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The K-theory of truncated polynomial algebras and coordinate axes

Banff International Research Station for Mathematical Innovation and Discovery

In this talk I will revisit the computation, originally 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 for topological cyclic homology I achieve the same result using only the homology of said cyclic bar construction, as well as the action of Connesâ operator. Time permitting, I will sketch how to use this method to make new computations of K-theory, in particular for 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. I also extend their computations to base rings which are smooth Q-algebras.

Mathematics; Category theory; homological algebra; K-theory; Algebraic topology

video/mp4

Author affiliation: UC Berkeley

2020-09-21

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/76110

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/