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# Arason invariant

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 H^{3}(*k*,**Z**/2**Z**). It was introduced by (Arason 1975, Theorem 5.7).

The Rost invariant is a generalization of the Arason invariant to other algebraic groups.

The Rost invariant is a generalization of the Arason invariant to other algebraic groups.

Definition

Suppose 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 *I*^{3} to the Galois cohomology group H^{3}(*k*,**Z**/2**Z**). 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 H^{1}(*k*,**Z**/2**Z**) = *k**/*k**^{2}. The Arason invariant vanishes on *I*^{4}, and it follows from the Milnor conjecture proved by Voevodsky that it is an isomorphism from *I*^{3}/*I*^{4} to H^{3}(*k*,**Z**/2**Z**).

References

Arason, Jón Kr. (1975), "Cohomologische Invarianten quadratischer Formen", J. Algebra (in German) 36 (3): 448–491, doi:10.1016/0021-8693(75)90145-3, ISSN 0021-8693, MR 0389761, Zbl 0314.12104

Esnault, Hélène; Kahn, Bruno; Levine, Marc; Viehweg, Eckart (1998), "The Arason invariant and mod 2 algebraic cycles", J. Amer. Math. Soc. 11 (1): 73–118, doi:10.1090/S0894-0347-98-00248-3, ISSN 0894-0347, MR 1460391, Zbl 1025.11009

Garibaldi, Skip; Merkurjev, Alexander; Serre, Jean-Pierre (2003), Cohomological invariants in Galois cohomology, University Lecture Series 28, Providence, RI: American Mathematical Society, ISBN 0-8218-3287-5, MR 1999383, Zbl 1159.12311

Knus, Max-Albert; Merkurjev, Alexander; Rost, Markus; Tignol, Jean-Pierre (1998), The book of involutions, Colloquium Publications 44, With a preface by J. Tits, Providence, RI: American Mathematical Society, p. 436, ISBN 0-8218-0904-0, Zbl 0955.16001

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