A motivic norm structure on equivariant algebraic K-theory
Okano, Tsutomu
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https://hdl.handle.net/2142/113270
Description
Title
A motivic norm structure on equivariant algebraic K-theory
Author(s)
Okano, Tsutomu
Issue Date
2021-07-08
Director of Research (if dissertation) or Advisor (if thesis)
Heller, Jeremiah
Doctoral Committee Chair(s)
McCarthy, Randy
Committee Member(s)
Rezk, Charles
Stojanoska, Vesna
Department of Study
Mathematics
Discipline
Mathematics
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Keyword(s)
Homotopy theory
Motivic homotopy theory
Algebraic K-theory
Algebraic geometry
Abstract
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 $$(\Sigma^\infty_+ \P(\U_A))[\beta^{-1}] \simeq \KGL_A$$ and use it to show that equivariant algebraic $K$-theory is a normed motivic ring spectrum.
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