Orientable homotopy modules

TL;DR

This paper proves a conjecture linking homotopy invariant sheaves with transfers to specific spectra in the stable homotopy category, using Rost's cycle modules, with applications to algebraic cobordism.

Contribution

It establishes a precise identification between Voevodsky's sheaves with transfers and certain spectra, advancing the understanding of their structure and relations to cycle modules.

Findings

Identification of sheaves with spectra concentrated in degree zero

Relation of these objects to Rost's cycle modules

Applications to algebraic cobordism and cycle class construction

Abstract

We prove a conjecture of Morel identifying Voevodsky's homotopy invariant sheaves with transfers with spectra in the stable homotopy category which are concentrated in degree zero for the homotopy t-structure and have a trivial action of the Hopf map. This is done by relating these two kind of objects to Rost's cycle modules. Applications to algebraic cobordism and construction of cycle classes are given.