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

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 $n,m \geq 0$.

💡 Research Summary

The paper addresses the reduced Grothendieck group (\widetilde K_{0}) of the coordinate rings of affine quadrics defined by a single quadratic equation of the form
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