Algebraic vs Geometric Picard groups

Question

Given a smooth manifold $P$. I found two definition of a Picard group. The first one is algebraic, $Pic(C^{\infty}(P))$ is defined as the set of self-equivalence bimodules (Morita equivalence) with tensor product of bimodules as group operation. The second one is geometric, $Pic(P)$ is defined as the set of isomorphism classes of line bundles over $P$ with tensor product of vector bundles as group operation. I know there is some relation between the two concepts via Serre-Swan Theorem. Yet I don't know what is the exact relation, I mean, are they isomorphic? Can one of them be seen as a (normal) subgroup of the other? I would like an answer avoiding use of algebraic geometric terminologies, like affine spaces or schemes, if possible.

Answer 1

They are different. The bimodule definition is for rings, not commutative rings; for commutative rings you should be using invertible modules rather than invertible bimodules, and then the two definitions agree by the Serre-Swan theorem. Invertible modules correspond to finitely generated projective modules of rank $1$, which are in turn identified with line bundles.