Framed and MW-transfers for homotopy modules

TL;DR

This paper develops a framework using framed correspondences to construct Milnor-Witt transfers on homotopy modules, linking stable $ ext{A}^1$-homotopy sheaves with MW-motivic complexes and establishing equivalences between certain homotopy categories.

Contribution

It introduces a novel approach to define Milnor-Witt transfers on homotopy modules and proves equivalences between categories of homotopy $t$-structures and Milnor-Witt motives.

Findings

Identification of zeroth stable $ ext{A}^1$-homotopy sheaves with MW-motivic complexes

Equivalence of hearts of homotopy $t$-structures on different categories

Construction of Milnor-Witt transfers using framed correspondences

Abstract

In the paper we use the theory of framed correpondences to construct Milnor-Witt transfers on homotopy modules. As a consequence we identify the zeroth stable $\mathbb{A}^1$-homotopy sheaves of smooth varieties with the zeroth homology of corresponding MW-motivic complexes and prove that the hearts of homotopy $t$-structures on the stable $\mathbb{A}^1$-derived category and the category of Milnor-Witt motives are equivalent.