The Stacks project

Proposition 37.70.7. Let $X$ be a nonempty quasi-compact and quasi-separated scheme with affine stratification number $n$. Then $H^ p(X, \mathcal{F}) = 0$, $p > n$ for every quasi-coherent $\mathcal{O}_ X$-module $\mathcal{F}$.

Proof. We will prove this by induction on the affine stratification number $n$. If $n = 0$, then $X$ is affine and the result is Cohomology of Schemes, Lemma 30.2.2. Assume $n > 0$. By Definition 37.70.4 there is an affine scheme $U$ and an affine open immersion $j : U \to X$ such that the complement $Z$ has affine stratification number $n - 1$. As $U$ and $j$ are affine we have $H^ p(X, j_*(\mathcal{F}|_ U)) = 0$ for $p > 0$, see Cohomology of Schemes, Lemmas 30.2.4 and 30.2.3. Denote $\mathcal{K}$ and $\mathcal{Q}$ the kernel and cokernel of the map $\mathcal{F} \to j_*(\mathcal{F}|_ U)$. Thus we obtain an exact sequence

\[ 0 \to \mathcal{K} \to \mathcal{F} \to j_*(\mathcal{F}|_ U) \to \mathcal{Q} \to 0 \]

of quasi-coherent $\mathcal{O}_ X$-modules (see Schemes, Section 26.24). A standard argument, breaking our exact sequence into short exact sequences and using the long exact cohomology sequence, shows it suffices to prove $H^ p(X, \mathcal{K}) = 0$ and $H^ p(X, \mathcal{Q}) = 0$ for $p \geq n$. Since $\mathcal{F} \to j_*(\mathcal{F}|_ U)$ restricts to an isomorphism over $U$, we see that $\mathcal{K}$ and $\mathcal{Q}$ are supported on $Z$. By Properties, Lemma 28.22.3 we can write these modules as the filtered colimits of their finite type quasi-coherent submodules. Using the fact that cohomology of sheaves on $X$ commutes with filtered colimits, see Cohomology, Lemma 20.19.1, we conclude it suffices to show that if $\mathcal{G}$ is a finite type quasi-coherent module whose support is contained in $Z$, then $H^ p(X, \mathcal{G}) = 0$ for $p \geq n$. Let $Z' \subset X$ be the scheme theoretic support of $\mathcal{G} \oplus \mathcal{O}_ Z$; we may and do think of $\mathcal{G}$ as a quasi-coherent module on $Z'$, see Morphisms, Section 29.5. Then $Z'$ and $Z$ have the same underlying topological space and hence the same affine stratification number, namely $n - 1$. Hence $H^ p(X, \mathcal{G}) = H^ p(Z', \mathcal{G})$ (equality by Cohomology of Schemes, Lemma 30.2.4) vanishes for $p \geq n$ by induction hypothesis. $\square$

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Your email address will not be published.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 0F2Y. Beware of the difference between the letter 'O' and the digit '0'.