Lemma 30.4.5. Let $f : X \to S$ be a morphism of schemes. Assume that $f$ is quasi-separated and quasi-compact.

1. For any quasi-coherent $\mathcal{O}_ X$-module $\mathcal{F}$ the higher direct images $R^ pf_*\mathcal{F}$ are quasi-coherent on $S$.

2. If $S$ is quasi-compact, there exists an integer $n = n(X, S, f)$ such that $R^ pf_*\mathcal{F} = 0$ for all $p \geq n$ and any quasi-coherent sheaf $\mathcal{F}$ on $X$.

3. In fact, if $S$ is quasi-compact we can find $n = n(X, S, f)$ such that for every morphism of schemes $S' \to S$ we have $R^ p(f')_*\mathcal{F}' = 0$ for $p \geq n$ and any quasi-coherent sheaf $\mathcal{F}'$ on $X'$. Here $f' : X' = S' \times _ S X \to S'$ is the base change of $f$.

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

$0 \to g_*\mathcal{F} \to a_*(\mathcal{F}|_ U) \oplus b_*(\mathcal{F}|_ V) \to c_*(\mathcal{F}|_{U \cap V}) \to R^1g_*\mathcal{F} \to \ldots$

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

$d_ j = \max \nolimits _{I \subset \{ 1, \ldots , t_ j\} } \left(|I| + t(\bigcap \nolimits _{i \in I} U_{ji})\right)$

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

$\bigcap \nolimits _{i \in I} S' \times _{S_ j} U_{ji} = S' \times _{S_ j} \bigcap \nolimits _{i \in I} U_{ji}$

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 $R^ pf'_*\mathcal{F}' = 0$. $\square$

Comment #7145 by Hao Peng on

I found that part $1$ can also be proven using Cech-to-sheaf cohomology(tag 0A3Z) much the same way as the proof of flat base change(tag 02KH). It amounts to prove in the affine case the presheaf $U\mapsto H^p(f^{-1}(U), \mathcal F)$ is in fact a sheaf, which is because localization $M\mapsto M_f$ is exact.

Comment #7192 by DatPham on

Typo (in the proof of (3)): for each $i$ (not $j$).

Comment #7319 by on

@#7145. Yes, you can prove this in some way like that.

@#7192: Thanks and fixed here.

Comment #7725 by Ryo Suzuki on

The proof of (1) is much the same as Lemma 30.4.1. I think it is better to use it.

Namely, we can assume $S$ is affine. I'm going to say $P(U)$ for a quasi-compact open $U\subset X$ when, for any quasi-coherent $O_U$-module $F$, and for any $p\geq 0$, $R^pg_*F$ is quasi-coherent, where $g$ is the composite of inclusion $U\subset X$ and $f$.

I will verify the conditions (1) and (2) of Lemma 30.4.1. First, let $U$ be affine open. $R^0g_*F = g_*F$ is quasi-coherent, because $g_*$ is a quasi-compact quasi-separated morphism. For $p>0$, $R^pg_*F = 0$ by Lemma 30.2.3. So (1) is verified.

Next, let $U$ be a quasi-compact open, $V$ an affine open, and assume $P(U)$, $P(V)$, $P(U\cap V)$ hold. Let $g\colon U\cup V\to S$, $a\colon U\to S$, $b\colon V\to S$, and $c\colon U\cap V\to S$ be the composite of an inclusion to $X$ and $f$. Then, for any quasi-coherent $O_{U\cup V}$-module $F$, we have the relative Mayer-Vietoris sequence . By hypothesis $Ra_*^p(F|_U)$, $Rb_*^p(F|_V)$, $Rc_*^p(F|_{U\cap V})$ are quasi-coherent for all $p\geq 0$. Hence $Rg_*^p(F)$ are quasi-coherent for all $p\geq 0$.

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