[Paper Review] Algebraic G-theory in motivic homotopy categories

Fangzhou Jin|arXiv (Cornell University)|Jun 11, 2018

Homotopy and Cohomology in Algebraic Topology8 references4 citations

TL;DR

This paper establishes that algebraic G-theory is representable in both unstable and stable motivic homotopy categories, identifying it with Borel-Moore motivic homology associated to K-theory. The key contribution is a natural isomorphism between the Borel-Moore theory of K-theory and G-theory, compatible with Quillen and Thomason's functorialities, including proper pushforwards and smooth pullbacks via Gysin maps.

ABSTRACT

We prove that algebraic G-theory in is representable in unstable and stable motivic homotopy categories; in the stable category we identify it with the Borel-Moore theory associated to algebraic K-theory, and show that such an identification is compatible with the functorialities defined by Quillen and Thomason.

Motivation & Objective

• To establish representability of algebraic G-theory in unstable and stable motivic homotopy categories.
• To identify G-theory with Borel-Moore motivic homology associated to K-theory in the stable category.
• To show that this identification is compatible with the functorial structures defined by Quillen and Thomason, including contravariant and proper functors.
• To extend the compatibility of Gysin morphisms and Chern character maps to singular schemes using motivic homotopy theory.

Proposed method

• Constructing a model for G-theory in the unstable motivic homotopy category using the category of spaces over a scheme.
• Proving representability of G-theory via the mapping spectrum from the suspension spectrum of a scheme to a universal G-theory object.
• Stabilizing the construction to obtain a spectrum-level identification in the stable motivic homotopy category.
• Using the six functors formalism and properties of Borel-Moore theories to relate G-theory to motivic cohomology with rational coefficients.
• Establishing compatibility of Gysin morphisms with contravariant functoriality via deformation to the normal cone and boundary maps.
• Leveraging the six functors formalism and motivic spectra to define and verify compatibility of the Chern character with proper and smooth functors.

Experimental results

Research questions

• RQ1 Is algebraic G-theory representable in the unstable motivic homotopy category?
• RQ2 Does the stable motivic homotopy category realize G-theory as a Borel-Moore theory associated to K-theory?
• RQ3 Is the identification of G-theory with Borel-Moore K-theory compatible with Quillen's contravariant functoriality?
• RQ4 Is the compatibility preserved under proper pushforwards and smooth pullbacks in the motivic setting?
• RQ5 Can the Chern character from G-theory to rational Borel-Moore motivic homology be defined and shown to commute with functorialities without using graph constructions?

Key findings

• Algebraic G-theory is representable in both the unstable and stable motivic homotopy categories.
• In the stable category, G-theory is isomorphic to the Borel-Moore motivic homology associated to K-theory, with an explicit isomorphism $\mathbf{KGL}^{BM}_{n,m}(X/S) \simeq G_{n-2m}(X)$.
• The proper functoriality of Borel-Moore theory corresponds exactly to the proper pushforward in G-theory.
• The Gysin morphism for lci morphisms in Borel-Moore theory corresponds to the contravariant functoriality of G-theory.
• The Chern character map $\mathrm{ch}: \mathbf{GGL} \to \oplus_{i\in\mathbb{Z}} \mathbf{H}\mathbb{Q}^{BM}(i)[2i]$ is compatible with proper and smooth functors, generalizing Fulde's Riemann-Roch theorem to singular schemes.
• The construction avoids MacPherson's graph method, relying instead on motivic homotopy theory and the six functors formalism.