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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-
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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.
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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)
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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)
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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)
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Documenta Mathematica Extra Volume Merkurjev (2015)
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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)
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Documenta Mathematica Extra Volume Merkurjev (2015)
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