Vigleik Angeltveit

Hopf algebra structure on topological Hochschild homology

The topological Hochschild homology THH(R) of a commu- tative S-algebra (E1 ring spectrum) R naturally has the structure of a commutative R-algebra in the strict sense, and of a Hopf algebra over R in the homotopy category. We show, under a flatness assumption, that this makes the Bokstedt spectral sequence converging to the mod p homology of THH(R) into a Hopf algebra spectral sequence. We then apply this additional structure to the study of some interesting examples, including the commutative S-algebras ku, ko, tmf, ju and j, and to calculate the homotopy groups of THH(ku) and THH(ko) after smashing with suitable finite complexes. This is part of a program to make systematic computa- tions of the algebraic K-theory of S-algebras, by means of the cyclotomic trace map to topological cyclic homology. AMS Classification 55P43, 55S10, 55S12, 57T05; 13D03, 55T15

Enriched Reedy categories

This research was partially conducted during the period the author was employed by the Clay Mathematics Institute as a Liftoff Fellow.

On the K-theory of truncated polynomial algebras over the integers

We show that the K_{2i}(Z[x]/(x^m),(x)) is finite of order (mi)!(i!)^{m-2} and that K_{2i+1}(Z[x]/(x^m),(x)) is free abelian of rank m-1. This is accomplished by showing that the equivariant homotopy groups of the topological Hochschild spectrum THH(Z) are finite, in odd degrees, and free abelian, in even degrees, and by evaluating their orders and ranks, respectively.

Uniqueness of Morava K -theory

We show that there is an essentially unique S-algebra structure on the Morava K-theory spectrum K(n), while K(n) has uncountably many MU or \hE{n}-algebra structures. Here \hE{n} is the K(n)-localized Johnson-Wilson spectrum. To prove this we set up a spectral sequence computing the homotopy groups of the moduli space of A-infinity structures on a spectrum, and use the theory of S-algebra k-invariants for connective S-algebras due to Dugger and Shipley to show that all the uniqueness obstructions are hit by differentials.

On the K-theory of truncated polynomial rings in non-commuting variables

We compute the algebraic K-theory of the non-commutative ring k /(m^a) when k is a perfect field of positive characteristic and m=(x_1,...,x_n). We express the answer in terms of the truncation poset Witt vectors developed in [1].