Timeline for What are the potential applications of perfectoid spaces to homotopy theory?
Current License: CC BY-SA 4.0
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| Sep 16, 2023 at 8:32 | comment | added | Z. M | I am not sure whether I understand correctly: the failure of this conjecture does not root out the possibility of existence of concept of "spherical" prisms $(A,I)$ such that, after base change along $\mathbb S\to\mathbb Z$, it becomes usual (animated/derived) prisms (everything $p$-completed)? | |
| Mar 15, 2021 at 23:49 | comment | added | Yuri Sulyma | Hi Peter, thanks for this answer! Since asking this question I've gone more into equivariant homotopy theory, and recently have been thinking about connections between that and prisms. I wrote up my thoughts in a separate answer. | |
| Mar 15, 2021 at 16:11 | history | edited | Peter Scholze | CC BY-SA 4.0 |
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| Mar 15, 2021 at 15:52 | comment | added | Jacob Lurie | Every $A$-module $M$ admits a canonical $\varphi_A$-semilinear endomorphism, given by $\varphi_M = 0$. | |
| Mar 15, 2021 at 15:36 | history | edited | Peter Scholze | CC BY-SA 4.0 |
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| Mar 15, 2021 at 15:34 | comment | added | Peter Scholze | I'm not sure if this conjecture should be true, so I'd be happy about a disproof, too :-). But for this construction, $M$ needs to be a $\phi$-module over $A$. Can one still use this argument to disprove the conjecture? | |
| Mar 15, 2021 at 14:31 | comment | added | Jacob Lurie | I don't think this conjecture can be true. Let $(A,I)$ be a perfect prism. Every free $A$-module $M$ of finite rank defines a prism $(A \oplus M, I \oplus IM)$. If you had such a functor, you could apply the "TP version" and quotient out $TP(A/I)$ to get a $TP(A/I)$-module $F(M)$, free of the same rank as $M$. Since $F$ is an additive functor this would need to come from a map associative ring spectra $A \rightarrow TP(A/I)$, which usually can't exist. | |
| S Mar 15, 2021 at 13:23 | history | answered | Peter Scholze | CC BY-SA 4.0 | |
| S Mar 15, 2021 at 13:23 | history | made wiki | Post Made Community Wiki by Peter Scholze |