Existence of a proper birational morphism from a Gorenstein scheme, with trivial higher direct images, implies Cohen-Macaulay?

Question

Let $R$ be a Noetherian excellent reduced local ring containing a field of characteristic $0$. If there exists a Gorenstein scheme $Y$ and a proper birational map $f: Y \to \text{Spec}(R)$ such that $R^i f_* \mathcal O_Y=0, \forall i>0$, then is it true that $R$ is Cohen-Macaulay?

Answer 1

Let $R = \{f : f(0,0) = f(0,1)\} \subset k[x,y] = S.$

Then $\operatorname{Spec} R$ has a nodal singularity and isn’t Cohen-Macaulay since it looks like two 2-planes meeting at a point. $\operatorname{Spec} S$ is smooth of course, hence Gorenstein.

The map $\operatorname{Spec} S \to \operatorname{Spec} R$ is proper and birational, and since the fibers are all zero-dimensional, the higher direct images vanish.