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

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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. ***