A motivic norm structure on equivariant algebraic K-theory

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.