Framed and MW-transfers for homotopy modules

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.