The Stacks project

Lemma 30.4.4. Let $X$ be a quasi-compact quasi-separated scheme. Let $X = U_1 \cup \ldots \cup U_ t$ be an affine open covering. Set

\[ d = \max \nolimits _{I \subset \{ 1, \ldots , t\} } \left(|I| + t(\bigcap \nolimits _{i \in I} U_ i)\right) \]

where $t(U)$ is the minimal number of affines needed to cover the scheme $U$. Then $H^ n(X, \mathcal{F}) = 0$ for all $n \geq d$ and all quasi-coherent sheaves $\mathcal{F}$.

Proof. Note that since $X$ is quasi-separated the numbers $t(\bigcap _{i \in I} U_ i)$ are finite. Let $\mathcal{U} : X = \bigcup _{i = 1}^ t U_ i$. By Cohomology, Lemma 20.11.5 there is a spectral sequence

\[ E_2^{p, q} = \check{H}^ p(\mathcal{U}, \underline{H}^ q(\mathcal{F})) \]

converging to $H^{p + q}(U, \mathcal{F})$. By Cohomology, Lemma 20.23.6 we have

\[ E_2^{p, q} = H^ p(\check{\mathcal{C}}_{alt}^\bullet ( \mathcal{U}, \underline{H}^ q(\mathcal{F})) \]

The alternating Čech complex with values in the presheaf $\underline{H}^ q(\mathcal{F})$ vanishes in high degrees by Lemma 30.4.2, more precisely $E_2^{p, q} = 0$ for $p + q \geq d$. Hence the result follows. $\square$

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