The Stacks project

30.4 Quasi-coherence of higher direct images

We have seen that the higher cohomology groups of a quasi-coherent module on an affine is zero. For (quasi-)separated quasi-compact schemes $X$ this implies vanishing of cohomology groups of quasi-coherent sheaves beyond a certain degree. However, it may not be the case that $X$ has finite cohomological dimension, because that is defined in terms of vanishing of cohomology of all $\mathcal{O}_ X$-modules.

Lemma 30.4.1 (Induction Principle). Let $X$ be a quasi-compact and quasi-separated scheme. Let $P$ be a property of the quasi-compact opens of $X$. Assume that

  1. $P$ holds for every affine open of $X$,

  2. if $U$ is quasi-compact open, $V$ affine open, $P$ holds for $U$, $V$, and $U \cap V$, then $P$ holds for $U \cup V$.

Then $P$ holds for every quasi-compact open of $X$ and in particular for $X$.

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 $U_ n$. $\square$

slogan

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. $\square$

Lemma 30.4.3. Let $X$ be a quasi-compact scheme with affine diagonal (for example if $X$ is separated). Then

  1. given a quasi-coherent $\mathcal{O}_ X$-module $\mathcal{F}$ there exists an embedding $\mathcal{F} \to \mathcal{F}'$ of quasi-coherent $\mathcal{O}_ X$-modules such that $H^ p(X, \mathcal{F}') = 0$ for all $p \geq 1$, and

  2. $\{ H^ n(X, -)\} _{n \geq 0}$ is a universal $\delta $-functor from $\mathit{QCoh}(\mathcal{O}_ X)$ to $\textit{Ab}$.

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

\[ \mathcal{F}_ x \to (j_*j^*\mathcal{F})_ x \to (j^*\mathcal{F})_{x'} = \mathcal{F}_ x \]

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. $\square$

Lemma 30.4.4. Let $X$ be a quasi-compact quasi-separated scheme. Let $X = U_1 \cup \ldots \cup U_ t$ be an affine open covering. Set

\[ d = \max \nolimits _{I \subset \{ 1, \ldots , t\} } \left(|I| + t(\bigcap \nolimits _{i \in I} U_ i)\right) \]

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 the numbers $t(\bigcap _{i \in I} U_ i)$ are finite. Let $\mathcal{U} : X = \bigcup _{i = 1}^ t U_ i$. By Cohomology, Lemma 20.11.5 there is a spectral sequence

\[ E_2^{p, q} = \check{H}^ p(\mathcal{U}, \underline{H}^ q(\mathcal{F})) \]

converging to $H^{p + q}(U, \mathcal{F})$. By Cohomology, Lemma 20.23.6 we have

\[ E_2^{p, q} = H^ p(\check{\mathcal{C}}_{alt}^\bullet ( \mathcal{U}, \underline{H}^ q(\mathcal{F})) \]

The alternating Čech complex with values in the presheaf $\underline{H}^ q(\mathcal{F})$ vanishes in high degrees by Lemma 30.4.2, more precisely $E_2^{p, q} = 0$ for $p + q \geq d$. Hence the result follows. $\square$

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 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 29.11.12 and (1) follows from Lemma 30.2.3. 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 26.21.7. 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 26.24 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 26.21.15). 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 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 $R^ pf'_*\mathcal{F}' = 0$. $\square$

Lemma 30.4.6. Let $f : X \to S$ be a morphism of schemes. Assume that $f$ is quasi-separated and quasi-compact. Assume $S$ is affine. For any quasi-coherent $\mathcal{O}_ X$-module $\mathcal{F}$ we have

\[ H^ q(X, \mathcal{F}) = H^0(S, R^ qf_*\mathcal{F}) \]

for all $q \in \mathbf{Z}$.

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. $\square$


Comments (3)

Comment #4556 by Weixiao Lu on

In Lemma 29.4.3, the requirement "X is quasi-compact" is not necessary.

Comment #4557 by Weixiao Lu on

In the first line of the proof of Lemma 29.3.2, should be replaced by .

Comment #4754 by on

@#4556. What you suggest does not work with the proof as given now. (Also, I'm a bit doubtful that Lemma 30.4.3 is true without the quasi-compactness assumption.) Namely, if isn't quasi-compact, then you cannot take a finite open covering by affines and hence you don't know that the morphism is affine. I have edited the proof to make this more apparant. See changes here.


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