The Stacks project

Lemma 18.29.2. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{F}$ be a $\mathcal{O}$-module. Let $\mathcal{G}, \eta , \epsilon $ be a left dual of $\mathcal{F}$ in the monoidal category of $\mathcal{O}$-modules, see Categories, Definition 4.43.5. Then

  1. 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,

  2. the map $e : \mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(\mathcal{F}, \mathcal{O}) \to \mathcal{G}$ sending a local section $\lambda $ to $(\lambda \otimes 1)(\eta )$ is an isomorphism,

  3. 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

\[ \mathcal{F} \xrightarrow {\eta \otimes 1} \mathcal{F} \otimes _\mathcal {O} \mathcal{G} \otimes _\mathcal {O} \mathcal{F} \xrightarrow {1 \otimes \epsilon } \mathcal{F} \quad \text{and}\quad \mathcal{G} \xrightarrow {1 \otimes \eta } \mathcal{G} \otimes _\mathcal {O} \mathcal{F} \otimes _\mathcal {O} \mathcal{G} \xrightarrow {\epsilon \otimes 1} \mathcal{G} \]

are the identity map. Let $U$ be an object of $\mathcal{C}$. After replacing $U$ by the members of a covering of $U$, we can find 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 $i$th 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 $\mathcal{F}|_ U$ factors through a finite free $\mathcal{O}_ U$-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 18.29.1. $\square$

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