Timeline for Valuations and (semi)norms on ring spectra
Current License: CC BY-SA 4.0
11 events
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| Mar 24, 2023 at 1:50 | history | edited | Emily | CC BY-SA 4.0 |
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| Mar 24, 2023 at 1:50 | comment | added | Emily | @Z.M Oh I see what you meant now, thanks for clarifying! Hmm... I don't know :/ I think it's also not clear whether the potential generalisations of valuation rings to 'valuation ring spectra' would agree depending on which definition of valuation ring you start with. E.g. I imagine two generalisations starting from the first and fourth definitions here might possibly disagree. | |
| Mar 22, 2023 at 12:01 | comment | added | Z. M | I was not only talking about a "theory for nonconnective rings" — the concept of valuations should be equivalent to the concept of valuation rings, and the question is whether there are nonconnective valuation rings? Let's be a bit simpler — what are fields? For example, would we consider Morava K-theories (not $E_2$, but quasi-commutative) being fields? | |
| Mar 22, 2023 at 10:33 | comment | added | Emily | Also there might be other conditions one would likely want of those objects, like asking that they also induce graded seminorms/valuations on $\pi_*E$. I'm hoping a theory of seminorms/valuations on ring spectra would be more interesting than just collapsing to the classical theory on $\pi_0$. | |
| Mar 22, 2023 at 10:33 | comment | added | Emily | @FernandoMuro Ah sorry (and thanks)! I've edited the question to correct that. Regarding $\pi_0R$, although that would certainly work I'm not sure if that would be the """right""" definition. E.g. for seminorms I wonder if a better definition would start with finding a homotopy-theoretic version $R$ of $\mathbb{R}_{\geq0}$ and considering maps of the form $f\colon E\to R$, as then the ordinary seminorm $|-|\colon E\to\mathbb{R}_{\geq0}$ would come out by applying $\pi_0$ to $f$. | |
| Mar 22, 2023 at 10:23 | history | edited | Emily | CC BY-SA 4.0 |
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| Mar 22, 2023 at 10:23 | comment | added | Emily | @Z.M I hadn't really though of this before, but apparently people do consider valuations on noncommutative rings (e.g. link), although I'm not sure how useful they are in that context. Probably we should indeed look at at least $\mathbb{E}_2$-rings. I'm also not sure about connectivity, although I think having such a theory for nonconnective ring spectra as well would certainly be nice | |
| Mar 22, 2023 at 6:06 | comment | added | Fernando Muro | In the definition of valuation, the disjoint union is not a group with the structure you give it. In any case, with the correct definitions, if you define valuations and semi norms on a ring spectrum $R$ as those in $\pi_0R$ then this seems to satisfy your conditions. | |
| Mar 22, 2023 at 5:09 | comment | added | Z. M | It does not seem reasonable to consider $E_1$-rings, as this definition does not seem to be reasonable for classical associative rings. On the other hand, it is unclear whether one should only look at connective guys — for example, if one looks at the sphere spectrum, do we need to see the "chromatic information" (for which one should pass to nonconnective guys)? | |
| Mar 22, 2023 at 1:47 | comment | added | Emily | (P.S. By "ring spectra" I mean either commutative algebras in $\mathrm{Sp}$ or $\mathbb{E}_k$-rings for some $1\leq k\leq\infty$, and by "extending" the classical notions I mean that a valuation/seminorm/norm on a ring spectrum $E$ should induce an ordinary valuation/seminorm/norm on $\pi_0(E)$.) | |
| Mar 22, 2023 at 1:47 | history | asked | Emily | CC BY-SA 4.0 |