[Paper Review] MW-motivic complexes

Frédéric Déglise, Jean Fasel|arXiv (Cornell University)|Aug 21, 2017

Homotopy and Cohomology in Algebraic Topology17 references20 citations

TL;DR

This paper develops a new theory of MW-motivic complexes based on finite Chow-Witt correspondences with coefficients in quadratic forms, constructing a motivic homotopy theory parallel to Voevodsky's framework. It establishes that MW-motivic cohomology groups are isomorphic to Chow-Witt groups via a hypercohomology spectral sequence, and proves that the cohomology sheaves are strictly A¹-invariant, enabling a full comparison with classical motivic cohomology and a ring isomorphism between MW-motivic cohomology and the graded Chow-Witt ring.

ABSTRACT

The aim of this work is to develop a theory parallel to that of motivic complexes based on cycles and correspondences with coefficients in quadratic forms. This framework is closer to the point of view of $\mathbb{A}^1$-homotopy than the original one envisioned by Beilinson and set up by Voevodsky.

Motivation & Objective

• To develop a motivic homotopy theory based on MW-correspondences, extending Voevodsky’s framework to incorporate quadratic forms.
• To construct the category of effective MW-motives and its stable completion, enabling a cohomological theory with coefficients in a ring R.
• To prove that the cohomology sheaves of MW-motivic complexes are strictly A¹-invariant, mirroring Voevodsky’s theorem for classical motives.
• To establish a comparison between MW-motivic cohomology and Chow-Witt groups, showing they are isomorphic in high degrees.
• To show that the hypercohomology spectral sequence for MW-motivic complexes induces an isomorphism between MW-motivic cohomology and the cohomology of Milnor-Witt K-theory sheaves.

Proposed method

• Uses the category of finite Chow-Witt correspondences from [CF14] as the foundation for defining MW-correspondences with coefficients in a ring R.
• Constructs sheaves and presheaves with MW-transfers, proving that the associated sheaf of a MW-presheaf is a MW-sheaf using a variant of Voevodsky’s method.
• Applies the theory of framed correspondences and leverages a result from [GP15] to prove that the sheaf associated to a homotopy invariant MW-presheaf is strictly A¹-invariant.
• Defines the A¹-derived category and the stable A¹-derived category of MW-sheaves, constructing the category of effective MW-motives as A¹-local objects.
• Uses the Suslin total complex to define an explicit A¹-localization functor, characterizing effective MW-motives via strictly A¹-invariant homology sheaves.
• Applies the hypercohomology spectral sequence for the complex $\tilde{\mathbb{Z}}(n)$ to compute cohomology sheaves and derive isomorphisms with Milnor-Witt K-theory sheaves.

Experimental results

Research questions

• RQ1 How can a motivic homotopy theory be built using MW-correspondences instead of classical finite correspondences?
• RQ2 What is the structure of the cohomology sheaves of MW-motivic complexes, and are they strictly A¹-invariant?
• RQ3 To what extent does MW-motivic cohomology coincide with Chow-Witt groups?
• RQ4 What is the role of the hypercohomology spectral sequence in relating MW-motivic cohomology to Milnor-Witt K-theory?
• RQ5 How does the Thom isomorphism in MW-motivic homotopy relate to Chow-Witt groups of vector bundles?

Key findings

• The cohomology sheaves $\mathbf{H}_{\mathrm{MW}}^{q,n}$ of the MW-motivic complex $\tilde{\mathbb{Z}}(n)$ vanish when $q > n$ or when $p \geq n$ and $q \neq n$, which restricts the support of the spectral sequence.
• For any smooth scheme $X$ and $n \in \mathbb{N}$, there is a canonical isomorphism $\mathrm{H}^{2n,n}_{\mathrm{MW}}(X,\mathbb{Z}) \simeq \widetilde{CH}^n(X)$, showing that MW-motivic cohomology in top degree is isomorphic to the Chow-Witt group.
• The hypercohomology spectral sequence for $\tilde{\mathbb{Z}}(n)$ induces an isomorphism $\mathrm{H}^{p,n}_{\mathrm{MW}}(X,\mathbb{Z}) \simeq \mathrm{H}^{p-n}(X, \mathbf{K}^\mathrm{MW}_n)$ for $p \geq 2n-1$, linking MW-motivic cohomology to Milnor-Witt K-theory.
• The isomorphism $\mathrm{H}^{2n,n}_{\mathrm{MW}}(X,\mathbb{Z}) \simeq \widetilde{CH}^n(X)$ is functorial and respects the ring structure, inducing a graded ring isomorphism between the MW-motivic cohomology and the Chow-Witt ring.
• The Thom space of a vector bundle $E$ over $X$ satisfies $\operatorname{Hom}_{\widetilde{\mathrm{DM}}(k,\mathbb{Z})}(\mathrm{Th}(E), \tilde{\mathbb{Z}}(n)[2n]) \simeq \widetilde{CH}^{n-r}(X, \det E)$, confirming the MW-motivic Thom isomorphism.