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 $q\ge 0$, we construct a new cycle complex $C^{\ast }\left(X,M,q\right)$ and we prove that in favorable cases, $C^{\ast }\left(X,M,q\right)$ is equivalent to the homotopy coniveau tower $M^{\left(q\right)}\left(X\right)$. 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 $q>0$, ${\underset{¯}{\pi }}_0\left(M^{\left(q\right)}\right)$ is a homotopy module, explain the significance of this conjecture for studying conservativity properties of the ${\mathbb{𝔾}}_m$–stabilization functor ${\mathcal{𝒮}\mathcal{\mathscr{H}}}^{S^1}\left(k\right)\to \mathcal{𝒮}\mathcal{\mathscr{H}}\left(k\right)$, and provide some evidence for the conjecture.