Vector bundles on algebraic 2-sphere Let $R=k[x,y,z]/(x^2+y^2+z^2-1)$ be an algebraic sphere over some field $k$. Is it true that any projective module of rank 1 is isomorphic to $R$? More generally, what is the structure of $\operatorname{K_0}(R)$?

This only answers the first part of your question. The ring $R$ is always regular if $k$ has charateristic different from $2$. It is known (see Swan: Vector bundles and projective modules, Theorem 5) that $R$ is a UFD if $k=\mathbb R$ and $R$ is not UFD if $k=\mathbb R$. As $R$ is integrally closed, this is equivalent to say Pic$(R)$ is trivial in the first case, and non-trivial in the second case. 
Over $\mathbb R$, Swan, op.cit, Theorem 3 implies that $K_0(R)$ is non-trivial. 
