$K$-theory of monoid algebras and a question of Gubeladze

TL;DR

This paper proves an isomorphism in K-theory for certain monoid algebras over regular rings containing Q, answers Gubeladze's question, and explores higher K-theory and applications to Serre dimension.

Contribution

It establishes the K_1 isomorphism for specific monoid algebras and computes their higher K-theory, addressing open questions in the field.

Findings

K_1 map is an isomorphism for the given monoid algebra

Higher K-theory does not extend the isomorphism beyond K_1

Applications to Serre dimension of monoid algebras

Abstract

We show that for any commutative noetherian regular ring $R$ containing $\Q$, the map $K_1(R) \to K_1(\frac{R[x_1, \cdots , x_4]}{(x_1x_2 - x_3x_4)})$ is an isomorphism. This answers a question of Gubeladze. We also compute the higher $K$-theory of this monoid algebra. In particular, we show that the above isomorphism does not extend to all higher $K$-groups. We give applications to a question of Lindel on the Serre dimension of monoid algebras.