Lemma 57.9.8 (0FZ7)—The Stacks project

Processing math: 100%

Table of contents
Part 3: Topics in Scheme Theory
Chapter 57: Derived Categories of Varieties
Section 57.9: Resolutions and bounds
Lemma 57.9.8 (cite)

numberstags

Lemma 57.9.8. 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 57.9.7 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. $\square$

Comments (0)

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$ ). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Comments (0)

Post a comment