The Stacks project

Example 20.46.6. 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 20.46.4. 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 20.38.5 for example). Then $K^\vee , \eta , \epsilon $ is a left dual for $K$ as in Categories, Definition 4.42.5. We omit the verification that $(1 \otimes \epsilon ) \circ (\eta \otimes 1) = \text{id}_ K$ and $(\epsilon \otimes 1) \circ (1 \otimes \eta ) = \text{id}_{K^\vee }$.


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