diagrams – Cohomology of the withdrawal of a coherent sheaf (considered as an abelian sheaf)

Let $ X $ to be a refined Noetherian system, $ Y $ a separate Noetherian schema, $ f: X rightarrow Y $ a morphism of patterns inducing a homeomorphism on the underlying topological spaces.

Let $ F $ to be a coherent sheaf on $ Y $; treat it like a sheaf of abelian groups and bring it back to $ X $. It's possible that $ H ^ 1 (X, f ^ {- 1} (F)) neq 0 $?

This question can be confusing because we essentially compute the sheaf cohomology of the same abelian sheaf on the same topological space, but the non-trivial information here is the existence of morphism. $ f $ (remember that the forgetful functor of schemas to spaces is not complete). Thus, a random coherent sheaf on a separate pattern with non-disappearing $ H ^ 1 $ is not going to do the business.