Let $k$ be a field of characteristic $0$. Let $R$ be a Noetherian local normal domain containing $k$. Also assume that $R$ is the homomorphic image of a Gorenstein ring of finite dimension, hence $R$ admits a Dualizing complex.
Let $Y$ be a Gorenstein normal scheme over Spec$(k)$ of finite Krull-dimension.


If there exists 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?
If this is not true in general, what if I also assume $Y$ is regular?


As $R$ is normal, $R$ is of course Cohen-Macaulay when $\dim R \leq 2$. So we only have to think about $\dim R \geq 3$.



Proper birational morphism from a Gorenstein normal scheme to a normal local domain, with trivial higher direct images, implies Cohen-Macaulay?
I started writing this last night, but didn't finish. In addition to [Jason Starr's answer](https://mathoverflow.net/a/450348/33088), you can use my new vanishing theorems to remove the assumption that $R$ is essentially of finite type over a field. Note that the special case when $R$ is essentially of finite type over a field of characteristic zero also follows from Kempf's criterion for rational singularities [[Kempf 1973](https://doi.org/10.1007/BFb0070319), p. 50].




---


In short, if $Y$ is assumed to be regular, then $R$ will be Cohen–Macaulay. In fact, we have the following stronger result.


**Theorem.** *Let $R$ be a Noetherian ring and let $f\colon Y \to \operatorname{Spec}(R)$ be a proper birational morphism such that $Y$ is regular and $R^if\_\*\mathcal{O}\_Y = 0$ for all $i > 0$. Suppose $R$ contains $\mathbf{Q}$. Then, every localization of $R$ is pseudo-rational, and in particular $R$ is Cohen–Macaulay.*


*Proof.* Since the conclusion can be checked locally, it suffices to consider the case when $(R,\mathfrak{m})$ is local. Set $\hat{Y} := Y \times\_{\operatorname{Spec}(R)} \operatorname{Spec}(\hat{R})$, and consider the Cartesian diagram
$$\require{AMScd}\begin{CD}
\hat{Y} @>>> Y\\
@V\hat{f}VV @VVfV\\
\operatorname{Spec}(\hat{R}) @>>> \operatorname{Spec}(R)
\end{CD}$$
By base change, $\hat{f}$ is proper, and by flat base change, $\hat{f}$ is birational [[EGAI$\_\text{new}$](https://mathscinet.ams.org/mathscinet/article?mr=3075000), Proposition 3.9.9] and $R^i\hat{f}\_\*\mathcal{O}\_{\hat{Y}} = 0$ for all $i > 0$. Note that $\hat{R}$ is normal by [[Lipman 1969](http://www.numdam.org/item/?id=PMIHES_1969__36__195_0), Remark 16.2] and that the pseudo-rationality of $\hat{R}$ would imply that $R$ is pseudo-rational [[Murayama 2022](https://doi.org/10.1112/S0010437X21007715), Proposition 4.20]. Since $\hat{Y}$ is regular by [[EGAIV$\_2$](http://www.numdam.org/item/PMIHES_1965__24__5_0/), Lemme 7.9.3.1], we may replace $R$ by $\hat{R}$ and $Y$ by $\hat{Y}$ to assume that $R$ is complete local.


We now prove the special case when $(R,\mathfrak{m})$ is complete local. First, by my version of Grauert-Riemenschneider vanishing [[Murayama](https://arxiv.org/abs/2101.10397), Theorem B(i)], we know that $R$ has rational singularities (see, e.g., [[Kollár 2013](https://doi.org/10.1017/CBO9781139547895), Definition 2.76] for the definition). Since $R$ having rational singularities is equivalent to saying that $R$ is pseudo-rational in our setting [[Murayama](https://arxiv.org/abs/2101.10397), Remark 7.4], we are done.


We can also show that $R$ is Cohen–Macaulay directly. Since Cohen–Macaulayness can be checked locally, it suffices to consider the case when $(R,\mathfrak{m})$ is local. Since $R$ is normal, the map $R \overset{\sim}{\to} f\_\*\mathcal{O}\_Y$ is an isomorphism [[EGAIII$\_1$](http://www.numdam.org/item/?id=PMIHES_1961__11__5_0), Corollaire 4.3.12]. Combined with the assumption that $R^if\_\*\mathcal{O}\_Y = 0$ for all $i > 0$, we have the quasi-isomorphism $R \overset{\sim}{\to} \mathbf{R}f\_\*\mathcal{O}\_Y$.
Applying $\mathbf{R}\Gamma\_{\mathfrak{m}}(-)$, we obtain
$$\mathbf{R}\Gamma\_{\mathfrak{m}}(R) \overset{\sim}{\longrightarrow} \mathbf{R}\Gamma\_{\mathfrak{m}}(\mathbf{R}f\_\*\mathcal{O}\_Y) \cong \mathbf{R}\Gamma\_{f^{-1}(\{\mathfrak{m}\})}(\mathcal{O}\_Y).$$
Now taking $i$-th cohomology modules, we obtain the isomorphisms
$$H^i\_{\mathfrak{m}}(R) \overset{\sim}{\longrightarrow} H^i\_{\mathfrak{m}}(\mathbf{R}f\_\*\mathcal{O}\_Y) \cong H^i\_{f^{-1}(\{\mathfrak{m}\})}(\mathcal{O}\_Y).$$
Since the local cohomology modules on the right vanish for all $i < \dim(R)$ by my version of Grauert–Riemenschneider vanishing (in its dual formulation) [[Murayama](https://arxiv.org/abs/2101.10397), Theorem B\*(i)], we see that $R$ is Cohen–Macaulay. $\blacksquare$


