A deformation of Borel equivariant homotopy

TL;DR

This paper introduces a deformation of the Borel $G$-spectra $$-category, offering new perspectives on motivic homotopy theory and spectral sequences, along with a novel computational spectral sequence for $RO(G)$-graded homotopy groups.

Contribution

It presents a deformation of the Borel $G$-spectra category, providing new models for motivic homotopy and a modified spectral sequence for computations.

Findings

New presentation of the $a$-complete real Artin--Tate motivic stable homotopy category.

A new interpretation of the $a$-completed $C_2$-effective slice spectral sequence.

A modified Adams--Novikov spectral sequence for $RO(G)$-graded Mackey functor homotopy computations.

Abstract

We describe a deformation of the $\infty$-category of Borel $G$-spectra for a finite group $G$. This provides a new presentation of the $a$-complete real Artin--Tate motivic stable homotopy category when $G=C_2$ and gives a new interpretation of the $a$-completed $C_2$-effective slice spectral sequence. As a new computational tool, we present a modified Adams--Novikov spectral sequence which computes the $RO(G)$-graded Mackey functor valued homotopy of Borel $G$-spectra.