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

Abstract

We show that for any commutative noetherian regular ring $R$ containing $\Q$, the map $K_1(R)\to K_1(\frac{R[x_1,\cdots ,x_4]}{(x_1{x}_2-x_3{x}_4)})$ 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.

30 pages, Final version, to appear in Journal of the Inst. of Math. Jussieu