A cellular absolute motivic ring spectrum representing Hermitian K-theory

TL;DR

This paper demonstrates that a cellular absolute motivic spectrum for Hermitian K-theory, constructed via orthogonal Grassmannians, matches a recently developed motivic ring spectrum, unifying different approaches in the motivic homotopy category.

Contribution

It proves the equivalence of two constructions of a motivic spectrum representing Hermitian K-theory, bridging geometric and algebraic methods.

Findings

The cellular absolute motivic spectrum coincides with the Calm ext`es-Harpaz-Nardin spectrum.

The construction applies over any scheme, not just specific cases.

Unifies geometric and algebraic approaches to Hermitian K-theory in motivic homotopy theory.

Abstract

In the Morel-Voevodsky motivic stable homotopy category of a quasi-compact quasi-separated scheme S, several candidates exist for a motivic spectrum representing hermitian K-theory. This note shows that the cellular absolute motivic spectrum constructed in the thesis of the first author via the geometry of orthogonal and hyperbolic Grassmannians over any scheme coincides with the motivic ring spectrum constructed recently by Calm\`es, Harpaz, and Nardin.