Let $S$ be a graded ring which is finitely generated by $S_1$ as an $S_0$-algebra. Let $X = \text{Proj}(S)$. Let $E$ be a vector bundle over $S$. Is $\oplus_{n \in \mathbb{Z}} H^0(X,E(n))$ a graded projective $S$-module?
The answer should be no. E.g. there exists indecomposable vector bundles of rank $> 1$ over complex projective space of dimension $> 1$. If the graded module (associated to such bundles) were projective then the module must be free (as graded projective modules over polynomials rings over a field are free) and hence the bundle must be a direct sum of line bundles.
My question is this: can someone give more insight into why $\oplus_{n \in \mathbb{Z}} H^0(X,E(n))$ is not projective module when $X$ is a projective variety.