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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