#### Volume 24, issue 4 (2020)

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

### Tom Bachmann and Maria Yakerson

Geometry & Topology 24 (2020) 1969–2034
##### 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 $M\phantom{\rule{-0.17em}{0ex}}$, 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 ${\mathsc{𝒮}\mathsc{ℋ}}^{{S}^{1}}\phantom{\rule{-0.17em}{0ex}}\left(k\right)\to \mathsc{𝒮}\mathsc{ℋ}\left(k\right)$, and provide some evidence for the conjecture.

##### Keywords
algebraic cycles, motivic cohomology, generalized motivic cohomology, motivic homotopy theory
##### Mathematical Subject Classification 2010
Primary: 14F42, 19E15