TITLE: Frobenius morphism and global sections of direct image of structure sheaf
QUESTION [5 upvotes]: Let $X$ be a proper scheme defined over an algebraically closed field of characteristic $p > 0$. Let $F : X\rightarrow X$ be the absolute Frobenius morphism. What is the dimension of $H^0(X, F_*\mathcal{O}_X)$?

REPLY [4 votes]: F is a finite morphism, so affine, so $H^i(X, \mathcal{O}_X) = H^i(X, F_*\mathcal{O}_X)$ for all i.