A Bloch-Wigner exact sequence over local rings

TL;DR

This paper extends the Bloch-Wigner exact sequence and Van der Kallen's theorem to local rings with larger residue fields, broadening their applicability beyond previous limitations.

Contribution

It generalizes key algebraic K-theory results to local rings with residue fields having more than nine and four elements, respectively.

Findings

Extended the Bloch-Wigner exact sequence to broader local rings.

Proved Van der Kallen's theorem for local rings with larger residue fields.

Results also applicable to semilocal rings with similar residue field properties.

Abstract

In this article we extend the Bloch-Wigner exact sequence over local rings, where their residue fields have more than nine elements. Moreover, we prove Van der Kallen's theorem on the presentation of the second $K$-group of local rings such that their residue fields have more than four elements. Note that Van der Kallen proved this result when the residue fields have more than five elements. Although we prove our results over local rings, all our proofs also work over semilocal rings where all their residue fields have similar properties as the residue field of local rings.