On invariants of quadratic forms


In this talk I will first give some recollection on the Witt ring W(k)of anisotropic quadratic forms over a field k of char not 2, and will explain the statement of Milnor's conjecture on the associated graded ring of the filtration of W(k) by the powers of its fundamental ideal I(k), proven some time ago by Orlov, Vishik and Voevodsky.

I will then explain some simple and new way to deduce this conjecture from Voevodsky's affirmation of Beilinson-Lichtenbaum conjecture at the prime 2. Our approach emphasizes elementary use of homological algebra in the category of (Zariski) sheaves on smooth k-varieties, and also leads in exactly the same spirit to a rather direct proof of some results of Arason-Elman producing a presentation of each of the powers I^n(k) of the fundamental ideal.