Mathematical Sciences: Algebraic K-Theory, Topological Cyclic Homology and Crystalline Cohomology

小行星9415615 该项目的一个目标是继续研究与代数K-理论密切相关但更易于计算的理论，特别是拓扑Hochschild同调和由此产生的各种理论。这些研究的一个主要焦点是W（p）（A），它是环A的拓扑Hochschild同调的不动点谱关于具有可被素数p整除的阶的圆的子群的同伦逆极限，其中结构映射是由等变映射到其不动点子空间的限制而产生的。 当A是交换的时，W（p）（A）的0阶同伦群由A的p阶Witt向量组成，并且存在从K（End（A））到W（p）（A）的自然映射，该自然映射在0阶同伦群上是期望映射（来自Almkvist的工作）。 如果我们使用W（p）（A）作为p阶Witt向量的推广，那么我们可以考虑W（p）（A）的同伦不动点谱（有一个圈作用）作为p阶DeRham-Witt复形的可能推广. 从K（A）到K（End（A））的复合映射（通过恒等自同态）通过W（p）（A）的同伦轨道谱分解到W（p）（A），希望这是Bloch的结晶特征的一个很好的修正. 这个项目对待一些代数仪器，已经开发了巨大的成功，在过去的几十年中，减少几何信息的计算对象。 所涉及的几何信息的性质是困难的关键。 虽然关于长度、面积、角度、体积等问题实际上迫切需要简化为计算，但它与所谓的几何对象的拓扑性质大不相同。 这些属性如连通性（一体化）、打结性、无孔性等等。 例如，对这些性质的所有系统研究，如何判断她的两个几何对象在这些性质中的一个方面是否真的不同，或者只是表面上的不同，或者如何对可能出现的各种差异进行分类，所有这些只有在它们被简化为计算问题时才真正被理解和掌握。 调查员的工作旨在更好地了解一些强大的代数工具之间的关系，现在可以做到这一点，明显的希望是，随着更好的理解来完善和锐化他们的能力。 如此基本的东西的潜在价值很难量化，因为拓扑性质和问题出现在各种各样的数学环境中，如控制卫星运动的微分方程和经济模型的平衡条件。 ***

9415615 McCarthy One goal of this project is to continue the study of theories closely related to algebraic K-theory but which are more accessible to computation, in particular, topological Hochschild homology and various theories which are generated from it. A major focal point for these investigations is W(p)(A), which is the homotopy inverse limit of the fixed point spectra of topological Hochschild homology of a ring A with respect to the subgroups of the circle having order divisible by the prime p, where the structure maps are those arising from the restriction of equivariant maps to their fixed point subspaces. When A is commutative, the 0-th homotopy group of W(p)(A) consists of the p-th Witt vectors of A, and there is a natural map from K(End(A)) to W(p)(A) which on the 0-th homotopy group is the expected map (from work of Almkvist). If we use W(p)(A) as a generalization of the p-th Witt vectors, one is led to consider the homotopy fixed point spectrum of W(p)(A) (there is a circle action) as a possible generalization of the p-th DeRham-Witt complex. The composite map from K(A) to K(End(A)) (by the identity endomorphism) to W(p)(A) factors through the homotopy orbit spectrum of W(p)(A), and it is the investigator's hope that this is a good modification of Bloch's crystalline chern character. This project treats some of the algebraic apparatus that has been developed with huge success over the past several decades for reducing geometric information to a subject for calculation. The nature of the geometric information involved is the crux of the difficulty. While questions about lengths, areas, angles, volumes, and so forth virtually cry out to be reduced to calculations, it is far different with what are known as topological properties of geometric objects. These are properties such as connectedness (being all in one piece), knottedness, having no holes, and so forth. All systematic study of such properties, for example, how to tell whet her two geometric objects really differ in respect to one of these properties or are only superficially different, or how to classify the variety of differences that can occur, all these have only truly been comprehended and mastered when they have been reduced to matters of calculation. The investigator's work aims to understand better the relations among some of the powerful algebraic tools now available for doing this, the obvious hope being that with better understanding comes the ability to perfect and sharpen them. The potential value of something so basic is hard to quantify, since topological properties and questions arise in such widely varied mathematical settings as the differential equations that govern satellite motions and the conditions for equilibrium of models of the ecomomy. ***