Timeline for Etale algebra whose local rank is constantly zero is the zero algebra
Current License: CC BY-SA 4.0
3 events
| when toggle format | what | by | license | comment | |
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| May 13, 2019 at 18:30 | review | Close votes | |||
| May 18, 2019 at 15:29 | |||||
| May 13, 2019 at 18:10 | comment | added | R. van Dobben de Bruyn | This doesn't use étale at all. If $\mathfrak q \subseteq B$ is a prime, then it has a preimage $\mathfrak p = f^{-1}(\mathfrak q) \subseteq A$. But then $\kappa(\mathfrak p) \otimes_A B$ maps to $\kappa(\mathfrak q)$, contradicting the hypothesis. We conclude that $\mathfrak q$ does not exist, i.e. $B$ is the zero ring. Geometrically, the assumption is that all fibres of $\operatorname{Spec} B \to \operatorname{Spec} A$ are empty, and the conclusion is that $\operatorname{Spec} B$ is empty. | |
| May 13, 2019 at 17:56 | history | asked | user223794 | CC BY-SA 4.0 |