Towards conservativity of 𝔾m–stabilization

Bachmann, Tom; Yakerson, Maria

doi:10.2140/gt.2020.24.1969

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

Tom Bachmann and Maria Yakerson

Geometry & Topology 24 (2020) 1969–2034

DOI: 10.2140/gt.2020.24.1969

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 $q \geq 0$ , we construct a new 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 $q > 0$ , ${\underline{π}}_{0} (M^{(q)})$ is a homotopy module, explain the significance of this conjecture for studying conservativity properties of the $𝔾_{m}$ –stabilization functor ${𝒮 ℋ}^{S^{1}} (k) \to 𝒮 ℋ (k)$ , 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

References

Publication

Received: 7 March 2019

Revised: 31 October 2019

Accepted: 18 December 2019

Published: 10 November 2020

Proposed: Mark Behrens

Seconded: Haynes R Miller, Stefan Schwede

Authors

Tom Bachmann
Department of Mathematics Massachusetts Institute of Technology Cambridge, MA United States
http://tom-bachmann.com
Maria Yakerson
Fakultät Mathematik Universität Regensburg Regensburg Germany
https://www.muramatik.com

Volume 24, issue 4 (2020)