The following theorem is sometimes called “Hilbert 90”.
Theorem 59.24.1. 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 59.20. By Cohomology on Sites, Lemma 21.6.1 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 59.17.4 or the more precise Descent, Proposition 35.8.9 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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