# Proper birational morphism from a Gorenstein normal scheme to a normal local domain, with trivial higher direct images, implies Cohen-Macaulay?

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 $$\dim R \geq 3$$.

• I believe that this is a theorem of Elkik when $R$ is the localization of a finitely generated $k$-algebra. Thus, it also holds if $R$ sits "between" such a localization and its completion (since you can check the Cohen-Macaulay property after passage to the localization). So probably you can deduce your case from the Cohen structure theorem. Jul 6 at 11:33
• @JasonStarr: Thank you for your comment. Could you please give a reference for the case of when $R$ is essentially of finite type over $k$? Also, Is Spec$(R)$ necessary, or can we just do it for any normal scheme essentially of finite type over $k$? Jul 7 at 0:27
• I learned about this in Koll'ar's paper, "Singularities of pairs," in the Santa Cruz volume, but he references an earlier paper of Elkik. I will try to track down the original article. Jul 7 at 11:09
• Elkik explains the proof in the discussion leading up to Definition 1 in the article, "Rationalit'e des singularit'es canoniques": eudml.org/doc/142810 Jul 7 at 11:16
• I read the proof, and I think it probably directly applies to your situation, without using the completion. I will try to write more soon. Jul 7 at 12:48

I started writing this last night, but didn't finish. In addition to Jason Starr's answer, 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, 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$$_\text{new}$$, 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, Remark 16.2] and that the pseudo-rationality of $$\hat{R}$$ would imply that $$R$$ is pseudo-rational [Murayama 2022, Proposition 4.20]. Since $$\hat{Y}$$ is regular by [EGAIV$$_2$$, 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, Theorem B(i)], we know that $$R$$ has rational singularities (see, e.g., [Kollár 2013, Definition 2.76] for the definition). Since $$R$$ having rational singularities is equivalent to saying that $$R$$ is pseudo-rational in our setting [Murayama, 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$$_1$$, 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, Theorem B*(i)], we see that $$R$$ is Cohen–Macaulay. $$\blacksquare$$

• Thank you for your answer. I will read it carefully, and will surely have many questions, but I quickly want to ask two things first: $Y$ is just a regular scheme with no other assumption? By $\widehat Y$, do you actually mean the fiber-product $Y \times_{Spec(R)} Spec(\widehat R)$? Jul 8 at 2:25
• I like your second argument... could you please spell out the Leray spectral sequence isomorphism ? Jul 8 at 3:59
• This is definitely better than the original version. Jul 8 at 10:10
• @SnakeEyes I edited my answer. For the regularity of $\hat{Y}$, see [Stacks Project, Tag 0BG6] for another reference. Jul 9 at 15:36

Edit. Thanks to user @Johan for pointing out the indexing mistake. It is now corrected.

I read more of Elkik's article. I am just expanding my comments into one answer. Edit. Assume that $$R$$ is essentially of finite type over $$\text{Spec}\ k$$. By hypothesis, the scheme $$X=\text{Spec}\ R$$ has a dualizing complex $$\omega^\bullet_R$$ that has nonzero homology sheaf $$\mathcal{H}^{\text{dim}\ R}(\omega^\bullet_R)$$, and has vanishing homology sheaves $$\mathcal{H}^i(\omega^\bullet_R)$$ for $$i>0$$ and for $$i<-\text{dim}\ R$$. The ring $$R$$ is Cohen-Macaulay if and only if the only integer $$i$$ with $$\mathcal{H}^i(\omega^\bullet_R)$$ nonzero is $$i=-\text{dim}\ R$$. It is Gorenstein if, further, this homology sheaf is a projective $$R$$-module of rank $$1$$.

Also by hypothesis, the scheme $$Y$$ has a dualizing complex $$\omega_Y^\bullet$$ such that the homology sheaf $$\mathcal{H}^i(\omega_Y^\bullet)$$ is nonzero if and only if $$i=-\text{dim}\ Y = -\text{dim}\ R$$. By relative duality, we have a natural quasi-isomorphism, $$\alpha:Rf_* \omega_Y^\bullet \cong R\text{Hom}_{\mathcal{O}_X}(Rf_*\mathcal{O}_Y,\omega_X^\bullet).$$ By hypothesis, the natural map $$\mathcal{O}_X\to Rf_*\mathcal{O}_Y$$ is a quasi-isomorphism. Thus, $$\alpha$$ is equivalent to a quasi-isomorphism, $$Rf_*\omega^\bullet_Y \cong \omega_X^\bullet.$$

Since the only integer $$i$$ with a nonvanishing homology sheaf $$\mathcal{H}^i(Rf_*\omega^\bullet_Y)$$ is $$i=-\text{dim}\ Y$$, the complex $$Rf_*\omega^\bullet_Y$$ has nonvanishing homology sheaves $$R^if_*\omega^\bullet_Y$$ only for $$-\text{dim}\ Y \leq i \leq 0$$.

Edit. Since $$R$$ is essentially of finite type and $$Y$$ is regular, by Grauert-Riemenschneider Vanishing the derived direct image $$R^if_*\omega^\bullet_Y$$ is nonzero only for $$i=-\text{dim}\ Y = -\text{dim}\ R$$, in which degree it equals $$f_*\mathcal{H}^{-\text{dim}\ Y}(\omega^\bullet_Y)$$.

Thus the complex $$\omega_X^\bullet$$ has nonvanishing homology sheaves only for $$i =- \text{dim}\ R = \text{dim}\ Y$$. Therefore $$R$$ is Cohen-Macaulay.

• Sorry, Jason, but there is something wrong with the numbering. Usually, people normalize the dualizing complex of R such that it has nonzero cohomology in degrees $[-\dim R, 0]$. When I use that normalization I have to use Grauert-Riemenschneider vanishing as in the article of Elkik. Cheers! Jul 7 at 23:14
• You are right: I misread the indexing convention. I will correct it now. Jul 8 at 0:57
• I know that $\omega_Y^{\bullet}$ is only supported at $\dim Y$, but could you please explain why $\mathcal H^i(Rf_* \omega_Y^{\bullet })=0$ for $i\neq -\dim Y$? If we can just say this, we would be done at this point, no? Because $X$ is Cohen-Macaulay if and only if $\omega_X^{\bullet}$ is supported in a single degree ... Jul 8 at 1:36
• As pointed out by user @Johan (and explained in the article by Elkik), this follows from Grauert-Riemenschneider Vanishing. Jul 8 at 1:43
• @JasonStarr: Thank you for your edits ... I do not have a reference in front of me right now, but for Grauert-Riemenschneider, don't you need both the schemes to be actually a variety over a characteristic $0$ field? Jul 8 at 1:55