Theorem 59.24.1. For any scheme $X$ we have canonical identifications

## 59.24 Picard groups

The following theorem is sometimes called “Hilbert 90”.

**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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