[Paper Review] MW-homotopy sheaves and Morel generalized transfers

Niels Feld|arXiv (Cornell University)|Jul 29, 2020

Algebraic structures and combinatorial models13 references4 citations

TL;DR

This paper proves Morel's conjecture on Bass-Tate transfers for MW-homotopy sheaves with rational coefficients, establishes an equivalence between homotopy sheaves with generalized transfers, MW-transfers, and framed transfers, and applies these results to verify the Bachmann-Yakerson conservativity conjecture rationally for d=1 and integrally for d=2 in $Γ^1$-stabilization.

ABSTRACT

We explore a conjecture of Morel about the Bass-Tate transfers defined on the contraction of a homotopy sheaf and prove that the conjecture is true with rational coefficients. Moreover, we study the relations between (contracted) homotopy sheaves, sheaves with Morel generalized transfers and MW-homotopy sheaves, and prove an equivalence of categories. As applications, we describe the essential image of the canonical functor that forgets MW-transfers and use theses results to discuss the conservativity conjecture in A^1-homotopy due to Bachmann and Yakerson.

Motivation & Objective

• To resolve Morel's conjecture regarding the functoriality of Bass-Tate transfers on the first contraction $M_{-1}$ of a homotopy sheaf.
• To clarify the relationship between homotopy sheaves with generalized transfers, MW-transfers, and framed transfers.
• To characterize the essential image of the forgetful functor from MW-homotopy sheaves to ordinary homotopy sheaves.
• To apply the results to the conservativity conjecture of Bachmann and Yakerson in $Γ^1$-stabilization.
• To establish categorical equivalences between sheaf categories with different transfer structures in motivic homotopy theory.

Proposed method

• Adapts Morel's axioms for generalized transfers to include twists by line bundles, defining sheaves with generalized transfers.
• Constructs the Rost-Schmid complex for sheaves with generalized transfers and proves functoriality of pushforwards, pullbacks, GW-actions, and residues.
• Proves that rationalized contracted homotopy sheaves $M_{-1,×}$ admit generalized transfers, confirming Morel's conjecture rationally.
• Establishes a canonical isomorphism between the presheaf $χ_*(M)$ and $M$ for sheaves with generalized transfers, showing $χ_*$ is an equivalence.
• Uses Calmès-Fasel's finite Milnor-Witt correspondences to define MW-transfers and proves equivalence between MW-homotopy sheaves and generalized transfer sheaves.
• Applies the equivalence to show that the forgetful functor $γ_*$ from MW-sheaves to ordinary sheaves has essential image characterized by generalized transfer structures.

Experimental results

Research questions

• RQ1 Does the Bass-Tate transfer on $M_{-1}$ define a well-defined generalized transfer structure for homotopy sheaves?
• RQ2 Are the categories of homotopy sheaves with generalized transfers, MW-transfers, and framed transfers equivalent?
• RQ3 What is the essential image of the forgetful functor from MW-homotopy sheaves to ordinary homotopy sheaves?
• RQ4 Does the $Γ^1$-stabilization functor $Σ^{∞}_{Γ_m}$ satisfy conservativity on bounded below objects for $d=1$ and $d=2$?
• RQ5 Can the intersection multiplicity formula for Bass-Tate transfers be established in full generality?

Key findings

• The Bass-Tate transfer on $M_{-1}$ defines a well-behaved generalized transfer structure when tensored with $×$, proving Morel's conjecture rationally.
• The category of MW-homotopy sheaves is equivalent to the category of homotopy sheaves with generalized transfers via the functors $γ_*$ and $χ^*$.
• The essential image of the forgetful functor $γ_*: Γ^{– ext{MW}}(k) → Γ^{ ext{gtr}}(k)$ consists precisely of those homotopy sheaves admitting a generalized transfer structure.
• The Bachmann-Yakerson conjecture holds rationally for $d=1$ and integrally for $d=2$, with the $Σ^{∞}_{Γ_m}$-stabilization functor conservative on bounded below objects.
• The category of homotopy sheaves with generalized transfers is equivalent to that of MW-homotopy sheaves and to that of framed transfers, establishing $Γ^{ ext{gtr}}(k) ≍ Γ^{ ext{MW}}(k) ≍ Γ^{ ext{fr}}(k)$.
• The canonical map $ℓ_0\Omega^d_{Γ^1}\Sigma^d_{Γ^1}\mathcal{X} \to \u03c0_0\Omega^{d+1}_{Γ^1}\Sigma^{d+1}_{Γ^1}\mathcal{X}$ is an isomorphism for $d=2$, resolving a question left open in [BCD+20].