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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 $\operatorname{K_0}(R)$?

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Is $k$ really arbitrary ? – user18119 Feb 5 '13 at 22:55
It looks natural to ask this question for real numbers because this is suppose to be some analog of a sphere, and also for complex numbers because it might be simpler. These are two examples I had in mind when asked, but if there is a dependence on a basic field, it would be interesting to understand that too. – Alex Feb 5 '13 at 23:50
The Picard group of $R$ is trivial over $\mathbb R$ but non-trivial over $\mathbb C$. I will come back to this later. – user18119 Feb 6 '13 at 9:26
up vote 1 down vote accepted

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.

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