K2 of localisations of local rings

TL;DR

This paper demonstrates that K2 groups of certain regular localizations of local rings can be described explicitly using the Steinberg presentation, employing an inductive proof based on co-Cartesian squares.

Contribution

It provides a new inductive method to describe K2 of localizations of local rings via Steinberg presentation, extending previous techniques.

Findings

K2 of regular localizations can be described by Steinberg presentation

Inductive proof using co-Cartesian squares

Applicable to localizations inverting regular parameters

Abstract

We show that K2 of "sufficiently regular" localisations of local rings (e.g. inverting a sequence of regular parameters) can be described by the Steinberg presentation. The proof is inductive on the number of irreducible elements being inverted, successively using a generalisation of a co-Cartesian square first exploited by Dennis and Stein.