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Fontaine's ring $B_{cris}^{\varphi=1}$ is a PID, and no expert in the field would have bet on it in the first place (this led to some very nice recent developments by Fargues and Fontaine).
Fontaine's ring $B_{cris}^{\varphi=1}$ is a PID, and no expert in the field would have bet on it in the first place (this led to some very nice recent developments by Fargues and Fontaine).