3:30 pm Thursday, May 10, 2007
Geometry Seminar: The algebraic K-theory of structured ring spectra by Andrew Blumberg (Stanford University) in RLM 9.166
The use of "rings up to homotopy" has been extremely fruitful in the study of algebraic K-theory. On the one hand, regarding ordinary rings as "rings up to homotopy" over the sphere spectrum enables the construction of sophisticated invariants such as topological cyclic homology and topological Hochschild homology. These theories are relatively computable and often approximate algebraic K-theory very well. On the other hand, the recent (mid 90's) construction of categories of structured ring spectra with well-behaved categories of modules allows the construction of the algebraic K-theory and related invariants associated to a "ring up to homotopy". The first hour of the talk will be a self-contained (albeit necessarily rapid) introduction to this collection of ideas --- I will discuss structured ring spectra (the modern technical implementation of "rings up to homotopy"), algebraic K-theory of Waldhausen categories, and topological Hochschild homology. Much of what we know about algebraic K-theory in the classical context comes from the interaction of localization, devissage, and approximation theorems --- localization theorems provide exact sequences in K-theory, and approximation and devissage theorems let us identify the terms of these sequences. Devissage phenomena turn out to be much harder to understand in the setting of the algebraic K-theory of structured ring spectra. Localization and approximation theorems turn out to be considerably more mysterious in the context of topological Hochschild homology, even for ordinary rings. In the second hour, I will survey recent results from my ongoing project with Michael Mandell to understand these issues. Submitted by
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