The Stacks project

Example 20.50.7. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $K$ be a perfect object of $D(\mathcal{O}_ X)$. Set $K^\vee = R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (K, \mathcal{O}_ X)$ as in Lemma 20.50.5. Then the map

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

is an isomorphism (by the lemma). Denote

\[ \eta : \mathcal{O}_ X \longrightarrow K \otimes _{\mathcal{O}_ X}^\mathbf {L} K^\vee \]

the map sending $1$ to the section corresponding to $\text{id}_ K$ under the isomorphism above. Denote

\[ \epsilon : K^\vee \otimes _{\mathcal{O}_ X}^\mathbf {L} K \longrightarrow \mathcal{O}_ X \]

the evaluation map (to construct it you can use Lemma 20.42.5 for example). Then $K^\vee , \eta , \epsilon $ is a left dual for $K$ as in Categories, Definition 4.43.5. We omit the verification that $(1 \otimes \epsilon ) \circ (\eta \otimes 1) = \text{id}_ K$ and $(\epsilon \otimes 1) \circ (1 \otimes \eta ) = \text{id}_{K^\vee }$.

Comments (0)

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Your email address will not be published.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.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0FPC. Beware of the difference between the letter 'O' and the digit '0'.