Here is something that isn't yet very clear to me. Say, I've got a commutative ring A. I consider the affine scheme from A, so it's a sheaf of rings over Spec A. 


**EDIT**: And additionally let's say Spec A is Hausdorff. 


Now additionally let's say I know an A-module M and from that I can make a sheave of modules over O\_Spec(A), call it M~. All standard stuffs till now. But now I want to have one more information, namely I have an open subset of Spec(A), say U, that is dense in Spec(A). And I know additionally that the stalks M~\_x are isomorphic to O\_x for all x in U.. Can one conclude that M and A are A-module isomorphic? (if so can one follow the same argument for general schemes with modules over them?) What are the conditions by which one can conclude this?



Dense section of sheaves of modules 
If U is an open set in X, but U isn't X, then there are non-zero sheaves on X whose support lies outside U. Now add O\_X to one of these to get a counterexample.


