On the topological cyclic homology of the integers
On the topological cyclic homology of the integers
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论整数的拓扑循环同调
DOI：
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发表时间：
1997
期刊：
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通讯作者：
S. Tsalidis
中科院分区：
文献类型：
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作者：
S. Tsalidis
本文给出了整数环的拓扑Hochschild同调的不动点在其阶为奇素数p的幂的圈群的任意有限子群作用下的模p同伦群的计算，从而导出了整数环的拓扑循环同调群的计算，并且还确定了p-adic整数的代数AT-理论的/7-adic完备化。1.导论.拓扑Hochschild同调J /（/_）是一个与每个代数环R，或者更一般地，与每个严格结合环谱T1泛函相关的谱（或无限循环空间）。它最初是由Bokstedt在（B）中引入的。谱THH（R）具有圆群T的自然作用。R的拓扑循环同调，TC（R）是以THH（R）的不动点谱THH（R）C为顶点的谱图在有限循环子群C作用下的同伦逆极限。这个图中的映射要么是不动点的包含，要么是拓扑Hochschild同调构造所特有的某些“Frobenius”映射（见（G））。R的拓扑Hochschild和循环同调都与代数jT有关，它们之间的关系可用交换谱图来描述
This article provides a computation of the mod p homotopy groups of the fixed points of the Topological Hochschild Homology of the ring of integers under the action of any finite subgroup of the cirle group whose order is a power of an odd prime p. This leads to a computation of the Topological Cyclic Homology groups of the ring of integers, and determines also the /7-adic completion of the algebraic AT-theory of the p-adic integers. 1. Introduction. The topological Hochschild homology J ////(/_) is a spec trum (or infinite loop space) associated functorially to each algebraic ring R, or, more generally, to each strictly associative ring spectrum Tl. It was introduced originally by Bokstedt in (B). The spectrum THH(R) comes equipped with a natural action of the circle group T. The topological cyclic homology of R, TC(R) is the homotopy inverse limit of a diagram of spectra with vertices the fixed point spectra THH(R)C of THH(R) under the action of the finite cyclic subgroups C of the circle. The maps in this diagram are either inclusions of fixed points or certain "Frobenius" maps particular to the topological Hochschild homology construction (see (G)). The topological Hochschild and cyclic homology of R are both related to the algebraic jT their relation can be depicted by a commutative diagram of spectra