The Stacks project

Lemma 69.8.1. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Assume that

  1. $f$ is quasi-compact and quasi-separated, and

  2. $Y$ is quasi-compact.

Then there exists an integer $n(X \to Y)$ such that for any algebraic space $Y'$, any morphism $Y' \to Y$ and any quasi-coherent sheaf $\mathcal{F}'$ on $X' = Y' \times _ Y X$ the higher direct images $R^ if'_*\mathcal{F}'$ are zero for $i \geq n(X \to Y)$.

Proof. Let $V \to Y$ be a surjective ├ętale morphism where $V$ is an affine scheme, see Properties of Spaces, Lemma 66.6.3. Suppose we prove the result for the base change $f_ V : V \times _ Y X \to V$. Then the result holds for $f$ with $n(X \to Y) = n(X_ V \to V)$. Namely, if $Y' \to Y$ and $\mathcal{F}'$ are as in the lemma, then $R^ if'_*\mathcal{F}'|_{V \times _ Y Y'}$ is equal to $R^ if'_{V, *}\mathcal{F}'|_{X'_ V}$ where $f'_ V : X'_ V = V \times _ Y Y' \times _ Y X \to V \times _ Y Y' = Y'_ V$, see Properties of Spaces, Lemma 66.26.2. Thus we may assume that $Y$ is an affine scheme.

Moreover, to prove the vanishing for all $Y' \to Y$ and $\mathcal{F}'$ it suffices to do so when $Y'$ is an affine scheme. In this case, $R^ if'_*\mathcal{F}'$ is quasi-coherent by Lemma 69.3.1. Hence it suffices to prove that $H^ i(X', \mathcal{F}') = 0$, because $H^ i(X', \mathcal{F}') = H^0(Y', R^ if'_*\mathcal{F}')$ by Cohomology on Sites, Lemma 21.14.6 and the vanishing of higher cohomology of quasi-coherent sheaves on affine algebraic spaces (Proposition 69.7.2).

Choose $U \to X$, $d$, $V_ p \to U_ p$ and $d_ p$ as in Lemma 69.7.3. For any affine scheme $Y'$ and morphism $Y' \to Y$ denote $X' = Y' \times _ Y X$, $U' = Y' \times _ Y U$, $V'_ p = Y' \times _ Y V_ p$. Then $U' \to X'$, $d' = d$, $V'_ p \to U'_ p$ and $d'_ p = d$ is a collection of choices as in Lemma 69.7.3 for the algebraic space $X'$ (details omitted). Hence we see that $H^ i(X', \mathcal{F}') = 0$ for $i \geq \max (p + d_ p)$ and we win. $\square$

Comments (0)

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 073G. Beware of the difference between the letter 'O' and the digit '0'.