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 H³(k,Z/2Z). It was introduced by (Arason 1975, Theorem 5.7).

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 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

• 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, vol. 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, vol. 44, With a preface by J. Tits, Providence, RI: American Mathematical Society, p. 436, ISBN 0-8218-0904-0, Zbl 0955.16001