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.
20.6 First cohomology and invertible sheaves
The Picard group of a ringed space is defined in Modules, Section 17.25.
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 $\mathcal{F}$. $\square$
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