Lemma 17.18.2. Let (X, \mathcal{O}_ X) be a ringed space. Let \mathcal{F} be an \mathcal{O}_ X-module. Let \mathcal{G}, \eta , \epsilon be a left dual of \mathcal{F} in the monoidal category of \mathcal{O}_ X-modules, see Categories, Definition 4.43.5. Then
\mathcal{F} is locally a direct summand of a finite free \mathcal{O}_ X-module,
the map e : \mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(\mathcal{F}, \mathcal{O}_ X) \to \mathcal{G} sending a local section \lambda to (\lambda \otimes 1)(\eta ) is an isomorphism,
we have \epsilon (f, g) = e^{-1}(g)(f) for local sections f and g of \mathcal{F} and \mathcal{G}.
Proof.
The assumptions mean that
\mathcal{F} \xrightarrow {\eta \otimes 1} \mathcal{F} \otimes _{\mathcal{O}_ X} \mathcal{G} \otimes _{\mathcal{O}_ X} \mathcal{F} \xrightarrow {1 \otimes \epsilon } \mathcal{F} \quad \text{and}\quad \mathcal{G} \xrightarrow {1 \otimes \eta } \mathcal{G} \otimes _{\mathcal{O}_ X} \mathcal{F} \otimes _{\mathcal{O}_ X} \mathcal{G} \xrightarrow {\epsilon \otimes 1} \mathcal{G}
are the identity map. Let x \in X. We can find an open neighbourhood U of x, a finite number of sections f_1, \ldots , f_ n and g_1, \ldots , g_ n of \mathcal{F} and \mathcal{G} over U such that \eta (1) = \sum f_ i g_ i. Denote
\mathcal{O}_ U^{\oplus n} \to \mathcal{F}|_ U
the map sending the ith basis vector to f_ i. Then we can factor the map \eta |_ U over a map \tilde\eta : \mathcal{O}_ U \to \mathcal{O}_ U^{\oplus n} \otimes _{\mathcal{O}_ U} \mathcal{G}|_ U. We obtain a commutative diagram
\xymatrix{ \mathcal{F}|_ U \ar[rr]_-{\eta \otimes 1} \ar[rrd]_-{\tilde\eta \otimes 1} & & \mathcal{F}|_ U \otimes \mathcal{G}|_ U \otimes \mathcal{F}|_ U \ar[r]_-{1 \otimes \epsilon } & \mathcal{F}|_ U \\ & & \mathcal{O}_ U^{\oplus n} \otimes \mathcal{G}|_ U \otimes \mathcal{F}|_ U \ar[u] \ar[r]^-{1 \otimes \epsilon } & \mathcal{O}_ U^{\oplus n} \ar[u] }
This shows that the identity on \mathcal{F} locally on X factors through a finite free module. This proves (1). Part (2) follows from Categories, Lemma 4.43.6 and its proof. Part (3) follows from the first equality of the proof. You can also deduce (2) and (3) from the uniqueness of left duals (Categories, Remark 4.43.7) and the construction of the left dual in Example 17.18.1.
\square
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