Here is a simple proof of Kempf's criterion for rational singularities:

**Theorem.** (Kempf)
Let $X$ be a normal variety over $\mathbb C$. Then $X$ has rational singularities (i.e., for a resolution $\phi:\widetilde X\to X$, $R^i\phi_*\mathcal O_{\widetilde X}=0$ for $i>0$) if and only if $X$ is Cohen-Macaulay and $\phi_*\omega_{\widetilde X}\simeq \omega_X$.

**Proof.** ($RHom$ stands for sheaf-RHom, $\omega^\cdot$ for the dualizing complex, $n=\dim X$).

First assume that $X$ has rational singularities. Then
\begin{multline}
\omega_X^\cdot\simeq RHom_X(\mathcal O_X,\omega_X^\cdot)\simeq RHom_X(R\phi_*\mathcal O_{\widetilde X},\omega_X^\cdot)
\simeq_{\text{by Grothendieck duality}}\\\ \simeq R\phi_*RHom(\mathcal O_{\widetilde X}, \omega_{\widetilde X}^\cdot) \simeq R\phi_*\omega_{\widetilde X}[n]\simeq \phi_*\omega_{\widetilde X}[n] 
\end{multline} 
The last isomorphism follows by Grauert-Riemenschneider vanishing. The two ends of the displayed isomorphism shows that $\omega_X^\cdot$ has only one non-zero cohomology sheaf and hence $X$ is Cohen-Macaulay and that non-zero cohomology sheaf is $\phi_*\omega_{\widetilde X}\simeq \omega_X$.

The other direction is similar:
\begin{multline}
\mathcal O_X\simeq RHom_X(\omega_X^\cdot,\omega_X^\cdot)\simeq RHom_X(R\phi_*\omega_{\widetilde X}^\cdot,\omega_X^\cdot)
\simeq_{\text{by Grothendieck duality}}\\\ \simeq R\phi_*RHom(\omega_{\widetilde X}^\cdot, \omega_{\widetilde X}^\cdot) \simeq R\phi_*\mathcal O_{\widetilde X}
\end{multline}