# 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.