Lemma 21.47.4. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $E$ be an object of $D(\mathcal{O})$. The following are equivalent

$E$ is perfect, and

$E$ is pseudo-coherent and locally has finite tor dimension.

Lemma 21.47.4. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $E$ be an object of $D(\mathcal{O})$. The following are equivalent

$E$ is perfect, and

$E$ is pseudo-coherent and locally has finite tor dimension.

**Proof.**
Assume (1). Let $U$ be an object of $\mathcal{C}$. By definition there exists a covering $\{ U_ i \to U\} $ such that $E|_{U_ i}$ is represented by a strictly perfect complex. Thus $E$ is pseudo-coherent (i.e., $m$-pseudo-coherent for all $m$) by Lemma 21.45.2. Moreover, a direct summand of a finite free module is flat, hence $E|_{U_ i}$ has finite Tor dimension by Lemma 21.46.3. Thus (2) holds.

Assume (2). Let $U$ be an object of $\mathcal{C}$. After replacing $U$ by the members of a covering we may assume there exist integers $a \leq b$ such that $E|_ U$ has tor amplitude in $[a, b]$. Since $E|_ U$ is $m$-pseudo-coherent for all $m$ we conclude using Lemma 21.47.3. $\square$

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