
## 21.45 Perfect complexes

In this section we discuss properties of perfect complexes on ringed sites.

Definition 21.45.1. 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.

Lemma 21.45.2. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $E$ be an object of $D(\mathcal{O})$.

1. 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.

2. If $E$ is perfect, then any complex representing $E$ is perfect.

Proof. Identical to the proof of Lemma 21.43.2. $\square$

Lemma 21.45.3. 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 $C \cong \mathcal{K}[a - 2]$ 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 21.18.14). 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 18.28.13. 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 21.45.4. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $E$ be an object of $D(\mathcal{O})$. The following are equivalent

1. $E$ is perfect, and

2. $E$ 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 21.43.2. Moreover, a direct summand of a finite free module is flat, hence $E|_{U_ i}$ has finite Tor dimension by Lemma 21.44.3. 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 21.45.3. $\square$

Lemma 21.45.5. 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 $D(\mathcal{O}_\mathcal {C})$.

Lemma 21.45.6. 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 21.45.4, 21.43.4, and 21.44.6. 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 21.42.8. Then $M|_ U$ is isomorphic to the cone of $\alpha$ which is strictly perfect by Lemma 21.42.2. $\square$

Lemma 21.45.7. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. If $K, L$ are perfect objects of $D(\mathcal{O})$, then so is $K \otimes _\mathcal {O}^\mathbf {L} L$.

Lemma 21.45.8. 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 $L$.

Lemma 21.45.9. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $K$ be a perfect object of $D(\mathcal{O})$. Then $K^\vee = R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (K, \mathcal{O})$ is a perfect object too and $(K^\vee )^\vee = K$. There are functorial isomorphisms

$K^\vee \otimes ^\mathbf {L}_\mathcal {O} M = R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(K, M)$

and

$H^0(\mathcal{C}, K^\vee \otimes _\mathcal {O}^\mathbf {L} M) = \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O})}(K, M)$

for $M$ in $D(\mathcal{O})$.

Proof. We will us without further mention that formation of internal hom commutes with restriction (Lemma 21.34.3). In particular we may check the first two statements locally, i.e., given any object $U$ of $\mathcal{C}$ it suffices to prove there is a covering $\{ U_ i \to U\}$ such that the statement is true after restricting to $\mathcal{C}/U_ i$ for each $i$. By Lemma 21.34.9 to see the final statement it suffices to check that the map (21.34.9.1)

$K^\vee \otimes ^\mathbf {L}_\mathcal {O} M \longrightarrow R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (K, M)$

is an isomorphism. This is a local question as well. Hence it suffices to prove the lemma when $K$ is represented by a strictly perfect complex.

Assume $K$ is represented by the strictly perfect complex $\mathcal{E}^\bullet$. Then it follows from Lemma 21.42.9 that $K^\vee$ is represented by the complex whose terms are $(\mathcal{E}^ n)^\vee = \mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(\mathcal{E}^ n, \mathcal{O})$ in degree $-n$. Since $\mathcal{E}^ n$ is a direct summand of a finite free $\mathcal{O}$-module, so is $(\mathcal{E}^ n)^\vee$. Hence $K^\vee$ is represented by a strictly perfect complex too. It is also clear that $(K^\vee )^\vee = K$ as we have $((\mathcal{E}^ n)^\vee )^\vee = \mathcal{E}^ n$. To see that (21.34.9.1) is an isomorphism, represent $M$ by a K-flat complex $\mathcal{F}^\bullet$. By Lemma 21.42.9 the complex $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (K, M)$ is represented by the complex with terms

$\bigoplus \nolimits _{n = p + q} \mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(\mathcal{E}^{-q}, \mathcal{F}^ p)$

On the other hand, the object $K^\vee \otimes ^\mathbf {L} M$ is represented by the complex with terms

$\bigoplus \nolimits _{n = p + q} \mathcal{F}^ p \otimes _\mathcal {O} (\mathcal{E}^{-q})^\vee$

Thus the assertion that (21.34.9.1) is an isomorphism reduces to the assertion that the canonical map

$\mathcal{F} \otimes _\mathcal {O} \mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(\mathcal{E}, \mathcal{O}) \longrightarrow \mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(\mathcal{E}, \mathcal{F})$

is an isomorphism when $\mathcal{E}$ is a direct summand of a finite free $\mathcal{O}$-module and $\mathcal{F}$ is any $\mathcal{O}$-module. This follows immediately from the corresponding statement when $\mathcal{E}$ is finite free. $\square$

Lemma 21.45.10. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $(K_ n)_{n \in \mathbf{N}}$ be a system of perfect objects of $D(\mathcal{O})$. Let $K = \text{hocolim} K_ n$ be the derived colimit (Derived Categories, Definition 13.31.1). Then for any object $E$ of $D(\mathcal{O})$ we have

$R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (K, E) = R\mathop{\mathrm{lim}}\nolimits E \otimes ^\mathbf {L}_\mathcal {O} K_ n^\vee$

where $(K_ n^\vee )$ is the inverse system of dual perfect complexes.

Proof. By Lemma 21.45.9 we have $R\mathop{\mathrm{lim}}\nolimits E \otimes ^\mathbf {L}_\mathcal {O} K_ n^\vee = R\mathop{\mathrm{lim}}\nolimits R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (K_ n, E)$ which fits into the distinguished triangle

$R\mathop{\mathrm{lim}}\nolimits R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (K_ n, E) \to \prod R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (K_ n, E) \to \prod R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (K_ n, E)$

Because $K$ similarly fits into the distinguished triangle $\bigoplus K_ n \to \bigoplus K_ n \to K$ it suffices to show that $\prod R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (K_ n, E) = R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (\bigoplus K_ n, E)$. This is a formal consequence of (21.34.0.1) and the fact that derived tensor product commutes with direct sums. $\square$

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