Title: Localization sequences in the algebraic K-theory of ring spectra
Speaker: David Gepner (Purdue University)
Abstract: The algebraic K-theory of the sphere spectrum, $K\left(S\right)$, encodes significant information in both homotopy theory and differential topology. In order to understand $K\left(S\right)$, one can apply the techniques of chromatic homotopy theory in an attempt to approximate K(S) by certain localizations $K\left({L}_{n}S\right)$. The ${L}_{n}S$ are in turn approximated by the Johnson-Wilson spectra E(n) = BP[v_n^{-1}], and it is not unreasonable to expect to be able to compute K(BP). This would lead inductively to information about K(E(n)) via the conjectural fiber sequence K(BP) --> K(BP) --> K(E(n)). In this talk, I will explain the basics of the K-theory of ring spectra, define the ring spectra of interest, and construct some actual localization sequences in their K-theory. I will then use trace methods to show that it the actual fiber of K(BP) --> K(E(n)) differs from K(BP), meaning that the situation is more complicated than was originally hoped. This is joint work with Ben Antieau and Tobias Barthel.