[Paper Review] Around Quillen's theorem A

Bruno Kahn|arXiv (Cornell University)|Aug 11, 2011

Homotopy and Cohomology in Algebraic Topology8 references3 citations

TL;DR

This paper introduces the notion of a cellular functor to provide a functorial, Theorem A-free proof of the Solomon-Tits theorem and to construct a rank spectral sequence converging to the homology of the $K'$-theory space of an integral scheme. The key contribution is a spectral sequence whose $E^1$-terms are expressed in terms of automorphism groups and reduced Steinberg modules, generalizing Quillen's exact sequences and linking them to homotopy cofibrations via Vogel's argument.

ABSTRACT

We reformulate some exact sequences of Quillen into a spectral sequence converging to the homology of certain $K$-theory spaces.

Motivation & Objective

• To provide a variant of Quillen’s proof of the Solomon-Tits theorem that avoids reliance on Theorem A.
• To generalize Quillen’s exact sequences for $K$-groups into a functorial spectral sequence framework.
• To construct a rank spectral sequence converging to the homology of the $K'$-theory space of an integral scheme using cellular functors and nerve constructions with coefficients.
• To reformulate Quillen’s $K$-theory exact sequences into a spectral sequence via homotopy cofibrations, dual to Theorem A’s role in fibrations.
• To establish a connection between the $E^1$-terms of the spectral sequence and the reduced Steinberg modules of general linear groups over function fields.

Proposed method

• Introduces the concept of a cellular functor, which is well-behaved under nerve constructions and enables the analysis of homotopy cofiber sequences.
• Applies Thomason’s theorem on the nerve of a Grothendieck construction to relate classifying spaces of categories to homotopy colimits.
• Uses nerves with coefficients in abelian group-valued functors to define homology theories on categories of sheaves.
• Constructs a rank spectral sequence via the filtration of torsion-free sheaves by rank, using the poset of subobjects in the generic fiber.
• Employs Vogel’s argument to show that the resulting spectral sequence coincides with Quillen’s original exact sequences, via isomorphisms between homology groups of mapping cones and relative homology.
• Applies the Solomon-Tits theorem to show that the nerve of the poset of proper subspaces of a vector space is weakly equivalent to a wedge of spheres, enabling the identification of $E^1$-terms.

Experimental results

Research questions

• RQ1 Can the Solomon-Tits theorem be reproved without invoking Quillen’s Theorem A?
• RQ2 How can Quillen’s exact sequences for $K$-groups be reinterpreted as a spectral sequence?
• RQ3 What is the relationship between homotopy cofibrations and the resulting spectral sequences in $K$-theory?
• RQ4 How do the $E^1$-terms of the rank spectral sequence relate to the reduced Steinberg modules of general linear groups?
• RQ5 Can the spectral sequence be constructed in a fully functorial way using cellular functors and nerve constructions with coefficients?

Key findings

• The paper constructs a rank spectral sequence converging to the homology of the $K'$-theory space of an integral scheme, with $E^1$-terms given by $\bigoplus_{E_{\alpha}} H_q(\operatorname{Aut}(E_{\alpha}), \operatorname{st}(E_{\alpha}))$.
• The $E^1$-terms are isomorphic to the homology of the automorphism group of a torsion-free sheaf of rank $p$ with coefficients in the reduced Steinberg module of its generic fiber.
• The spectral sequence arises from a filtration by rank of torsion-free sheaves and is compatible with Quillen’s original exact sequences via Vogel’s argument.
• The proof of the Solomon-Tits theorem is rederived without using Theorem A, relying instead on the cellular functor framework and the homotopy cofiber structure.
• The spectral sequence is shown to be equivalent to Quillen’s exact sequences through an isomorphism between relative homology and cone homology, established via Vogel’s argument.
• The construction is functorial and generalizes Quillen’s method to a broader context, including torsion-free sheaves on integral schemes.