I ask this question yesterday, i wish it is more clear now.
Let $X=Spec(R)$ the spectrum of a ring $R$
In the article "Modules projectifs et espaces fibrés à fibre vectorielle". Jean pierre serre says that $R$ (if it noetherian) is a product of rings $R_{i}$ such the spectrum of each $R_{i}$ is connected component of $X$, this will allow us is the study of modules over $R$ to to restrict ourselves to the case that $X$ is connected.
For example in proposition 6, he says that every semi-local ring $R$ is a finite product of indecomposable semi rings $R_{i}$, hence every projective module over $R$ is direct sum of one of $R_{i}$.
I think the proof of that follow from this question "Projective modules over a direct product of rings".
My question is if this is true for an infinite product of indecomposable semi rings, i think it's false.
More generally do you know some example when we learn something about a module $M$ by using base change in all of the connected components or by some covering of the space $X$. This look to me like local to global principle in topology.