Lemma 36.40.3 (0GEL)—The Stacks project

Processing math: 100%

Table of contents
Part 2: Schemes
Chapter 36: Derived Categories of Schemes
Section 36.40: Detecting Boundedness
Lemma 36.40.3 (cite)

numberstags

Lemma 36.40.3. Let $X$ be a quasi-compact and quasi-separated scheme. Let $P \in D_{perf}(\mathcal{O}_ X)$ and $E \in D_{\mathit{QCoh}}(\mathcal{O}_ X)$ . Let $a \in \mathbf{Z}$ . The following are equivalent

$\mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}_ X)}(P[-i], E) = 0$ for $i \gg 0$ , and
$\mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}_ X)}(P[-i], \tau _{\geq a} E) = 0$ for $i \gg 0$ .

Proof. Using the triangle $\tau _{< a} E \to E \to \tau _{\geq a} E \to$ we see that the equivalence follows if we can show

$\mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}_ X)}(P[-i], \tau _{< a} E) = \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}_ X)}(P, (\tau _{< a} E)[i]) = 0$

for $i \gg 0$ . As $P$ is perfect this is true by Lemma 36.13.5. $\square$

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