Pushforward of an invertible sheaf under certain hypothesis on the Higher direct images

Question

Assume that $f:X\to Y$ is a flat and proper morphism between intergral noetherian schemes. Assume that $L$ is an invertible sheaf such that $R^i f_\ast L=0$ for $i>0$.

Can we conclude that $f_\ast L$ is an invertible sheaf on $Y$?

Answer 1

No; you should roughly think of $f_* L$ as bundling together the global sections on each fiber of $f$. So for example, if $f$ is a finite morphism of degree $d$, $f_* L$ is a rank $d$ vector bundle.