Lemma 20.6.1. Let $(X, \mathcal{O}_ X)$ be a ringed space. If all stalks $\mathcal{O}_{X, x}$ are local rings, then 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 #8241 by Ryo Suzuki on

It seems that locally ringed space is not yet defined. It is defined in Definition 01HB. It might be better to say something like "a ringed space that all of its stalks are local rings".

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