# The Stacks Project

## Tag 0BQR

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. $\square$

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\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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