TWO-PRIMARY ALGEBRAIC K-THEORY OF RINGS OF INTEGERS IN NUMBER FIELDS
TWO-PRIMARY ALGEBRAIC K-THEORY OF RINGS OF INTEGERS IN NUMBER FIELDS
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DOI：
10.1090/s0894-0347-99-00317-3
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发表时间：
1999-08
影响因子：
3.9
通讯作者：
J. Rognes;C. Weibel;appendix by M. Kolster
中科院分区：
文献类型：
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作者：
J. Rognes;C. Weibel;appendix by M. Kolster
20世纪70年代初，Lichtenbaum [L1， L2]对全实数域F的代数k理论、<s:1>上同调与zeta函数之间的关系提出了几个不同的猜想。本文证实了Lichtenbaum关于2-初等k理论与F的<s:1>上同调（当Gal（F/Q）是阿贝尔函数时）与zeta函数之间的猜想联系。在系数为21的情况下，我们得到Lichtenbaum在[L2, 2.4和2.6]中推测的关系。在特殊情况F = Q下，此结果在[W3]中得到。我们的方法依赖于Voevodsky [V2]， Suslin和Voevodsky [SV]， Bloch和Lichtenbaum [BL]最近的惊人结果。它们与本文的附录B一起，得到了一个从任意特征为0的域的<s:1>上同调开始并收敛到它的2初级k理论的谱序列。对于数域，这本质上是Quillen在[Q4]中推测的谱序列。本文克服了该谱序列的主要技术困难，即当F具有实嵌入时，它不会在E2处退化，并且它没有已知的乘法结构。为了描述我们的结果，我们引入了一些符号。如果A是一个阿贝尔群，设A{2}表示它的2-主扭转子群，设#A表示当A有限时它的阶。我们用Kn(R)表示环R的第n个代数k群，H （R；M）表示系数在M中的Spec(R)的第n个代数k群。定理0.1。设F是一个全实数域，有r1个实数嵌入。设R = OF[12]表示F中的2整数环。那么即使我> 0 21·# K2i−2 (R) {2} # K2i−1 (R) {2} = # H et (R; Z2 (i)) # H1 et (R; Z2 (i))。
In the early 1970’s, Lichtenbaum [L1, L2] made several distinct conjectures about the relation between the algebraic K-theory, étale cohomology and zeta function of a totally real number field F . This paper confirms Lichtenbaum’s conjectural connection between the 2-primary K-theory and étale cohomology of F , and (when Gal(F/Q) is Abelian) to the zeta function. Up to a factor of 21 , we obtain the relationship conjectured by Lichtenbaum in [L2, 2.4 and 2.6]. In the special case F = Q, this result was obtained in [W3]. Our methods depend upon the recent spectacular results of Voevodsky [V2], Suslin and Voevodsky [SV], and Bloch and Lichtenbaum [BL]. Together with Appendix B to this paper, they yield a spectral sequence starting with the étale cohomology of any field of characteristic zero and converging to its 2-primary K-theory. For number fields, this is essentially the spectral sequence whose existence was conjectured by Quillen in [Q4]. The main technical difficulties with this spectral sequence, overcome in this paper, are that it does not degenerate at E2 when F has a real embedding, and that it has no known multiplicative structure. To describe our result we introduce some notation. If A is an Abelian group, we let A{2} denote its 2-primary torsion subgroup, and let #A denote its order when A is finite. We write Kn(R) for the nth algebraic K-group of a ring R, and H ét(R;M) for the nth étale cohomology group of Spec(R) with coefficients in M . Theorem 0.1. Let F be a totally real number field, with r1 real embeddings. Let R = OF [ 12 ] denote the ring of 2-integers in F . Then for all even i > 0 21 · #K2i−2(R){2} #K2i−1(R){2} = #H ét(R; Z2(i)) #H1 ét(R; Z2(i)) .