B studerar vi tensorprodukter av abelska affina grupp- scheman ¨over en perfekt kropp k. Vi bevisar f¨orst att tensorprodukten G1 ⊗ G2 av tv˚aabelska affina grupp scheman G1,G2 ¨over en perfekt kropp k existerar. Vi best¨ammer den multiplikativa och unipotenta delen av gruppschemat G1 ⊗ G2. Den multiplikativa delen beskrivs i termer av Galoismoduler ¨over den absoluta Galoisgrouppen av k. I karakteristik noll ¨ar den unipotenta delen av G1 ⊗ G2 det unipoten- ta gruppschemat vars primitiva element ¨ar P (G1) ⊗ P (G2). I positiv karakteristik, s˚ager vi en formel f¨or tensorprodukten i termer av Di- eudonn´eteori. I Artikel C anv¨ander vi metoder fr˚anhomotopiteori f¨or att definiera nya obstruktioner till l¨osningar av inb¨addningsproblem och ber¨aknar ´etalekohomologiringen av ringen av heltal av en totalt imagin¨ar tal- kropp med koefficienter i Z/2Z. Som en till¨ampning av obstruktionste- orin ger vi en o¨andlig familj av totalt imagin¨ara kvadratiska talkroppar s˚aatt Aut(PSL(2, q2)), f¨or q en udda primtalspotens, inte kan reali- seras som en oramifierad Galoisgrupp ¨over K, medan dess maximalt l¨osbara kvot kan realiseras som en oramifierad Galoisgroupp ¨over K. I Artikel D ber¨aknar vi ´etalekohomologiringen av en godtycklig talkropp med koefficienter i Z/nZ, f¨or n ett godtyckligt positivt heltal. Detta generaliserar ber¨akningen i Artikel C. Som en till¨ampning ger vi en formel f¨or en invariant definierad av Minhyong Kim. Contents Contents vii Acknowledgements ix Part I: Introduction and Summary 1 Introduction 1 1.1 Summary of Paper A . 1 1.2 Summary of Paper B . 8 1.3 Summary of Paper C . 15 1.4 Summary of Paper D . 24 References 31 Part II: Scientific Papers Paper A Classification of plethories in characteristic zero Submitted. Preprint: http://arxiv.org/abs/1701.01314 Paper B Tensor products of affine and formal abelian groups (joint with Tilman Bauer) Preprint: http://arxiv.org/abs/1804.10153 vii viii CONTENTS Paper C The unramified inverse Galois problem and cohomology rings of totally imaginary number fields (joint with Tomer Schlank) Preprint: http://arxiv.org/abs/1612.01766 Paper D The ´etalecohomology ring of the ring of integers of a number field (joint with Eric Ahlqvist) Preprint: http://arxiv.org/abs/1803.08437 Acknowledgements I am grateful to my advisor Tilman Bauer for taking me on as his PhD student almost five years ago. Tilman has always supported my work and encouraged me to work with mathematics that I found exciting. He has been open-minded and listened to my mathematical ideas throughout the years. His advice has been invaluable. I also would like to thank my friend and collaborator Tomer Schlank. We began speaking more than 5 years ago, back when I knew almost nothing about mathematics. He introduced me to the field of arithmetic topology, and we have had many good discussions over the years. Many of the ideas found in this thesis stem from conversations I have had with Tomer. I am lucky to have been part of the Stockholm Topology Center. The members of the group have created an intellectually vibrant atmosphere, where I have had the opportunity to learn topology. Through the Stockholm Topology Center I got the chance to organize the Young Topologists Meeting 2017 together with other graduate students in topology, which was a very rewarding experience. I am also grateful to the members of the Mathematics Departments of KTH and SU. My undergraduate studies were at SU, where I got the chance to pursue the courses that I found interesting. I especially would like to thank Rikard Bøgvad and Alexander Berglund. At KTH wish to mention Eric Ahlqvist, David Rydh and Wojciech Chacholski. To Eric, thank you for a wonderful collaboration. I thank David for sharing his immense knowledge of algebraic geometry and Wojciech for teaching me abstract homotopy theory. Andreas Holmstr¨omwas the unofficial supervisor of my master’s thesis and always took his time to answer my questions. He was also very kind ix x ACKNOWLEDGEMENTS in inviting me to IHES in Paris. Andreas’s enthusiasm for mathematics is inspiring, and he introduced me to many mysterious arithmetic questions. It has been a joy discussing plethories and mathematics in general with Jim Borger. He was very encouraging of my classification results of plethories in characteristic zero, which helped lift my spirits up when it was much- needed. I wish to thank Johan S¨orlinfor being such a great friend. To mom and dad: I am so thankful for the fact that you have throughout my life always encouraged me to do what I am passionate about. Morfar, you inspired me to educate myself. I miss you. Lastly, with the utmost of admiration, I wish to thank my beloved Anja. You never fail to make me happy. Having you as my companion has imbued my life with meaning. I love you. 1 Introduction In this introductory chapter we give an informal exposition of the four papers contained in this thesis and summarize their main results. Background results and motivating examples are given. Two of the Papers, A and B, can very broadly be said to belong to a field of mathematics known as algebra, while Paper C and D belong to a part of mathematics known as (algebraic) number theory. Paper A and B share a common theme and arose out of trying to understand an algebraic object known as a plethory. Paper C and D are also closely related. They can broadly be said to center around two questions: Given a finite group G and a number field K, what are the obstructions to realizing G as an unramified Galois group over K? Given a number field K and OK its ring of integers, what is the struc- ∗ ture of the ´etalecohomology ring H (Spec OK , Z/nZ) for n an arbi- trary positive integer? The reader should be aware that no proofs are given in the introduction, we delegate them to the main text. 1.1 Summary of Paper A In Paper A we classify algebraic objects known as “plethories” over a field of characteristic zero. In this section, we start by giving some motivation for plethories and then define them. We continue by giving examples of plethories, the most prominent example being the plethory which governs the theory of Witt vectors. We then move on to stating the results of Paper 1 2 CHAPTER 1. INTRODUCTION A. Any ring or algebra appearing in this section is assumed to be unital and associative. Witt vectors and plethories Let k be a commutative ring, Modk the category of k-modules, and consider the category of representable endofunctors Modk → Modk .
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