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

Tom Bachmann and Maria Yakerson

Geometry & Topology 24 (2020) 1969–2034

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 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 M2 is a homotopy module. Finally we conjecture that for q > 0, π¯0(M(q)) is a homotopy module, explain the significance of this conjecture for studying conservativity properties of the 𝔾m–stabilization functor 𝒮S1 (k) 𝒮(k), and provide some evidence for the conjecture.

algebraic cycles, motivic cohomology, generalized motivic cohomology, motivic homotopy theory
Mathematical Subject Classification 2010
Primary: 14F42, 19E15
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
Tom Bachmann
Department of Mathematics
Massachusetts Institute of Technology
Cambridge, MA
United States
Maria Yakerson
Fakultät Mathematik
Universität Regensburg