Let $X$ be a normal, of finite type over a field of characteristic zero and regular scheme. Let $s \in X$ be a closed point. Assume we have a proper birational map
$f: Y \to X$
with the property that the restriction $$f \vert _{Y \backslash f^{-1}(s)}: Y \backslash f^{-1}(s) \to X \backslash s$$ is an isomorphism.
My question is how to deduce that the higher direct images of $f_{*}$ applied to $O_Y$ vanish for $i>0$. That is,
$$ R^{i}f_{*}{\mathcal {O}}_{Y}=0$$
for $i>0$.
Obviously this is true on the restruction $X \backslash s$. But what about $ R^{i}f_{*}{\mathcal {O}}_{Y} \vert _U$ for an open neighbourhood of $s$?
Does anybody see an argument?
Remark: Background of this problem is following former thread of mine: Resolution of Rational Singularities
where I tried to deduce that the property of a rational singularity at a isolated singular point $s$ of a normal scheme $W$ for proper birational map (=the resolution of the singularity)
$h: Z \to W$
from a regular scheme $Z$ that the higher direct images vanish, so
(*) $$ R^{i}h_{*}{\mathcal {O}}_{Z}=0$$
for $i>0$ is independent of the choice of the resolution.
Namely, if I choose another resolution of the singularity $h': Z' \to W$ then (*) holds also for $h'$.
With help of KReiser and Youngsu the problem could be reduced to following problem (via "dominating argument"... see comments in the linked thread):
Assume $h:Z \to W$ has a dominating resolution $g: D \to Z$ with $k:= h \circ g, D \to W$ and by a symmetry argument it suffice to show that
$$R^{i}h_{*}{\mathcal {O}}_{Z}= R^{i}k_{*}{\mathcal {O}}_{D}$$
Using Leray Serre spectral sequence the problem reduces to show that
$$R^jg_*{\mathcal {O}}_{D}=0$$
for $j >0$ (see again the linked thread).
Since $Z$ is regular we are now in the above setting for $f:Y \to X$.