Definition 21.47.1. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{E}^\bullet $ be a complex of $\mathcal{O}$-modules. We say $\mathcal{E}^\bullet $ is perfect if for every object $U$ of $\mathcal{C}$ there exists a covering $\{ U_ i \to U\} $ such that for each $i$ there exists a morphism of complexes $\mathcal{E}_ i^\bullet \to \mathcal{E}^\bullet |_{U_ i}$ which is a quasi-isomorphism with $\mathcal{E}_ i^\bullet $ strictly perfect. An object $E$ of $D(\mathcal{O})$ is perfect if it can be represented by a perfect complex of $\mathcal{O}$-modules.
21.47 Perfect complexes
In this section we discuss properties of perfect complexes on ringed sites.
If $\mathop{\mathit{Sh}}\nolimits (\mathcal{C})$ is quasi-compact (Sites, Section 7.17), then a perfect object of $D(\mathcal{O})$ is in $D^ b(\mathcal{O})$. But this need not be the case otherwise.
Lemma 21.47.2. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $E$ be an object of $D(\mathcal{O})$.
If $\mathcal{C}$ has a final object $X$ and there exist a covering $\{ U_ i \to X\} $, strictly perfect complexes $\mathcal{E}_ i^\bullet $ of $\mathcal{O}_{U_ i}$-modules, and isomorphisms $\alpha _ i : \mathcal{E}_ i^\bullet \to E|_{U_ i}$ in $D(\mathcal{O}_{U_ i})$, then $E$ is perfect.
If $E$ is perfect, then any complex representing $E$ is perfect.
Proof. Identical to the proof of Lemma 21.45.2. $\square$
Lemma 21.47.3. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $E$ be an object of $D(\mathcal{O})$. Let $a \leq b$ be integers. If $E$ has tor amplitude in $[a, b]$ and is $(a - 1)$-pseudo-coherent, then $E$ is perfect.
Proof. Let $U$ be an object of $\mathcal{C}$. After replacing $U$ by the members of a covering and $\mathcal{C}$ by the localization $\mathcal{C}/U$ we may assume there exists a strictly perfect complex $\mathcal{E}^\bullet $ and a map $\alpha : \mathcal{E}^\bullet \to E$ such that $H^ i(\alpha )$ is an isomorphism for $i \geq a$. We may and do replace $\mathcal{E}^\bullet $ by $\sigma _{\geq a - 1}\mathcal{E}^\bullet $. Choose a distinguished triangle
\[ \mathcal{E}^\bullet \to E \to C \to \mathcal{E}^\bullet [1] \]
From the vanishing of cohomology sheaves of $E$ and $\mathcal{E}^\bullet $ and the assumption on $\alpha $ we obtain $C \cong \mathcal{K}[2 - a]$ with $\mathcal{K} = \mathop{\mathrm{Ker}}(\mathcal{E}^{a - 1} \to \mathcal{E}^ a)$. Let $\mathcal{F}$ be an $\mathcal{O}$-module. Applying $- \otimes _\mathcal {O}^\mathbf {L} \mathcal{F}$ the assumption that $E$ has tor amplitude in $[a, b]$ implies $\mathcal{K} \otimes _\mathcal {O} \mathcal{F} \to \mathcal{E}^{a - 1} \otimes _\mathcal {O} \mathcal{F}$ has image $\mathop{\mathrm{Ker}}(\mathcal{E}^{a - 1} \otimes _\mathcal {O} \mathcal{F} \to \mathcal{E}^ a \otimes _\mathcal {O} \mathcal{F})$. It follows that $\text{Tor}_1^\mathcal {O}(\mathcal{E}', \mathcal{F}) = 0$ where $\mathcal{E}' = \mathop{\mathrm{Coker}}(\mathcal{E}^{a - 1} \to \mathcal{E}^ a)$. Hence $\mathcal{E}'$ is flat (Lemma 21.17.15). Thus there exists a covering $\{ U_ i \to U\} $ such that $\mathcal{E}'|_{U_ i}$ is a direct summand of a finite free module by Modules on Sites, Lemma 18.29.3. Thus the complex
\[ \mathcal{E}'|_{U_ i} \to \mathcal{E}^{a + 1}|_{U_ i} \to \ldots \to \mathcal{E}^ b|_{U_ i} \]
is quasi-isomorphic to $E|_{U_ i}$ and $E$ is perfect. $\square$
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. $\square$
Lemma 21.47.5. Let $(f, f^\sharp ) : (\mathcal{C}, \mathcal{O}_\mathcal {C}) \to (\mathcal{D}, \mathcal{O}_\mathcal {D})$ be a morphism of ringed sites. Let $E$ be an object of $D(\mathcal{O}_\mathcal {D})$. If $E$ is perfect in $D(\mathcal{O}_\mathcal {D})$, then $Lf^*E$ is perfect in $D(\mathcal{O}_\mathcal {C})$.
Proof. This follows from Lemma 21.47.4, 21.46.5, and 21.45.3. $\square$
Lemma 21.47.6. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $(K, L, M, f, g, h)$ be a distinguished triangle in $D(\mathcal{O})$. If two out of three of $K, L, M$ are perfect then the third is also perfect.
Proof. First proof: Combine Lemmas 21.47.4, 21.45.4, and 21.46.6. Second proof (sketch): Say $K$ and $L$ are perfect. Let $U$ be an object of $\mathcal{C}$. After replacing $U$ by the members of a covering we may assume that $K|_ U$ and $L|_ U$ are represented by strictly perfect complexes $\mathcal{K}^\bullet $ and $\mathcal{L}^\bullet $. After replacing $U$ by the members of a covering we may assume the map $K|_ U \to L|_ U$ is given by a map of complexes $\alpha : \mathcal{K}^\bullet \to \mathcal{L}^\bullet $, see Lemma 21.44.8. Then $M|_ U$ is isomorphic to the cone of $\alpha $ which is strictly perfect by Lemma 21.44.2. $\square$
Lemma 21.47.7. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. If $K, L$ are perfect objects of $D(\mathcal{O})$, then so is $K \otimes _\mathcal {O}^\mathbf {L} L$.
Proof. Follows from Lemmas 21.47.4, 21.45.5, and 21.46.7. $\square$
Lemma 21.47.8. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. If $K \oplus L$ is a perfect object of $D(\mathcal{O})$, then so are $K$ and $L$.
Proof. Follows from Lemmas 21.47.4, 21.45.6, and 21.46.8. $\square$
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