The Stacks project

Proof. Recall that as $X$ has an ample invertible sheaf, it is quasi-compact and separated (Properties, Definition 28.26.1 and Lemma 28.26.7). Thus we may apply Proposition 36.7.5 and represent $K$ by a complex $\mathcal{F}^\bullet $ of quasi-coherent modules. Pick any $p$ such that $\mathcal{H}^ p = \mathop{\mathrm{Ker}}(\mathcal{F}^ p \to \mathcal{F}^{p + 1})/ \mathop{\mathrm{Im}}(\mathcal{F}^{p - 1} \to \mathcal{F}^ p)$ 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 28.17.2 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. $\square$