Global sections of pushforwards

Question

Let $X$ and $Y$ be projective schemes over $\mathrm{Spec}A$, where $A$ is a ring. Let $\pi:X\rightarrow Y$ be a morphism. Let $\mathscr F$ be a coherent sheaf on $X$. When is it true that $\Gamma(X,\mathscr F)\simeq \Gamma(Y,\pi_*\mathscr F)$?

Answer 1

Let $\mathcal{F}$ denote a sheaf of Abelian groups (not strictly necessary) on a topological space $X$. Let a continuous map $\pi:X\to Y$ be given. $\Gamma(Y,\pi_*\mathcal{F})=(\pi_*\mathcal{F})(Y)=\mathcal{F}(\pi^{-1}(Y))=\mathcal{F}(X)=\Gamma(X,\mathcal{F}).$ So, the answer is that this is true quite generally, and comes down to the definition of the pushforward of a sheaf.