Nikita A. Karpenko: Holes in $I^n$

Submission: 2003, Dec 12

We prove Vishik's conjecture stating that for any positive integer n and any anisotropic quadratic form q over a field lying in the n-th power I^n of the fundamental ideal I of the Witt ring of the field, either dim(q) is at least 2^{n+1} or dim(q)=2^{n+1}-2^i for some i. This provides a complete solution of an old-standing problem in the algebraic theory of quadratic forms. The proof is based on computations in the Chow groups of direct products of projective quadrics (involving the Steenrod operations); the method developed can be also applied to other types of algebraic varieties.

2000 Mathematics Subject Classification: 11E04; 14C25

Keywords and Phrases: Quadratic forms, Witt indices, Chow groups, Steenrod operations, correspondences.

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