周期性与循环同调。仿S-模与扰动引理
Periodicity and Cyclic Homology. Para-S-Modules and Perturbation Lemmas
摘要：
在本文中，我们引入了一个准循环版本的$S$ -模。这些新的对象称为para- $S$ -模块。准循环模和准链复形产生准$S$ -模的方式与循环模和混合复形产生$S$ -模的方式非常相似。更一般地说，对- $S$ -模为我们提供了一个自然的框架，以获得各种结构的仿循环模和仿链复形的类似物以及循环模或混合复形的等价结果。一个仿$S$ -模的基准并不提供给我们一个链复形，因此同调和拟同构的概念没有意义。建立了微分同调代数的基本扰动引理的仿$S$ -模和仿链复形的一些推广。这些推广为我们提供了将Hoschild链复形的形变收缩核转化为对- $S$ -模的形变收缩核的一般方法。利用Kassel的思想，我们可以得到与仿预循环模相关的各种仿$S$ -模之间的比较结果，以及它们与Connes循环链复形之间的比较结果.这些比较结果导致我们的康纳斯的周期性算子的替代描述。这在周期循环同调中有一些应用。我们还描述了这些结果在循环上同调的对应。特别地，我们得到了一个明确的方式来转换周期$(b,B)$ -上循环到上同调周期循环上循环。
英文摘要：
In this paper, we introduce a paracyclic version of $S$-modules. These new objects are called para-$S$-modules. Paracyclic modules and parachain complexes give rise to para-$S$-modules much in the same way as cyclic modules and mixed complexes give rise to $S$-modules. More generally, para-$S$-modules provide us with a natural framework to get analogues for paracyclic modules and parachain complexes of various constructions and equivalence results for cyclic modules or mixed complexes. The datum of a para-$S$-module does not provide us with a chain complex, and so notions of homology and quasi-isomorphisms do not make sense. We establish some generalizations for para-$S$-modules and parachain complexes of the basic perturbation lemma of differential homological algebra. These generalizations provide us with general recipes for converting deformation retracts of Hoschschild chain complexes into deformation retracts of para-$S$-modules. By using ideas of Kassel this then allows us to get comparison results between the various para-$S$-modules associated with para-precyclic modules, and between them and Connes' cyclic chain complex. These comparison results lead us to alternative descriptions of Connes' periodicity operator. This has some applications in periodic cyclic homology. We also describe the counterparts of these results in cyclic cohomology. In particular, we obtain an explicit way to convert a periodic $(b,B)$-cocycle into a cohomologous periodic cyclic cocycle.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
