The Stacks project

Lemma 20.46.3. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $\mathcal{F}^\bullet $ be a complex of $\mathcal{O}_ X$-modules. If $\mathcal{F}^\bullet $ has a left dual in the monoidal category of complexes of $\mathcal{O}_ X$-modules (Categories, Definition 4.42.5) then $\mathcal{F}^\bullet $ is a locally bounded complex whose terms are locally direct summands of finite free $\mathcal{O}_ X$-modules and the left dual is as constructed in Example 20.46.2.

Proof. By uniqueness of left duals (Categories, Remark 4.42.7) we get the final statement provided we show that $\mathcal{F}^\bullet $ is as stated. Let $\mathcal{G}^\bullet , \eta , \epsilon $ be a left dual. Write $\eta = \sum \eta _ n$ and $\epsilon = \sum \epsilon _ n$ where $\eta _ n : \mathcal{O}_ X \to \mathcal{F}^ n \otimes _{\mathcal{O}_ X} \mathcal{G}^{-n}$ and $\epsilon _ n : \mathcal{G}^{-n} \otimes _{\mathcal{O}_ X} \mathcal{F}^ n \to \mathcal{O}_ X$. Since $(1 \otimes \epsilon ) \circ (\eta \otimes 1) = \text{id}_{\mathcal{F}^\bullet }$ and $(\epsilon \otimes 1) \circ (1 \otimes \eta ) = \text{id}_{\mathcal{G}^\bullet }$ by Categories, Definition 4.42.5 we see immediately that we have $(1 \otimes \epsilon _ n) \circ (\eta _ n \otimes 1) = \text{id}_{\mathcal{F}^ n}$ and $(\epsilon _ n \otimes 1) \circ (1 \otimes \eta _ n) = \text{id}_{\mathcal{G}^{-n}}$. Hence we see that $\mathcal{F}^ n$ is locally a direct summand of a finite free $\mathcal{O}_ X$-module by Modules, Lemma 17.17.2. Since the sum $\eta = \sum \eta _ n$ is locally finite, we conclude that $\mathcal{F}^\bullet $ is locally bounded. $\square$


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