# Stalks of Higher direct images of structure sheaf at smooth points

Let $$k$$ be a field of Characteristic zero, and we will consider normal separated schemes of finite type over $$k$$.

Let $$X$$ be such a scheme and $$f: Y\to X$$ be a proper birational map where $$Y$$ is a regular scheme. If $$x$$ is a smooth (closed) point of $$X$$ i.e. if $$\mathcal O_{X,x}$$ is a regular local ring, then is it true that the stalk at $$x$$ of the higher direct images of $$f_*$$ applied to $$\mathcal O_Y$$ are trivial i.e. is it true that $$\left (R^i f_* \mathcal O_Y\right)_x=0, \forall i>0$$ ?

• If $U \subset X$ is such that $f^{-1} U \to U$ is an isomorphism, than this is clearly true for all $x \in U$, since the definition of $R^if_* \mathcal{O}_Y$ is local in $X$. I'm not sure about the general case though. – red_trumpet Jul 21 '20 at 14:44

• Could you please mention where in Hironaka's paper can I locate the fact ? Also, the paper you link only talks about projective morphism between regular schemes ... my morphisms are only proper and my $X$ is not regular .... – Louis Jul 21 '20 at 21:37
• The result is on page 153 of the first volume of the paper, numbered corollary 2.26. And you can easily reduce to the situation where $X$ and $Y$ are regular: given that your point $x$ is smooth, there's an open neighborhood of $U\subset X$ so that $x\in U$ and $U$ is smooth, so base-changing by the inclusion we get a proper birational morphism of regular schemes. Projective to proper should be no trouble - by the standard method of taking a resolution of singularities simultaneously dominating both $f$ + a resolution obtained by a blowup, we see that the relevant higher direct images vanish. – KReiser Jul 21 '20 at 23:29