TAILIEUCHUNG - Đề tài " An exact sequence for KM /2 with applications to quadratic forms "

An exact sequence for KM /2 ∗ with applications to quadratic forms By D. Orlov,∗ A. Vishik,∗∗ and V. Voevodsky∗∗* Contents 1. Introduction 2. An exact sequence for KM /2 ∗ 3. Reduction to points of degree 2 4. Some applications . Milnor’s Conjecture on quadratic forms . The Kahn-Rost-Sujatha Conjecture . The J-filtration conjecture 1. Introduction Let k be a field of characteristics zero. For a sequence a = (a1 , . . . , an ) of invertible elements of k consider the homomorphism M M K∗ (k)/2 → K∗+n (k)/2 in Milnor’s K-theory modulo elements divisible by 2 defined by. | Annals of Mathematics An exact sequence for KMU 2 with applications to quadratic forms By D. Orlov A. Vishik and V. Voevodsky Annals of Mathematics 165 2007 1 13 An exact sequence for K 2 with applications to quadratic forms By D. Orlov a. Vishik and V. VoEVODSKy Contents 1. Introduction 2. An exact sequence for KM 2 3. Reduction to points of degree 2 4. Some applications . Milnor s Conjecture on quadratic forms . The Kahn-Rost-Sujatha Conjecture . The J-filtration conjecture 1. Introduction Let k be a field of characteristics zero. For a sequence a a1 . an of invertible elements of k consider the homomorphism kM k 2 KM n k 2 in Milnor s K-theory modulo elements divisible by 2 defined by multiplication with the symbol corresponding to a. The goal of this paper is to construct a four-term exact sequence 18 which provides information about the kernel and cokernel of this homomorphism. The proof of our main theorem Theorem consists of two independent parts. Let Qa be the norm quadric defined by the sequence a see below . First we use the techniques of 13 to establish a four term exact sequence 1 relating the kernel and cokernel of multiplication by a with Milnor s K-theory of the closed and the generic points of Qa respectively. This is done in the first section. Then using elementary geometric arguments we show that the sequence can be rewritten in its final form 18 which involves only the generic point and the closed points with residue fields of degree 2. Supported by NSF grant DMS-97-29992. Supported by NSF grant DMS-97-29992 and RFFI-99-01-01144. Supported by NSF grants DMS-97-29992 and DMS-9901219 and the Ambrose Monell Foundation. 2 D. ORLOV A. VISHIK AND V. VOEVODSKY As an application we establish for fields of characteristics zero the validity of three conjectures in the theory of quadratic forms - the Milnor conjecture on the structure of the Witt ring the Khan-Rost-Sujatha conjecture and the J-filtration conjecture. All these results require .

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