[Paper Review] The trivial fiber topology and framed motives over the integers

A. Druzhinin, Håkon Kolderup+1 more|arXiv (Cornell University)|Dec 14, 2020

Homotopy and Cohomology in Algebraic Topology4 citations

TL;DR

This paper introduces the trivial fiber (tf) topology on schemes to study fibrant replacements and strict A¹-invariance in the stable motivic homotopy category over one-dimensional base schemes. By establishing tf-Nisnevich strict A¹-invariance and extending Voevodsky's and Morel's theorems, it enables new computations of motivic invariants over arithmetic base schemes, particularly through framed motives and motivic infinite loop spaces.

ABSTRACT

This paper introduces the trivial fiber topology on schemes. For one-dimensional base schemes, we use it to describe fibrant replacements in the stable motivic homotopy category and motivic infinite loop spaces. We also extend the Garkusha-Panin and Voevodsky strict $\mathbb{A}^{1}$-invariance theorems to one-dimensional base schemes. The trivial fiber topology plays a central role in the proof of refined localization results for motivic homotopy categories. Moreover, we extend Morel's $\mathbb{A}^{1}$-connectivity theorem on Nisnevich sheaves of stable motivic homotopy groups. These results open new vistas for computations of motivic invariants over deeper base schemes of arithmetic interest.

Motivation & Objective

• To develop a new topological framework—the trivial fiber (tf) topology—for studying motivic homotopy theory over one-dimensional base schemes.
• To extend Garkusha-Panin and Voevodsky's strict A¹-invariance theorems from fields to one-dimensional base schemes.
• To establish motivic fibrant replacements and infinite loop space structures in the stable motivic homotopy category using framed correspondences.
• To generalize Morel’s A¹-connectivity theorem to Nisnevich sheaves of stable motivic homotopy groups over one-dimensional bases.
• To enable new computational tools for motivic invariants over arithmetic base schemes, such as the integers.

Proposed method

• Introduce the trivial fiber (tf) topology as a Grothendieck topology on schemes, defined via tf-coverings that refine Nisnevich and A¹-homotopy coverings.
• Use the tf-topology to define tf-localization functors and establish tf-Nisnevich strict A¹-invariance for abelian group-valued sheaves.
• Apply deformation techniques and pro-étale limits to reduce problems on smooth schemes with closed subschemes to local henselian neighborhoods.
• Construct fibrant replacements in the stable motivic homotopy category via the endofunctor Ω_{Gm}L_{A¹}, leveraging framed correspondences.
• Prove that motivic infinite loop spaces arise as tf-local sheaves in the category of framed presheaves over a base scheme B.
• Use the tf-topology to establish a localization theorem for tf-sheaves, enabling reduction to smooth affine schemes with trivial tangent bundles.

Experimental results

Research questions

• RQ1 How can the stable motivic homotopy category over one-dimensional base schemes be described using fibrant replacements via the tf-topology?
• RQ2 To what extent does strict A¹-invariance hold for Nisnevich sheaves of stable motivic homotopy groups over one-dimensional bases?
• RQ3 Can Voevodsky’s and Garkusha-Panin’s strict A¹-invariance theorems be extended beyond fields to more general base schemes?
• RQ4 What role does the trivial fiber topology play in refining localization and homotopy invariance in motivic homotopy theory?
• RQ5 How do framed motives and motivic infinite loop spaces behave under tf-localization over arithmetic base schemes?

Key findings

• The trivial fiber topology enables a full description of fibrant replacements in the stable motivic homotopy category over one-dimensional base schemes.
• The paper establishes tf-Nisnevich strict A¹-invariance for abelian group-valued sheaves, generalizing Voevodsky’s and Morel’s theorems to one-dimensional bases.
• The motivic infinite loop space structure is realized as a tf-local sheaf in the category of framed presheaves over a base scheme B.
• The A¹-connectivity theorem for Nisnevich sheaves of stable motivic homotopy groups is extended to one-dimensional base schemes.
• Counterexamples show that Nisnevich strict A¹-invariance fails in general, but the tf-topology restores it in the required contexts.
• The localization theorem for tf-sheaves allows reduction of motivic problems to smooth affine schemes with trivial vector bundles, simplifying computations.