Title: The May spectral sequence for topological Hochschild homology
Speaker: Andrew Salch (Wayne State University)
Abstract: I give the construction of a "THH-May spectral sequence" which takes as input the topological Hochschild homology of the "associated graded ring spectrum" of a filtration of a ring spectrum by "ring spectrum ideals," and which outputs THH of the original ring spectrum. The construction of this spectral sequence turns out to be quite nontrivial and involves some new ideas. I describe in general terms how this spectral sequence looks for THH(K(F_q)), the topological Hochschild homology of the K-theory spectrum of certain finite fields; the detailed computation of the spectral sequence (computation of the differentials, etc.) is quite nontrivial, and is the topic of G. Angelini-Knoll's talk this same week (on Thursday). I also describe, in general terms, how this spectral sequence looks for THH of a connective model for the K(2)-local sphere, which involves new work of Bruner and Rognes on THH of the spectrum of topological modular forms.
These two THH computations are the input one needs for a trace method computation of the topological cyclic homology of connective models for the K(1)-local and K(2)-local sphere spectrum, which by a theorem of C. Ogle on the Goodwillie derivative of algebraic K-theory of ring spectra, computes the (relative) algebraic K-groups of these connective models for the K(1)-local and K(2)-local sphere spectrum. These are the first two presently-unknown lines in the Gersten spectral sequence E_1-term for the sphere spectrum; the computation of this Gersten spectral sequence is an old problem posed by Waldhausen. I will describe this point of view and its relationship to higher Chow groups and algebraic cycles in ring spectra.
This project is joint work with C. Ogle and G. Angelini-Knoll.
Seminar URL: https://research.math.osu.edu/topology/#7180440
