K_0 of hypersurfaces defined by x_1^2+ ... + x_n^2 = \pm 1

Abstract: Let $k$ be a field of characteristic $\ne 2$ and let $Q_{n,m}(x_1, ...,x_n,y_1, ...,y_m)=x_1^2+... +x_n^2-(y_1^2+... +y_m^2)$ be a quadratic form over $k$. Let $R(Q_{n,m})=R_{n,m}=k[x_1, ...,x_n,y_1, ...,y_m]/(Q_{n,m}-1)$. In this note we will calculate $\wt K_0(R_{n,m})$ for every $n,m \geq 0$.