2003/12/15 by Nikita A. Karpenko, Karpenko, Nikita A. · 1 citation
Mathematics · Physics and Astronomy · #11E04 #14C25 #Algebraic Geometry (math.AG) #Black Holes and Theoretical Physics #Commutative Algebra (math.AC) #FOS: Mathematics #math.AC #math.AG #msc:11E04 #msc:14C25
paper · pdf · doi:10.48550/arxiv.math/0312273
29 pages
arxiv created 2003/12/15 · openalex publication_date 2003/12/15 · arxiv updated 2009/12/01 · openalex created_date 2024/04/11 · openalex updated_date 2026/07/28
Let F be an arbitrary field of characteristic not 2. We write W(F) for the Witt ring of F, consisting of the isomorphism classes of all anisotropic quadratic forms over F. For any element x of W(F), dimension dim x is defined as the dimension of a quadratic form representing x. The elements of all even dimensions form an ideal denoted I(F). The filtration of the ring W(F) by the powers I(F)n of this ideal plays a fundamental role in the algebraic theory of quadratic forms. The Milnor conjectures, recently proved by Voevodsky and Orlov-Vishik-Voevodsky, describe the successive quotients I(F)n/I(F)n+1 of this filtration, identifying them with Galois cohomology groups and with the Milnor K-groups modulo 2 of the field F. In the present article we give a complete answer to a different old-standing question concerning I(F)n, asking about the possible values of dim x for x in I(F)n. More precisely, for any positive integer n, we prove that the set dim In of all dim x for all x in I(F)n and all F consisists of 2n+1-2i, i=1,2,...,n+1 together with all even integers greater or equal to 2n+1. Previously available partial informations on dim In include the classical Arason-Pfister theorem, saying that no integer between 0 and 2n lies in dim In, as well as a recent Vishik's theorem, saying the same on the integers between 2n and 2n+2n-1 (the case n=3 is due to Pfister, n=4 to Hoffmann). Our proof is based on computations in Chow groups of powers of projective quadrics (involving the Steenrod operations); the method developed can be also applied to other types of algebraic varieties.