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Tag 0BQR

Chapter 35: Derived Categories of Schemes > Section 35.15: An example generator

Lemma 35.15.1. Let $X$ be a scheme and $\mathcal{L}$ an ample invertible $\mathcal{O}_X$-module. If $K$ is a nonzero object of $D_\textit{QCoh}(\mathcal{O}_X)$, then for some $n \geq 0$ and $p \in \mathbf{Z}$ the cohomology group $H^p(X, K \otimes_{\mathcal{O}_X}^\mathbf{L} \mathcal{L}^{\otimes n})$ is nonzero.

Proof. Recall that as $X$ has an ample invertible sheaf, it is quasi-compact and separated (Properties, Definition 27.26.1 and Lemma 27.26.7). Thus we may apply Proposition 35.6.5 and represent $K$ by a complex $\mathcal{F}^\bullet$ of quasi-coherent modules. Pick any $p$ such that $\mathcal{H}^p = \mathop{\rm Ker}(\mathcal{F}^p \to \mathcal{F}^{p + 1})/ \mathop{\rm Im}(\mathcal{F}^{p - 1} \to \mathcal{F}^p)$ is nonzero. Choose a point $x \in X$ such that the stalk $\mathcal{H}^p_x$ is nonzero. Choose an $n \geq 0$ and $s \in \Gamma(X, \mathcal{L}^{\otimes n})$ such that $X_s$ is an affine open neighbourhood of $x$. Choose $\tau \in \mathcal{H}^p(X_s)$ which maps to a nonzero element of the stalk $\mathcal{H}^p_x$; this is possible as $\mathcal{H}^p$ is quasi-coherent and $X_s$ is affine. Since taking sections over $X_s$ is an exact functor on quasi-coherent modules, we can find a section $\tau' \in \mathcal{F}^p(X_s)$ mapping to zero in $\mathcal{F}^{p + 1}(X_s)$ and mapping to $\tau$ in $\mathcal{H}^p(X_s)$. By Properties, Lemma 27.17.2 there exists an $m$ such that $\tau' \otimes s^{\otimes m}$ is the image of a section $\tau'' \in \Gamma(X, \mathcal{F}^p \otimes \mathcal{L}^{\otimes mn})$. Applying the same lemma once more, we find $l \geq 0$ such that $\tau'' \otimes s^{\otimes l}$ maps to zero in $\mathcal{F}^{p + 1} \otimes \mathcal{L}^{\otimes (m + l)n}$. Then $\tau''$ gives a nonzero class in $H^p(X, K \otimes^\mathbf{L}_{\mathcal{O}_X} \mathcal{L}^{(m + l)n})$ as desired. $\square$

    The code snippet corresponding to this tag is a part of the file perfect.tex and is located in lines 3179–3187 (see updates for more information).

    \begin{lemma}
    \label{lemma-nonzero-some-cohomology}
    Let $X$ be a scheme and $\mathcal{L}$ an ample invertible
    $\mathcal{O}_X$-module. If $K$ is a nonzero object of
    $D_\QCoh(\mathcal{O}_X)$, then for some $n \geq 0$ and $p \in \mathbf{Z}$
    the cohomology group
    $H^p(X, K \otimes_{\mathcal{O}_X}^\mathbf{L} \mathcal{L}^{\otimes n})$
    is nonzero.
    \end{lemma}
    
    \begin{proof}
    Recall that as $X$ has an ample invertible sheaf, it is quasi-compact
    and separated (Properties, Definition \ref{properties-definition-ample} and
    Lemma \ref{properties-lemma-affine-s-opens-cover-quasi-separated}).
    Thus we may apply
    Proposition \ref{proposition-quasi-compact-affine-diagonal}
    and represent $K$ by a complex $\mathcal{F}^\bullet$ of
    quasi-coherent modules. Pick any $p$ such that
    $\mathcal{H}^p = \Ker(\mathcal{F}^p \to \mathcal{F}^{p + 1})/
    \Im(\mathcal{F}^{p - 1} \to \mathcal{F}^p)$ is nonzero.
    Choose a point $x \in X$ such that the stalk $\mathcal{H}^p_x$ is
    nonzero. Choose an $n \geq 0$ and $s \in \Gamma(X, \mathcal{L}^{\otimes n})$
    such that $X_s$ is an affine open neighbourhood of $x$.
    Choose $\tau \in \mathcal{H}^p(X_s)$ which maps to a nonzero
    element of the stalk $\mathcal{H}^p_x$; this is possible
    as $\mathcal{H}^p$ is quasi-coherent and $X_s$ is affine.
    Since taking sections over $X_s$ is an exact functor on
    quasi-coherent modules, we can find a section $\tau' \in \mathcal{F}^p(X_s)$
    mapping to zero in $\mathcal{F}^{p + 1}(X_s)$ and mapping to
    $\tau$ in $\mathcal{H}^p(X_s)$. By
    Properties, Lemma \ref{properties-lemma-invert-s-sections}
    there exists an $m$ such that $\tau' \otimes s^{\otimes m}$
    is the image of a section
    $\tau'' \in \Gamma(X, \mathcal{F}^p \otimes \mathcal{L}^{\otimes mn})$.
    Applying the same lemma once more, we find $l \geq 0$ such that
    $\tau'' \otimes s^{\otimes l}$ maps to zero in
    $\mathcal{F}^{p + 1} \otimes \mathcal{L}^{\otimes (m + l)n}$.
    Then $\tau''$ gives a nonzero class in
    $H^p(X, K \otimes^\mathbf{L}_{\mathcal{O}_X} \mathcal{L}^{(m + l)n})$
    as desired.
    \end{proof}

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