The Stacks project

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 4.42.5. Then

  1. $\mathcal{F}$ is locally a direct summand of a finite free $\mathcal{O}_ X$-module,

  2. 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,

  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

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

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 $i$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 $\mathcal{F}$ locally on $X$ factors through a finite free module. This proves (1). Part (2) follows from Categories, Lemma 4.42.6 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 4.42.7) and the construction of the left dual in Example 17.18.1. $\square$


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