The Stacks project

Proof. The implication (2) $\Rightarrow $ (1) is true on any ringed site. Namely, assume (2) holds. Recall that a perfect object of the derived category is pseudo-coherent, see Cohomology on Sites, Lemma 21.47.4. Then it follows from the definitions that $\tau _{\geq -n}K_ n$ is $(-n + 1)$-pseudo-coherent and hence $\tau _{\geq -n}K$ is $(-n + 1)$-pseudo-coherent, hence $K$ is $(-n + 1)$-pseudo-coherent. This is true for all $n$, hence $K$ is pseudo-coherent, see Cohomology on Sites, Definition 21.45.1.

Assume (1). We start by choosing an approximation $K_1 \to K$ of $(X, K, -2)$ by a perfect complex $K_1$, see Definitions 75.14.1 and 75.14.2 and Theorem 75.14.7. Suppose by induction we have

with $K_ i$ perfect such that such that $\tau _{\geq -i}K_ i \to \tau _{\geq -i}K$ is an isomorphism for all $1 \leq i \leq n$. Then we pick $a \leq b$ as in Lemma 75.17.2 for the perfect object $K_ n$. Choose an approximation $K_{n + 1} \to K$ of $(X, K, \min (a - 1, -n - 1))$. Choose a distinguished triangle

Then we see that $C \in D_\mathit{QCoh}(\mathcal{O}_ X)$ has $H^ i(C) = 0$ for $i \geq a$. Thus by our choice of $a, b$ we see that $\mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}_ X)}(K_ n, C) = 0$. Hence the composition $K_ n \to K \to C$ is zero. Hence by Derived Categories, Lemma 13.4.2 we can factor $K_ n \to K$ through $K_{n + 1}$ proving the induction step.

We still have to prove that $K = \text{hocolim} K_ n$. This follows by an application of Derived Categories, Lemma 13.33.8 to the functors $H^ i( - ) : D(\mathcal{O}_ X) \to \textit{Mod}(\mathcal{O}_ X)$ and our choice of $K_ n$. $\square$