Towards conservativity of $\mathbb{G}_m$-stabilization

Abstract: We study the interplay of the homotopy coniveau tower, the Rost-Schmid complex of a strictly homotopy invariant sheaf, and homotopy modules. For a strictly homotopy invariant sheaf $M$, smooth $k$-scheme $X$ and $q \geqslant 0$ we construct a novel cycle complex $C^*(X, M, q)$ and we prove that in favorable cases, $C^*(X, M, q)$ is equivalent to the homotopy coniveau tower $M^{(q)}(X)$. To do so we establish moving lemmas for the Rost-Schmid complex. As an application we deduce a cycle complex model for Milnor-Witt motivic cohomology. Furthermore we prove that if $M$ is a strictly homotopy invariant sheaf, then $M_{-2}$ is a homotopy module. Finally we conjecture that for $q>0$, $\underline{\pi}_0(M^{(q)})$ is a homotopy module, explain the significance of this conjecture for studying conservativity properties of the $\mathbb{G}_m$-stabilization functor $\mathcal{SH}^{S^1}!(k) \to \mathcal{SH}(k)$, and provide some evidence for the conjecture.