Timeline for Are rigid-analytic spaces obsolete, since adic spaces exist?
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
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| May 22 at 5:07 | comment | added | Alex Youcis | But, to say that the rigid geometry done in Tate language is obsolete is like saying any algebraic geometry done before schemes is useless. Some people might say this, but those people are frankly not very well educated. | |
| May 22 at 5:07 | comment | added | Alex Youcis | (d) it very much clarifies the Raynaud perspective, as in the adic world as for a (quasi-compact quasi-separated) rigid space $X$, one has that $(X,\mathcal{O}_X^+)$ is literally the limit in the category of topologically ringed spaces of $(\mathfrak{X},\mathcal{O}_\mathfrak{X})$ as $\mathfrak{X}$ travels over formal models of $X$ (and of course $\mathcal{O}_X$ is obtained by inverting a uniformizer in $\mathcal{O}_X^+$). | |
| May 22 at 5:05 | comment | added | Alex Youcis | To amplify what David Loeffler is saying, the passage from $\mathrm{Sp}(A)$ to $\mathrm{Spa}(A,A^\circ)$ is morally similar to the passage of $\mathrm{MaxSpec}(R)$ to $\mathrm{Spec}(R)$ for $R$ finite type over an algebraically closed field. This makes many things clearer, especially those of a 'topological nature', e.g., (a) the notion of a 'wide open subset' in adic language is just 'an open subset closed under specialization', (b) the fact that any map of rigid spaces is 'generalizing' is a powerful tool, (c) the existence of universal compactifications which leave the Tate world | |
| Jul 3, 2023 at 7:57 | comment | added | David Loeffler | Incidentally, I was working in p-adic automorphic forms / eigenvarieties roughly between 2005 and 2012. During this time, I read many, many papers, and wrote a fair number myself, in which Tate-style rigid spaces appeared. I can count on one hand the number of times I encountered Berkovich spaces during this period. (Do not fall into the trap of judging what was "generally known" at some point in the past from a few landmark papers of that time that are widely read nowadays; these are, by definition, unrepresentative of the time they were written.) | |
| Jul 3, 2023 at 7:39 | comment | added | David Loeffler | I disagree with your claim that "Berkovich spaces were what was supplanted by adic spaces after Scholze popularized them, rigid spaces as originally defined had already fallen out of favor for many". During the decade or so prior to Scholze's arrival on the scene, Berkovich spaces were used in some areas as an alternative to classical rigid spaces, but they never came near supplanting classical rigid spaces across the board, as adic spaces later did. | |
| Jul 1, 2023 at 15:13 | comment | added | Jack J. Garzella | One thing that is slightly glossed over--while adic spaces were in the literature for many years, Berkovich spaces appeared at about the same time (e.g. Harris-Taylor's proof of LLC, Kedlaya's work, etc). It seems to me that Berkovich spaces were what was supplanted by adic spaces after Scholze popularized them, rigid spaces as originally defined had already fallen out of favor for many. Though really I think there was just and influx of new people learning adic spaces to understand Scholze's work, many Berkovich space users still seem to use them. | |
| Jun 2, 2022 at 13:36 | comment | added | Z. M | It seems that life is too short to say "both" in every similar situation. There are similar issues on the language of $\infty$-categories (in comparison to the language of model categories, derivators, etc.) say. | |
| May 31, 2021 at 13:52 | vote | accept | Wojowu | ||
| May 28, 2021 at 13:30 | comment | added | Wojowu | Thanks for the answer. This is an interesting pragmatic point, and does answer a somewhat implicit question whether it is still worth it to learn the theory, or at least the language, of rigid spaces. | |
| May 27, 2021 at 20:59 | history | answered | David Loeffler | CC BY-SA 4.0 |