Let $f: X\rightarrow Y$ be a morphism of varieties. Let $0\rightarrow F\rightarrow E\rightarrow G\rightarrow 0$ be a short exact sequence of locally free sheave of finite
rank. If direct images of above sheaves are locally free, then is it true that it
induces a short exact sequence $0\rightarrow f_*F\rightarrow f_*E\rightarrow f_*G\rightarrow 0$?
The method of forcing certainly fits here. Before, set theorists expected that independence results would be obtained by building non-standard, ill-founded models, and model theoretic methods would be key to achieve this. Cohen's method begins with a transitive model and builds another transitive one, and the construction is very different from all the techniques being tried before. 

This was completely unexpected. Of course, in hindsight, we see that there are similar approaches in recursion theory and elsewhere happening before or at the same time. 

But it was the fact that nobody could imagine you would be able to obtain transitive models that mostly had us stuck.
