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.
$\square$

## Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like `$\pi$`

). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

## Comments (0)