Abstract

AbstractWe establish a kind of ‘degree $0$ Freudenthal ${\mathbb {G}_m}$ -suspension theorem’ in motivic homotopy theory. From this we deduce results about the conservativity of the $\mathbb P^1$ -stabilization functor.In order to establish these results, we show how to compute certain pullbacks in the cohomology of a strictly homotopy-invariant sheaf in terms of the Rost–Schmid complex. This establishes the main conjecture of [2], which easily implies the aforementioned results.

Highlights

  • We describe these two main sections in reverse order, and sketch their relation

  • Stable motivic homotopy theory is concerned with the category obtained by making ΣP1 := ∧P1 into an equivalence

  • Preliminaries We recall some well-known results from motivic homotopy theory

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Summary

Introduction

We describe these two main sections in reverse order, and sketch their relation. Stable motivic homotopy theory is concerned with the category obtained by making ΣP1 := ∧P1 into an equivalence. It is this context in which algebraic cycles and motivic cohomology naturally appear. This is an immediate consequence of Corollary 4.9 and, for example, Morel’s Hurewicz theorem [18, Theorem 6.37]. It is similar to the fact that stabilization is conservative on connected topological spaces. Our main results in the form of Corollary 4.9, Theorem 4.14, and Corollary 4.15 can be understood without reading the rest of the article (except perhaps for taking a glance at §4.1, where some notation is introduced)

Pullbacks and the Rost–Schmid complex
From pullbacks to stabilization
Notation and conventions
Strictly homotopy-invariant sheaves
Cousin and Rost–Schmid complexes
A ‘formula’ for closed pullback
Applications
Canonical resolutions
Full Text
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