On torsion spaces

TL;DR

This paper constructs examples of compact torsion objects in the stable homotopy category over any field, using algebraic methods to analyze their levels and distinguish points in the Balmer spectrum.

Contribution

It adapts Mitchell's approach to algebraic settings and characterizes levels via $v_m$-elements acting on $MGL$-motives, refining the understanding of the Balmer spectrum.

Findings

Constructed examples of torsion objects of every p-level over arbitrary fields.

Established that the p-level is determined by $v_m$-action on $MGL$-motives.

Provided a refinement to distinguish isotropic Morava points.

Abstract

The purpose of this note is to construct examples of compact torsion objects of ${\cal SH}(F)$ of every $p$-level over an arbitrary field of characteristic different from $p$. We adapt the approach of Mitchell to the algebraic situation. We show that the respective level is determined by the action of the $v_m$-elements on the $MGL$-motive, and also prove the refinement which permits to distinguish isotropic Morava points of the Balmer spectrum of ${\cal SH}(F)$.