[论文解读] Algebraic Jf-theory and Trace Invariants

Lars Hesselholt|arXiv (Cornell University)|Jan 1, 2002

Algebraic Geometry and Number Theory参考文献 26被引用 9

一句话总结

本文研究了从代数K-理论到拓扑限制同伦（TR）的分圆迹映射，证明对于正则Fp-代数以及具有分离闭合剩余域的henselian混合特征离散赋值环，该映射诱导同构K*(A, ℤ/pᵛ) ≅ TR*(A; p, ℤ/pᵛ)^F=1。本文确立了TR*(A; p)由de Rham-Witt复形控制，从而使得K-理论计算可通过一种刚性、基于算子的结构中的代数不变量实现。

ABSTRACT

The cyclotomic trace of Bokstedt-Hsiang-Madsen, the subject of Bokstedt's lecture at the congress in Kyoto, is a map of pro-abelian groups K-fiA) ^.TR;(A;p) from Quillen's algebraic A-theory to a topological refinement of Connes' cyclic homology. Over the last decade, our understanding of the target and its relation to A-theory has been significantly advanced. This and possible future development is the topic of my lecture. The cyclotomic trace takes values in the subset fixed by an operator F called the Frobenius. It is known that the induced map K*(A,Z/pv) -^ TR;(A;P,Z/PV)F=1 is an isomorphism, for instance, if A is a regular local Fp-algebra, or if A is a henselian discrete valuation ring of mixed characteristic (0,p) with a separably closed residue field. It is possible to evaluate A-theory by means of the cyclotomic trace for a wider class of rings, but the precise connection becomes slightly more complicated to spell out. The pro-abelian groups TR*(A;p) are typically very large. But they come equipped with a number of operators, and the combined algebraic structure is quite rigid. There is a universal example of this structure — the de Rham-Witt complex — which was first considered by Bloch-Deligne -Illusie in connection with Grothendieck's crystalline cohomology. In general, the canonical map W.QqA^TR-q(A-p) is an isomorphism, if q < 1, and the higher groups, too, can often be expressed in terms of the de Rham-Witt groups. This is true, for example, if A is a regular Fp-algebra, or if A is a smooth algebra over the ring of integers in a local number field. The calculation in the latter case verifies the LichtenbaumQuillen conjecture for focal number fields, or more generally, for henselian discrete valuation fields of geometric type.

研究动机与目标

• 阐明分圆迹在连接代数K-理论与p-进环的拓扑限制同伦（TR）之间的作用。
• 理解pro-阿贝尔群TR*(A; p)及其在Frobenius算子F作用下的不动点结构。
• 确定分圆迹在更广范围的环类中是否诱导同构K*(A, ℤ/pᵛ) ≅ TR*(A; p, ℤ/pᵛ)^F=1。
• 确立de Rham-Witt复形作为TR*(A; p)的普遍模型，在有利情况下具有普遍性。
• 通过TR-理论计算验证局部数域的Lichtenbaum-Quillen猜想。

提出的方法

• 利用Quillen代数K-理论K*(A)到pro-阿贝尔群TR*(A; p)的分圆迹映射，后者是Connes循环同伦的拓扑提升。
• 分析Frobenius算子F在TR*(A; p)上的不动点，特别关注子群TR*(A; p, ℤ/pᵛ)^F=1。
• 将de Rham-Witt复形用作控制TR*(A; p)的普遍结构，尤其在A为Fp上正则或局部数域整数环上光滑代数的情况下。
• 应用已知的W_q(A)与TR_q(A; p)在q < 1时的同构关系，并在适当的正则性条件下将其推广至更高q。
• 利用TR*(A; p)上代数结构的刚性——包括Frobenius、Verschiebung与Teichmüller映射——推导出其结构控制。
• 应用关于具有分离闭合剩余域的henselian离散赋值环的结果，以在几何情形下验证Lichtenbaum-Quillen猜想。

实验结果

研究问题

• RQ1 在何种条件下，分圆迹会诱导同构K*(A, ℤ/pᵛ) ≅ TR*(A; p, ℤ/pᵛ)^F=1？
• RQ2 de Rham-Witt复形如何作为TR*(A; p)在代数K-理论计算中的普遍模型？
• RQ3 在正则Fp-代数之外的环类中，代数K-理论在多大程度上可通过TR-理论计算？
• RQ4 TR*(A; p)与de Rham-Witt群W_q(A)（q ≥ 1）之间的精确关系为何？
• RQ5 TR-理论框架是否验证了几何类型的henselian离散赋值域的Lichtenbaum-Quillen猜想？

主要发现

• 当A为正则局部Fp-代数时，分圆迹诱导同构K*(A, ℤ/pᵛ) ≅ TR*(A; p, ℤ/pᵛ)^F=1。
• 若A为混合特征(0,p)的henselian离散赋值环且剩余域分离闭合，则同构关系依然成立。
• 对所有q < 1，自然映射W_q(A) → TR_q(A; p)是同构，为de Rham-Witt与TR-理论之间建立了基础联系。
• 对于局部数域整数环上光滑代数，TR*(A; p)可表示为de Rham-Witt群的组合，从而实现K-理论计算。
• 对这类环计算TR*(A; p)验证了焦点数域情形下的Lichtenbaum-Quillen猜想。
• TR*(A; p)的结构是刚性的，由de Rham-Witt复形控制，后者作为该类pro-阿贝尔群的普遍模型。