K-theory of one-dimensional rings via pro-excision

TL;DR

This paper advances the understanding of algebraic K-theory for one-dimensional rings by proving Geller's conjecture in certain cases, establishing finiteness results, and exploring pro-excision techniques.

Contribution

It introduces new results on pro-excision in K-theory, proves Geller's conjecture for equal characteristic rings, and provides initial results in mixed characteristic.

Findings

Proved Geller's conjecture for equal characteristic rings over perfect fields of finite characteristic.

Established finiteness results for K-groups of singularities in various contexts.

Developed new techniques in pro-excision for one-dimensional rings.

Abstract

This paper studies "pro-excision" for the K-theory of one-dimensional (usually semi-local) rings and its various applications. In particular, we prove Geller's conjecture for equal characteristic rings over a perfect field of finite characteristic, give the first results towards Geller's conjecture in mixed characteristic, and we establish various finiteness results for the K-groups of singularities (covering both orders in number fields and singular curves over finite fields).