EDIT. Karl Schwede has given a different approach to solve this question, which is very clear. However, I still want to figure out how to deal with this problem along Ein's line, and waiting for the answer about my QUSETION A. B. below.

This is a duplication of a question from MSE.

I am currently reading the notes Lectures on singularities and adjoint linear systems written by Ein and encounter some difficulties as follows, see Prop1.12.

**Definition (Rational Singularity).** A morphism $f: Y \rightarrow X$ is said to be a rational resolution if $Y$ is smooth and $f$ is a proper and birational morphism such that $R^{i} f_{*} \mathscr{O}_{Y}=0$ for $i>0$.

**Proposition.** Let $f: Y \rightarrow X$ be a rational resolution and $f^{\prime}: Y^{\prime} \rightarrow X$ be another resolution. Then $f^{\prime}: Y^{\prime} \rightarrow X$ is also a rational resolution.

*Proof.* We have a birational map $\varphi: Y -\to Y^{\prime}$. Successively blowing up the undefined locus of $\varphi$, we get a variety $Z$ and two proper birational morphisms $g: Z \rightarrow Y$ and $g^{\prime}: Z \rightarrow Y^{\prime}$ such that $h:=f \circ g=f^{\prime} \circ g^{\prime}$. Since $g$ is the composition of blowing-ups. Then $R^{q} g_{*}\left(\mathscr{O}_{Z}\right)=0$ for $q>0$. Apply the Leray spectral sequence
$$
E_{2}^{p, q}=R^{p} f_{*}\left(R^{q} g_{*}(\mathscr{F})\right) \Rightarrow R^{p+q}(f \circ g)_{*}(\mathscr{F}) .
$$
It follows that $R^{i} h_{*} \mathscr{O}_{Z}=0$ for $i>0$. Apply the Leray spectral sequence to $f^{\prime} \circ g^{\prime}$. It is easy to see that $R^{1} f_{*}^{\prime} \mathscr{O}_{Y^\prime}=0$. In fact, it fits in the following exact sequence
$$
0 \rightarrow R^{1} f_{*}^{\prime} \mathscr{O}_{Y^\prime} \rightarrow R^{1} h_{*} \mathscr{O}_{Z} \rightarrow f_{*}^{\prime} R^{1} g^{\prime} \mathscr{O}_{Z}
$$
Since $Y^{\prime}$ is smooth hence $Y^{\prime}$ has a rational resolution. Now $Z$ is another resolution of $Y^{\prime}$. By the above argument, we can conclude that
$R^{1} g_{*}^{\prime} \mathscr{O}_{Z}=0$. **Apply the Leray spectral sequence to $p+q=2$.** We see that $R^{2} f_{*}^{\prime} \mathscr{O}_{Y^\prime}=0$. Hence $R^{2} g_{*}^{\prime} \mathscr{O}_{Z}=0$. By induction, we conclude that $R^{p} f_{*}^{\prime} \mathscr{O}_{Y^\prime}=0$ for $p>0$.

Here is my question:

**QUESTION A.** Suppose we know that $R^{1} f_{*}^{\prime} \mathscr{O}_{Y^\prime}=0$ and $R^{1} g_{*}^{\prime} \mathscr{O}_{Z}=0,$ how to deduce $R^{2} g_{*}^{\prime} \mathscr{O}_{Z}=0$ by applying the Leray spectral sequence as Ein said rather than using the five-term sequence, that is
$$
0 \rightarrow R^{1} f_{*}^{\prime} \mathscr{O}_{Y^\prime} \rightarrow R^{1} h_{*} \mathscr{O}_{Z} \rightarrow f_{*}^{\prime} R^{1} g_*^{\prime} \mathscr{O}_{Z}\rightarrow R^{2} f_{*}^{\prime} \mathscr{O}_{Y^\prime}\rightarrow R^{2} h_{*} \mathscr{O}_{Z}?
$$

We can say $R^{2} f_{*}^{\prime} \mathscr{O}_{Z}=0$ since its prior one and posterior one both vanish.

I wonder how to use Leray spectral sequence at $p+q=2$, since now, we have $$0=R^{2} h_{*} \mathscr{O}_{Z}=R^{2} f_{*}^{\prime} \mathscr{O}_{Y^\prime}\oplus R^{1} f_{*}^{\prime}(R^1 g_{*}^{\prime} \mathscr{O}_{Z})\oplus R^{2} g_{*}^{\prime} \mathscr{O}_{Z}$$ with the middle term vanishing, but we cannot say anything about the vanishing of $R^{2} f_{*}^{\prime} \mathscr{O}_{Y^\prime}$ or $R^{2} g_{*}^{\prime} \mathscr{O}_{Z}$.

**QUESTION B.** How to do the induction if we cannot use the five-term sequence for larger $p$, $q$?