Lemma 36.40.4 (0GEM)—The Stacks project

Table of contents
Part 2: Schemes
Chapter 36: Derived Categories of Schemes
Section 36.40: Detecting Boundedness
Lemma 36.40.4 (cite)

Lemma 36.40.4. 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 $i \ll 0$, and
$\mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}_ X)}(P[-i], \tau _{\leq a} E) = 0$ for $i \ll 0$.

Proof. Using the triangle $ \tau _{\leq a} E \to E \to \tau _{> 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 $i \ll 0$. As $P$ is perfect this is true by Lemma 36.13.5. $\square$

The Stacks project

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