Lemma 75.29.1 (0GFF)—The Stacks project

Table of contents
Part 4: Algebraic Spaces
Chapter 75: Derived Categories of Spaces
Section 75.29: Detecting Boundedness
Lemma 75.29.1 (cite)

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Lemma 75.29.1. Let $S$ be a scheme. Let $X$ be a quasi-compact and quasi-separated algebraic space over $S$. 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 75.17.2. $\square$

The Stacks project

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