The Stacks project

Lemma 29.4.5. 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 25.24.1. 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 may assume $S$ is affine.

Assume $S$ is affine and $f$ quasi-compact and separated. Let $t \geq 1$ be the minimal number of affine opens needed to cover $X$. We will prove this case of (1) by induction on $t$. If $t = 1$ then the morphism $f$ is affine by Morphisms, Lemma 28.11.12 and (1) follows from Lemma 29.2.3. 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, see Schemes, Lemma 25.21.7. We will apply the relative Mayer-Vietoris sequence

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

see Cohomology, Lemma 20.8.3. By induction we see that $R^ pa_*\mathcal{F}$, $R^ pb_*\mathcal{F}$ and $R^ pc_*\mathcal{F}$ are all quasi-coherent. This implies that each of the sheaves $R^ pf_*\mathcal{F}$ 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. Using the results on quasi-coherent sheaves in Schemes, Section 25.24 we see conclude $R^ pf_*\mathcal{F}$ is quasi-coherent.

Assume $S$ is affine and $f$ quasi-compact and quasi-separated. Let $t \geq 1$ be the minimal number of affine opens needed to cover $X$. We will prove (1) by induction on $t$. In case $t = 1$ the morphism $f$ is separated and we are back in the previous case (see previous paragraph). If $t > 1$ write $X = U \cup V$ with $V$ affine open and $U$ a union of $t - 1$ open affines. Note that in this case $U \cap V$ is an open subscheme of an affine scheme and hence separated (see Schemes, Lemma 25.21.15). We will apply the relative Mayer-Vietoris sequence

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

see Cohomology, Lemma 20.8.3. By induction and the result of the previous paragraph we see that $R^ pa_*\mathcal{F}$, $R^ pb_*\mathcal{F}$ and $R^ pc_*\mathcal{F}$ are quasi-coherent. As in the previous paragraph this implies each of sheaves $R^ pf_*\mathcal{F}$ is quasi-coherent.

Next, we prove (3) and a fortiori (2). Choose a finite affine open covering $S = \bigcup _{j = 1, \ldots m} S_ j$. For each $i$ 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 29.4.4. 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 29.4.4 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 $R^ pf'_*\mathcal{F}' = 0$. $\square$


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