Morel homotopy modules and Milnor-Witt cycle modules (Q2031836)

Summary: We study the cohomology theory and the canonical Milnor-Witt cycle module associated to a motivic spectrum. We prove that the heart of Morel-Voevodsky stable homotopy category over a perfect field (equipped with its homotopy t-structure) is equivalent to the category of Milnor-Witt cycle modules, thus generalising \textit{F. Déglise}'s thesis [C. R., Math., Acad. Sci. Paris 336, No. 1, 41--46 (2003; Zbl 1042.19001)]. As a corollary, we recover a theorem of \textit{A. Ananyevskiy} and \textit{A. Neshitov} [Sel. Math., New Ser. 25, No. 2, Paper No. 26, 41 p. (2019; Zbl 1436.14042)], and we prove that the Milnor-Witt \(K\)-theory groups are birational invariants.