On page 17 of this paper there is the following claim.

For $f: Y \rightarrow X$ a proper birational map with $Y$ smooth (i.e. a desingularization of $X$) and $X$ is a normal variety, $R^i f_* \mathcal{O}_Y = 0$ for $i>0$ is equivalent to $Rf_* \mathcal{O}_Y \simeq \mathcal{O}_X$ in the derived category.

I can't see why this is true. Does anyone know why, or know a good reference where I could find the answer?

Bonus question: is it true that for $f: Y \rightarrow X$ a proper birational map with $Y$ smooth, that $X$ is normal if and only if $f_* \mathcal{O}_Y = \mathcal{O}_X$? (nonderived pushforward)


I found an answer. Higher derived structure is not relevant. Let $f: Y \rightarrow X$ be a proper birational resolution of singularities, with $X$ normal. This problem is local on the base so we may assume $X$ is affine.

Since $f$ is birational, $\Gamma(Y, \mathcal{O}_Y) \subset \text{Frac}(\mathcal{O}_X)$.

Since $f$ is proper, pushforward of coherent sheaf is coherent, so $\Gamma(Y, \mathcal{O}_Y)$ is a finite $\mathcal{O}_X$-module.

Since $X$ is normal, a finite $\mathcal{O}_X$-module in its field of fractions is itself. (Converse is also true).

  • $\begingroup$ I think the reference is Weyman's "Cohomology and Syzygy of Vector Bundles" but it's been awhile so I forgot. $\endgroup$
    – user148177
    May 29 '14 at 10:59

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