Lemma 30.4.4. Let $X$ be a quasi-compact quasi-separated scheme. Let $X = U_1 \cup \ldots \cup U_ t$ be an open covering with each $U_ i$ quasi-compact and separated (for example affine). Set
\[ d = \max \nolimits _{I \subset \{ 1, \ldots , t\} } \left(|I| + t(\bigcap \nolimits _{i \in I} U_ i) - 1\right) \]
where $t(U)$ is the minimal number of affines needed to cover the scheme $U$. Then $H^ n(X, \mathcal{F}) = 0$ for all $n \geq d$ and all quasi-coherent sheaves $\mathcal{F}$.
Proof.
Note that since $X$ is quasi-separated and $U_ i$ quasi-compact the numbers $t(\bigcap _{i \in I} U_ i)$ are finite. Proof using induction on $t$. If $t = 1$ then the result follows from Lemma 30.4.2. If $t > 1$, write $X = U \cup V$ with $U = U_1 \cup \ldots \cup U_{t - 1}$ and $V = U_ t$. 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. Since $V$ is affine, we have $H^ i(V, \mathcal{F}) = 0$ for $i \geq 0$. By induction hypothesis we have $H^ i(U, \mathcal{F}) = 0$ for
\[ i \geq \max \nolimits _{I \subset \{ 1, \ldots , t - 1\} } \left(|I| + t(\bigcap \nolimits _{i \in I} U_ i) - 1\right) \]
and the bound on the right is less than the bound in the statement of the lemma. Finally we may use our induction hypothesis for the open $U \cap V = (U_1 \cap U_ t) \cup \ldots \cup (U_{t - 1} \cap U_ t)$ to get the vanishing of $H^ i(U \cap V, \mathcal{F}) = 0$ for
\[ i \geq \max \nolimits _{I \subset \{ 1, \ldots , t - 1\} } \left(|I| + t(U_ t \cap \bigcap \nolimits _{i \in I} U_ i) - 1\right) \]
Since the bound on the right is at least $1$ less than the bound in the statement of the lemma, the lemma follows.
$\square$
Comments (4)
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