How to compute the K-group of this affine scheme?

Question

By Bott periodicity, we know that $$K_{0}^{top}(S^2)=\mathbb{Z} \oplus \mathbb{Z}$$ But $S^2$ is defined by the equation $x^2+y^2+z^2=1$. If we write this abstractly as $$X:=Spec\mathbb{R}[x,y,z]/(x^2+y^2+z^2-1) ,$$ then how can we compute $$K_{0}^{alg}(X)$$ the $K_0$ is the same definition as Hartshorne's Ex II.6.10

Answer 1

The algebraic K-theory of smooth affine or projective quadrics is quite well understood (as opposed to their Chow groups). There is a nice paper of Swan called K-theory of quadric hypersurfaces, where not only the $K_0$ but also the higher $K_i$ are computed in term of the K-theory of the base ring. You will see that it also depends on the Clifford algebra of the quadratic form.

This paper also consider the $K_0$ of spheres over $\mathbb{R, C}$ and $\mathbb{H}$. This is theorem 3. It shows that there is a one to one correspondence between stable classes of algebraic and topological vector bundles.

Hence, for the 2-sphere : $K_0^{alg}(\mathbb{R}[x,y,z]/(x^2+y^2+z^2-1))=KO^0(S^2)=\mathbb{Z}\oplus\mathbb{Z}/2$.

Note that you could have said that $S^2$ is homeomorphic to $\mathbb{CP}^1$ and ask for the algebraic K-theory of $\mathbb{P}^1_\mathbb{C}$. In that case $K_0^{alg}(\mathbb{P}^1_\mathbb{C})=K^0(S^2)=\mathbb{Z}\oplus\mathbb{Z}$