The Stacks project

Proof. First we argue by induction that $P$ holds for separated quasi-compact opens $W \subset X$. Namely, such an open can be written as $W = U_1 \cup \ldots \cup U_ n$ and we can do induction on $n$ using property (2) with $U = U_1 \cup \ldots \cup U_{n - 1}$ and $V = U_ n$. This is allowed because $U \cap V = (U_1 \cap U_ n) \cup \ldots \cup (U_{n - 1} \cap U_ n)$ is also a union of $n - 1$ affine open subschemes by Schemes, Lemma 26.21.7 applied to the affine opens $U_ i$ and $U_ n$ of $W$. Having said this, for any quasi-compact open $W \subset X$ we can do induction on the number of affine opens needed to cover $W$ using the same trick as before and using that the quasi-compact open $U_ i \cap U_ n$ is separated as an open subscheme of the affine scheme $U_ n$. $\square$

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

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. $\square$

Proof. Let $X = \bigcup U_ i$ be a finite affine open covering. Set $U = \coprod U_ i$ and denote $j : U \to X$ the morphism inducing the given open immersions $U_ i \to X$. Since $U$ is an affine scheme and $X$ has affine diagonal, the morphism $j$ is affine, see Morphisms, Lemma 29.11.11. For every $\mathcal{O}_ X$-module $\mathcal{F}$ there is a canonical map $\mathcal{F} \to j_*j^*\mathcal{F}$. This map is injective as can be seen by checking on stalks: if $x \in U_ i$, then we have a factorization

where $x' \in U$ is the point $x$ viewed as a point of $U_ i \subset U$. Now if $\mathcal{F}$ is quasi-coherent, then $j^*\mathcal{F}$ is quasi-coherent on the affine scheme $U$ hence has vanishing higher cohomology by Lemma 30.2.2. Then $H^ p(X, j_*j^*\mathcal{F}) = 0$ for $p > 0$ by Lemma 30.2.4 as $j$ is affine. This proves (1). Finally, we see that the map $H^ p(X, \mathcal{F}) \to H^ p(X, j_*j^*\mathcal{F})$ is zero and part (2) follows from Homology, Lemma 12.12.4. $\square$

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

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

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

Since the bound on the right is at least $1$ less than the bound in the statement of the lemma, the lemma follows. $\square$

Proof. We first prove (1). Note that under the hypotheses of the lemma the sheaf $R^0f_*\mathcal{F} = f_*\mathcal{F}$ is quasi-coherent by Schemes, Lemma 26.24.1. Using Cohomology, Lemma 20.7.4 we see that forming higher direct images commutes with restriction to open subschemes. Since being quasi-coherent is local on $S$ we reduce to the case discussed in the next paragraph.

Proof of (1) in case $S$ is affine. We will use the induction principle. Since $f$ quasi-compact and quasi-separated we see that $X$ is quasi-compact and quasi-separated. For $U \subset X$ quasi-compact open and $a = f|_ U$ we let $P(U)$ be the property that $R^ pa_*\mathcal{F}$ is quasi-coherent on $S$ for all quasi-coherent modules $\mathcal{F}$ on $U$ and all $p \geq 0$. Since $P(X)$ is (1), it suffices the prove conditions (1) and (2) of Lemma 30.4.1 hold. If $U$ is affine, then $P(U)$ holds because $R^ pa_*\mathcal{F} = 0$ for $p \geq 1$ (by Lemma 30.2.3 and Morphisms, Lemma 29.11.12) and we've already observed the result holds for $p = 0$ in the first paragraph. Next, let $U \subset X$ be a quasi-compact open, $V \subset X$ an affine open, and assume $P(U)$, $P(V)$, $P(U \cap V)$ hold. Let $a = f|_ U$, $b = f|_ V$, $c = f|_{U \cap V}$, and $g = f|_{U \cup V}$. Then for any quasi-coherent $\mathcal{O}_{U \cup V}$-module $\mathcal{F}$ we have the relative Mayer-Vietoris sequence

see Cohomology, Lemma 20.8.3. By $P(U)$, $P(V)$, $P(U \cap V)$ we see that $R^ pa_*(\mathcal{F}|_ U)$, $R^ pb_*(\mathcal{F}|_ V)$ and $R^ pc_*(\mathcal{F}|_{U \cap V})$ are all quasi-coherent. Using the results on quasi-coherent sheaves in Schemes, Section 26.24 this implies that each of the sheaves $R^ pg_*\mathcal{F}$ is quasi-coherent since it sits in the middle of a short exact sequence with a cokernel of a map between quasi-coherent sheaves on the left and a kernel of a map between quasi-coherent sheaves on the right. Whence $P(U \cup V)$ and the proof of (1) is complete.

Next, we prove (3) and a fortiori (2). Choose a finite affine open covering $S = \bigcup _{j = 1, \ldots m} S_ j$. For each $j$ choose a finite affine open covering $f^{-1}(S_ j) = \bigcup _{i = 1, \ldots t_ j} U_{ji} $. Let

be the integer found in Lemma 30.4.4. We claim that $n(X, S, f) = \max d_ j$ works.

Namely, let $S' \to S$ be a morphism of schemes and let $\mathcal{F}'$ be a quasi-coherent sheaf on $X' = S' \times _ S X$. We want to show that $R^ pf'_*\mathcal{F}' = 0$ for $p \geq n(X, S, f)$. Since this question is local on $S'$ we may assume that $S'$ is affine and maps into $S_ j$ for some $j$. Then $X' = S' \times _{S_ j} f^{-1}(S_ j)$ is covered by the open affines $S' \times _{S_ j} U_{ji}$, $i = 1, \ldots t_ j$ and the intersections

are covered by the same number of affines as before the base change. Applying Lemma 30.4.4 we get $H^ p(X', \mathcal{F}') = 0$. By the first part of the proof we already know that each $R^ qf'_*\mathcal{F}'$ is quasi-coherent hence has vanishing higher cohomology groups on our affine scheme $S'$, thus we see that $H^0(S', R^ pf'_*\mathcal{F}') = H^ p(X', \mathcal{F}') = 0$ by Cohomology, Lemma 20.13.6. Since $R^ pf'_*\mathcal{F}'$ is quasi-coherent we conclude that $R^ pf'_*\mathcal{F}' = 0$. $\square$

Proof. Consider the Leray spectral sequence $E_2^{p, q} = H^ p(S, R^ qf_*\mathcal{F})$ converging to $H^{p + q}(X, \mathcal{F})$, see Cohomology, Lemma 20.13.4. By Lemma 30.4.5 we see that the sheaves $R^ qf_*\mathcal{F}$ are quasi-coherent. By Lemma 30.2.2 we see that $E_2^{p, q} = 0$ when $p > 0$. Hence the spectral sequence degenerates at $E_2$ and we win. See also Cohomology, Lemma 20.13.6 (2) for the general principle. $\square$