If a resolution $f:Y\to X$ satisfies $R^if_*\omega_Y=0$ and $R^if_*\mathcal{O}_Y=0$ for all $i>0$, then do we have $f_*\omega_Y=\omega_X$?

Question

Let $X$ be a normal projective variety over an algebraically closed field of arbitrary characteristic (but I'm mainly interested in positive characteristic). Assume that $X$ has rational singularities, which by definition shall mean that there exists a resolution of singularities $f:Y\to X$ such that for all $i>0$ we have $R^if_*\mathcal{O}_Y=0$ and $R^if_*\omega_Y=0$. Then is it true that $f_*\omega_Y=\omega_X$, i.e. the trace is an isomorphism?

I can envision a proof when $X$ is Cohen-Macaulay: let $\mathcal{L}$ be an arbitrary line bundle on $X$. By the vanishing of $R^if_*\mathcal{O}_Y$ and the projection formula we obtain that $$ H^d(X,\mathcal{L}^{-1})\cong H^d(Y,f^*\mathcal{L}^{-1}) $$ where $d=\dim X$. As $X$ is Cohen-Macaulay (and $Y$ is regular), applying Serre-duality on both sides yields $$ H^0(X,\omega_X\otimes\mathcal{L})\cong H^0(Y,\omega_Y\otimes f^*\mathcal{L}). $$ Now from the vanishing of $R^if_*\omega_Y$ and the projection formula we may conclude $H^0(Y,\omega_Y\otimes f^*\mathcal{L})\cong H^0(X,f_*\omega_Y\otimes \mathcal{L})$, and as $\mathcal{L}$ was arbitrary we obtain indeed $\omega_X= f_*\omega_Y$.

Now it seems to be well known that rational singularities are Cohen-Macaulay, but every proof I saw is either in characteristic $0$ (where one may use Kodaira vanishing), or already starts with $\omega_X\cong Rf_*\omega_Y$ in the derived category, which as far as I understand already relies on $\omega_X\cong f_*\omega_Y$. What am I missing? I guess I'm just very confused, the material is very new to me.

Edit: I think I found a proof now:

We have the isomorphism $Rf_*\mathcal{O}_Y\cong\mathcal{O}_X$ by definition. Now applying $R\mathcal{Hom}_{\mathcal{O}_X}(-,\omega_X^{\bullet})$ to both sides, we find $$ R\mathcal{Hom}_{\mathcal{O}_X}(Rf_*\mathcal{O}_Y,\omega_X^{\bullet})\cong \underbrace{R\mathcal{Hom}_{\mathcal{O}_X}(\mathcal{O}_X,\omega_X^{\bullet})}_{\cong\omega_X^{\bullet}}. $$ The LHS can then be written as $$R\mathcal{Hom}_{\mathcal{O}_X}(Rf_*\mathcal{O}_Y,\omega_X^{\bullet})\cong Rf_* R\mathcal{Hom}_{\mathcal{O}_Y}(\mathcal{O}_Y,\underbrace{f^{!}\omega_X^{\bullet}}_{\cong\omega_Y})\cong Rf_*\omega_Y.$$

Hence we obtain $Rf_*\omega_Y\cong \omega_X^{\bullet}$, which by the vanishing of $R^i f_*\omega_Y$ implies that $X$ is Cohen Macaulay and also $f_*\omega_Y\cong \omega_X$.

Is this proof correct? And if yes, can we conclude that the trace is an isomorphism?

Answer 1

Yes, this is true. Aizenbud and Avni's Representation Growth and Rational Singularities of the Moduli Space of Local Systems (arxiv link) collects the following list of equivalent characterizations of rational singularities in appendix B section 7:

Proposition B.7.2. Let $X$ be an algebraic variety (i.e. reduced scheme of finite type over a field). The following are equivalent:

1. $X$ has rational singularities (for one (or equivalently, for some) resolution of singularities $p:\widetilde{X}\to X$, the natural morphism $Rp_*(\mathcal{O}_{\widetilde{X}})\to \mathcal{O}_X$ is an isomorphism).
2. For any (or equivalently, for some) resolution of singularities $p:\widetilde{X}\to X$, the trace map $Tr_p:Rp_*(\omega_{\widetilde{X}})\to \omega_X$ is an isomorphism.
3. $X$ is Cohen-Macaulay and for any (or equivalently, for some) resolution of singularities $p:\widetilde{X}\to X$, the trace map $Tr_p:Rp_*(\omega_{\widetilde{X}})\to \omega_X$ is an isomorphism.
4. $X$ is Cohen-Macaulay and for any (or equivalently, for some) resolution of singularities $p:\widetilde{X}\to X$, the trace map $Tr_p:Rp_*(\omega_{\widetilde{X}})\to \omega_X$ is onto.
5. $X$ is Cohen-Macaulay, normal (or equivalently, regular in codimension one), and for any (or equivalently, for some) resolution of singularities $p:\widetilde{X}\to X$, the morphism $p_*(\omega_X)\to i_*(\omega_{X^{sm}})$ which is the composition of the trace map and the natural isomorphism $\omega_X\to i_*(\omega_{X^{sm}})$ is an isomorphism. Here, $i:X^{sm}\to X$ is the embedding of the regular locus.

This is originally all stated in characteristic zero, but the only part that they need that for is the Grauert-Riemenschneider vanishing theorem used in the equivalence of 2 and 3, which states that if $\pi:X\to Y$ is a proper birational morphism of varieties over $k$, then $R\pi_*\omega_X = \pi_*\omega_X$. This result is now available in arbitrary characteristic, see for instance Chatzistamatiou and Rülling's Higher Direct Images of the Structure Sheaf in Positive Characteristic (arxiv link).

The proof of the equivalence of 1 and 2, which is what you're interested in, follows either from Elkik's early paper Singularities rationelles et deformations (relying on Grauert-Riemenschneider) or from a duality argument (not explained in the paper, but from my recollections, it should look very similar to what you've written here).