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. $\square$


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