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*}

54.24 Picard groups

The following theorem is sometimes called “Hilbert 90”.

Theorem 54.24.1. For any scheme $X$ we have canonical identifications

\begin{align*} H_{fppf}^1(X, \mathbf{G}_ m) & = H^1_{syntomic}(X, \mathbf{G}_ m) \\ & = H^1_{smooth}(X, \mathbf{G}_ m) \\ & = H_{\acute{e}tale}^1(X, \mathbf{G}_ m) \\ & = H^1_{Zar}(X, \mathbf{G}_ m) \\ & = \mathop{\mathrm{Pic}}\nolimits (X) \\ & = H^1(X, \mathcal{O}_ X^*) \end{align*}

Proof. Let $\tau $ be one of the topologies considered in Section 54.20. By Cohomology on Sites, Lemma 21.7.1 we see that $H^1_\tau (X, \mathbf{G}_ m) = H^1_\tau (X, \mathcal{O}_\tau ^*) = \mathop{\mathrm{Pic}}\nolimits (\mathcal{O}_\tau )$ where $\mathcal{O}_\tau $ is the structure sheaf of the site $(\mathit{Sch}/X)_\tau $. Now an invertible $\mathcal{O}_\tau $-module is a quasi-coherent $\mathcal{O}_\tau $-module. By Theorem 54.17.4 or the more precise Descent, Proposition 34.8.11 we see that $\mathop{\mathrm{Pic}}\nolimits (\mathcal{O}_\tau ) = \mathop{\mathrm{Pic}}\nolimits (X)$. The last equality is proved in the same way. $\square$


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