# On the K-theory of coordinate axes in affine space

## TL;DR

This paper computes the relative K-groups of coordinate axes in affine space over perfect fields of positive characteristic and extends results to ind-smooth algebras over the rationals, linking to Witt vectors and de Rham forms.

## Contribution

It provides explicit calculations of K_q(A_d, I_d) in terms of Witt vectors and de Rham forms, generalizing previous results for specific cases and characteristics.

## Key findings

- K_q(A_d, I_d) expressed via p-typical Witt vectors for characteristic p>0
- Explicit computation of K_q(A_d, I_d) over ind-smooth algebras in characteristic zero
- Extension of known results to higher dimensions and more general base fields

## 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(A_d,I_d) 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(A_d,I_d) 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.