Semistable Symmetric Spectra in $A1$-homotopy theory

TL;DR

This paper investigates semistability in symmetric spectra within a broad class of monoidal model categories, extending known characterizations to motivic spectra and demonstrating preservation under localization.

Contribution

It generalizes Schwede's semistability characterizations to motivic symmetric spectra and proves that key spectra like motivic Eilenberg-MacLane and algebraic cobordism are semistable.

Findings

Motivic Eilenberg-MacLane spectrum is semistable.

Algebraic cobordism spectrum is semistable.

Semistability is preserved under localization under certain conditions.

Abstract

We study semistable symmetric spectra based on quite general monoidal model categories, including motivic examples. In particular, we establish a generalization of Schwede's list of equivalent characterizations of semistability in the case of motivic symmetric spectra. We also show that the motivic Eilenberg-MacLane spectrum and the algebraic cobordism spectrum are semistable. Finally, we show that semistability is preserved under localization if some reasonable conditions - which often hold in practice - are satisfied.