Holes in I^n
| dc.creator | Karpenko, Nikita A. | |
| dc.date | 2003-12-15 | |
| dc.date.accessioned | 2026-07-07T05:03:54Z | |
| dc.date.available | 2026-07-07T05:03:54Z | |
| dc.description | 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 I^n of all dim x for all x in I(F)^n and all F consisists of 2^{n+1}-2^i, i=1,2,...,n+1 together with all even integers greater or equal to 2^{n+1}. Previously available partial informations on dim I^n include the classical Arason-Pfister theorem, saying that no integer between 0 and 2^n lies in dim I^n, as well as a recent Vishik's theorem, saying the same on the integers between 2^n and 2^n+2^{n-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. | |
| dc.description | 29 pages | |
| dc.identifier | https://arxiv.org/abs/math/0312273 | |
| dc.identifier | http://arxiv.org/abs/math/0312273 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/69597 | |
| dc.subject | Algebraic Geometry | |
| dc.subject | Commutative Algebra | |
| dc.subject | 11E04; 14C25 | |
| dc.title | Holes in I^n | |
| dc.type | text |