Direct image of structure sheaf under blow-up along non-singular subvariety

Question

I'm trying to prove the following statement:

Theorem A Let $X$ be a non-singular variety over a field $k$ and let $Y \subset X$ be a smooth subvariety. Consider the blow-up $f : \widetilde X = Bl_Y(X) \to X$. Then for $i > 0$: $$R^i f_* \mathcal O_{\widetilde X} = 0.$$

This is mentioned for example in Hironaka, Resolution of Singularities of an Algebraic Variety Over a Field of Characteristic Zero I, p. 153 without a reference or a proof. My attempt (following the proof of Proposition V.3.4 in Hartshorne's Algebraic Geometry):

let $\mathcal F^i := R^i f_* \mathcal O_{\widetilde X}$ and let $y$ be the generic point of $Y$. Then the support of $\mathcal F^i$ is contained in $Y$ and using the Formal Functions Theorem, we get:

$$ \mathcal F^i_y = \lim_{\leftarrow} H^i(E_n, \mathcal O_{E_n}),$$

where $E_1 = E = f^{-1}(Y)$ and $E_n$ is given by the ideal sheaf $\mathcal J^n$ (where $\mathcal J$ is the ideal sheaf of $E$ in $\widetilde X$). Thus the above statement should be equivalent to:

$$ H^i(E_n, \mathcal O_{E_n}) = 0 \qquad \text{for all } i, n \ge 1.$$

Also, we have an exact sequence:

$$ 0 \to \mathcal J^n/\mathcal J^{n + 1} = \mathcal O_E(n) \to \mathcal O_{E_{n+1}} \to \mathcal O_{E_{n}} \to 0 \qquad (*)$$

Thus, it seems to me that the Theorem A is equivalent to the statement that $$H^i(E, \mathcal O_{E}(n)) = 0 \qquad \text{for all } i, n > 0. $$

On the other hand, $E = \mathbb P(\mathcal I/\mathcal I^2)$ is a projective bundle over $Y$ (where $\mathcal I$ is the ideal sheaf of $Y$ in $X$). Thus $$ R^i g_* \mathcal O_E (d) = 0 $$ for $i, d > 0$ (where $g = f|_E : E \to Y$) - see e.g. Stacks. Therefore by Leray spectral sequence we obtain: $$ H^i(E, \mathcal O_{E}(n)) = H^i(Y, g_* \mathcal O_{E}(n)) = H^i(Y, S^n(\mathcal I/\mathcal I^2)). $$ The right hand side seems to be non-zero in general.

Question: where's the mistake? How to fix it? Alternatively, what is a reference for the proof of Theorem A?

Answer 1

This is largely correct, except you need a slightly better version of the theorem on formal functions. Here's how it's stated in Hartshorne:

Theorem on Formal Functions (Hartshorne III.11.1): Let $f:X\to Y$ be a projective morphism of noetherian schemes, let $\mathcal{F}$ be a coherent sheaf on $X$, let $y\in Y$, let $X_n = X\times_Y \operatorname{Spec} \mathcal{O}_y/\mathfrak{m}_y^n$, and let $\mathcal{F}_n = v_n^*\mathcal{F}$ where $v_n: X_n\to X$ is the natural map.

Then $R^if_*(\mathcal{F})_y^{\wedge} \cong \lim_{\leftarrow} H^i(X_n,\mathcal{F}_n)$ is an isomorphism for all $i\geq 0$.

A better version replaces $y\in Y$ by a closed subscheme $Z\subset Y$: upgrading to a closed subscheme instead of a closed point, Hartshorne's proof works verbatim to show that $$\varprojlim (R^if_*\mathcal{F})\otimes_{\mathcal{O}_Y} \mathcal{O}_Y/\mathcal{I}_Z^n \cong\varprojlim R^if_*(\mathcal{F}\otimes_{\mathcal{O}_X} \mathcal{O}_Y/\mathcal{I}_Z^n)$$ and then one can use the exact same argument as the proof of proposition V.3.4 to conclude that $R^\bullet f_*\mathcal{O}_{\widetilde{X}}=\mathcal{O}_X$ as complexes. (For a reference for the improved version of the theorem, see EGAIII, theorem 4.1.5.)