| 释义 |
- Definition
- References
In mathematics, the Arason invariant is a cohomological invariant associated to a quadratic form of even rank and trivial discriminant and Clifford invariant over a field k of characteristic not 2, taking values in H3(k,Z/2Z). It was introduced by {{harvs|last=Arason|year=1975|loc=Theorem 5.7}}. The Rost invariant is a generalization of the Arason invariant to other algebraic groups. DefinitionSuppose that W(k) is the Witt ring of quadratic forms over a field k and I is the ideal of forms of even dimension. The Arason invariant is a group homomorphism from I3 to the Galois cohomology group H3(k,Z/2Z). It is determined by the property that on the 8-dimensional diagonal form with entries 1, –a, –b, ab, -c, ac, bc, -abc (the 3-fold Pfister form«a,b,c») it is given by the cup product of the classes of a, b, c in H1(k,Z/2Z) = k*/k*2. The Arason invariant vanishes on I4, and it follows from the Milnor conjecture proved by Voevodsky that it is an isomorphism from I3/I4 to H3(k,Z/2Z). References- {{citation|mr=0389761|last=Arason|first= Jón Kr.|title=Cohomologische Invarianten quadratischer Formen|journal=J. Algebra |volume=36 |year=1975|issue= 3|pages= 448–491|doi=10.1016/0021-8693(75)90145-3 | zbl=0314.12104 | language=German | issn=0021-8693}}
- {{citation|first1= Hélène|last1= Esnault|first2= Bruno |last2=Kahn|first3=Marc|last3=Levine|first4= Eckart|last4= Viehweg |journal= J. Amer. Math. Soc.|volume= 11 | number=1 |year=1998|pages= 73–118 |mr=1460391 |doi=10.1090/S0894-0347-98-00248-3 |title=The Arason invariant and mod 2 algebraic cycles | zbl=1025.11009| issn=0894-0347}}
- {{citation | last1=Garibaldi | first1=Skip | author1-link=Skip Garibaldi | last2=Merkurjev | first2=Alexander | author2-link=Alexander Merkurjev | last3=Serre | first3=Jean-Pierre | author3-link=Jean-Pierre Serre | title=Cohomological invariants in Galois cohomology | series=University Lecture Series | volume=28 | location=Providence, RI | publisher=American Mathematical Society | year=2003 | isbn=0-8218-3287-5 | zbl=1159.12311 | mr=1999383 }}
- {{citation | last1=Knus | first1=Max-Albert | last2=Merkurjev | first2=Alexander | author2-link=Alexander Merkurjev | last3=Rost | first3=Markus | author3-link=Markus Rost | last4=Tignol | first4=Jean-Pierre | title=The book of involutions | others=With a preface by J. Tits | zbl=0955.16001 | series=Colloquium Publications | publisher=American Mathematical Society | volume=44 | location=Providence, RI | year=1998 | isbn=0-8218-0904-0 | page=436 }}
{{algebra-stub}} 1 : Algebraic groups |