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

数学科学：代数 K 理论、拓扑循环同调和晶体上同调

• 批准号: 9415615
• 负责人: Randy McCarthy
• 金额: $ 6.85万
• 依托单位: University of Illinois at Urbana-Champaign
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1994
• 资助国家: 美国
• 起止时间: 1994-07-15 至 1997-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9415615&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Algebraic; Theory; Topological

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. ***

9415615 McCarthy 该项目的目标之一是继续研究与代数 K 理论密切相关但更易于计算的理论，特别是拓扑 Hochschild 同调以及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 同调的定点谱的同伦逆极限，该环的子群的阶数可被素数 p 整除，其中结构映射是由等变映射对其定点子空间的限制而产生的。 当 A 可交换时，W(p)(A) 的第 0 个同伦群由 A 的第 p 个维特向量组成，并且存在从 K(End(A)) 到 W(p)( 的自然映射A) 第 0 个同伦群上的预期映射是（来自 Almkvist 的工作）。 如果我们使用 W(p)(A) 作为第 p 个维特向量的推广，则会导致将 W(p)(A)（存在圆作用）的同伦不动点谱视为一种可能的推广第 p 个 DeRham-Witt 复合体。 从 K(A) 到 K(End(A))（通过恒等自同态）到 W(p)(A) 的复合映射通过 W(p)(A) 的同伦轨道谱因子，它是研究者的希望这是对布洛赫晶莹剔透性格的一个很好的修饰。 该项目处理了过去几十年来取得巨大成功的一些代数装置，用于将几何信息简化为计算主题。 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、体积等的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有很大不同。 这些属性包括连通性（全部为一体）、打结性、无孔等。 所有对这些性质的系统研究，例如，如何判断两个几何对象在这些性质之一上是否确实存在差异或只是表面上的差异，或者如何对可能出现的各种差异进行分类，所有这些都只有真正的意义。当它们被简化为计算问题时，它们就被理解和掌握了。 研究人员的工作旨在更好地理解现在可用于执行此操作的一些强大代数工具之间的关系，显然希望通过更好的理解来完善和提高它们的能力。 如此基本的东西的潜在价值很难量化，因为拓扑特性和问题出现在如此广泛不同的数学环境中，例如控制卫星运动的微分方程和经济模型的平衡条件。 ***