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36.19 Characterizing pseudo-coherent complexes, I

We can use the methods above to characterize pseudo-coherent objects as derived homotopy limits of approximations by perfect objects.

Lemma 36.19.1. Let $X$ be a quasi-compact and quasi-separated scheme. Let $K \in D(\mathcal{O}_ X)$. The following are equivalent

  1. $K$ is pseudo-coherent, and

  2. $K = \text{hocolim} K_ n$ where $K_ n$ is perfect and $\tau _{\geq -n}K_ n \to \tau _{\geq -n}K$ is an isomorphism for all $n$.

Proof. The implication (2) $\Rightarrow $ (1) is true on any ringed space. Namely, assume (2) holds. Recall that a perfect object of the derived category is pseudo-coherent, see Cohomology, Lemma 20.46.5. 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, Definition 20.44.1.

Assume (1). We start by choosing an approximation $K_1 \to K$ of $(X, K, -2)$ by a perfect complex $K_1$, see Definitions 36.14.1 and 36.14.2 and Theorem 36.14.6. Suppose by induction we have

\[ K_1 \to K_2 \to \ldots \to K_ n \to K \]

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 36.18.2 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

\[ K_{n + 1} \to K \to C \to K_{n + 1}[1] \]

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 $K_ n$. $\square$

Lemma 36.19.2. Let $X$ be a quasi-compact and quasi-separated scheme. Let $T \subset X$ be a closed subset such that $X \setminus T$ is quasi-compact. Let $K \in D(\mathcal{O}_ X)$ supported on $T$. The following are equivalent

  1. $K$ is pseudo-coherent, and

  2. $K = \text{hocolim} K_ n$ where $K_ n$ is perfect, supported on $T$, and $\tau _{\geq -n}K_ n \to \tau _{\geq -n}K$ is an isomorphism for all $n$.

Proof. The proof of this lemma is exactly the same as the proof of Lemma 36.19.1 except that in the choice of the approximations we use the triples $(T, K, m)$. $\square$


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