A motivic norm structure on equivariant algebraic K-theory

Equivariant motivic homotopy theory is a homotopy theory of schemes with algebraic group actions. This thesis is mainly divided into two parts. In the first part, we define four model categories of motivic spectra that present the $\infty$-category \SH^G(S). We use the model categorical setup to reproduce the motivic norm functors, which were originally defined in \cite{BH}. In the second part, we define a theory of orientation in $A$-equivariant motivic homotopy theory, at least for a finite abelian group $A$. As an application, we prove the equivariant motivic analogue of the Snaith theorem $(\Sigma^\infty₊ \P(\U_A))[β^-1] \simeq \KGL_A$ and use it to show that equivariant algebraic $K$-theory is a normed motivic ring spectrum.

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Mathematics

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2022

Author dc:creator

• Okano, Tsutomu

Contributors dc:contributor

• Heller, Jeremiah
• McCarthy, Randy
• Rezk, Charles
• Stojanoska, Vesna

• Homotopy theory
• Motivic homotopy theory
• Algebraic K-theory
• Algebraic geometry

Statement dc:rights

• Copyright 2021 Tsutomu Okano

Language dc:language

en

Handle dc:identifier

http://hdl.handle.net/2142/113270

OAI identifier oai:identifier

oai:www.ideals.illinois.edu:2142/113270