Arithmetic Geometry and Number Theory RTG Seminar: The K-theory of truncated polynomial algebras and coordinate axes.

Seminar | October 21 | 3:10-5 p.m. | 740 Evans Hall

 Martin Speirs, Berkeley

 Department of Mathematics

Algebraic K-theory is a fundamental invariant of rings and schemes and is connected with several areas of number theory such as p-adic cohomology theories and special values of zeta functions. Unfortunately, explicit computations of K-theory are rare and typically hard. However, certain trace maps from K-theory to more computable invariants have led to computations.

In this talk I will revisit such a computation, originally due to Hesselholt and Madsen, of the K-theory of truncated polynomial algebras over perfect fields of positive characteristic. After introducing some of the relevant tools from algebraic topology (in particular topological cyclic homology) I will present a simpler proof of the Hesselholt-Madsen result. This may be viewed as a concrete application of recent work by Nikolaus and Scholze. Time permitting, I will sketch how to use the same method to make new computations of K-theory, in particular for the coordinate axes in affine d-space over perfect fields of positive characteristic.

In the pretalk I will define algebraic K-theory in some detail, give a crash course on spectra, define topological cyclic homology and give some background on the connections with number theory.