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Chapter 29: Cohomology of Schemes > Section 29.4: Quasi-coherence of higher direct images

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 20.8.4 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 25.21.8. 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 20.9.3. 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 25.21.6). 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 20.9.3. 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 20.14.6. Since $R^pf'_*\mathcal{F}'$ is quasi-coherent we conclude that $R^pf'_*\mathcal{F}' = 0$. $\square$

    The code snippet corresponding to this tag is a part of the file coherent.tex and is located in lines 679–695 (see updates for more information).

    \begin{lemma}
    \label{lemma-quasi-coherence-higher-direct-images}
    Let $f : X \to S$ be a morphism of schemes.
    Assume that $f$ is quasi-separated and quasi-compact.
    \begin{enumerate}
    \item For any quasi-coherent $\mathcal{O}_X$-module $\mathcal{F}$ the
    higher direct images $R^pf_*\mathcal{F}$ are quasi-coherent on $S$.
    \item 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 $X$.
    \item 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 $f$.
    \end{enumerate}
    \end{lemma}
    
    \begin{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 \ref{schemes-lemma-push-forward-quasi-coherent}.
    Using
    Cohomology, Lemma \ref{cohomology-lemma-localize-higher-direct-images}
    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.
    
    \medskip\noindent
    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 \ref{morphisms-lemma-morphism-affines-affine}
    and (1) follows from
    Lemma \ref{lemma-relative-affine-vanishing}.
    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 \ref{schemes-lemma-characterize-separated}.
    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 \ref{cohomology-lemma-relative-mayer-vietoris}.
    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 \ref{schemes-section-quasi-coherent} we see
    conclude $R^pf_*\mathcal{F}$ is quasi-coherent.
    
    \medskip\noindent
    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 \ref{schemes-lemma-affine-separated}).
    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 \ref{cohomology-lemma-relative-mayer-vietoris}.
    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.
    
    \medskip\noindent
    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 \ref{lemma-vanishing-nr-affines-quasi-separated}.
    We claim that $n(X, S, f) = \max d_j$ works.
    
    \medskip\noindent
    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 \ref{lemma-vanishing-nr-affines-quasi-separated}
    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 \ref{cohomology-lemma-apply-Leray}.
    Since $R^pf'_*\mathcal{F}'$ is quasi-coherent
    we conclude that $R^pf'_*\mathcal{F}' = 0$.
    \end{proof}

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