Real-analytic manifolds as algebraifolds

Establish whether, for any real-analytic manifold M, the R-algebra C^\omega(M) of real-analytic functions is an R-algebraifold; equivalently, show that the C^\omega(M)-module of R-derivations Der_R(C^\omega(M)) is finitely generated projective.

Background

For open subsets of \mathbb{R}^n, derivations of C^\omega(\mathbb{R}^n) are given by real-analytic vector fields, and the module is free of rank n. The paper cites results indicating that for a real-analytic manifold M, Der_R(C^\omega(M)) corresponds to real-analytic vector fields.

However, the authors do not yet have a proof that this derivation module is finitely generated projective over C^\omega(M) for general M, which is needed to conclude that C^\omega(M) is an algebraifold.