The Stacks project

Lemma 57.10.8. Let $k$ be a field. Let $X$ be a proper smooth scheme over $k$. There exists integers $m, n \geq 1$ and a finite locally free $\mathcal{O}_ X$-module $\mathcal{G}$ such that every coherent $\mathcal{O}_ X$-module is contained in $smd(add(\mathcal{G}[-m, m])^{\star n})$ with notation as in Derived Categories, Section 13.35.

Proof. In the proof of Lemma 57.10.7 we have shown that there exist $m', n \geq 1$ such that for any coherent $\mathcal{O}_ X$-module $\mathcal{F}$,

\[ \mathcal{F} \in smd(add(\mathcal{G}[-m' + a, m' + b])^{\star n}) \]

for any $a \leq b$ such that $H^ i(X, \mathcal{F})$ is nonzero only for $i \in [a, b]$. Thus we can take $a = 0$ and $b = \dim (X)$. Taking $m = \max (m', m' + b)$ finishes the proof. $\square$


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