# Group schemes and motivic spectra

## TL;DR

This paper introduces various types of motivic spectra, such as linear and orthogonal, and demonstrates that their stable homotopy theories are equivalent, extending classical results to the motivic setting.

## Contribution

It generalizes the stable homotopy theory from classical spectra to new motivic spectra types and shows their equivalence, providing a framework for motivic homotopy theory.

## Key findings

- Stable homotopy theory of motivic spectra is recovered from each spectrum type.
- Introduces and studies linear, symplectic, orthogonal motivic spectra.
- Application to localization functor converting motivic homotopy theory to framed bispectra.

## Abstract

By a theorem of Mandell-May-Schwede-Shipley the stable homotopy theory of classical $S^1$-spectra is recovered from orthogonal spectra. In this paper general linear, special linear, symplectic, orthogonal and special orthogonal motivic spectra are introduced and studied. It is shown that the stable homotopy theory of motivic spectra is recovered from each of these types of spectra. An application is given for the localization functor $C_*\mathcal Fr:SH_{nis}(k)\to SH_{nis}(k)$ in the sense of [15] that converts the Morel-Voevodsky stable motivic homotopy theory $SH(k)$ into the equivalent local theory of framed bispectra [15].