The Stacks project

Example 20.50.2. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $\mathcal{F}^\bullet $ be a locally bounded complex of $\mathcal{O}_ X$-modules such that each $\mathcal{F}^ n$ is locally a direct summand of a finite free $\mathcal{O}_ X$-module. In other words, there is an open covering $X = \bigcup U_ i$ such that $\mathcal{F}^\bullet |_{U_ i}$ is a strictly perfect complex. Consider the complex

as in Section 20.41. Let

be $\eta = \sum \eta _ n$ and $\epsilon = \sum \epsilon _ n$ where $\eta _ n : \mathcal{O}_ X \to \mathcal{F}^ n \otimes _{\mathcal{O}_ X} \mathcal{G}^{-n}$ and $\epsilon _ n : \mathcal{G}^{-n} \otimes _{\mathcal{O}_ X} \mathcal{F}^ n \to \mathcal{O}_ X$ are as in Modules, Example 17.18.1. Then $\mathcal{G}^\bullet , \eta , \epsilon $ is a left dual for $\mathcal{F}^\bullet $ as in Categories, Definition 4.43.5. We omit the verification that $(1 \otimes \epsilon ) \circ (\eta \otimes 1) = \text{id}_{\mathcal{F}^\bullet }$ and $(\epsilon \otimes 1) \circ (1 \otimes \eta ) = \text{id}_{\mathcal{G}^\bullet }$. Please compare with More on Algebra, Lemma 15.72.2.