On the $K$-theory of the coordinate axes in the plane

Abstract

Let $k$ a regular noetherian $\mathbb{F}_p$-algebra, let $A = k[x,y]/(xy)$ be the coordinate ring of the coordinate axes in the affine $k$-plane, and let $I = (x,y)$ be the ideal that defines the intersection point. We evaluate the relative $K$-groups $K_q(A,I)$ completely in terms of the big de Rham-Witt groups of $k$. This generalizes a formula for $K_1(A,I)$ and $K_2(A,I)$ by Dennis and Krusemeyer.