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

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.

• Do you know that $R$-modules are the same as locally free sheaves on $\operatorname{Spec}(R)$? Also, if the fibers of $X \to \operatorname{Spec}(R)$ are connected, then $f_* \mathcal O_X = R$ (since $k$ has characteristic $0$). Commented Mar 27 at 7:49
• @red_trumpet: $R$ is not regular, so I am not sure what you mean when you say that $R$-modules are locally free sheaves on $\text{Spec}(R)$... Commented Mar 27 at 9:19
• Sorry, I was missing the word "projective". So projective $R$-modules are locally free sheaves. Even if $R$ is regular this is not true for arbitrary $R$-modules. Commented Mar 27 at 9:46
• @red_trumpet You need to assume that $R$ is normal (i.e., integrally closed). In general, $f_* \mathcal{O}_X$ will be the normalization of $R$. Commented Mar 29 at 19:36

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