On the $K$–theory of coordinate axes in affine space

Abstract

Let $k$ be a perfect field of characteristic $p>0$, let $A_d$ be the coordinate ring of the coordinate axes in affine $d$–space over $k$, and let $I_d$ be the ideal defining the origin. We evaluate the relative $K$–groups $K_q\left(A_d,I_d\right)$ in terms of $p$–typical Witt vectors of $k$. When $d=2$ the result is due to Hesselholt, and for $K_2$ it is due to Dennis and Krusemeyer. We also compute the groups $K_q\left(A_d,I_d\right)$ in the case where $k$ is an ind-smooth algebra over the rationals, the result being expressed in terms of algebraic de Rham forms. When $k$ is a field of characteristic zero this calculation is due to Geller, Reid and Weibel.