In his book, the corollary states as
Let $f:X\rightarrow Y $ be a projective morphism of schemes of finite type over a field $k$. Let $\mathscr{F}$ be a coherent sheaf on $X$. Then $f_*\mathscr{F}$ is coherent on $Y$.
My first question is that in the statement what exactly is defined over field $k$. From the definitions, I can't see where $k$ plays a role.
Second, in his proof, he uses $\widetilde{\Gamma(Y,f_* \mathscr{F}})=\widetilde{\Gamma(X, \mathscr{F}})$. How could we deduce this? Is this always true for arbitrary morphisms and schemes?
Any suggestions are welcomed. Thanks.