Support of the pushforward of structure sheaf of a smooth scheme along proper birational morphism

Question

Let $k$ be a field of characteristic $0$ and $R$ be a finite type $k$-algebra. Let $X$ be a smooth $k$-scheme and $f: X \to \text{Spec}(R)$ be a proper birational morphism. Then, is the $R$-module $f_* \mathcal O_X$ supported at every maximal ideal of $R$, i.e., is the localization of $f_* \mathcal O_X$ non-zero at every maximal ideal of $R$ ? If needed, I am willing to assume that $f_* \mathcal O_X$ is a projective $R$-module.

Answer 1

Proper + birational tells you that $f$ is surjective, so yes, $f_* \mathcal{O}_X$ will be supported on all of ${\rm Spec}(R)$.

The characteristic 0 assumption is not needed.