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

### 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.Comment: 30 pages, Final version, to appear in Journal of the Inst. of Math. Jussie

Topics: Mathematics - Algebraic Geometry, Primary 19D50, Secondary 13F15, 14F35
Publisher: 'Cambridge University Press (CUP)'
Year: 2017
DOI identifier: 10.1017/S1474748017000317
OAI identifier: oai:arXiv.org:1610.01825

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