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Tag 02O5

Chapter 29: Cohomology of Schemes > Section 29.19: Higher direct images of coherent sheaves

Proposition 29.19.1. Let $S$ be a locally Noetherian scheme. Let $f : X \to S$ be a proper morphism. Let $\mathcal{F}$ be a coherent $\mathcal{O}_X$-module. Then $R^if_*\mathcal{F}$ is a coherent $\mathcal{O}_S$-module for all $i \geq 0$.

Proof. Since the problem is local on $S$ we may assume that $S$ is a Noetherian scheme. Since a proper morphism is of finite type we see that in this case $X$ is a Noetherian scheme also. Consider the property $\mathcal{P}$ of coherent sheaves on $X$ defined by the rule $$ \mathcal{P}(\mathcal{F}) \Leftrightarrow R^pf_*\mathcal{F}\text{ is coherent for all }p \geq 0 $$ We are going to use the result of Lemma 29.12.6 to prove that $\mathcal{P}$ holds for every coherent sheaf on $X$.

Let $$ 0 \to \mathcal{F}_1 \to \mathcal{F}_2 \to \mathcal{F}_3 \to 0 $$ be a short exact sequence of coherent sheaves on $X$. Consider the long exact sequence of higher direct images $$ R^{p - 1}f_*\mathcal{F}_3 \to R^pf_*\mathcal{F}_1 \to R^pf_*\mathcal{F}_2 \to R^pf_*\mathcal{F}_3 \to R^{p + 1}f_*\mathcal{F}_1 $$ Then it is clear that if 2-out-of-3 of the sheaves $\mathcal{F}_i$ have property $\mathcal{P}$, then the higher direct images of the third are sandwiched in this exact complex between two coherent sheaves. Hence these higher direct images are also coherent by Lemma 29.9.2 and 29.9.3. Hence property $\mathcal{P}$ holds for the third as well.

Let $Z \subset X$ be an integral closed subscheme. We have to find a coherent sheaf $\mathcal{F}$ on $X$ whose support is contained in $Z$, whose stalk at the generic point $\xi$ of $Z$ is a $1$-dimensional vector space over $\kappa(\xi)$ such that $\mathcal{P}$ holds for $\mathcal{F}$. Denote $g = f|_Z : Z \to S$ the restriction of $f$. Suppose we can find a coherent sheaf $\mathcal{G}$ on $Z$ such that (a) $\mathcal{G}_\xi$ is a $1$-dimensional vector space over $\kappa(\xi)$, (b) $R^pg_*\mathcal{G} = 0$ for $p > 0$, and (c) $g_*\mathcal{G}$ is coherent. Then we can consider $\mathcal{F} = (Z \to X)_*\mathcal{G}$. As $Z \to X$ is a closed immersion we see that $(Z \to X)_*\mathcal{G}$ is coherent on $X$ and $R^p(Z \to X)_*\mathcal{G} = 0$ for $p > 0$ (Lemma 29.9.9). Hence by the relative Leray spectral sequence (Cohomology, Lemma 20.14.8) we will have $R^pf_*\mathcal{F} = R^pg_*\mathcal{G} = 0$ for $p > 0$ and $f_*\mathcal{F} = g_*\mathcal{G}$ is coherent. Finally $\mathcal{F}_\xi = ((Z \to X)_*\mathcal{G})_\xi = \mathcal{G}_\xi$ which verifies the condition on the stalk at $\xi$. Hence everything depends on finding a coherent sheaf $\mathcal{G}$ on $Z$ which has properties (a), (b), and (c).

We can apply Chow's Lemma 29.18.1 to the morphism $Z \to S$. Thus we get a diagram $$ \xymatrix{ Z \ar[rd]_g & Z' \ar[d]^-{g'} \ar[l]^\pi \ar[r]_i & \mathbf{P}^n_S \ar[dl] \\ & S & } $$ as in the statement of Chow's lemma. Also, let $U \subset Z$ be the dense open subscheme such that $\pi^{-1}(U) \to U$ is an isomorphism. By the discussion in Remark 29.18.2 we see that $i' = (i, \pi) : Z' \to \mathbf{P}^n_Z$ is a closed immersion. Hence $$ \mathcal{L} = i^*\mathcal{O}_{\mathbf{P}^n_S}(1) \cong (i')^*\mathcal{O}_{\mathbf{P}^n_Z}(1) $$ is $g'$-relatively ample and $\pi$-relatively ample (for example by Morphisms, Lemma 28.37.7). Hence by Lemma 29.16.2 there exists an $n \geq 0$ such that both $R^p\pi_*\mathcal{L}^{\otimes n} = 0$ for all $p > 0$ and $R^p(g')_*\mathcal{L}^{\otimes n} = 0$ for all $p > 0$. Set $\mathcal{G} = \pi_*\mathcal{L}^{\otimes n}$. Property (a) holds because $\pi_*\mathcal{L}^{\otimes}|_U$ is an invertible sheaf (as $\pi^{-1}(U) \to U$ is an isomorphism). Properties (b) and (c) hold because by the relative Leray spectral sequence (Cohomology, Lemma 20.14.8) we have $$ E_2^{p, q} = R^pg_* R^q\pi_*\mathcal{L}^{\otimes n} \Rightarrow R^{p + q}(g')_*\mathcal{L}^{\otimes n} $$ and by choice of $n$ the only nonzero terms in $E_2^{p, q}$ are those with $q = 0$ and the only nonzero terms of $R^{p + q}(g')_*\mathcal{L}^{\otimes n}$ are those with $p = q = 0$. This implies that $R^pg_*\mathcal{G} = 0$ for $p > 0$ and that $g_*\mathcal{G} = (g')_*\mathcal{L}^{\otimes n}$. Finally, applying the previous Lemma 29.16.3 we see that $g_*\mathcal{G} = (g')_*\mathcal{L}^{\otimes n}$ is coherent as desired. $\square$

    The code snippet corresponding to this tag is a part of the file coherent.tex and is located in lines 4723–4730 (see updates for more information).

    \begin{proposition}
    \label{proposition-proper-pushforward-coherent}
    Let $S$ be a locally Noetherian scheme.
    Let $f : X \to S$ be a proper morphism.
    Let $\mathcal{F}$ be a coherent $\mathcal{O}_X$-module.
    Then $R^if_*\mathcal{F}$ is a coherent $\mathcal{O}_S$-module
    for all $i \geq 0$.
    \end{proposition}
    
    \begin{proof}
    Since the problem is local on $S$ we may assume that $S$ is
    a Noetherian scheme. Since a proper morphism is of finite type
    we see that in this case $X$ is a Noetherian scheme also.
    Consider the property $\mathcal{P}$ of coherent sheaves
    on $X$ defined by the rule
    $$
    \mathcal{P}(\mathcal{F}) \Leftrightarrow
    R^pf_*\mathcal{F}\text{ is coherent for all }p \geq 0
    $$
    We are going to use the result of
    Lemma \ref{lemma-property} to prove that
    $\mathcal{P}$ holds for every coherent sheaf on $X$.
    
    \medskip\noindent
    Let
    $$
    0 \to \mathcal{F}_1 \to \mathcal{F}_2 \to \mathcal{F}_3 \to 0
    $$
    be a short exact sequence of coherent sheaves on $X$.
    Consider the long exact sequence of higher direct images
    $$
    R^{p - 1}f_*\mathcal{F}_3 \to
    R^pf_*\mathcal{F}_1 \to
    R^pf_*\mathcal{F}_2 \to
    R^pf_*\mathcal{F}_3 \to
    R^{p + 1}f_*\mathcal{F}_1
    $$
    Then it is clear that if 2-out-of-3 of the sheaves $\mathcal{F}_i$
    have property $\mathcal{P}$, then the higher direct images of the
    third are sandwiched in this exact complex between two coherent
    sheaves. Hence these higher direct images are also coherent by
    Lemma \ref{lemma-coherent-abelian-Noetherian} and
    \ref{lemma-coherent-Noetherian-quasi-coherent-sub-quotient}.
    Hence property $\mathcal{P}$ holds for the third as well.
    
    \medskip\noindent
    Let $Z \subset X$ be an integral closed subscheme.
    We have to find a coherent sheaf $\mathcal{F}$ on $X$ whose support is
    contained in $Z$, whose stalk at the generic point $\xi$ of $Z$ is a
    $1$-dimensional vector space over $\kappa(\xi)$ such that $\mathcal{P}$
    holds for $\mathcal{F}$. Denote $g = f|_Z : Z \to S$ the restriction of $f$.
    Suppose we can find a coherent sheaf $\mathcal{G}$ on $Z$ such
    that
    (a) $\mathcal{G}_\xi$ is a $1$-dimensional vector space over $\kappa(\xi)$,
    (b) $R^pg_*\mathcal{G} = 0$ for $p > 0$, and
    (c) $g_*\mathcal{G}$ is coherent. Then we can consider
    $\mathcal{F} = (Z \to X)_*\mathcal{G}$. As $Z \to X$ is a closed immersion
    we see that $(Z \to X)_*\mathcal{G}$ is coherent on $X$
    and $R^p(Z \to X)_*\mathcal{G} = 0$ for $p > 0$
    (Lemma \ref{lemma-finite-pushforward-coherent}).
    Hence by the relative Leray spectral sequence
    (Cohomology, Lemma \ref{cohomology-lemma-relative-Leray})
    we will have $R^pf_*\mathcal{F} = R^pg_*\mathcal{G} = 0$ for $p > 0$
    and $f_*\mathcal{F} = g_*\mathcal{G}$ is coherent.
    Finally $\mathcal{F}_\xi = ((Z \to X)_*\mathcal{G})_\xi = \mathcal{G}_\xi$
    which verifies the condition on the stalk at $\xi$.
    Hence everything depends on finding a coherent sheaf $\mathcal{G}$
    on $Z$ which has properties (a), (b), and (c).
    
    \medskip\noindent
    We can apply Chow's Lemma \ref{lemma-chow-Noetherian}
    to the morphism $Z \to S$. Thus we get a diagram
    $$
    \xymatrix{
    Z \ar[rd]_g & Z' \ar[d]^-{g'} \ar[l]^\pi \ar[r]_i & \mathbf{P}^n_S \ar[dl] \\
    & S &
    }
    $$
    as in the statement of Chow's lemma. Also, let $U \subset Z$ be
    the dense open subscheme such that $\pi^{-1}(U) \to U$ is an isomorphism.
    By the discussion in Remark \ref{remark-chow-Noetherian} we see that
    $i' = (i, \pi) : Z' \to \mathbf{P}^n_Z$ is
    a closed immersion. Hence
    $$
    \mathcal{L} = i^*\mathcal{O}_{\mathbf{P}^n_S}(1) \cong
    (i')^*\mathcal{O}_{\mathbf{P}^n_Z}(1)
    $$
    is $g'$-relatively ample and $\pi$-relatively ample (for example by
    Morphisms, Lemma \ref{morphisms-lemma-characterize-ample-on-finite-type}).
    Hence by Lemma \ref{lemma-kill-by-twisting}
    there exists an $n \geq 0$ such that
    both $R^p\pi_*\mathcal{L}^{\otimes n} = 0$ for all $p > 0$ and
    $R^p(g')_*\mathcal{L}^{\otimes n} = 0$ for all $p > 0$.
    Set $\mathcal{G} = \pi_*\mathcal{L}^{\otimes n}$.
    Property (a) holds because $\pi_*\mathcal{L}^{\otimes}|_U$ is
    an invertible sheaf (as $\pi^{-1}(U) \to U$ is an isomorphism).
    Properties (b) and (c) hold because by the relative Leray
    spectral sequence
    (Cohomology, Lemma \ref{cohomology-lemma-relative-Leray})
    we have
    $$
    E_2^{p, q} = R^pg_* R^q\pi_*\mathcal{L}^{\otimes n}
    \Rightarrow
    R^{p + q}(g')_*\mathcal{L}^{\otimes n}
    $$
    and by choice of $n$ the only nonzero terms in $E_2^{p, q}$ are
    those with $q = 0$ and the only nonzero terms of
    $R^{p + q}(g')_*\mathcal{L}^{\otimes n}$ are those with $p = q = 0$.
    This implies that $R^pg_*\mathcal{G} = 0$ for $p > 0$ and
    that $g_*\mathcal{G} = (g')_*\mathcal{L}^{\otimes n}$.
    Finally, applying the previous
    Lemma \ref{lemma-locally-projective-pushforward}
    we see that $g_*\mathcal{G} = (g')_*\mathcal{L}^{\otimes n}$ is
    coherent as desired.
    \end{proof}

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