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monoid algebras

Journal of the Institute of Mathematics of Jussieu (1)

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$K$-THEORY OF MONOID ALGEBRAS AND A QUESTION OF GUBELADZE
Part of
Amalendu Krishna, Husney Parvez Sarwar
Journal:

Journal of the Institute of Mathematics of Jussieu / Volume 18 / Issue 5 / September 2019

Published online by Cambridge University Press:

10 August 2017, pp. 1051-1085

Print publication:

September 2019
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We show that for any commutative Noetherian regular ring $R$ containing $\mathbb{Q}$, the map $K_{1}(R)\rightarrow K_{1}\left(\frac{R[x_{1},\ldots ,x_{4}]}{(x_{1}x_{2}-x_{3}x_{4})}\right)$ 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.