Algebraic K-Theory and Modular Symbols

TL;DR

This paper extends algebraic K-theory calculations to general Noetherian rings using spectral sequences, Tits buildings, and modular symbols, providing new insights into the homotopy types of module categories.

Contribution

It generalizes Quillen's spectral sequence from Dedekind domains to all Noetherian rings and computes the differential using modular symbols and Tits buildings.

Findings

Calculated the differential d^1 of the rank spectral sequence.

Generalized Quillen's spectral sequence to Noetherian rings.

Connected homotopy types of module categories with modular symbols.

Abstract

In this paper, we calculate the differential $d^1$ of the rank spectral sequence. We generalize Quillen's spectral sequence from Dedekind domain to general integral Noetherian ring $A$ by considering the Q-construction $Q^{tf}(A)$ of the category of finitely generated torsion-free modules. In particular, by resolution theorem, if $A$ is regular, then $Q^{tf}(A)$ is homotopy equivalent to $QP(A)$ where $P(A)$ is the category of finitely generated projective $A$-modules. We deduce the differential $d^1$ by using Tits buildings, Steinberg modules, and modular symbols in the sense of Ash-Rudolph. The spirit of Quillen's categorical homotopy theory will be used intensively throughout this paper.