The Stacks project

\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

18.31 Invertible modules

Here is the definition.

Definition 18.31.1. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site.

  1. A finite locally free $\mathcal{O}$-module $\mathcal{F}$ is said to have rank $r$ if for every object $U$ of $\mathcal{C}$ there exists a covering $\{ U_ i \to U\} $ of $U$ such that $\mathcal{F}|_{U_ i}$ is isomorphic to $\mathcal{O}_{U_ i}^{\oplus r}$ as an $\mathcal{O}_{U_ i}$-module.

  2. An $\mathcal{O}$-module $\mathcal{L}$ is invertible if the functor

    \[ \textit{Mod}(\mathcal{O}) \longrightarrow \textit{Mod}(\mathcal{O}),\quad \mathcal{F} \longmapsto \mathcal{F} \otimes _\mathcal {O} \mathcal{L} \]

    is an equivalence.

  3. The sheaf $\mathcal{O}^*$ is the subsheaf of $\mathcal{O}$ defined by the rule

    \[ U \longmapsto \mathcal{O}^*(U) = \{ f \in \mathcal{O}(U) \mid \exists g \in \mathcal{O}(U)\text{ such that }fg = 1\} \]

    It is a sheaf of abelian groups with multiplication as the group law.

Lemma 18.39.7 below explains the relationship with locally free modules of rank $1$.

Lemma 18.31.2. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{L}$ be an $\mathcal{O}$-module. The following are equivalent:

  1. $\mathcal{L}$ is invertible, and

  2. there exists an $\mathcal{O}$-module $\mathcal{N}$ such that $\mathcal{L} \otimes _\mathcal {O} \mathcal{N} \cong \mathcal{O}$.

In this case $\mathcal{L}$ is flat and of finite presentation and the module $\mathcal{N}$ in (2) is isomorphic to $\mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(\mathcal{L}, \mathcal{O})$.

Proof. Assume (1). Then the functor $- \otimes _\mathcal {O} \mathcal{L}$ is essentially surjective, hence there exists an $\mathcal{O}$-module $\mathcal{N}$ as in (2). If (2) holds, then the functor $- \otimes _\mathcal {O} \mathcal{N}$ is a quasi-inverse to the functor $- \otimes _\mathcal {O} \mathcal{L}$ and we see that (1) holds.

Assume (1) and (2) hold. Since $- \otimes _\mathcal {O} \mathcal{L}$ is an equivalence, it is exact, and hence $\mathcal{L}$ is flat. Denote $\psi : \mathcal{L} \otimes _\mathcal {O} \mathcal{N} \to \mathcal{O}$ the given isomorphism. Let $U$ be an object of $\mathcal{C}$. We will show that the restriction $\mathcal{L}$ to the members of a covering of $U$ is a direct summand of a free module, which will certainly imply that $\mathcal{L}$ is of finite presentation. By construction of $\otimes $ we may assume (after replacing $U$ by the members of a covering) that there exists an integer $n \geq 1$ and sections $x_ i \in \mathcal{L}(U)$, $y_ i \in \mathcal{N}(U)$ such that $\psi (\sum x_ i \otimes y_ i) = 1$. Consider the isomorphisms

\[ \mathcal{L}|_ U \to \mathcal{L}|_ U \otimes _{\mathcal{O}_ U} \mathcal{L}|_ U \otimes _{\mathcal{O}_ U} \mathcal{N}|_ U \to \mathcal{L}|_ U \]

where the first arrow sends $x$ to $\sum x_ i \otimes x \otimes y_ i$ and the second arrow sends $x \otimes x' \otimes y$ to $\psi (x' \otimes y)x$. We conclude that $x \mapsto \sum \psi (x \otimes y_ i)x_ i$ is an automorphism of $\mathcal{L}|_ U$. This automorphism factors as

\[ \mathcal{L}|_ U \to \mathcal{O}_ U^{\oplus n} \to \mathcal{L}|_ U \]

where the first arrow is given by $x \mapsto (\psi (x \otimes y_1), \ldots , \psi (x \otimes y_ n))$ and the second arrow by $(a_1, \ldots , a_ n) \mapsto \sum a_ i x_ i$. In this way we conclude that $\mathcal{L}|_ U$ is a direct summand of a finite free $\mathcal{O}_ U$-module.

Assume (1) and (2) hold. Consider the evaluation map

\[ \mathcal{L} \otimes _\mathcal {O} \mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(\mathcal{L}, \mathcal{O}_ X) \longrightarrow \mathcal{O}_ X \]

To finish the proof of the lemma we will show this is an isomorphism. By Lemma 18.27.4 we have

\[ \mathop{\mathrm{Hom}}\nolimits _\mathcal {O}(\mathcal{O}, \mathcal{O}) = \mathop{\mathrm{Hom}}\nolimits _\mathcal {O} (\mathcal{N} \otimes _\mathcal {O} \mathcal{L}, \mathcal{O}) \longrightarrow \mathop{\mathrm{Hom}}\nolimits _\mathcal {O} (\mathcal{N}, \mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(\mathcal{L}, \mathcal{O})) \]

The image of $1$ gives a morphism $\mathcal{N} \to \mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(\mathcal{L}, \mathcal{O})$. Tensoring with $\mathcal{L}$ we obtain

\[ \mathcal{O} = \mathcal{L} \otimes _\mathcal {O} \mathcal{N} \longrightarrow \mathcal{L} \otimes _\mathcal {O} \mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(\mathcal{L}, \mathcal{O}) \]

This map is the inverse to the evaluation map; computation omitted. $\square$

Lemma 18.31.3. Let $f : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}_\mathcal {C}) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}_\mathcal {D})$ be a morphism of ringed topoi. The pullback $f^*\mathcal{L}$ of an invertible $\mathcal{O}_\mathcal {D}$-module is invertible.

Proof. By Lemma 18.31.2 there exists an $\mathcal{O}_\mathcal {D}$-module $\mathcal{N}$ such that $\mathcal{L} \otimes _{\mathcal{O}_\mathcal {D}} \mathcal{N} \cong \mathcal{O}_\mathcal {D}$. Pulling back we get $f^*\mathcal{L} \otimes _{\mathcal{O}_\mathcal {C}} f^*\mathcal{N} \cong \mathcal{O}_\mathcal {C}$ by Lemma 18.26.1. Thus $f^*\mathcal{L}$ is invertible by Lemma 18.31.2. $\square$

Lemma 18.31.4. Let $(\mathcal{C}, \mathcal{O})$ be a ringed space.

  1. If $\mathcal{L}$, $\mathcal{N}$ are invertible $\mathcal{O}$-modules, then so is $\mathcal{L} \otimes _\mathcal {O} \mathcal{N}$.

  2. If $\mathcal{L}$ is an invertible $\mathcal{O}$-module, then so is $\mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(\mathcal{L}, \mathcal{O})$ and the evaluation map $\mathcal{L} \otimes _\mathcal {O} \mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(\mathcal{L}, \mathcal{O}) \to \mathcal{O}$ is an isomorphism.

Proof. Part (1) is clear from the definition and part (2) follows from Lemma 18.31.2 and its proof. $\square$

Lemma 18.31.5. Let $(\mathcal{C}, \mathcal{O})$ be a ringed space. There exists a set of invertible modules $\{ \mathcal{L}_ i\} _{i \in I}$ such that each invertible module on $(\mathcal{C}, \mathcal{O})$ is isomorphic to exactly one of the $\mathcal{L}_ i$.

Proof. Omitted, but see Sheaves of Modules, Lemma 17.22.8. $\square$

Lemma 18.31.5 says that the collection of isomorphism classes of invertible sheaves forms a set. Lemma 18.31.4 says that tensor product defines the structure of an abelian group on this set with inverse of $\mathcal{L}$ given by $\mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(\mathcal{L}, \mathcal{O})$.

In fact, given an invertible $\mathcal{O}$-module $\mathcal{L}$ and $n \in \mathbf{Z}$ we define the $n$th tensor power $\mathcal{L}^{\otimes n}$ of $\mathcal{L}$ as the image of $\mathcal{O}$ under applying the equivalence $\mathcal{F} \mapsto \mathcal{F} \otimes _\mathcal {O} \mathcal{L}$ exactly $n$ times. This makes sense also for negative $n$ as we've defined an invertible $\mathcal{O}$-module as one for which tensoring is an equivalence. More explicitly, we have

\[ \mathcal{L}^{\otimes n} = \left\{ \begin{matrix} \mathcal{O} & \text{if} & n = 0 \\ \mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(\mathcal{L}, \mathcal{O}) & \text{if} & n = -1 \\ \mathcal{L} \otimes _\mathcal {O} \ldots \otimes _\mathcal {O} \mathcal{L} & \text{if} & n > 0 \\ \mathcal{L}^{\otimes -1} \otimes _\mathcal {O} \ldots \otimes _\mathcal {O} \mathcal{L}^{\otimes -1} & \text{if} & n < -1 \end{matrix} \right. \]

see Lemma 18.31.4. With this definition we have canonical isomorphisms $\mathcal{L}^{\otimes n} \otimes _\mathcal {O} \mathcal{L}^{\otimes m} \to \mathcal{L}^{\otimes n + m}$, and these isomorphisms satisfy a commutativity and an associativity constraint (formulation omitted).

Definition 18.31.6. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. The Picard group $\mathop{\mathrm{Pic}}\nolimits (\mathcal{O})$ of the ringed site is the abelian group whose elements are isomorphism classes of invertible $\mathcal{O}$-modules, with addition corresponding to tensor product.


Comments (2)

Comment #1188 by Mohamed Hashi on

In definition 18.31.4, the second sentence is missing a 'of'.


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