PhD Defense: Martin P. Speirs

Title: On the Algebraic K-theory of coordinate axes and truncated polynomial algebras

Abstract:
This thesis is concerned with computations of algebraic K-theory using the cyclotomic trace map. We use the framework for cyclotomic spectra due to Nikolaus and Scholze, which avoids the use of genuine equivariant homotopy theory. The thesis contains an introduction followed by two papers.

The first paper computes the K-theory of the coordinate axes in affine d-space over perfect fields of positive characteristic. This extends work by Hesselholt in the case d = 2. The analogous results for fields of characteristic zero were found by Geller, Reid and Weibel in 1989. We also extend their computations to base rings which are smooth Q-algebras.

In the second paper we revisit the computation, due to Hesselholt and Madsen, of the K-theory of truncated polynomial algebras for perfect fields of positive characteristic. The original proof relied on an understanding of cyclic polytopes in order to determine the genuine equivariant homotopy type of the cyclic bar construction for a suitable monoid. Using the Nikolaus-Scholze framework we achieve the same result using only the homology of said cyclic bar construction, as well as the action of Connes’ operator.

Supervisor:
Prof. Lars Hesselholt, MATH, University of Copenhagen

Assessment Committee:
Prof. (Chairman), Ryszard Nest, University of Copenhagen
Prof. Marcel Bökstedt, Aarhus University
Ass. Prof, Christian Ausoni, Université Paris XIII