Homeomorphism between Schemes is Affine Morphism Let $f: X \to Y$ be a proper morphism between Noetherian $k$-schemes with property that $f$ - considered as continuous map between underlying topological spaces $X$ and $Y$ is a homeomorphism.
My question is why in this case $f$ is already a affine morphism?
Attempts:
I intend to use Serre's criterion for affinity:
Let $X$ Noetherian scheme. Then
$$X \text{ affine } \Leftrightarrow H^i(X, \mathcal{F})=0 \text{ for all quasicoherent sheaves of modules } \mathcal{F} \text{and } i > 0$$
Therefore it suffice to solve following problems:

*

*If $\mathcal{F}$ quasicohernt how to show that $f_*\mathcal{F}$ is also quasicoherent?
(here I know an argument only in the case that $f$ is already affine =) )


*Why holds $H^i(f^{-1}(U), \mathcal{F}) = H^i(U, f_* \mathcal{F})$ for given conditions above?
 A: You can find answers to both of these questions in Hartshorne's Algebraic Geometry.
Your first question is Proposition II.5.8(c).  As a sketch of the proof, since $X$ is Noetherian, it can be covered by finitely many affine open sets $U_i$, and each intersection $U_i\cap U_j$ can also be covered by finitely many affine open sets $U_{ijk}$.  Now, the pushforward of $\mathcal{F}$ restricted to any $U_i$ or $U_{ijk}$ is quasicoherent.  But the pushforward of $\mathcal{F}$ is just the kernel of a certain morphism between products of these restrictions, by the sheaf gluing axiom (a section of $f_*\mathcal{F}$ over $U$ is the same as sections over $U\cap U_i$ for each $i$ such that the restrictions to the $U_{ijk}$ agree).  Since the kernel of a morphism of quasicoherent sheaves is quasicoherent, this implies $f_*\mathcal{F}$ is quasicoherent.  (The Noetherian hypothesis is used to guarantee that the products of the restricted sheaves are finite, since an infinite product of quasicoherent sheaves may not be quasicoherent.)
Your second question follows from the fact that cohomology of sheaves can be defined as derived functors of the functor of global sections on the category of sheaves of abelian groups on the space.  The category of sheaves of abelian groups depends only on the topological structure (not the ringed space structure), and so this definition is obviously preserved by homeomorphisms.  The equivalence of cohomology defined for sheaves of abelian groups and cohomology defined for sheaves of $O_X$-modules is Proposition III.2.6 in Hartshorne.  As a sketch of the proof, you can show that flasque sheaves are acyclic in both categories, and so you can compute the cohomology of an $O_X$-module in either category by taking a resolution by flasque $O_X$-modules.
