Lemma 30.4.4. Let $X$ be a quasi-compact quasi-separated scheme. Let $X = U_1 \cup \ldots \cup U_ n$ be an open covering with each $U_ i$ quasi-compact and separated (for example affine). Set

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

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

**Proof.**
Note that since $X$ is quasi-separated and $U_ i$ quasi-compact the numbers $t(\bigcap _{i \in I} U_ i)$ are finite. Proof using induction on $n$. If $n = 1$ then the result follows from Lemma 30.4.2. If $n > 1$, write $X = U \cup V$ with $U = U_1 \cup \ldots \cup U_{n - 1}$ and $V = U_ n$. We apply the Mayer-Vietoris long exact sequence

\[ 0 \to H^0(X, \mathcal{F}) \to H^0(U, \mathcal{F}) \oplus H^0(V, \mathcal{F}) \to H^0(U \cap V, \mathcal{F}) \to H^1(X, \mathcal{F}) \to \ldots \]

see Cohomology, Lemma 20.8.2. To finish the proof for $q \geq d$ we will show that $H^ q(V, \mathcal{F})$, $H^ q(U, \mathcal{F})$, and $H^{q - 1}(U \cap V, \mathcal{F})$ vanish. By the case $n = 1$ we have $H^ q(V, \mathcal{F}) = 0$ for $q \geq t(V) = t(U_ n)$. Since $t(V) \leq d$ this proves what we want. By induction hypothesis we have $H^ q(U, \mathcal{F}) = 0$ for

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

Since the integer on the right is less than or equal to $d$, this proves what we want. Finally we may use our induction hypothesis for the open $U \cap V = (U_1 \cap U_ n) \cup \ldots \cup (U_{n - 1} \cap U_ n)$ to get the vanishing of $H^ q(U \cap V, \mathcal{F}) = 0$ for

\[ q \geq \max \nolimits _{I \subset \{ 1, \ldots , n - 1\} } \left(|I| + t(U_ n \cap \bigcap \nolimits _{i \in I} U_ i) - 1\right) \]

Since the integer on the right is strictly less than $d$ the lemma follows.
$\square$

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