tailieunhanh - Đề tài " On the K-theory of local fields "

Topological Hochschild homology and localization 2. The homotopy groups of T (A|K) 3. The de Rham-Witt complex and TR· (A|K; p) ∗ 4. Tate cohomology and the Tate spectrum 5. The Tate spectral sequence for T (A|K) 6. The pro-system TR· (A|K; p, Z/pv ) ∗ Appendix A. Truncated polynomial algebras References Introduction In this paper we establish a connection between the Quillen K-theory of certain local fields and the de Rham-Witt complex of their rings of integers with logarithmic poles at the maximal ideal. The fields K we consider are complete discrete valuation fields of characteristic zero with perfect residue. | Annals of Mathematics On the X-theory of local fields By Lars Hesselholt and Ib Madsen Annals of Mathematics 158 2003 1 113 On the Â-theory of local fields By Lars Hesselholt and IB Madsen Contents Introduction 1. Topological Hochschild homology and localization 2. The homotopy groups of T A K 3. The de Rham-Witt complex and TR A K p 4. Tate cohomology and the Tate spectrum 5. The Tate spectral sequence for T A K 6. The pro-system TR A K p L pv Appendix A. Truncated polynomial algebras References Introduction In this paper we establish a connection between the Quillen K-theory of certain local fields and the de Rham-Witt complex of their rings of integers with logarithmic poles at the maximal ideal. The fields K we consider are complete discrete valuation fields of characteristic zero with perfect residue field k of characteristic p 2. When K contains the pv-th roots of unity the relationship between the K-theory with z pv-coefficients and the de Rham-Witt complex can be described by a sequence . . . . 1 F . KJ w V A M z pv Apv w V A M z pv Apv which is exact in degrees 1. Here A Ok is the valuation ring and w V A M is the de Rham-Witt complex of A with log poles at the maximal ideal. The factor sz pv Pp is the symmetric algebra of Pp considered as a z pv-module located in degree two. Using this sequence we evaluate the K -theory with TL p -coefficients of K. The result which is valid also if K does not con- The first named author was supported in part by NSF Grant and the Alfred P. Sloan Foundation. The second named author was supported in part by The American Institute of Mathematics. 2 LARS HESSELHOLT AND IB MADSEN tain the pv-th roots of unity verifies the Lichtenbaum-Quillen conjecture for K 26 38 Theorem a. There are natural isomorphisms for s 1 TS ĨP 7 íT-VẦ u0ÍTT s tj2 o s 1 K2s K p H K Tpv H K Tpv K2s-i K pv H1 K fpry The Galois cohomology on the right can be effectively calculated when k is finite or equivalently when K is a finite extension of Qp 42

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