Given a graded ring S and a graded S-module M http://latex.mathoverflow.net/png?M we can carry out a construction in order to get \tilde{M} http://latex.mathoverflow.net/png?%5Ctilde%7BM%7D, which is a sheaf over the scheme Proj S. With this in view, I have an equivalence of categories between the category of (quasi-finite generate) modules and the category of q-coherent sheaves over Proj S.

On the other hand, given a locally free sheaf \mathcal{F} http://latex.mathoverflow.net/png?%5Cmathcal%7BF%7D of rank n http://latex.mathoverflow.net/png?n over Proj S we can get a vector bundle out of it and further we have a 1-1 correspondence between isomorphism classes of free sheaves of rank n and isomorphism classes of vector bundles of rank n.

With the last two facts in view, my question is the following. if I start with a finite generated module M http://latex.mathoverflow.net/png?M then the sheaf \tilde{M} http://latex.mathoverflow.net/png?%5Ctilde%7BM%7D is locally free? If so, I can get from M http://latex.mathoverflow.net/png?M a locally free sheaf \tilde{M} http://latex.mathoverflow.net/png?%5Ctilde%7BM%7D and from such a sheaf a vector bundle (and perhaps backwards as well). Therefore, Is it the same having a vector bundle over Proj S than a S-Module? or what are the limits of such a relation described here among Vector Bundles & S-Modules?. By "Is it the same" I mean, We have the same amount of information in such objects.

Given a graded ring $S$ and a graded S-module $M$ we can carry out a construction in order to get $\tilde{M}$, which is a sheaf over the scheme $\mathrm{Proj}~ S$. With this in view, I have an equivalence of categories between the category of (quasi-finite generate) modules and the category of $q$-coherent sheaves over $\mathrm{Proj}~S$.

On the other hand, given a locally free sheaf $\mathcal{F}$ of rank $n$ over $\mathrm{Proj}~S$ we can get a vector bundle out of it and further we have a 1-1 correspondence between isomorphism classes of free sheaves of rank $n$ and isomorphism classes of vector bundles of rank $n$.

With the last two facts in view, my question is the following. if I start with a finite generated module $M$ then the sheaf $\tilde{M}$ is locally free? If so, I can get from $M$ a locally free sheaf $\tilde{M}$ and from such a sheaf a vector bundle (and perhaps backwards as well). Therefore, Is it the same having a vector bundle over $\mathrm{Proj}~ S$ than a $S$-Module? or what are the limits of such a relation described here among Vector Bundles & $S$-Modules?. By "Is it the same" I mean, We have the same amount of information in such objects.