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麦卡锡这个项目的一个目标是继续研究与代数K理论密切相关的理论，但这些理论更容易获得计算，尤其是拓扑的Hochschild同源性和从中产生的各种理论。 这些研究的一个主要焦点是W（p）（a），这是环A拓扑点a的固定点光谱的同质逆极限，相对于由prime P划分的圆的子组的圆圈A的固定点光谱，其中结构映射是由于对固定点限制了其固定点的限制而产生的结构图。 当A的交换性时，W（p）（a）的第0-同型组由A的p-th Witt矢量组成，并且有一个从k（末端（a））到w（p）（a）的自然图，而在0-同型组上是预期的图（almkvist的工作）。 如果我们使用w（p）（a）作为p-th witt载体的概括，则被引导考虑w（p）（a）（有一个圆圈动作）的同型固定点光谱作为P-th derham-witt复合物的可能概括。 从k（a）到k（末端（a））（通过身份内态）到W（p）（a）通过W（p）（a）的同型轨道谱的因素的复合图，这是研究者希望这是Bloch Croch的Crystalline Chern特征的良好修改。 该项目处理了过去几十年中成功开发的一些代数设备，以将几何信息减少到主题进行计算。 涉及的几何信息的性质是难度的关键。 虽然关于长度，区域，角度，体积等的问题实际上呼唤到计算中，但与几何对象的拓扑特性相差很大。 这些都是连接性（一件一件），打结，没有孔等的属性。 例如，所有对此类属性的系统研究，例如，如何告诉WHET她的两个几何对象在这些特性中的一个或仅表面上的不同，或者仅在表面上不同，或者如何将可能发生的差异进行分类，只有在减少计算问题时才真正理解和掌握了所有这些差异。 调查人员的工作旨在更好地了解现在可以做到这一点的一些强大的代数工具之间的关系，显而易见的希望是，更好地理解能够完善和培养它们。 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是在众多的数学设置中出现的，例如控制卫星运动的微分方程以及生态学模型平衡的条件。 ***