20.6 First cohomology and invertible sheaves

The Picard group of a ringed space is defined in Modules, Section 17.25.

Lemma 20.6.1. Let $(X, \mathcal{O}_ X)$ be a locally ringed space. There is a canonical isomorphism

$H^1(X, \mathcal{O}_ X^*) = \mathop{\mathrm{Pic}}\nolimits (X).$

of abelian groups.

Proof. Let $\mathcal{L}$ be an invertible $\mathcal{O}_ X$-module. Consider the presheaf $\mathcal{L}^*$ defined by the rule

$U \longmapsto \{ s \in \mathcal{L}(U) \text{ such that } \mathcal{O}_ U \xrightarrow {s \cdot -} \mathcal{L}_ U \text{ is an isomorphism}\}$

This presheaf satisfies the sheaf condition. Moreover, if $f \in \mathcal{O}_ X^*(U)$ and $s \in \mathcal{L}^*(U)$, then clearly $fs \in \mathcal{L}^*(U)$. By the same token, if $s, s' \in \mathcal{L}^*(U)$ then there exists a unique $f \in \mathcal{O}_ X^*(U)$ such that $fs = s'$. Moreover, the sheaf $\mathcal{L}^*$ has sections locally by Modules, Lemma 17.25.4. In other words we see that $\mathcal{L}^*$ is a $\mathcal{O}_ X^*$-torsor. Thus we get a map

$\begin{matrix} \text{invertible sheaves on }(X, \mathcal{O}_ X) \\ \text{ up to isomorphism} \end{matrix} \longrightarrow \begin{matrix} \mathcal{O}_ X^*\text{-torsors} \\ \text{ up to isomorphism} \end{matrix}$

We omit the verification that this is a homomorphism of abelian groups. By Lemma 20.4.3 the right hand side is canonically bijective to $H^1(X, \mathcal{O}_ X^*)$. Thus we have to show this map is injective and surjective.

Injective. If the torsor $\mathcal{L}^*$ is trivial, this means by Lemma 20.4.2 that $\mathcal{L}^*$ has a global section. Hence this means exactly that $\mathcal{L} \cong \mathcal{O}_ X$ is the neutral element in $\mathop{\mathrm{Pic}}\nolimits (X)$.

Surjective. Let $\mathcal{F}$ be an $\mathcal{O}_ X^*$-torsor. Consider the presheaf of sets

$\mathcal{L}_1 : U \longmapsto (\mathcal{F}(U) \times \mathcal{O}_ X(U))/\mathcal{O}_ X^*(U)$

where the action of $f \in \mathcal{O}_ X^*(U)$ on $(s, g)$ is $(fs, f^{-1}g)$. Then $\mathcal{L}_1$ is a presheaf of $\mathcal{O}_ X$-modules by setting $(s, g) + (s', g') = (s, g + (s'/s)g')$ where $s'/s$ is the local section $f$ of $\mathcal{O}_ X^*$ such that $fs = s'$, and $h(s, g) = (s, hg)$ for $h$ a local section of $\mathcal{O}_ X$. We omit the verification that the sheafification $\mathcal{L} = \mathcal{L}_1^\#$ is an invertible $\mathcal{O}_ X$-module whose associated $\mathcal{O}_ X^*$-torsor $\mathcal{L}^*$ is isomorphic to $\mathcal{F}$. $\square$

Comment #5992 by Gabriel Ribeiro on

I don't think the proof uses in any way that $(X,\mathscr{O}_X)$ is a locally ringed space.

Comment #5993 by on

Actually, it does. Namely, our definition of invertible modules on general ringed spaces doesn't imply that they are locally generated by 1 section. You can see this: the proof doesn't work if the stalks of $\mathcal{O}_X$ aren't local because this is used in Lemma 17.25.4.

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