Preface
The invariant s we are going to discuss ar e the analogue s for Galoi s cohomolog y
of th e characteristi c classe s o f topolog y (Cher n classes , Stiefel-Whitne y classes ,
etc.), wher e th e topologica l space s ar e replace d b y th e schem e Spec(/c ) fo r k a
field. Historically , on e of the first example s o f cohomological invariant s o f the typ e
considered her e was the Hasse-Wit t invarian t o f quadratic forms , define d i n Witt' s
seminal pape r [Wi37] . Later , mor e invariant s o f quadrati c form s wer e defined , fo r
example Stiefel-Whitne y classe s an d th e Araso n invariant . Th e first wor k i n thi s
volume classifie s invariant s o f quadrati c form s an d etal e algebra s wit h value s i n
Galois cohomolog y modul o 2 o r i n th e Wit t ring . Th e invariant s o f som e othe r
algebraic structure s ar e als o determined . A principa l too l i s th e notio n o f versa l
torsor, whic h i s an analogu e o f the universa l bundl e i n topology .
For G a simple simpl y connecte d algebrai c group , Ros t prove d th e existenc e of
a canonica l an d nontrivia l invarian t o f G-torsors wit h value s in dimension 3 Galois
cohomology. Th e Araso n invarian t o f quadrati c form s i s a specia l cas e o f Rost' s
invariant, se e p . 107. Th e secon d wor k i n thi s volum e give s detaile d proof s o f th e
existence and basic properties of the Rost invariant . Thi s is the first time that mos t
of this materia l appear s i n print .
The two parts of this book are really separate works. Th e first bega n a s lectur e
notes tha t Serr e gav e a t UCL A i n Januar y 200 1 as par t o f th e Gil l Distinguishe d
Lecture Series . The y hav e bee n expande d an d no w include man y result s tha t wer e
not mentione d durin g th e lectures , especiall y o n etal e algebra s o f lo w ran k an d
their application s t o Noether's problem o n invariants. Th e second part wa s writte n
independently b y Merkurjev (wit h one section by Garibaldi). A s a consequence, th e
definitions an d notation s use d i n th e tw o part s diffe r somewhat . Fo r th e reader' s
convenience, w e have included commo n indexe s an d a common bibliograph y a t th e
end o f the book .
The secon d autho r i s gratefu l t o th e NS F fo r partia l suppor t (gran t DM S
0098111).
This boo k i s dedicated t o Marku s Rost . Th e influenc e o f his idea s ca n b e see n
throughout.
Skip Garibald i Alexande r Merkurje v Jean-Pierr e Serr e
Atlanta, Georgi a Lo s Angeles, Californi a Pari s
USA US A Franc e
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