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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 $\left (R^i f_* \mathcal O_Y\right)_x=0, \forall i>0$ ?

(https://en.m.wikipedia.org/wiki/Direct_image_functor).

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  • $\begingroup$ 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. $\endgroup$ – red_trumpet Jul 21 '20 at 14:44
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Yes, this is true. It's originally a result of Hironaka, in his 1964 Annals paper Resolution of singularities of an algebraic variety over a field of characteristic zero. A modern generalization to arbitrary characteristic may be found here on the arXiv.

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  • $\begingroup$ 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 .... $\endgroup$ – Louis Jul 21 '20 at 21:37
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    $\begingroup$ 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. $\endgroup$ – KReiser Jul 21 '20 at 23:29
  • $\begingroup$ One should note that the second step needs some additional work in positive characteristic - I'm not sure that if there's a proper resolution of singularities that there has to be a projective resolution of singularities. There might be by some version of Chow's lemma, but I don't recall the correct fix off the top of my head. Also, if you're interested in a hands-on proof for the case of a blowup, you may wish to consult this. $\endgroup$ – KReiser Jul 21 '20 at 23:31

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