The Multiplicative Structures on Motivic Homotopy Groups

TL;DR

This paper clarifies and reconciles different definitions of multiplication on motivic homotopy groups, highlighting potential conflicts and their implications in motivic homotopy theory.

Contribution

It identifies and resolves discrepancies in the multiplicative structures on motivic homotopy groups used by Voevodsky and Dugger.

Findings

Reconciliation of multiplicative structures on motivic homotopy groups

Identification of conflicting definitions and their potential consequences

Elementary connection similar to phenomena in supersymmetry

Abstract

We reconcile the multiplications on the homotopy rings of motivic ring spectra used by Voevodsky and Dugger. While the connection is elementary and similar phenomena have been observed in situations like supersymmetry, neither we nor other researchers we consulted were aware of the conflicting definitions and the potential consequences. Hence this short note.