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Documenta Mathematica Journal der Deutschen Mathematiker-Vereinigung Gegru¨ndet 1996 Extra Volume 2015 · A Collection of Manuscripts Written in Honour of Alexander S. Merkurjev on the Occasion of His Sixtieth Birthday Editors: P. Balmer, V. Chernousov, I. Fesenko, E. Friedlander, S. Garibaldi, U. Rehmann, Z. Reichstein Documenta Mathematica, Journal der Deutschen Mathematiker-Vereinigung, ver¨offentlicht Forschungsarbeiten aus allen mathematischen Gebieten und wird in traditionellerWeisereferiert. EswirdindiziertdurchMathematicalReviews,Science Citation IndexExpanded,Zentralblatt fu¨r Mathematik. Artikel k¨onnen als TEX-Dateien per E-Mail bei einem der Herausgeber eingereicht werden. Hinweisefu¨rdieVorbereitungderArtikelk¨onnenunterderuntenangegebe- nen WWW-Adressegefunden werden. Documenta Mathematica, Journal der Deutschen Mathematiker-Vereinigung, publishes research manuscripts out of all mathematical fields and is refereed in the traditional manner. It is indexed in Mathematical Reviews, Science Citation Index Expanded,Zentralblatt fu¨r Mathematik. Manuscripts should besubmittedas TEX -filesbye-mailto oneof theeditors. Hints for manuscript preparation can be found underthefollowing web address. http://www.math.uni-bielefeld.de/documenta Gescha¨ftsfu¨hrende Herausgeber / Managing Editors: Ulf Rehmann (techn.),Bielefeld [email protected] Stefan Teufel, Tu¨bingen [email protected] Otmar Venjakob,Heidelberg [email protected] Herausgeber / Editors: Christian B¨ar, Potsdam [email protected] Don Blasius, Los Angeles [email protected] Joachim Cuntz, Mu¨nster [email protected] Patrick Delorme, Marseille [email protected] Gavril Farkas, Berlin (HU) [email protected] Friedrich G¨otze, Bielefeld [email protected] Ursula Hamenst¨adt, Bonn [email protected] Max Karoubi, Paris [email protected] Stephen Lichtenbaum,Providence Stephen [email protected] Alfred K. Louis, Saarbru¨cken [email protected] Eckhard Meinrenken, Toronto [email protected] AlexanderS. Merkurjev, Los Angeles [email protected] Anil Nerode, Ithaca [email protected] Thomas Peternell, Bayreuth [email protected] Takeshi Saito, Tokyo [email protected] Stefan Schwede,Bonn [email protected] Heinz Siedentop,Mu¨nchen (LMU) [email protected] ISSN1431-0635 (Print),ISSN 1431-0643 (Internet) SPARC Documenta MathematicaisaLeadingEdgePartnerofSPARC, theScholarlyPublishingandAcademicResourceCoalitionoftheAs- Leading Edge sociationofResearchLibraries(ARL),WashingtonDC,USA. Address of Technical Managing Editor: Ulf Rehmann, Fakulta¨t fu¨r Mathematik, Universit¨at Bielefeld, Postfach 100131, D-33501 Bielefeld, Copyright c 2015 for Layout: Ulf Rehmann. (cid:13) TypesettinginTEX. Documenta Mathematica Extra Volume: Alexander S. Merkurjev’s Sixtieth Birthday, 2015 Preface 1 Merkurjev’s Faves 5 Aravind Asok and Jean Fasel Secondary Characteristic Classes and the Euler Class 7–29 Asher Auel, R. Parimala, and V. Suresh Quadric Surface Bundles over Surfaces 31–70 E. Bayer-Fluckiger, V. Emery, and J. Houriet Hermitian Lattices and Bounds in K-Theory of Algebraic Integers 71–83 Mikhail Borovoi and Boris Kunyavski˘ı Stably Cayley Semisimple Groups 85–112 Baptiste Calm`es, Kirill Zainoulline, Changlong Zhong Equivariant Oriented Cohomology of Flag Varieties 113–144 Denis-Charles Cisinski, Fr´ed´eric D´eglise Integral Mixed Motives in Equal Characteristic 145–194 Jean-Louis Colliot-Th´el`ene Descente galoisienne sur le second groupe de Chow : mise au point et applications 195–220 Haruzo Hida Limit Mordell–Weil Groups and their p-Adic Closure 221–264 Detlev W. Hoffmann Motivic Equivalence and Similarity of Quadratic Forms 265–275 Bruno Kahn, R. Sujatha Birational Geometry and Localisation of Categories With Appendices by Jean-Louis Colliot-Th´el`ene and Ofer Gabber 277–334 D. Kaledin K-Theory as an Eilenberg-Mac Lane Spectrum 335–365 Nikita A. Karpenko Minimal Canonical Dimensions of Quadratic Forms 367–385 Max-Albert Knus and Jean-Pierre Tignol Triality and algebraic groups of type 3D 387–405 4 Marc Levine, Girja Shanker Tripathi Quotients of MGL, Their Slices and Their Geometric Parts 407–442 iii Roland Lo¨tscher Essential Dimension of Separable Algebras Embedding in a Fixed Central Simple Algebra 443–459 A. S. Merkurjev Divisible Abelian Groups are Brauer Groups (Translation of an article originally published in Russian in Uspekhi Mat. Nauk, vol. 40 (1985), no. 2(242), 213–214) 461–463 Matthew Morrow Zero Cycles on Singular Varieties and Their Desingularisations 465–486 Manuel Ojanguren Wedderburn’s Theorem for Regular Local Rings 487–490 I. Panin, V. Petrov Rationally Isotropic Exceptional Projective Homogeneous Varieties Are Locally Isotropic 491–500 Alena Pirutka and Nobuaki Yagita Note on the Counterexamples for the Integral Tate Conjecture over Finite Fields 501–511 Vladimir L. Popov Around the Abhyankar–Sathaye Conjecture 513–528 Anne Qu´eguiner-Mathieu and Jean-Pierre Tignol The Arason Invariant of Orthogonal Involutions of Degree 12 and 8, and Quaternionic Subgroups of the Brauer Group 529–576 David J. Saltman Finite u Invariant and Bounds on Cohomology Symbol Lengths 577–590 iv Documenta Math. 1 Preface Alexander Sergeevich Merkurjev – or just Sasha to his friends – was born in 1955 in Leningrad (now St. Petersburg) Russia. His mathematical talents manifested themselves at an early age. In 1972 he was a part of the eight member Sovietteamthat wonthe firstprize at the InternationalMathematics Olympiad for high school students. (Sasha also won a silver medal for his individual performance.) In the early 1980s Sasha burst on the research scene, first with a proof of a conjectureofJohnTateaboutthe K-theoryoflocalfields,thenwithaproofof a long-standing conjecture relating K of a field to the 2-torsion in its Brauer 2 group. Then, still in his 20s, Sasha (jointly with Andrei Suslin) strengthended the latter result to settle a key conjecture in the theory of central simple alge- bras. The theorem they proved, now known as the Merkurjev-Suslin theorem, isgenerallyrecognizedasahighpointof20thcenturyalgebra. Itcanbe found inmanytextbooksandhasopenedthedoortomanysubsequentdevelopments, includingVladimirVoevodsky’sFieldsmedalwinningproofoftheMilnorCon- jecture in the 1990s. In the subsequent three decades Sasha has firmly established himself as one of the world’s leading algebraists. He has made fundamental contributions in a number of areas, including algebraic K-theory, quadratic forms, Galois coho- mology, algebraic groups, arithmetic and algebraic geometry (including higher classfieldtheoryandintersectiontheory),andessentialdimension. Hisresearch accomplishments, too numerous to detail here, have been recognized with a prize of the St. Petersburg Mathematical Society (1982), a sectional lecture at the International Congress of Mathematicians (1986), the Humboldt Prize (1995),aplenarylectureattheEuropeanCongressofMathematics(1996),the AMS Cole Prize in algebra (2012) and a Guggenheim Fellowship (2013-14). At 60,Sasha is full of creative energy. His lectures are crystalclear and effort- lessly delivered, his papers are efficiently written and uniformly of the highest quality. The three research monographs he has coauthored are standard ref- erences in the subject. Sasha has been an inspiring thesis advisor to many graduate students, both at St. Petersburg University and at UCLA, where he has been on the faculty since 1997. According to the Mathematics Geneal- ogy Project, eight students have written their Ph.D. dissertations under his supervision at St. Petersburg University and fourteen at UCLA. Throughout his career Sasha devoted a great deal of his time to organizing and running highschoolmathematicalcompetitions. Heservedasamemberoftheorganiz- ingcommitteefortheSt.Petersburgmathematicalolympiad(in 1980-1999)as well as of the national Soviet – and then Russian – olympiad (8 times). Documenta Mathematica Extra Volume Merkurjev (2015) · 2 Preface We are happy to dedicate this volume to Sasha on the occasion of his 60th birthday. Documenta Mathematica is a particularly appropriateforum for this volume in view of Sasha’s nearly 20 years of service as an editor, since the first issue of Documenta in 1996. In addition to peer-reviewed papers submitted by his friends and colleagues, this issue includes a new crossword by one of Sasha’s PhD students who has published puzzles in venues such as the New York Times, and also the first English translation of a brief note by Merkurjev that has previously appeared only in Russian. Happy birthday, Sasha! P. Balmer, V. Chernousov, I. Fesenko, E. Friedlander, S. Garibaldi, U. Rehmann, Z. Reichstein Documenta Mathematica Extra Volume Merkurjev (2015) · Preface 3 At Mathematisches Forschungsinstitut Oberwolfach1 in 1982 Lecturing at the Fields Institute thematic programTorsors, Nonassociative Algebras and Cohomological Invariants in 2013.2 1Author: George M.Bergman; Source: Archivesofthe Mathematisches Forschungsinsti- tutOberwolfach 2Author: NikolaiVavilov Documenta Mathematica Extra Volume Merkurjev (2015) · 4 Documenta Mathematica Extra Volume Merkurjev (2015) · Documenta Math. 5 Merkurjev’s Faves Alex Boisvert ACROSS 1 2 3 4 5 6 7 8 9 10 11 12 13 1. "___ mia!" 6. Burden 14 15 16 10. Singer Stefani 17 18 19 14. Sparkle, as in an eye 15. Prefix meaning "all" 20 21 16. "This is terrible!" 17. Merkurjev's favorite beer? 22 23 24 25 26 27 28 19. Encourage 20. Pelvic region 29 30 31 21. Mock-innocent question 32 33 34 22. Elman, Karpenko, and Merkurjev, e.g. 35 36 37 38 39 40 41 25. Immediately 28. Princess's bane, in a fairy 42 43 44 tale 29. "The Annotated Flatland" 45 46 47 48 49 50 author Stewart 30. Merkurjev's favorite toolbox 51 52 53 item? 54 55 56 57 32. Ringmaster, for example 34. Lennart Carleson, for one 58 59 60 61 62 63 35. Brought into being 38. One doing the jitterbug, 64 65 66 maybe 42. Goods for sale 67 68 69 44. Chutzpah 45. Merkurjev's favorite rural (Published via Across Lite) pastime? 3. Voice actor Blanc of 36. "Etale Homotopy of 50. Slippery swimmer "Looney Tunes" Simplicial Schemes" author 51. "___ to it!" 4. Tropical smoothie staple Friedlander 52. Where we meet the 5. Love, in Latin 37. Section: Abbr. characters in a play 6. Nabokov novel 39. Increased in size 53. Berserk 7. Foreboding 40. At any time 54. Wedding locale, at times 8. Little worker 41. Depend (on) 56. "Man, it's sweltering today!" 9. Cube referenced in 43. "Snape kills Dumbledore", 58. Svelte probability classes e.g. 59. Merkurjev's favorite formal 10. Vincent van ___ 45. Pure event? 11. Actress Goldberg of "Ghost" 46. Empty on the inside 64. Sao ___ and Principe 12. Tooth covering 47. They may be global or local 65. Iron, Bronze, and Space, 13. "I reject your o!er!" 48. Deepest notably 18. Lion's yell 49. Soft drink brand with a 66. Root systems may be 21. Glass-stomping occasion "Blue Ice Cream" flavor simply, doubly, or triply 22. Knots, as shoes 53. Plate appearance 67. Actor McGregor 23. Wheelchair-friendly feature 55. Prayer ender 68. Left, on a ship 24. Ancient Peruvian 57. Site of an annual prize 69. Marine mammal that floats 26. Goings-on announcement on its back 27. Workers on a ship 59. Something a proof should 30. "Hit Me With Your Best not have DOWN Shot" singer Pat 60. In the past 1. ___ Grand (Las Vegas 31. Descartes's first name 61. Make a move casino) 33. "Groooooooss!" 62. Jeans brand 2. "Float like a butterfly, sting 63. One in charge, for short like a bee" speaker Documenta Mathematica Extra Volume Merkurjev (2015) · 6 Documenta Mathematica Extra Volume Merkurjev (2015) ·

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Extra Volume: Alexander S. Merkurjev's Sixtieth Birthday, 2015. Preface. 1 . well as of the national Soviet – and then Russian – olympiad (8 times). Princess's bane, in a fairy tale. 29. "The Annotated Flatland" author Stewart. 30.
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