Finite-dimensionality of $H^1(X,\mathscr O_X)$ for a projective curve

Question

A well-known result by Serre is that properness of a noetherian scheme $(X,\mathscr O_X)$ over $k$ implies finite dimensionality of $H^i(X,\mathscr O_X)$ for all $i \geq 0$. For a projective variety it is easy to prove that $H^0(X,\mathscr O_X) = \mathscr O_X(X) = k$. I was wondering if there's an elementary proof for finite-dimensionality of higher cohomology groups. Maybe, for the easiest case of $X$ being an smooth curve and finite-dimensionality of $H^1(X,\mathscr O_X)$ one could reduce it to computing it for $\mathbb{P}^1$?

Answer 1

Elementary can mean many things, but here is a way to reduce to projective space case. Given a projective variety $X$ of dimension $n$, you can find a finite map $f:X\to\mathbb{P}^n$. Since finite maps are affine, one checks that for any quasi-coherent sheaf $F$ on $X$, $H^i(X,F)=H^i(\mathbb{P}^n, f_*F)$. Finally, using finiteness, for any coherent sheaf $F$ on $X$, one checks $f_*F$ is coherent and thus you are reduced to the projective space case.