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UNIVERSITY of CALIFORNIA Los Angeles Degree Three UNIVERSITY OF CALIFORNIA Los Angeles Degree Three Cohomological Invariants and Motivic Cohomology of Reductive Groups A dissertation submitted in partial satisfaction of the requirements for the degree Doctor of Philosophy in Mathematics by Donald Joseph Laackman III 2018 c Copyright by Donald Joseph Laackman III 2018 ABSTRACT OF THE DISSERTATION Degree Three Cohomological Invariants and Motivic Cohomology of Reductive Groups by Donald Joseph Laackman III Doctor of Philosophy in Mathematics University of California, Los Angeles, 2018 Professor Alexander Sergee Merkurjev, Chair This dissertation is concerned with calculating the group of degree three cohomological invariants of a reductive group over a field of arbitrary characteristic. We prove a formula for the group of degree three cohomological invariants of a split reductive group G with coefficients in Q=Z(2) over a field F of arbitrary characteristic. As an application, we then use this to define the group of reductive invariants of split semisimple groups, and compute these groups in all (almost) simple cases. We additionally prove the existence of a discrete relative motivic complex for any reductive group, which could be used to compute the degree two and three invariants of arbitrary reductive groups. ii The dissertation of Donald Joseph Laackman III is approved. Paul Balmer Richard S. Elman Milos D. Ercegovac Alexander Sergee Merkurjev, Committee Chair University of California, Los Angeles 2018 iii TABLE OF CONTENTS 1 Introduction :::::::::::::::::::::::::::::::::::::: 1 2 Preliminaries ::::::::::::::::::::::::::::::::::::: 4 2.1 Algebraic Groups . .4 2.1.1 Linear Representations . .8 2.1.2 Borel Subgroups . 10 2.2 Torsors . 11 2.2.1 Galois Cohomology . 11 2.2.2 Classifying Varieties and Versal Torsors . 13 2.3 Hi+1(F; Q=Z(i)).................................. 14 2.4 Cohomological Invariants . 15 2.4.1 Cohomological Invariants . 17 2.5 K-Theory, Cycle Modules, and K-Cohomology . 18 2.5.1 Milnor K−theory . 18 2.5.2 Cycle Modules . 18 2.6 K-cohomology of split tori and split simply connected algebraic groups . 20 2.7 Cohomological invariants of algebraic groups . 23 3 Degree Three Cohomological Invariants of Split Reductive Groups ::: 25 3.1 K-Cohomological Background . 25 3.1.1 Classifying Spaces . 28 3.2 Motivic Cohomology of Weight at Most 2 . 28 3.3 Cohomology of the classifying space . 33 3.4 Degree 3 Invariants . 34 iv 3.5 Reductive Invariants . 35 4 The Relative Motivic Complex :::::::::::::::::::::::::: 40 4.1 Motivation . 40 4.2 The Extended Picard Complex . 42 4.3 Defining N(G) . 42 4.3.1 Cohomology of Zf (1) . 43 • 4.3.2 Comparing Zf (1) with C (Xsep;K1)................... 44 4.4 The Weight 2 Relative Motivic Complex . 46 4.5 Hypothetical Calculations . 50 References ::::::::::::::::::::::::::::::::::::::::: 53 v ACKNOWLEDGMENTS First, I owe incredible thanks to advisor, Professor Alexander Merkurjev, for his support, insight, and dedication. His guidance and perspective brought me to worlds of mathematics I had never previously dared. His presentation was always an invitation to consider questions in a new light, and his writing provokes questions of grand underlying structure - two of which led to the material he helped me to develop in this dissertation. The first question he ever asked me resulted in my coming to UCLA, and the opportunity to work with him has been one of the great joys of my life. My interest in mathematics was fostered by a number of exceptional teachers and mentors through my life. Diane Herrmann and Paul Sally encouraged me to pursue graduate studies, both in conversation and through their teaching. Patrick Cully provided me with countless outlets for engagement, filling my life with math at a time when I could never have enough. Dan Zaharopol taught me the beauty of abstract thinking, and the importance of sharing that beauty with others. David Hartigan took my skill for math and turned it into a passion that is still burning to this day. And, at the very beginning, Allyson Laackman patiently sat with a little boy who just wanted to understand why the two sides of an equation were equal. I am so thankful to my family for the love and support they have given me through the years. My parents have always wanted me to pursue what made me happy, and have made my pursuit of mathematics possible in so many ways. My sister Emily is one of my closest friends, and has always been my biggest supporter. The person that I am, I owe to them. Thanks to my fianc´eeElizabeth, for making every day happier and more meaningful than I ever believed possible. I could not imagine making it through this without you. Finally, my thanks to Susan Durst, Yuval Wigderson, Ruthi Hortsch, Josie Bailey, Chris Ohrt, and Ian Coley for being wonderful friends. I can never express how lucky I am to know you. vi VITA 2012 B.S. (Mathematics), University of Chicago 2014{2018 Teaching Assistant, Mathematics Department, UCLA. 2013{2017 Research Assistant, Mathematics Department, UCLA. PUBLICATIONS Degree Three Cohomological Invariants of Reductive Groups. vii CHAPTER 1 Introduction If A is any sort of algebraic object, a great deal of data about A can be recovered from its automorphism group Aut(A). If, in particular, A is an algebraic object defined over a field, its automorphism group can have the structure of an algebraic group. In this case, there is a correspondence between the twisted forms of A - those objects that become isomorphic to A when passing to a field extension - and the Aut(A)-torsors; if you send a twisted form B to the Aut(A)-torsor Iso(B; A), you get an bijection between the set of isomorphism classes of twisted form as the set of isomorphism classes of Aut(A)-torsors. One classic example of this correspondence is the relationship between torsors of the Orthogonal group and nondegenerate quadratic forms; since if q is a nondegenerate quadratic form of dimension n, then Aut(q) ' On, and the twisted forms of q are precisely the quadratic forms of dimension n, so we can study all nondegenerate quadratic forms of dimension n by studying On-torsors. Another frequently studied case is that of P GLn; since P GLn ' Aut(Mn), the torsors of P GLn classify the twisted forms of the algebra Mn, which are all central simple algebras of degree n. In order to take advantage of this correspondence, the concept of a cohomological invari- ant is extremely useful. Let G be a linear algebraic group over a field F . Consider a functor G-torsors : F ieldsF ! Sets from F ieldsF , the category of field extensions of F , taking a field K to the set of isomor- 1 phism classes of G-torsors over SpecK. Let Φ: F ieldsF ! Abelian Groups be some other functor. Then a Φ-invariant of G is a natural transformation I : G-torsors ! Φ with Φ viewed as going to Sets. The group of Φ-invariants of G is written Inv(G; Φ). An Invariant I 2 Inv(G; Φ) is normalized if I(E) = 0 for all trivial G-torsors E. The normalized invariants, Inv(G; Φ)norm, form a subgroup, and Inv(G; Φ) ' Φ(F ) ⊕ Inv(G; Φ)norm An invariant is called a cohomological invariant if the functor Φ is Galois cohomology; when Φ = Hn(−;A), then the standard notation is Inv(G; Φ) = Invn(G; A). One class of examples of cohomological invariants are the Stiefel-Whitney classes, which are invariants for the orthogonal group of a quadratic form in characteristic other than two with values in Z=2Z; in dimension 1, this is the discriminant of the quadratic form, and in dimension 2 it is the Hasse-Witt invariant. Here, we will consider the cohomology functors Φ taking a field K=F to the Galois coho- n mology Hn(K; Q=Z(j)) and write Inv (G; Q=Z(j)) for this group of cohomological invariants of G of degree n with coefficients in Q=Z(j). These are of particular interest because if a group G has a non-constant unramified invariant with values in Q=Z(j), then the classifying space BG is not stably rational; to date, no such examples have been found over algebraically closed fields, and the study of these invariants is one of the most promising avenues of attack. 1 2 If G is connected, then Inv (G; Q=Z(j))norm = 0. For reductive G, Inv (G; Q=Z(1))norm = P ic(G), the Picard group of G. [7] 3 The degree 2 cohomological invariants with coefficients in Q=Z(1), and Inv (T; Q=Z(2)) 3 where T is an algebraic torus, were computed by Merkurjev and Blinstein [2]. Inv (G; Q=Z(2)) 2 was computed by Rost when G is simply connected [7] and by Merkurjev for G an arbitrary semisimple group[18]. In this dissertation, we are interested in computing the group of degree three cohomologi- cal invariants of reductive groups. The next chapter contains needed background, including a discussion of torsors, classifying spaces, motivic cohomology, K−cohomology, and the struc- ture of reductive groups. In the next chapter, we relate the degree 3 invariants of a split reductive group G to the dual lattice T ∗ of a split maximal torus T ⊆ G and the dual lattice of a particular finite group C; in particular, C∗ ' P ic(G), so we are building up the degree three invariants out of the degree 1 invariants and the root data of G: Theorem Let G be a split reductive group, T ⊂ G a split maximal torus, W the Weyl group, and C the kernel of the universal cover of the commutator subgroup of G.
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