For schemes with affine diagonal, the cohomology of quasi-coherent modules vanishes in degrees bigger than the number of affine opens needed in a covering.

Lemma 30.4.2. Let $X$ be a quasi-compact scheme with affine diagonal (for example if $X$ is separated). Let $t = t(X)$ be the minimal number of affine opens needed to cover $X$. Then $H^ n(X, \mathcal{F}) = 0$ for all $n \geq t$ and all quasi-coherent sheaves $\mathcal{F}$.

Proof. First proof. By induction on $t$. If $t = 1$ the result follows from Lemma 30.2.2. If $t > 1$ write $X = U \cup V$ with $V$ affine open and $U = U_1 \cup \ldots \cup U_{t - 1}$ a union of $t - 1$ open affines. Note that in this case $U \cap V = (U_1 \cap V) \cup \ldots (U_{t - 1} \cap V)$ is also a union of $t - 1$ affine open subschemes. Namely, since the diagonal is affine, the intersection of two affine opens is affine, see Lemma 30.2.5. 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. By induction we see that the groups $H^ i(U, \mathcal{F})$, $H^ i(V, \mathcal{F})$, $H^ i(U \cap V, \mathcal{F})$ are zero for $i \geq t - 1$. It follows immediately that $H^ i(X, \mathcal{F})$ is zero for $i \geq t$.

Second proof. Let $\mathcal{U} : X = \bigcup _{i = 1}^ t U_ i$ be a finite affine open covering. Since $X$ is has affine diagonal the multiple intersections $U_{i_0 \ldots i_ p}$ are all affine, see Lemma 30.2.5. By Lemma 30.2.6 the Čech cohomology groups $\check{H}^ p(\mathcal{U}, \mathcal{F})$ agree with the cohomology groups. By Cohomology, Lemma 20.23.6 the Čech cohomology groups may be computed using the alternating Čech complex $\check{\mathcal{C}}_{alt}^\bullet (\mathcal{U}, \mathcal{F})$. As the covering consists of $t$ elements we see immediately that $\check{\mathcal{C}}_{alt}^ p(\mathcal{U}, \mathcal{F}) = 0$ for all $p \geq t$. Hence the result follows. $\square$

Comment #3023 by Brian Lawrence on

Suggested slogan: Coherent cohomology on a scheme vanishes in degree at least the number of affine opens needed to cover the scheme, provided the scheme is quasi-compact with open diagonal.

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