Example 18.29.1. Let (\mathcal{C}, \mathcal{O}) be a ringed site. Let \mathcal{F} be an \mathcal{O}-module such that for every object U of \mathcal{C} there exists a covering \{ U_ i \to U\} such that \mathcal{F}|_{U_ i} is a direct summand of a finite free \mathcal{O}|_{U_ i}-module. Then the map
is an isomorphism. Namely, this is a local question, it is true if \mathcal{F} is finite free, and it holds for any summand of a module for which it is true (details omitted). Denote
the map sending 1 to the section corresponding to \text{id}_\mathcal {F} under the isomorphism above. Denote
the evaluation map. Then we see that \mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(\mathcal{F}, \mathcal{O}), \eta , \epsilon is a left dual for \mathcal{F} as in Categories, Definition 4.43.5. We omit the verification that (1 \otimes \epsilon ) \circ (\eta \otimes 1) = \text{id}_\mathcal {F} and (\epsilon \otimes 1) \circ (1 \otimes \eta ) = \text{id}_{\mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(\mathcal{F}, \mathcal{O})}.
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