18.29 Duals
Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. The category of $\mathcal{O}$-modules endowed with the tensor product constructed in Section 18.26 is a symmetric monoidal category. For an $\mathcal{O}$-module $\mathcal{F}$ the following are equivalent
$\mathcal{F}$ has a left dual in the monoidal category of $\mathcal{O}$-modules,
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, and
$\mathcal{F}$ is of finite presentation and flat as an $\mathcal{O}$-module.
This is proved in Example 18.29.1 and Lemmas 18.29.2 and 18.29.3 of this section.
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
\[ \mathcal{F} \otimes _\mathcal {O} \mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(\mathcal{F}, \mathcal{O}) \longrightarrow \mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(\mathcal{F}, \mathcal{F}) \]
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
\[ \eta : \mathcal{O} \longrightarrow \mathcal{F} \otimes _\mathcal {O} \mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(\mathcal{F}, \mathcal{O}) \]
the map sending $1$ to the section corresponding to $\text{id}_\mathcal {F}$ under the isomorphism above. Denote
\[ \epsilon : \mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(\mathcal{F}, \mathcal{O}) \otimes _\mathcal {O} \mathcal{F} \longrightarrow \mathcal{O} \]
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 $(\epsilon \otimes 1) \circ (1 \otimes \eta ) = \text{id}_{\mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(\mathcal{F}, \mathcal{O})}$.
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
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,
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,
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.
$\square$
Lemma 18.29.3. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{F}$ be locally of finite presentation and flat. Then given an 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.
Proof.
Choose an object $U$ of $\mathcal{C}$. After replacing $U$ by the members of a covering, we may assume there exists a presentation
\[ \mathcal{O}_ U^{\oplus r} \to \mathcal{O}_ U^{\oplus n} \to \mathcal{F}|_ U \to 0 \]
By Lemma 18.28.14 we may, after replacing $U$ by the members of a covering, assume there exists a factorization
\[ \mathcal{O}_ U^{\oplus n} \to \mathcal{O}_ U^{\oplus n_1} \to \mathcal{F}|_ U \]
such that the composition $\mathcal{O}_ U^{\oplus r} \to \mathcal{O}_ U^{\oplus n} \to \mathcal{O}_ U^{\oplus n_ r}$ is zero. This means that the surjection $\mathcal{O}_ U^{\oplus n_ r} \to \mathcal{F}|_ U$ has a section and we win.
$\square$
Comments (0)