On the algebraic K-theory of $R[X,Y,Z]/(X^2+Y^2+Z^2-1)$

Abstract

It is well known after R. Swan that $\tilde K_0(R[X,Y,Z]/(X^2+Y^2+Z^2-1))$ is isomorphic to the integers $\mathbb Z$, whenever $R$ is a field of characteristic not two which contains the squared root of $-1$. \par First, we give explicit idempotent matrices $\gamma^p$ of order two, corresponding to the integer $p,$ in the isomorphism above, if $R$ is a field of characteristic zero. Then, we use the algebraic de Rham cohomology of Kähler differentials to define Brouwer degree for polynomial homomorphisms of $R[X,Y,Z]/(X^2+Y^2+Z^2-1)$ to itself, and relate the problem of finding hermitian representatives for $R=K(i),$ $K$ a field not containing $i,$ to some unsolved problems of representing Brouwer degrees by polynomial maps.