Let $R$ be the ring $R[X,Y]/(X^2+Y^2−1)$. The space of $\mathbb{R}$-rational points of the affine scheme associated to $R$ is the topological circle $S^1$.


An algebraic vector bundle over $R$ is an $R$-algebra $A$ with certain properties or equivalently a finitely generated projective $R$-module $A$.


My first question is: What is the explicit projective $R$-module $A$ corresponding to the topological Moebius bundle over $S^1$? (By 'corresponding' I mean in particular that the $\mathbb{R}$-rational points should induce the topological Moebius bundle and that it has rank one.)


My second question is motivated by the fact that topologically there are only two non-isomorphic rank one bundles over $S^1$: the trivial bundle and the Moebius bundle. What is the analogous algebraic situation? Are there more than two non-isomorphic and rank one projective modules over $R$?


Thank you!


(I apologize that I first asked this question on math.stackexchange, I deleted it there.)



Algebraic analogue of the Moebius bundle over the circle
The internal note "Le ruban de Moebius comme représentation d'un idéal non principal" (Moebius band, as a non-principal ideal), by Daniel Ferrand, contains more or less the material you want. Daniel is retired and doesn't have a website, but, thank you Google!,
the note is downloadable from <http://www.math.unibas.ch/~giordano/Moebius.pdf>


The non-principal ideal you want is the ideal $(1+x,y)$ in the ring $R[x,y]/(x^2+y^2-1)$.


