Filtrations on homotopy invariant sheaves with transfers

TL;DR

This paper develops filtrations on homotopy invariant sheaves with transfers and demonstrates their alignment with conjectured filtrations under certain motivic conjectures, extending Pelaez's work.

Contribution

It introduces a new filtration on homotopy invariant sheaves with transfers and relates it to existing conjectures in motivic homotopy theory.

Findings

Filtration constructed on homotopy invariant sheaves with transfers.

Under Ayoub's conjectures, the filtration matches the conjectured one by Ayoub and Barbieri-Viale.

The construction is motivated by Pelaez's work.

Abstract

We construct filtrations on homotopy invariant sheaves with transfers and show that under Ayoub's conjectures on $n$-motives, our filtration agrees with the one conjectured by Ayoub and Barbieri-Viale if the latter exists. Our construction is directly motivated by the work of Pelaez.